Our goal is to develop computer algebra methods and software
for solving functional equations, i.e. equations where the unknowns represent
functions rather than numerical values, as well as to foster the use
of such methods in engineering by producing the programs and tools
necessary to apply them to industrial problems.
We study in particular linear and nonlinear differential and
Algorithms based on algebraic theories are developed to investigate the structure of the solution set of general differential systems. Different algebraic and geometric theories are the sources of our algorithms and making bridges between them is our challenge.
Formal integrability
is the first problem that our algorithms address.
The idea is to complete a system of partial differential
equation so as to be in a position to compute the
Hilbert differential dimension polynomial or
equivalently, its coefficients,
the Cartan characters. Those provide an accurate
measure of the arbitrariness that comes in the solution set
(how many arbitrary functions of so many variables).
Closely related is the problem of determining the initial conditions
that can be freely chosen for having a well-posed problem
(i.e. that lead to existence and uniqueness of solutions).
This is possible if we can compute all the differential relations up
to a given order, meaning that we cannot obtain equations of lower order by
combining the existing equations in the system. Such a system
is called formally integrable and
numerous algorithms for making systems of partial differential
equations formally integrable have been developed using different
approaches by E. Cartan
Differential elimination is the second problem that our
algorithms deal with. One typically wants to determine what are the
lowest differential equations that
vanish on the solution set of a given differential system.
The sense in which lowest has to be understood is to be
specified. It can first be order-wise, as it is of use in the formal
integrability problem. But one can also wish to find differential
equations in a subset of the variables, allowing the model
to be reduced.
The radical differential ideal generated by a set of differential
polynomials
In the nonlinear case the best we can hope for is to have
information outside of some hypersurface.
Actually, the radical differential
differential ideal can be decomposed into components on which
the answers to formal integrability and eliminations are different.
For each component the characteristic set
delivers the information about the singular hypersurface
together with the quasi-generating set and membership test.
Triangulation-decomposition algorithms perform the task of
computing a characteristic set for all the components of the radical
differential ideal generated by a finite set of differential polynomials.
References for those algorithms are the book
chapters written by E. Hubert
The objectives for future research in the branch of triangulation-decomposition is the improvement of the algorithms, the development of alternative approaches to certain class of differential systems and the study of the intrinsic complexity of differential systems.
Another problem, specific to the nonlinear case, is the understanding
and algorithmic classification of the different behaviors of
interference of the locus of one component on the locus of another.
The problem becomes clear in the specific case of
radical differential ideals generated by a single differential
polynomial. One then wishes to understand the behavior of
non singular solutions in the vicinity of singular solutions.
Only the case of the first order differential polynomial equations
is clear. Singular solutions are either the envelope or the limit case
of the non singular solutions and the classification is
algorithmic
When the set
In addition, thanks to the works of
B. Malgrange
Though not a major subject of expertise, the topic is at the crossroads of the algorithmic themes developed in the team.
The Lie group, or symmetry group, of a differential
system is the (biggest) group of point transformations
leaving the solution set invariant.
Besides the group structure, a Lie group
has the structure of a differentiable manifold.
This double structure allows to concentrate on studying
the tangent space at the origin, the Lie algebra.
The Lie group and the Lie algebra
thus capture the geometry of a differential system.
This geometric knowledge is exploited to solve nonlinear
differential systems.
The Lie algebra is described by the solution set of a system of linear partial differential equations, whose determination is algorithmic. The dimension of the solution space of that linear differential system is the dimension of the Lie group and can be determined by the tools described in Section . Explicit subalgebras can be determined thanks to the methods developed within the context of Section .
For a given group of transformations on a
set of independent and dependent variables
there exist invariant derivations
and a finite set of differential invariants
that generate all the differential invariants
Differential Galois theory, developed first by Picard and Vessiot, then
algebraically by Kolchin, associates a linear algebraic group to a linear
ordinary differential equation or system. Many properties of its solutions,
in particular the existence of closed-form solutions,
are then equivalent to group-theoretic properties of the associated Galois
group
An exciting application of differential Galois theory to dynamical
systems is the Morales-Ramis theory, which arose
as a development of the Kovalevskaya-Painlevé analysis and Ziglin's
integrability theory
defined on a complex
We can decrease the order of that system by considering the
induced system on the normal bundle
where
In his two fundamental papers
There is a problem however in making that theory algorithmic:
the monodromy group is known only for a few
differential equations. To overcome that problem,
Morales-Ruiz and Ramis recently generalized
Ziglin's approach by replacing the monodromy group
When applying the Morales-Ramis criterion, our first step is to
find a non-equilibrium particular solution, which often lies on an
invariant submanifold. Next, we calculate the corresponding VEs and NVEs.
If we know that our Hamiltonian system possesses
Our main objectives in that field are: (i) to apply the Morales-Ramis theory to various dynamical systems occurring in mechanics and astronomy; (ii) to develop algorithms that carry out effectively all the steps of that theory; (iii) to extend it by making use of non-homogeneous variational equations; (iv) to generalize it to various non-Hamiltonian systems, e.g. for systems with certain tensor invariants; (v) to formulate theorems about partial integrability of dynamical systems and about real integrability (for real dynamical systems) in the framework of the Morales-Ramis theory;
The general theme of this aspect of our work is to develop tools that make it possible to share mathematical knowledge or algorithms between different software systems running at arbitrary locations on the web.
