`OreModules (click here)`

is a `Maple`

package dedicated to *module theory* and
*homological algebra* for finitely presented modules defined over an *Ore
algebra* of functional operators (e.g., ordinary or partial
differential operators, shift operators, time-delay operators,
difference operators) available in the Maple package
`Ore_algebra`

, and to their
applications in *mathematical systems theory* and *mathematical physics*.

Within this algebraic unified framework, `OreModules`

handles the
following classes of linear functional systems:

- ordinary differential systems,
- partial differential systems,
- multidimensional discrete systems,
- differential time-delay systems,
- multidimensional convolutional codes...

The `OreModules`

package is an implementation of algorithms
which compute different invariants or
algebraic objects associated with a finitely presented left module
over an Ore algebra such that:

- (shortest) free resolution,
- projective dimension,
- Hilbert series,
- extension modules with values in the ring,
- (minimal, chain, injective) parametrizations...

It can also be used to check whether or not a finitely presented module admits torsion elements and if so, to compute a generating set, or if it is torsion-free, reflexive, projective, stably free or free. If the module is torsion-free, then a parametrization which parametrizes its solutions space can be computed.

In the context of *linear control systems*, the main features of `OreModules`

are
the following:

- compute a family of generators for the autonomous elements,
- decide controllability and parametrizability,
- compute (minimal, chain, injective) parametrizations,
- compute Monge parametrizations,
- compute Bezout identities (left/right/generalized inverses),
- decide flatness (also pi-freeness),
- study linear quadratic problems...

For more details, see:

- F. Chyzak, A. Quadrat, D. Robertz,
*Effective algorithms for parametrizing linear control systems over Ore algebras*, Applicable Algebra in Engineering, Communications and Computing, 16 (2005), 319-376. - F. Chyzak, A. Quadrat, D. Robertz,
*OreModules: A symbolic package for the study of multidimensional linear systems*, in Applications of Time-Delay Systems, J. Chiasson, J.-J. Loiseau (Eds.), Lecture Notes in Control and Information Sciences 352, Springer, 2007, 233-264.

The package `OreModules`

was initiated by
F. Chyzak and
A. Quadrat, and
further developed by
D. Robertz and A. Quadrat.

A Mathematica version of the `OreModules`

package, called `OreAlgebraicAnalysis`

, is also available. It was developed by
Maris Tõnso,
T. Cluzeau and A. Quadrat
within the PHC
Parrot project CASCAC. The Mathematica version of the
`OreModules`

package is based on the
implementation of Gröbner bases over Ore algebras available in the
Mathematica HolonomicFunctions package developed by Christoph Koutschan.