‎in this paper, we study some kinds of majorizations on $textbf{m}_{n}$ and their linear or strong linear preservers. also, we find the structure of linear or strong linear preservers which are multiplicative, i.e.  linear or strong linear preservers like $phi $ with the property $phi (ab)=phi (a)phi (b)$ for every $a,bin textbf{m}_{n}$.

let a and b be n × m matrices. the matrix b is said to be g-row majorized (respectively g-column majorized) by a, if every row (respectively column) of b, is g-majorized by the corresponding row (respectively column) of a. in this paper all kinds of g-majorization are studied on mn,m, and the possible structure of their linear preservers will be found. also all linear operators t : mn,m ---> mn...

Abstract    Historical monument of Menar-Jonban (shaking tower) is located in the famous city of Isfahan in central Iran. Initial construction of this interesting and unique masonry monument belongs to 700 years ago. This monument has two vibrating circular towers of 7.5 m height. These towers are separated from each other by a distance of 9.2 m and constructed on top of an ancient tomb of 10 m...