Communications in Mathematical Research CMR Volume 32, Number 1

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Volume 32, Number 1

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Triple Crossing Numbers of Graphs

Tanaka Hiroyuki & Teragaito Masakazu

10.13447/j.1674-5647.2016.01.01

Commun. Math. Res., 32 (2016), pp. 1-38.

Preview Full PDF 2229 25681 Abstract

Abstract

We introduce the triple crossing number, a variation of the crossing number, of a graph, which is the minimal number of crossing points in all drawings of the graph with only triple crossings. It is defined to be zero for planar graphs, and to be infinite for non-planar graphs which do not admit a drawing with only triple crossings. In this paper, we determine the triple crossing numbers for all complete multipartite graphs which include all complete graphs.

A New Generalized FB Complementarity Function for Symmetric Cone Complementarity Problems

Yunsheng Zhang & Leifu Gao

10.13447/j.1674-5647.2016.01.02

Commun. Math. Res., 32 (2016), pp. 39-46.

Preview Full PDF 2024 25587 Abstract

Abstract

We establish that the generalized Fischer-Burmeister (FB) function and penalized Generalized Fischer-Burmeister (FB) function defined on symmetric cones are complementarity functions (C-functions), in terms of Euclidean Jordan algebras, and the Generalized Fischer-Burmeister complementarity function for the symmetric cone complementarity problem (SCCP). It provides an affirmative answer to the open question by Kum and Lim (Kum S H, Lim Y. Penalized complementarity functions on symmetric cones. $J$. $Glob$. $Optim$.. 2010, 46: 475–485) for any positive integer.

Normality Criteria of Meromorphic Functions Concerning Shared Analytic Function

Qi Yang

10.13447/j.1674-5647.2016.01.03

Commun. Math. Res., 32 (2016), pp. 47-56.

Preview Full PDF 1869 25561 Abstract

Abstract

In this paper, we use Pang-Zalcman lemma to investigate the normal family of meromorphic functions concerning shared analytic function, which improves some earlier related results.

L-Octo-Algebras

Huihui An & Zhichun Wang

10.13447/j.1674-5647.2016.01.04

Commun. Math. Res., 32 (2016), pp. 57-69.

Preview Full PDF 1841 24819 Abstract

Abstract

L-octo-algebra with 8 operations as the Lie algebraic analogue of octo-algebra such that the sum of 8 operations is a Lie algebra is discussed. Any octo-algebra is an L-octo-algebra. The relationships among L-octo-algebras, L-quadri-algebras, L-dendriform algebras, pre-Lie algebras and Lie algebras are given. The close relationships between L-octo-algebras and some interesting structures like Rota-Baxter operators, classical Yang-Baxter equations and some bilinear forms satisfying certain conditions are given also.

Ore Extensions over Weakly 2-Primal Rings

Yao Wang, Meimei Jiang & Yanli Ren

10.13447/j.1674-5647.2016.01.05

Commun. Math. Res., 32 (2016), pp. 70-82.

Preview Full PDF 1732 24569 Abstract

Abstract

A weakly 2-primal ring is a common generalization of a semicommutative ring, a 2-primal ring and a locally 2-primal ring. In this paper, we investigate Ore extensions over weakly 2-primal rings. Let $α$ be an endomorphism and $δ$ an $α$-derivation of a ring $R$. We prove that (1) If $R$ is an $(α, δ)$-compatible and weakly 2-primal ring, then $R[x; α, δ]$ is weakly semicommutative; (2) If $R$ is $(α, δ)$-compatible, then $R$ is weakly 2-primal if and only if $R[x; α, δ]$ is weakly 2-primal.

An Identity with Skew Derivations on Lie Ideals

Zhengping Wang, Rehman Ur Nadeem & Shuliang Huang

10.13447/j.1674-5647.2016.01.06

Commun. Math. Res., 32 (2016), pp. 83-87.

Preview Full PDF 2150 25181 Abstract

Abstract

Let $R$ be a 2-torsion free prime ring and $L$ a noncommutative Lie ideal of $R$. Suppose that $(d, σ)$ is a skew derivation of $R$ such that $x^s d(x)x^t = 0$ for all $x ∈ L$, where $s, t$ are fixed non-negative integers. Then $d = 0$.

Normality Criteria of Meromorphic Functions

Qiong Wang, Wenjun Yuan, Wei Chen & Honggen Tian

10.13447/j.1674-5647.2016.01.07

Commun. Math. Res., 32 (2016), pp. 88-96.

Preview Full PDF 1817 24677 Abstract

Abstract

In this paper, we consider normality criteria for a family of meromorphic functions concerning shared values. Let $\mathcal{F}$ be a family of meromorphic functions defined in a domain $D, m, n, k$ and $d$ be four positive integers satisfying $m ≥ n + 2$ and $d ≥ \frac{k + 1}{m − n − 1}$, and $a(≠0), b$ be two finite constants. Suppose that every $f ∈ \mathcal{F}$ has all its zeros and poles of multiplicity at least $k$ and $d$, respectively. If $(f^n)^{(k)}−af^m$ and $(g^n)^{(k)}−ag^m$ share the value $b$ for every pair of functions $(f, g)$ of $\mathcal{F}$, then $\mathcal{F}$ is normal in $D$. Our results improve the related theorems of Schwick (Schwick W. Normality criteria for families of meromorphic function. $J$. $Anal$. $Math$., 1989, 52: 241–289), Li and Gu (Li Y T, Gu Y X. On normal families of meromorphic functions. $J$. $Math$. $Anal$. $Appl$., 2009, 354: 421–425).

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