Algebraic analysis

We develop effective methods of algebraic analysis, particularly algebraic D-modules, their implementations in dedicated computer algebra packages (Maple, Mathematica, GAP), and its applications to mathematical systems theory (study of the factorization, (Serre/Stafford) reduction, and decomposition problems for linear functional systems, Stafford theorems of linear PD systems), control theory (study of the structural properties of linear functional systems, system equivalences, synthesis problems), signal processing (parameter estimation problem, demodulation problems), and mathematical physics (existence of parametrizations/potentials/conversation laws, symmetries).

To do that, using of rewriting (Gröbner or Janet basis methods, Spencer’s formal integrability theory), we develop effective versions of module theory (classification of module properties, invariants, standard theorems) and homological algebra (free resolutions, extension and torsion functors, Baer’s extensions) for classes of noncommutative polynomial rings of functional operators (e.g., Weyl algebras of PD operators with polynomial coefficients, rings of difference operators with polynomial coefficients, rings of differential constant/varying/distributed delay, rings of integro-differential operators).

Recent publications

  • [PDF] [DOI] T. Cluzeau, C. Pinto, and A. Quadrat, “Further results on the computation of the annihilators of integro-differential operators,” in 2023 International Symposium on Symbolic and Algebraic Computation, Tromso, Norway, 2023, p. 9.
    [Bibtex] 
    @inproceedings{cluzeau:hal-04203853,
TITLE = {{Further results on the computation of the annihilators of integro-differential operators}},
AUTHOR = {Cluzeau, Thomas and Pinto, Camille and Quadrat, Alban},
URL = {https://inria.hal.science/hal-04203853},
BOOKTITLE = {{2023 International Symposium on Symbolic and Algebraic Computation}},
ADDRESS = {Tromso, Norway},
PAGES = {9},
YEAR = {2023},
MONTH = Jul,
DOI = {10.1145/3597066.3597083},
KEYWORDS = {Non-commutative operators ; integro-differential operators ; annihilator ; coherence},
PDF = {https://inria.hal.science/hal-04203853/file/article_issac_publie%CC%81.pdf},
HAL_ID = {hal-04203853},
HAL_VERSION = {v1},
}
  • [PDF] [DOI] M. Chartouny, T. Cluzeau, and A. Quadrat, “Algorithmic study of the algebraic parameter estimation problem for a class of perturbations,” Maple Transactions, vol. 3, 2023.
    [Bibtex] 
    @article{chartouny:hal-04203089,
TITLE = {{Algorithmic study of the algebraic parameter estimation problem for a class of perturbations}},
AUTHOR = {Chartouny, Maya and Cluzeau, Thomas and Quadrat, Alban},
URL = {https://inria.hal.science/hal-04203089},
JOURNAL = {{Maple Transactions}},
PUBLISHER = {{Western Libraries Western University}},
VOLUME = {3},
YEAR = {2023},
MONTH = Feb,
DOI = {10.5206/mt.v2i2.14467},
KEYWORDS = {Parameter estimation problem ; Inverse Cauchy problem ; Algebraic systems ; Elimination ; Annihilators ; Rings of ordinary differential operators},
PDF = {https://inria.hal.science/hal-04203089/file/CCQ_MapleTransactionsFinal-1.pdf},
HAL_ID = {hal-04203089},
HAL_VERSION = {v1},
}
  • [PDF] [DOI] A. Quadrat and R. Ushirobira, “On the Ore extension ring of differential time-varying delay operators,” in Accounting for Constraints in Delay Systems, Advances in Delays and Dynamics (ADD), volume 12, Springer, pp. 87-107, Springer, 2022, vol. ADD-2, p. pp. 87-107.
    [Bibtex] 
    @incollection{quadrat:hal-03908643,
TITLE = {{On the Ore extension ring of differential time-varying delay operators}},
AUTHOR = {Quadrat, Alban and Ushirobira, Rosane},
URL = {https://inria.hal.science/hal-03908643},
BOOKTITLE = {{Accounting for Constraints in Delay Systems, Advances in Delays and Dynamics (ADD), volume 12, Springer, pp. 87-107}},
PUBLISHER = {{Springer}},
SERIES = {Advances in Delays and Dynamics},
