As the development of minimum mean variance criterion,the minimum mean entropy difference criterion is introduced by means of data entropy and data entropy difference and is applied to data analysis and pattern cluster.

As the development of minimum mean variance criterion,the minimum mean entropy difference criterion is introduced by means of data entropy and data entropy difference and is applied to data analysis and pattern cluster.

Also,some relations of measures of non-compactness and entropy numbers,approximation numbers be discussed in this paper.

Second part: Follo (Ricerche di Matematica, 56(2007) 89-105) established rela-tions between entropy numbers and box-counting dimension.

Then that the total entropy criterion and the inequality of Chausius were all the same in essence was indicated.

The relationship between η=-W/Q 1 of the irreversible heat engine and its cyclic direction is proved according to Clauius statement of second law of thermodynamics and entropy criterion.

According to Whalley's activation entropy criterion and Zuckcr-Hammett's hypothesis and Taft's entropy criterion, A-2 mechanisms is attained.

From the entropy criterion of isolated system,the balance equilibrium condition of multi-compound poly-phase system C_V>0,(P[]V)_(T,N_i)<0,∑l[]i=1∑l[]j=1(μ_i[]N_j)_(T,P,N_k)δN_jδN_i>0 is intensively deduced,and the physical significance of the balance equilibrium condition is given and proved.