NDLI: The computation of purity filtrations over commutative noetherian rings of operators and their application to behaviors

Content Provider	Springer Nature Link
Author	Oberst, Ulrich
Copyright Year	2013
Abstract	Due to the theoretical work and computer implementations of, for instance, Barakat, Quadrat and Robertz and their coauthors the theory of finitely generated (f.g.) modules over non-commutative regular noetherian rings of partial differential operators with variable coefficients like the Weyl algebras and over other similar rings has become constructive in recent years. In particular these authors compute the purity or grade filtration of a f.g. module by homological means and discuss its significance for the associated behavior. Pommaret and Quadrat noted this significance already in 1999. In this note it is shown that over an arbitrary commutative noetherian ring of operators the purity filtration of a finitely generated module can be easily computed by means of the the primary decomposition of its zero submodule and indeed, with smaller complexity, by inductively computing equidimensional parts. In most books on Constructive Commutative Algebra the connection between this primary decomposition, the equidimensional parts and the purity filtration is implicitly stated for cyclic modules. For many commutative rings of operators the standard signal modules are injective cogenerators. In this case the purity filtration of the module gives rise to a corresponding filtration of the dual behavior, and the primary decomposition induces additional sum representations of the pure dimensional factors of this filtration. For non-commutative rings of operators the standard signal modules are in general neither injective nor cogenerators, and for such signal modules the usefulness of the purity filtration of the module for the determination of the behavior and its structural properties is not obvious. It is also shown by a counter-example that dimensional purity of the module or behavior does not imply dimensional purity of the initial conditions according to Riquier of the associated homogeneous Cauchy problem.
Starting Page	389
Ending Page	404
Page Count	16
File Format	PDF
ISSN	09236082
Journal	Multidimensional Systems and Signal Processing
Volume Number	26
Issue Number	2
e-ISSN	15730824
Language	English
Publisher	Springer US
Publisher Date	2013-10-29
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Purity filtration decomposition Multidimensional behavior filtration Riquier’s initial condition Circuits and Systems Electrical Engineering Signal, Image and Speech Processing Artificial Intelligence (incl. Robotics)
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Artificial Intelligence Signal Processing Information Systems Computer Science Applications Software Hardware and Architecture

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