Most computer algebra systems deal with a lot of non algorithmic knowledge, represented directly in their source code. Typical examples are the values of particular integrals or sums. A very natural idea is to group this knowledge into a database. Unfortunately, common database systems are not capable to support the kind of mathematical manipulations that are needed for an efficient retrieval (doing pattern-matching, taking into account commutativity, etc.). The design and implementation of a suitable database raise some interesting problems at the frontier of computer algebra. We are currently developing a prototype for such a database that is capable of doing some deductions. Part of our prototype could be applied to the general problem of searching through mathematical texts, a problem that we plan to address in the near future.
The computer algebra community recognized more than ten years ago that in order to share knowledge such as the above database on the web, it was first necessary to develop a standard for communicating mathematical objects (via interprocess communication, e-mail, archiving in databases). We actively participated in the definition of such a standard, OpenMath (partly in the course of a European project). We were also involved in the definition of MathML by the World Wide Web Consortium. The availability of these two standards are the first step needed to develop rich mathematical services and new architectures for computer algebra and scientific computation in general enabling a transparent and dynamic access to mathematical components. We are now working towards this goal by experimenting with our mathematical software and emerging technologies (Web Services) and participating in the further development of OpenMath.
We have applied our algorithms and programs for computing differential Galois groups to determine necessary conditions for integrability in mechanical modeling and astronomy (see ). We also apply our research on partial linear differential equations to control theory, for example for parameterization and stabilization of linear control systems (see and ).
The diffalg library
A new release as a package will be available in early 2004. The high point of the new release is the implementation of algorithms for differential polynomial rings where derivations satisfy nontrivial commutation rules. That follows the investigations on the algebra of the differential invariants but bears already on several other fields of applications in mathematics. This new version also incorporates a specific treatment of parameters as well as improved algorithms for higher degree polynomials.
A library OreModules of
MgfunOreModules is to use the recent development
of the Gröbner basis over some Ore algebras
(non-commutative polynomial rings) in order to effectively check
some properties of
module theory (e.g. torsion/torsion-free/reflexive/projective/free
modules) and homological algebra
(e.g. free resolutions, split exact sequences, duality, extension
functor, dimensions).
A library of examples is in development but several examples
(two pendulum mounted on a car, a time-varying system of algebraic
equations, a wind tunnel model,
a two reflector antenna, an electric transmission line,
Einstein equations, Lie-Poisson structures)
are already freely available with
OreModules
The libaldor libraryAldor library, distributed with
the compiler back in 2001. During 2003, we have added new data
structures and enhanced the I/O features in order to facilitate
the porting to libaldor of external Aldor projects.
The new library, version 1.0.2 is expected to be officially
released in December 2003.
The Algebra libraryAlgebra library is being distributed together with the
Aldor compiler
The Bernina
Together with A. Maciejewski (Zielona Góra), we studied in detail
the Morales-Ramis theory from the point of view of its applications to
integrability analysis. We used it to study the integrability of the
following systems: a rigid satellite moving under influence of
geomagnetic field
The Morales-Ramis theory concerns only complex integrability.
We showed how combining some results of that theory and the Ziglin
theory makes it possible to prove real non-integrability for
real dynamical systems
We study the effective and efficient computation of power series solutions of partial differential systems.
For a finite set of differential polynomials, triangulation-decomposition algorithms output a finite set of regular differential chains. The set of zeros of the original system is the union of the non singular zeros of the output regular differential chains. For each regular differential chain we can determine the generic initial conditions that lead to existence and uniqueness of a power series solution. The proof of this formal integrability of regular differential chains is constructive and leads to a first algorithm to compute the power series. The coefficients of the power series are obtained by successive derivations of the differential polynomials in the regular differential chains.
We proposed another approach to the computation of the power series
solution of regular differential chains. The method is of Newton type:
at each step, the linearisation of the system is used to
double the size of the approximation so far. More precisely,
if the power series is
known up to order
Our results were presented at the conference ISSAC 2003 in
Philadelphia
We seek to further improve the computation performance by using modular computation followed by a lifting of Hensel type. For that, we must work out a bound on the coefficients of the power series solutions.
The basic assumption of classical differential algebra and differential elimination is that the derivations do commute, which is the standard case arising from systems of partial differential equations. We developed the a generalization of the theory to the case where the derivations satisfy nontrivial commutation rules. That situation arises, for instance, when we consider a system of equations on the differential invariants of a Lie group action.
It had already been noted
The results are written in diffalg Maple library (cf. ).
The reduction developed in 2002 with B.M. Trager (IBM Research)
has been applied to the lazy Hermite integration of algebraic functions,
which has been implemented in Maple by Damien Jamet. A new improvement,
computing a good initial basis of integral elements via Newton sums, was
discovered and implemented, yielding very efficient results for
integration. This work has been presented at the MEGA'2003
conference Journal of Symbolic Computation.
