OreMorphisms: A Homological Algebraic Package for Factoring, Reducing and Decomposing Linear Functional Systems

Cluzeau, Thomas; Quadrat, Alban

doi:10.1007/978-3-642-02897-7_16

Cited by 20 publications

(30 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the constructive methods developed in this paper are dedicated to Serre's reduction, and thus are more effective than the ones developed in [7] as we shall explicitly show with different examples (e.g., compare Examples 4.7 and 4.11 with the example studied in Sect. 2 of [8]). …”

mentioning

confidence: 92%

“…Finally, using the computation of solutions ∈ D p×q of the algebraic Riccati equation XRX = X , Serre's reduction was also studied in [7,8] as a particular case of the decomposition problem which aims at studying when there exist two unimodular matrices V and W such that VRW = diag(R 11 , R 22 ) (R 11 being I r in the case of Serre's reduction). However, the constructive methods developed in this paper are dedicated to Serre's reduction, and thus are more effective than the ones developed in [7] as we shall explicitly show with different examples (e.g., compare Examples 4.7 and 4.11 with the example studied in Sect.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Serre’s Reduction of Linear Functional Systems

Boudellioua

Quadrat

2010

Math.Comput.Sci.

View full text Add to dashboard Cite

Serre's reduction aims at reducing the number of unknowns and equations of a linear functional system. Finding an equivalent presentation of a linear functional system containing fewer equations and fewer unknowns can generally simplify both the study of the structural properties of the linear functional system and of different numerical analysis issues, and it can sometimes help solving the linear functional system. The purpose of this paper is to present a constructive approach to Serre's reduction for determined and underdetermined linear functional systems.

show abstract

mentioning

confidence: 92%

mentioning

confidence: 99%

Serre’s Reduction of Linear Functional Systems

Boudellioua

Quadrat

2010

Math.Comput.Sci.

View full text Add to dashboard Cite

show abstract

“…From a more general perspective, the constructive approach to algebraic analysis initiated in [1, 3,4,7,11,12,13,14,15,16,31,32,33,34] relies on the solvability of inhomogeneous linear systems in the ring under consideration. Within category theory, this idea was axiomatized in [2] and applied, for instance, to sheaf theory.…”

Section: If Rank D (M )mentioning

confidence: 99%

“…Now, multiplying R from the left to (14), we obtain R = R G T because µ ∈ ker D (R.). Using R = C T , we get C T = R G T , i.e., (C − R G) T = 0 and using ker D (.T ) = D 1×t T 2 , there exists H ∈ D q×t such that:…”

Section: Algorithm 1 (Unimodularelement)mentioning

confidence: 99%

See 1 more Smart Citation

A Constructive Study of the Module Structure of Rings of Partial Differential Operators

Quadrat

Robertz

2014

Acta Appl Math

View full text Add to dashboard Cite

To cite this version:Alban Quadrat, Daniel Robertz. A constructive study of the module structure of rings of partial differential operators. [ Abstract:The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two to two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r + 1 elements but no fewer. These results are implemented in the Stafford package for D = A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series. Key-words:Weyl algebras, Stafford's theorems, linear systems of partial differential equations, D-modules, mathematical systems theory, constructive algebra, symbolic computation. A constructive study of the module structure of rings of partial differential operators 4

show abstract

Effective Algebraic Analysis Approach to Linear Systems over Ore Algebras

Cluzeau¹,

Koutschan²,

Quadrat³

et al. 2020

Algebraic and Symbolic Computation Methods in Dynamical Systems

View full text Add to dashboard Cite

The purpose of this paper is to present a survey on the effective algebraic analysis approach to linear systems theory with applications to control theory and mathematical physics. In particular, we show how the combination of effective methods of computer algebra − based on Gröbner basis techniques over a class of noncommutative polynomial rings of functional operators called Ore algebras − and constructive aspects of module theory and homological algebra enables the characterization of structural properties of linear functional systems. Algorithms are given and a dedicated implementation, called OREALGEBRAIC-ANALYSIS, based on the Mathematica package HOLONOMICFUNCTIONS, is demonstrated.

show abstract

OreMorphisms: A Homological Algebraic Package for Factoring, Reducing and Decomposing Linear Functional Systems

Cited by 20 publications

References 12 publications

Serre’s Reduction of Linear Functional Systems

Serre’s Reduction of Linear Functional Systems

A Constructive Study of the Module Structure of Rings of Partial Differential Operators

Effective Algebraic Analysis Approach to Linear Systems over Ore Algebras

Contact Info

Product

Resources

About