Various parallel and diversive aspects of the mathematical fluctuations theory with the related standing issues

Abstract

This presentation is devoted to explain the important aspects of the Mathematical Fluctuations Theory. Although its name started to be used basically by us quite recently it is not a completely independent issue from the other existing ones. It uses many well known concepts of linear algebra, especially Hilbert spaces and matrix representations. This paper bears the characteristics of a review article together with the recent developments in the concrete applications and abstract ideas related to this theory. Main emphasis focuses on the definition and meaning of the fluctuation concept which is related to somehow truncation errors. Although all of them are originated from the same idea, the fluctuations in functions, in operators, and in matrix representations are tried to be distinguished from each others. To this end certain simple illustrations are given to easily explain the authors' ideas about these concepts and their utilization in some applications.