About the Authors

Juan Zhang

Affiliation Department of Mathematics, North University of China, Taiyuan, Shan'xi, People's Republic of China

Zhen Jin

Affiliation Department of Mathematics, North University of China, Taiyuan, Shan'xi, People's Republic of China

Gui-Quan Sun

Affiliation Department of Mathematics, North University of China, Taiyuan, Shan'xi, People's Republic of China

Tao Zhou

Affiliations Web Sciences Center, University of Electronic Science and Technology of China, Chengdu, Sichuan, People's Republic of China, Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui, People's Republic of China

Shigui Ruan

ruan@math.miami.edu

Affiliation Department of Mathematics, University of Miami, Coral Gables, Florida, United States of America

Competing Interests

The authors have declared that no competing interests exist.

Author Contributions

Conceived and designed the experiments: JZ ZJ GS TZ SR. Performed the experiments: JZ ZJ GS SR. Analyzed the data: JZ ZJ GS SR. Contributed reagents/materials/analysis tools: JZ ZJ GS TZ SR. Wrote the paper: JZ ZJ GS SR.