We proved the intermediate value theorem for continuous function at closed interval by constructing auxiliary sequence ingeniously and applying compact theorem as well as Cauchy convergence criterion.

In this contribution we study the torsional properties for the different chiral single-walled carbon nanotube(SWCNT) based on the constitutive model established by the higher-order Cauchy-Born rule.

To counter hydrogen atom system,by expressing the correspondence principle with two mathematics criterions,supplementing the stationary state hypothesis,and uniting them with the transition radiation hypothesis,the ladder criterion is obtained.