VOLUME = {ADD-2},
PAGES = {pp. 87-107},
YEAR = {2022},
MONTH = Apr,
DOI = {10.1007/978-3-030-89014-8\_5},
PDF = {https://inria.hal.science/hal-03908643/file/decod-2018-final.pdf},
HAL_ID = {hal-03908643},
HAL_VERSION = {v1},
}
  • [PDF] C. Chenavier, T. Cluzeau, and A. Quadrat, “Computation of Koszul homology and application to involutivity of partial differential systems,” in SSSC 2022 – 8th IFAC Symposium on System Structure and Control, Montréal, Canada, 2022.
    [Bibtex] 
    @inproceedings{chenavier:hal-03908688,
TITLE = {{Computation of Koszul homology and application to involutivity of partial differential systems}},
AUTHOR = {Chenavier, Cyrille and Cluzeau, Thomas and Quadrat, Alban},
URL = {https://inria.hal.science/hal-03908688},
BOOKTITLE = {{SSSC 2022 - 8th IFAC Symposium on System Structure and Control}},
ADDRESS = {Montr{\'e}al, Canada},
YEAR = {2022},
MONTH = Sep,
KEYWORDS = {Systems of partial differential equations ; Multidimensional systems ; Behaviours ; Cartan's involutivity ; Formal integrability ; Koszul homology ; Spencer cohomology},
PDF = {https://inria.hal.science/hal-03908688/file/Koszul_reduced2.pdf},
HAL_ID = {hal-03908688},
HAL_VERSION = {v1},
}
  • [PDF] [DOI] A. Quadrat, “An Integro-differential Operator Approach to Linear State-space Systems,” in SSSC 2022 – 8th IFAC Symposium on System Structure and Control, Montreal, Canada, 2022.
    [Bibtex] 
    @inproceedings{quadrat:hal-03908550,
TITLE = {{An Integro-differential Operator Approach to Linear State-space Systems}},
AUTHOR = {Quadrat, Alban},
URL = {https://inria.hal.science/hal-03908550},
BOOKTITLE = {{SSSC 2022 - 8th IFAC Symposium on System Structure and Control}},
ADDRESS = {Montreal, Canada},
YEAR = {2022},
MONTH = Sep,
DOI = {10.1016/j.ifacol.2022.11.299},
KEYWORDS = {Linear systems ; Continuous-time linear state-space models ; Polynomial methods ; Algebraic analysis ; Rings of integro-differential operators ; System equivalence ; Behaviours},
PDF = {https://inria.hal.science/hal-03908550/file/main_final.pdf},
HAL_ID = {hal-03908550},
HAL_VERSION = {v1},
}
  • [PDF] [DOI] A. Quadrat, “An Integro-differential-delay Operator Approach to Transformations of Linear Differential Time-delay Systems,” in SSSC 2022 – 8th IFAC Symposium on System Structure and Control, Montréal, Canada, 2022.
    [Bibtex] 
    @inproceedings{quadrat:hal-03908561,
TITLE = {{An Integro-differential-delay Operator Approach to Transformations of Linear Differential Time-delay Systems}},
AUTHOR = {Quadrat, Alban},
URL = {https://inria.hal.science/hal-03908561},
BOOKTITLE = {{SSSC 2022 - 8th IFAC Symposium on System Structure and Control}},
ADDRESS = {Montr{\'e}al, Canada},
YEAR = {2022},
MONTH = Sep,
DOI = {10.1016/j.ifacol.2022.11.301},
KEYWORDS = {Linear systems ; Systems with time-delays ; Polynomial methods ; Delay compensation},
PDF = {https://inria.hal.science/hal-03908561/file/Integro-diff-delay-final-reduced_corrected.pdf},
HAL_ID = {hal-03908561},
HAL_VERSION = {v1},
}
  • [PDF] [DOI] A. Quadrat, “An integro-differential operator approach to linear differential systems,” in MTNS 2022 – 25th International Symposium on Mathematical Theory of Networks and Systems, Bayreuth, Germany, 2022.
    [Bibtex] 
    @inproceedings{quadrat:hal-03908541,
TITLE = {{An integro-differential operator approach to linear differential systems}},
AUTHOR = {Quadrat, Alban},
URL = {https://inria.hal.science/hal-03908541},
BOOKTITLE = {{MTNS 2022 - 25th International Symposium on Mathematical Theory of Networks and Systems}},
ADDRESS = {Bayreuth, Germany},
YEAR = {2022},
MONTH = Sep,
DOI = {10.1016/j.ifacol.2022.11.054},
KEYWORDS = {Linear systems ; Polynomial methods ; Algebraic analysis ; Rings of integro-differential operators ; Parametrization ; Reachability ; Behaviour theory},
PDF = {https://inria.hal.science/hal-03908541/file/MTNS_final.pdf},
HAL_ID = {hal-03908541},
HAL_VERSION = {v1},
