the second law of thermodynamic (entropy principle) is of special theo-philosophical importance. it seems that this law, which is renowned as “the death of software design” and despite the order in the world, is like an incompatible universal. this article examines the concept of entropy in the three temperature, statistic, and information applications and addresses the contradiction of the sec...

The generalized inverse has numerous important applications in aspects of the theoretic research matrices and statistics. One core problems is finding necessary sufficient conditions for reverse (or forward) order laws matrix products. In this paper, by using extremal ranks Schur complement, some are given forward law A1{1,2}A2{1,2}…An{1,2}⊆(A1A2…An){1,2}.

A weak group element is introduced in a proper *-ring. Several equivalent conditions of elements are investigated. We prove that an pseudo core invertible if it both partial isometry and invertible. Reverse order law additive property the inverse presented. Finally, under certain assumption on a, a◯Wa∗=a∗a◯W presented by using normality part its group-EP decomposition.