Polynomial Operator in the Shifts in Discrete Algebraic Dynamical Systems
Ramamonjy Andriamifidisoa *
Department of Mathematics and Computer Science, Faculty of Sciences, P.O.B. 906, University of Antananarivo, 101 Antananarivo, Madagascar and Higher Polytechnics Institute of Madagascar (ISPM), Ambatomaro - Antsobolo, 101 Antananarivo, Madagascar.
Juanito Andrianjanahary
Central Bank of Madagascar, P.O.B. 550 Antaninarenina, 101 Antananarivo, Madagascar.
*Author to whom correspondence should be addressed.
Abstract
The vector space of the multi-indexed sequences over a field and the vector space of the sequences with finite support are dual to each other, with respect to an appropriate scalar product. It follows that the polynomial operator in the shift which U. Oberst and J. C. Willems have introduced to define time invariant discrete linear dynamical systems can be explained as the adjoint of the polynomial multiplication.
Keywords: Dynamical system, behavior, polynomial operator in the shift, scalar product, adjoint of a linear mapping, categories and functors.