}
  • [PDF] [DOI] M. Chartouny, T. Cluzeau, and A. Quadrat, “On the inverse Cauchy problem for linear ordinary differential equations,” in GAMM 2021 – 92nd Annual Meeting of the International Association of Applied Mathematics and Mechanics, Kassel, Germany, 2021.
    [Bibtex] 
    @inproceedings{chartouny:hal-03530281,
TITLE = {{On the inverse Cauchy problem for linear ordinary differential equations}},
AUTHOR = {Chartouny, Maya and Cluzeau, Thomas and Quadrat, Alban},
URL = {https://inria.hal.science/hal-03530281},
BOOKTITLE = {{GAMM 2021 - 92nd Annual Meeting of the International Association of Applied Mathematics and Mechanics}},
ADDRESS = {Kassel, Germany},
EDITOR = {Wiley},
SERIES = {Applied Mathematics and Mechanics PAMM 2021},
VOLUME = {21},
NUMBER = {1},
YEAR = {2021},
MONTH = Mar,
DOI = {10.1002/pamm.202100214},
PDF = {https://inria.hal.science/hal-03530281/file/GAMM_Alban.pdf},
HAL_ID = {hal-03530281},
HAL_VERSION = {v1},
}
  • [DOI] A. Quadrat and E. Zerz, Algebraic and Symbolic Computation Methods in Dynamical Systems, E. Z. Alban Quadrat, Ed., Springer, 2020, vol. 9.
    [Bibtex] 
    @book{quadrat:hal-03070784,
TITLE = {{Algebraic and Symbolic Computation Methods in Dynamical Systems}},
AUTHOR = {Quadrat, Alban and Zerz, Eva},
URL = {https://inria.hal.science/hal-03070784},
EDITOR = {Alban Quadrat, Eva Zerz},
PUBLISHER = {{Springer}},
SERIES = {Advances in Delays and Dynamics},
VOLUME = {9},
PAGES = {311},
YEAR = {2020},
DOI = {10.1007/978-3-030-38356-5},
KEYWORDS = {Algebraic Theories ; Control Theory ; Computer Algebra System ; Symbolic Computation Method ; Algebraic Computation Methods},
HAL_ID = {hal-03070784},
HAL_VERSION = {v1},
}
  • T. Cluzeau, C. Koutschan, A. Quadrat, and M. Tõnso, “Effective algebraic analysis approach to linear systems over Ore algebras,” in Algebraic and Symbolic Computation Methods in Dynamical Systems, Springer, 2020, vol. 9, pp. 4-52.
    [Bibtex] 
    @incollection{cluzeau:hal-02436985,
TITLE = {{Effective algebraic analysis approach to linear systems over Ore algebras}},
AUTHOR = {Cluzeau, Thomas and Koutschan, Christoph and Quadrat, Alban and Tõnso, Maris},
URL = {https://hal.science/hal-02436985},
BOOKTITLE = {{Algebraic and Symbolic Computation Methods in Dynamical Systems}},
PUBLISHER = {{Springer}},
SERIES = {Advances in Delays and Dynamics},
VOLUME = {9},
PAGES = {4-52},
YEAR = {2020},
HAL_ID = {hal-02436985},
HAL_VERSION = {v1},
}
  • T. Cluzeau and A. Quadrat, “Equivalences of linear functional systems,” in Algebraic Methods and Symbolic-Numeric Computation in Systems Theory, Springer, 2020, vol. 9, pp. 53-86.
    [Bibtex] 
    @incollection{cluzeau:hal-03070672,
TITLE = {{Equivalences of linear functional systems}},
AUTHOR = {Cluzeau, Thomas and Quadrat, Alban},
URL = {https://inria.hal.science/hal-03070672},
BOOKTITLE = {{Algebraic Methods and Symbolic-Numeric Computation in Systems Theory}},
PUBLISHER = {{Springer}},
VOLUME = {9},
PAGES = {53-86},
YEAR = {2020},
KEYWORDS = {Linear systems theory ; Equivalence problem ; Control theory ; Algebraic analysis ; Computer algebra},
HAL_ID = {hal-03070672},
HAL_VERSION = {v1},
}
  • A. Quadrat and G. Regensburger, “Computing polynomial solutions and annihilators of integro-differential operators with polynomial coefficients,” in Algebraic and Symbolic Computation Methods in Dynamical Systems, Springer, 2020, vol. 9.
    [Bibtex] 
    @incollection{quadrat:hal-02436978,
TITLE = {{Computing polynomial solutions and annihilators of integro-differential operators with polynomial coefficients}},
AUTHOR = {Quadrat, Alban and Regensburger, Georg},
URL = {https://hal.science/hal-02436978},
BOOKTITLE = {{Algebraic and Symbolic Computation Methods in Dynamical Systems}},
PUBLISHER = {{Springer}},
SERIES = {Advances in Delays and Dynamics},
VOLUME = {9},
NUMBER = {87-114},
YEAR = {2020},
HAL_ID = {hal-02436978},
HAL_VERSION = {v1},
}

