vbackgrounds needed for later chapters. Chapter 3 provides the needed mathematical founda-
tions for PID, PRI and the geometry for the unified fields. Morephysical oriented readers can
jump directly to later chapters, Chapters 4 - 7 , for the main physical developments.
The research presented in this book was supported in part by grants from the US NSF
and the Chinese NSF. The authors are most grateful for their advice, encouragement, and
consistent support from Louis Nirenberg, Roger Temam, and Wenyuan Chen throughout our
scientific career. Our warm thanks to Jie Shen, Mickael Chekroun, Xianling Fan, Kevin
Zumbrun, Wen Masters and Reza Malek-Madani for their support, insightful discussions and
appreciation for his work.
Last but not least, we would like to express our gratitude to our wives, Li and Ping, and
our children, Jiao, Melinda and Wayne, for their love and appreciation. In particular, we are
most grateful for Li and Ping’s unflinching support and understanding during the long course
of our academic career.
March 30, 2015
Chengdu, China Tian Ma
Bloomington, IN Shouhong Wang