Abstract
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.
Highlights
Ëèíåéíûå äèôôåðåíöèàëüíûå îïåðàòîðûÀíàëîãè÷íîå ïðåäñòàâëåíèå ñóùåñòâóåò äëÿ ñå÷åíèé ðàññëîåíèÿ ζ, êîîðäèíàòû â ñëîÿõ êîòîðîãî áóäåì îáîçíà÷àòü ÷åðåç q1,
Èññëåäóþòñÿ îáðàòèìûå ëèíåéíûå äèôôåðåíöèàëüíûå îïåðàòîðû ñ îäíîé íåçàâèñèìîé ïåðåìåííîé, îáðàòíûå ê êîòîðûì òàêæå ÿâëÿþòñÿ äèôôåðåíöèàëüíûìè
The results obtained can be useful in the transformation and classification of control systems, in particular to describe flat systems
Summary
Àíàëîãè÷íîå ïðåäñòàâëåíèå ñóùåñòâóåò äëÿ ñå÷åíèé ðàññëîåíèÿ ζ, êîîðäèíàòû â ñëîÿõ êîòîðîãî áóäåì îáîçíà÷àòü ÷åðåç q1, . Ìíîæåñòâî âñåõ ëèíåéíûõ äèôôåðåíöèàëüíûõ îïåðàòîðîâ ïîðÿäêà íå âûøå k, äåéñòâóþùèõ èç P â Q, ïðåäñòàâëÿåò ñîáîé A-ìîäóëü îòíîñèòåëüíî ýòîãî óìíîæåíèÿ. Îáúåäèíåíèå Diff+k (P, Q) äëÿ âñåõ k ≥ 0 ïðåäñòàâëÿåò ñîáîé A-ìîäóëü áåñêîíå÷íîé ðàçìåðíîñòè. Äèôôåðåíöèàëüíûé îïåðàòîð ∆ : P → Q íàçûâàþò (äâóñòîðîííå) îáðàòèìûì, åñëè ñóùåñòâóåò òàêîé äèôôåðåíöèàëüíûé îïåðàòîð ∆−1 : Q → P, ÷òî êîìïîçèöèÿ ∆−1 ◦ ∆ åñòü òîæäåñòâåííîå îòîáðàæåíèå ìîäóëÿ P, à êîìïîçèöèÿ ∆ ◦ ∆−1 | òîæäåñòâåííîå îòîáðàæåíèå ìîäóëÿ Q.  ýòîì ñëó÷àå îïåðàòîð ∆−1 íàçûâàþò îáðàòíûì ê ∆.  êîîðäèíàòàõ ýëåìåíòû P, Q óäîáíî ïðåäñòàâëÿòü â âèäå ñòîëáöîâ ôóíêöèé, à ëèíåéíûé îïåðàòîð ∆ : P → Q | â âèäå ìàòðèöû ñêàëÿðíûõ îïåðàòîðîâ. Íåòðóäíî äîêàçàòü, ÷òî îáðàòèìûé ëèíåéíûé äèôôåðåíöèàëüíûé îïåðàòîð çàäàåòñÿ êâàäðàòíîé ìàòðèöåé. Íåòðóäíî óáåäèòüñÿ â òîì, ÷òî äàííûå îïåðàòîðû ÿâëÿþòñÿ âçàèìíî îáðàòíûìè
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