Open AccessInvertible Linear Ordinary Differential Operators Represented as a Composition of the Simplest Operators
Author(s) -
В. Н. Четвериков
Publication year - 2018
Publication title -
matematika i matematičeskoe modelirovanie
Language(s) - English
Resource type - Journals
ISSN - 2412-5911
DOI - 10.24108/mathm.0418.0000138
Subject(s) - invertible matrix , mathematics , operator (biology) , linear map , operator theory , pure mathematics , differential operator , matrix (chemical analysis) , fourier integral operator , operator norm , algebra over a field , discrete mathematics , biochemistry , chemistry , materials science , repressor , transcription factor , composite material , gene
This article is a sequel to the earlier articles, which describe the invertible ordinary differential operators and their generalizations. The generalizations are invertible mappings of filtered modules generated by one differentiation, and are called invertible D-operators. In particular, invertible ordinary linear differential operators, invertible linear difference operators with periodic coefficients, maps defined by unimodular matrices, and C-transformations of control systems are invertible D-operators. C-Transformations are those invertible transformations for which the variables of one system are expressed in terms of the variables of the other system and their derivatives.In the article we consider the invertible D-operators whose inverses are D-operators of the same type. In previous papers, a classification of invertible D-operators was obtained. Namely, a table of integers was associated to each invertible D-operator. These tables were described in a clear elementary-geometric language. Thus, to each invertible D-operator one assigns an elementary-geometric model, which is called a d-scheme of squares. The class of invertible D-operators having the same d-scheme was also described. In this paper, the invertible D-operators whose d-schemes consist of a single square are called unicellular. It is proved that any unicellular operator in some bases is given by an upper triangular matrix that differs from the identity matrix only by the first row. The main result is representation of the arbitrary invertible D-operator as a composition of unicellular operators. The minimum number of unicellular operators in such a composition is equal to the number of squares of the d-scheme of the original D-operator. As in previous papers, the used method is based on the description of d-schemes in the language of spectral sequences of algebraic complexes.The results obtained can be useful in the transformation and classification of control systems, in particular to describe flat systems.