Chinese Quarterly Journal of Mathematics

Distance Integral Complete Multipartite Graphs with s = 5; 6

YANG Ruo-song, WANG Li-gong

2016, 31(2): 111-117. doi:10.13371/j.cnki.chin.q.j.m.2016.02.001

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Let D（G） =(d_ij)_n×n denote the distance matrix of a connected graph G with order n, where d_ij is equal to the distance between vertices viand vjin G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In 2014, Yang and Wang gave a sufficient and necessary condition for complete r-partite graphs K_{p1,p2,···,pr}=K_{a1·p1,a2·p2,···,as···ps} to be distance integral and obtained such distance integral graphs with s = 1, 2, 3, 4. However distance integral complete multipartite graphs K_{a1·p1,a2·p2,···,as·ps} with s > 4 have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs K_{a1·p1,a2·p2,···,as·ps}with s = 5, 6. The problem of the existence of such distance integral graphs K_{a1·p1,a2·p2,···,as·ps} with arbitrarily large number s remains open.

Cyclic Codes with Complementary Duals over F_p + vF_p

ZHANG Guang-hui, LIU Qing-qing

2016, 31(2): 118-124. doi:10.13371/j.cnki.chin.q.j.m.2016.02.002

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In this paper,we characterize the necessary and sufficient conditions for a cyclic code of length n over F_p+ vF_pto be an LCD code,where p is an odd prime.

Uniform Blow-up Behavior for Degenerate and Singular Parabolic Equations

LIU Bing-chen, ZHANG Chang-cheng

2016, 31(2): 125-138. doi:10.13371/j.cnki.chin.q.j.m.2016.02.003

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This paper deals with the degenerate and singular parabolic equations coupled via nonlinear nonlocal reactions, subject to zero-Dirichlet boundary conditions. After giving the existence and uniqueness of local classical nonnegative solutions, we show critical blowup exponents for the solutions of the system. Moreover, uniform blow-up behaviors near the blow-up time are obtained for simultaneous blow-up solutions, divided into four subcases.

Exact Solutions of the Wick-type KdV-Burgers Equation

LIU Shao-qing, GAO Guo-cheng

2016, 31(2): 139-146. doi:10.13371/j.cnki.chin.q.j.m.2016.02.004

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In this paper,we consider the wick-type Kd V-Burgers equation with variable coefficients. By using Tanh method with the aid of Hermite transformation, we deduce the exact solutions which include hyperbolic-exponential, trigonometric-exponential and exponential function solutions for the considered equation.

Vertex-distinguishing IE-total Colorings of Complete Bipartite Graphs K_8;n

SHI Jin, CHEN Xiang-en

2016, 31(2): 147-154. doi:10.13371/j.cnki.chin.q.j.m.2016.02.005

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Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C（x） be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C（u） ≠ C（v） for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χ_vt^ie（G） and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K_8,nare discussed in this paper. Particularly, the VDIET chromatic number of K_8,nare obtained.

Common Fixed Point Theorems in Non-normal Cone Metric Spaces with Banach Algebras

HUANG Hua-ping, XU Shao-yuan, LIU Qiu-hua, MING Wei

2016, 31(2): 155-161. doi:10.13371/j.cnki.chin.q.j.m.2016.02.006

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In this paper, we obtain a class of common fixed point theorems for generalized Lipschitz mappings in cone metric spaces with Banach algebras without the assumption of normality of cones. The results greatly generalize some results in the literature. Moreover,we give an example to support the main assertions.

L_r Convergence for Arrays of Rowwise Negatively Superadditive Dependent Random Variables

ZHU Hua-yan, SHEN Ai-ting, ZHANG Ying

2016, 31(2): 162-170. doi:10.13371/j.cnki.chin.q.j.m.2016.02.007

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Let {X_nk, k ≥ 1, n ≥ 1} be an array of rowwise negatively superadditive dependent random variables and {a_n, n ≥ 1} be a sequence of positive real numbers such that a_n↑∞. Under some suitable conditions,L_r convergence of 1/a_n max 1≤j≤n |j∑k=1 X_nk| is studied. The results obtained in this paper generalize and improve some corresponding ones for negatively associated random variables and independent random variables.

Some Inequalities for the General Radial Bodies

ZHANG Ping, WANG Wei-dong

2016, 31(2): 171-177. doi:10.13371/j.cnki.chin.q.j.m.2016.02.008

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Lutwak showed the radial bodies combining with the radial Minkowski linear combinations of star bodies. In this paper, the general radial bodies are introduced and its some properties and inequalities are established.

Likelihood Inference under Generalized Hybrid Censoring Scheme with Competing Risks

MAO Song, SHI Yi-min

2016, 31(2): 178-188. doi:10.13371/j.cnki.chin.q.j.m.2016.02.009

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Statistical inference is developed for the analysis of generalized type-Ⅱ hybrid censoring data under exponential competing risks model. In order to solve the problem that approximate methods make unsatisfactory performances in the case of small sample size,we establish the exact conditional distributions of estimators for parameters by conditional moment generating function（CMGF）. Furthermore, confidence intervals（CIs） are constructed by exact distributions, approximate distributions as well as bootstrap method respectively,and their performances are evaluated by Monte Carlo simulations. And finally, a real data set is analyzed to illustrate all the methods developed here.

Unbounded Motions in Asymmetric Oscillators Depending on Derivatives

WANG Li-xia, MA Shi-wang, WANG Xiao-ming

2016, 31(2): 189-200. doi:10.13371/j.cnki.chin.q.j.m.2016.02.010

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In this paper, we consider the unboundedness of solutions for the asymmetric equation x^’’+ax⁺-bx^-+（x）ψ（x^’）+f（x）+g（x’）=p（t）,where x⁺= max{x, 0}, x^-= max{-x, 0}, a and b are two different positive constants,f（x） is locally Lipschitz continuous and bounded, （x）, ψ（x）, g（x） and p（t） are continuous functions, p（t） is a 2π-periodic function. We discuss the existence of unbounded solutions under two classes of conditions: the resonance case 1/a^1/2+1/b^1/2∈Q and the nonresonance case 1/a^1/2+1/b^1/2∈Q

Self-consistent Sources and Conservation Laws for Super-Geng Equation Hierarchy

WEI Han-yu, CUI Zhong-yuan, XIA Tie-cheng

2016, 31(2): 201-210. doi:10.13371/j.cnki.chin.q.j.m.2016.02.011

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Based on the matrix Lie super algebra and supertrace identity, the integrable super-Geng hierarchy with self-consistent is established. Furthermore, we establish the infinitely many conservation laws for the integrable super-Geng hierarchy. The methods derived by us can be generalized to other nonlinear equation hierarchies.

Growth Theorem and Distortion Theorem for a Subclass of Parabolic Starlike Mappings

ZHANG Xiao-fei, YAN Chun-yan

2016, 31(2): 211-220. doi:10.13371/j.cnki.chin.q.j.m.2016.02.012

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In this article, we firstly introduce a subclass of parabolic starlike mappings writen as PS(β; ρ).Secondly,the growth theorem for PS(β; ρ) on the unit ball in complex Banach space is obtained. Finaly, as the application of the growth theorem of PS(β; ρ), the distortion theorem along a unit direction is also established.

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