ICMS 2016 Session: Software for the Symbolic Study of Functional Equations

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Aim and Scope

The aim of this session is to present softwares or packages dedicated to the symbolic or symbolic-numeric treatment of (systems of) functional equations such as ordinary or partial differential equations, (q-)difference equations,  differential time-delay equations, discrete-time equations, ... The presentations are expected to enhance interactions between developers and potential users on the one hand, and amongst developers/researchers on the other hand. Survey talks of research groups, reviews of the state of the art, works which make substantial use of such existing softwares for miscellaneous applications, and demonstrations of softwares under development are also welcomed.

Topics (including, but not limited to)

Submission Guidelines

The duration of each talk is 30 minutes. For more details, we kindly refer to the daily program.

First slot :
  1. Suzy S. Maddah
    Overview talk

    In this talk, we give an overview of the packages which have been developed within the Computer Algebra (Calcul Formel) group at the University of Limoges, for the symbolic treatment of (systems of) functional equations. We illustrate with examples on the analysis of singularities and the computation of local formal, rational, and hyper-exponential solutions, of certain classes of ordinary (perturbed) and partial linear differential systems.

  2. Thomas Cluzeau
    Algorithms and related Maple packages for integrable connections and planar polynomial vector fields

    In the previous talk, Suzy S. Maddah presents algorithms (and related packages) handling first order systems of linear differential equations. In this talk, I will consider two kind of generalizations of linear differential systems.

    (a) Integrable connections, i.e., a class of systems of first order linear partial differential systems. I will present algorithms for computing closed form (e.g., rational and hyperexponential) solutions of  such systems and illustrate a dedicated Maple package. This work has been done within the Computer Algebra group at the University of Limoges (France) in collaboration with M. A. Barkatou, C. El Bacha and J.-A. Weil. (Ref: Computing Closed-Form Solutions of Integrable Connections, Proc. of ISSAC 2012: 43-50)

    (b)  Planar polynomial vector fields, i.e., a class of systems of two non linear (polynomial) differential equations in two dependent variables of one independent variable (e.g., time). I will give the idea of an efficient algorithm for computing rational first integrals of such vector fields. A Maple implementation of this algorithm within a dedicated package will be demonstrated. This is a joint work with A. Bostan, G. Chèze and J.-A. Weil.  (Ref: Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields, Math. of Comp. 85: 1393-1425, 2016)

  3. Jamal Hossein Poor
    Normal forms for operators via Gröbner bases in tensor algebras

    We propose a general algorithmic approach to noncommutative operator algebras generated by linear operators. Many cases arising in applications, like differential and difference operators, are covered by skew polynomials. Integral operators, however, can not be treated in this framework since their normal forms do not match the normal forms of skew polynomials. In our approach, we model operator algebras as suitable quotients of tensor algebras instead. They naturally capture the multilinearity of composition of linear operators and make a basis-free treatment of multiplication operators possible. In order to work with reduction systems in tensor algebras, Bergman’s setting provides a tensor analog of Gröbner bases. Given a Gröbner basis, the structure of normal forms can be determined via a combinatorial word problem.
    We introduce a refinement of Bergman’s setting, which allows for smaller reduction systems and tends to make computations more efficient. Verification of the confluence criterion based on S-polynomials defined in terms of pure tensors has been implemented as a Mathematica package. Our implementation can also be used for computer-assisted construction of Gröbner bases starting from basic identities of operators. We illustrate our approach and the software using integro-differential operators as an example.
    Joint work with Clemens G. Raab, and Georg Regensburger.

    Second slot:

  4. Albert Heinle
    Factoring Elements in G-Algebras with 'ncfactor.lib'

    The library `ncfactor.lib' initiated the support for non-commutative factorization in the computer algebra system Singular. The last major update added functions to factor elements in linear partial differential, difference and certain q-differential operators in n variables (Giesbrecht, Heinle, Levandovskyy, ISSAC'14 and the follow-up paper in JSC 2015).  Now we announce an algorithm to factor elements in any given G-algebra, with minor assumptions on the underlying field.  We are going to present our implementation of this algorithm, and discuss how our findings enable us to generalize methods formerly only defined for commutative rings, like e.g. the factorized Groebner basis algorithm.
    Unsurprisingly, for certain choices of G-algebras, there exist better factorization techniques that leverage specific structural properties. We will show certain characteristics of a given algebra that can be used to cut down computational costs while computing factorizations.
    The key mathematical result, providing the termination on our factorization algorithm is a recent discovery (Bell, Heinle, Levandovskyy, Trans. AMS 2015) that a big class of algebras, including G-algebras, are finite factorization domains.  We will outline further consequences and future research possibilities.
    Joint work with Viktor Levandovskyy.

  5. Viktor Levandovskyy
    Algorithms for systems of linear functional equations and their implementation in Singular

    A system Singular:Plural provides a user with the wide range of implemented algorithms for noncommutative G-algebras. In particular, classical algebras of linear functional operators as well as their tensor products are supported.
    We give an overview of recent developments both in algorithms and in the implementation in Singular:Plural, focusing on Ore localization, dimensions, Weyl closure and various constructions with (not only) differential modules with applications to the study of their solution spaces.

  6. Thomas Cluzeau and Christoph Koutschan
    Effective algebraic analysis approach to linear systems over Ore algebras

    Physical systems are typically modeled by systems of functional equations; these can be differential equations, differential time-delay systems, discrete-time systems, etc.  The algebraic analysis approach studies such systems from a purely algebraic viewpoint, using D-module theory and homological algebra techniques. The different types of systems can be uniformly described by using the general concept of Ore algebras.  Recently, we have developed the Mathematica package OreAlgebraicAnalysis that implements (1) Groebner basis techniques for finitely presented left modules over Ore algebras, (2) algorithms for deciding module-theoretic properties, such as torsion-freeness, reflexivity, projectiveness, freeness, and (3) algorithms from homological algebra, such as computation of free resolutions and projective dimension. The OreAlgebraicAnalysis package makes use of the very generic implementation of Ore algebras in the package HolonomicFunctions, that also provides the computation of Groebner bases in such rings. We present and demonstrate these packages by means of examples from control theory.
    Joint work with Alban Quadrat, Maris Tonso.