We study the Multi-critical Point Principle (MPP) in a complex singlet scalar extension of Standard Model (CxSM). The MPP discussed this selects model parameters so that two low-energy vacua realized by fields are degenerate. further note may inhibit electroweak phase transition (EWPT) certain class models where tree-level potential plays an essential role its realization. Despite that, we show...

Consider the sub-Riemannian Heisenberg group $\mathbb{H}$. In this paper, we answer following question: given a compact set $K \subseteq \mathbb{R}$ and continuous map $f\colon K \to \mathbb{H}$, when is there horizontal $C^m$ curve $F\colon \mathbb{R} \mathbb{H}$ such that $F|\_K = f$? Whitney originally answered question for real valued mappings, Fefferman provided complete functions defined ...

The functional type extension of Ekeland’s variational principle [7] due to Zhong [18] is logical equivalent with it.

Huili Li, Kebo Xie, Wenjun Yu, Liejie Hu, Haiyan Huang, Huijun Xie, and 4 Shuning Wang 5 6 State Key Laboratory of Microbial Technology, Shandong University, Jinan, People’s 7 Republic of China, Institute of Basic Medicine, Shandong Academy of Medical 8 Science, Jinan, People’s Republic of China, and Environment Research Institute, 9 Shandong University, Jinan, People’s Republic of China 10 11 ...