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Junjie Zhao, Xiquan Liang and Li Yan

Several Higher Differentiation Formulas of Special Functions

In this paper, we proved some basic properties of higher differentiation, and higher differentiation formulas of special functions [4].

MML identifier: HFDIFF 1, version: 7.8.10 4.100.1011

Open access

Li Yan, Xiquan Liang and Junjie Zhao

Gauss Lemma and Law of Quadratic Reciprocity

In this paper, we defined the quadratic residue and proved its fundamental properties on the base of some useful theorems. Then we defined the Legendre symbol and proved its useful theorems [14], [12]. Finally, Gauss Lemma and Law of Quadratic Reciprocity are proven.

MML identifier: INT 5, version: 7.8.05 4.89.993

Open access

Xiquan Liang, Li Yan and Junjie Zhao

Linear Congruence Relation and Complete Residue Systems

In this paper, we defined the congruence relation and proved its fundamental properties on the base of some useful theorems. Then we proved the existence of solution and the number of incongruent solution to a linear congruence and the linear congruent equation class, in particular, we proved the Chinese Remainder Theorem. Finally, we defined the complete residue system and proved its fundamental properties.

Open access

Tao Sun, Junjie Zhao and Xiquan Liang

BCI-algebras with Condition (S) and their Properties

In this article we will first investigate the elementary properties of BCI-algebras with condition (S), see [8]. And then we will discuss the three classes of algebras: commutative, positive-implicative and implicative BCK-algebras with condition (S).

MML identifier: BCIALG 4, version: 7.8.09 4.97.1001

Open access

Cao Jianfang, Chen Junjie and Zhao Qingshan


In order to solve the problems of security threats on workflow scheduling in cloud computing environments, the security of tasks and virtual machine resources are quantified using a cloud model, and the users’ satisfaction degree with the security of tasks assigned to the virtual resources is measured through the similarity of the security cloud. On this basis, combined with security, completion time and cost constraints, an optimized cloud workflow scheduling algorithm is proposed using a discrete particle swarm. The particle in the particle swarm indicates a different cloud workflow scheduling scheme. The particle changes its velocity and position using the evolution equation of the standard particle swarm algorithm, which ensures that it is a feasible solution through the feasible solution adjustment strategies. The simulation experiment results show that the algorithm has better comprehensive performance with respect to the security utility, completion time, cost and load balance compared to other similar algorithms.