The conditions for the sums and differences of two idempotent operators to be idempotent are generalized from a finite dimensional space to an infinite dimensional space by means of operator theory.

The author diffcussed the monotonicities of difference generated by the inequalities of the Chebyshev type, and gave some the applications.

Through the investigation and analysis of the reduplicative forms “of cha(差)” in Luotian dialect of Hubei Province,this paper sums up the semantic and pragmatic rules of using the two basic reduplicative forms of “cha(差)”:“cha(差)+X+cha(差)” and “cha(差) +cha(差)+X”,and meanwhile makes a by-probing into Luotian dialect.