Packages

OreModules  A symbolic package for the study of linear functional systems over Ore algebras (Maple)

OreMorphisms A homological algebra package for factoring, reducing and decomposing linear functional systems (Maple)

OreAlgebraicAnalysis A Mathematica implementation of the OreModules and OreMorphisms packages (Mathematica)

Stafford A package dedicated to Stafford’s results on Weyl algebras and their applications (Maple)

PurityFiltration A package dedicated to the computation of the purity (codimension/bidualizing) filtration of modules and reduction to equidimensional block-triangular forms (Maple)

Foundational publications

T. Cluzeau, A. Quadrat, A new insight into Serre’s reduction problem, Linear Algebra and its Applications, vol. 483, 2015, pp. 40-100.

A. Quadrat, D. Robertz, A constructive study of the module structure of rings of partial differential operators, Acta Applicandæ Mathematicæ, vol. 133, 2014, pp. 187-234.

A. Quadrat, Grade filtration of linear functional systems, Acta Applicandæ Mathematicæ, vol. 127, 2013, pp. 27-86.

T. Cluzeau, A. Quadrat, Factoring and decomposing a class of linear functional systems, Linear Algebra and Its Applications, vol. 428, 2008, pp. 324-381.

A. Quadrat, D. Robertz, Computation of bases of free modules over the Weyl algebras, Journal of Symbolic Computation, vol. 42, 2007, pp. 1113-1141

F. Chyzak, A. Quadrat, D. Robertz, Effective algorithms for parametrizing linear control systems over Ore algebras, Applicable Algebra in Engineering, Communications and Computing, vol. 16, 2005, pp. 319-376.

J.-F. Pommaret, A. Quadrat, Algebraic analysis of linear multidimensional control systems, IMA J. Mathematical Control & Information, vol. 16, 1999, pp. 275-297

 

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