In addition, since the 2002 implementation of the genus computation
has shown that computing the local exponents at singularities is
a significant computing step, a hybrid symbolic/numeric approach
that approximates those exponents using LAPACK has been implemented
in Aldor by Rohit Shrivastava.
In collaboration with E. Compoint
In addition, we have also completed our implementation in the
We continued our work towards developing effective
algorithms and programs to factor systems of linear partial
differential or (
This year, we generalized Beke's approach to factoring
finite-dimensional
The equation
In OreModules (),
these results are illustrated on different
systems (e.g. two pendulum mounted on a car, a
time-varying system of algebraic equations, a wind tunnel model,
a two reflector antenna, an electric transmission line,
Einstein equations, Lie-Poisson structures).
Let us remark that the problem of parameterizing all the solutions of
linear controllable multidimensional systems
has been extensively studied by the school of M. Fliess in France and
J. C. Willems in the Netherlands
In
For linear systems of partial differential equations, we
have shown
In infinite-dimensional linear systems the fractional representation approach to systems
In Youla-Kuera
parameterization of all stabilizing controllers
We have also shown stable range, introduced in algebra
by H. Bass, plays a central role in the
strong stabilization problem (stabilization of a plant
by means of a stable controller)
We have also shown
All these results have been summarized in some lecture notes for
the summer school ``International School in Automatic Control of Lille''
entitled ``Control of Distributed Parameter Systems: Theory and
Applications'', organized by M. Fliess, 02-06/09/02,
Ecole Nationale de Lille (France).
They will appear
Through this one-year contract,
WMI (makers of the computer algebra system Maple)
supports the collaboration between Café and Prof. Abramov's group
at the Russian academy of science, by paying for travel and living expenses.
In exchange, WMI gets early reports on the results of that collaboration
as well as a faster track between algorithmic developments and their
distribution as part of Maple.
A prolongation for 2004 is awaiting signature.
A. Quadrat is a member of the working group ``Systèmes à Retards''
of the GdR Automatique.
Café continues its participation in the OpenMath
(IST-2000-28719)
Thematic Network which is a follow on from the earlier ESPRIT Project.
The network's main activities are to organize workshops bringing together
people working on OpenMath from around the world, to provide a continued
focus-point for the development of the OpenMath Standard, to facilitate
European participation in the W3C Math Working group, to coordinate the
development of OpenMath and MathML tools, to coordinate the development of
OpenMath and MathML applications and to disseminate information about
OpenMath and MathML.
The current membership of the network is NAG Ltd (UK, coordinator), the University of Bath (UK), Stilo Technology Ltd (UK), INRIA, the University of St Andrews (UK), the Technical University of Eindhoven (Netherlands), Springer Verlag (Germany), the University of Nice Sophia Antipolis, ZIB (Germany), Explo-IT Research (Italy), RISC (Austria), German Research Center for Artificial Intelligence (Germany), and the University of Helsinki (Finland).
Our PAI Alliance project
moving frames and differential systems continued
to support the collaboration
between E. Hubert and Elisabeth Mansfield (University of Canterbury at Kent)
during 2003.
K. Avratchenkov (MISTRAL), P.A. Bliman (SOSSO) and A. Quadrat (Café)
have a collaboration with Prof. K. Galkowski's group at the University
of Zielona Góra (Poland)
within the framework of an exchange research program PAI Polonium
entitled ``Theory and applications of
Under the supervision of A. Quadrat, the project
``Computational methods in linear control systems''
of Daniel Robertz, PhD student at the
University of Aachen (Germany), has been granted from the
This was the second and last year of our supported collaboration with
the Ontario Research Center for Computer Algebra on the topic of
extension of classical differential algebra and related software.
A meeting between Greg Reid, Evelyne Hubert and Nicolas Le Roux was held in
Philadelphia at the occasion of the ISSAC'2003 conference.
Gregory Reid plans to spend his sabbatical term in our project in the
fall of 2004.
Our collaboration with Prof. S.A. Abramov (Moscow) continues to be supported
by the Liapunov institute within the project
Hypergeometric Computer Algebra.
During 2003, we improved the LinearFunctionalSystems Maple
package, which implements the equation solvers that we developed in
the previous years, and performed some extensive benchmarks
showing its superiority to all the other known solvers. We also
explained in a report the differences with the other algorithms and
the reasons for the superior benchmark results
Following the termination of our PRA with Z. Li (Academia Sinica), our collaboration continues with the co-direction of the thesis of Min Wu, who is alternating 6-month stays in our project and in Beijing.
Stéphane Dalmas was a member of the W3C Math Working Group. This group was responsible for defining MathML, an XML application for describing mathematical notation and capturing both its structure and content. The goal of MathML is to enable mathematics to be served, received, and processed on the World Wide Web, just as HTML has enabled this functionality for text.
In 2003 the Working Group released
Under the sponsorship of the Control Training Site (Sect. ),
Daniel Robertz visited our project for three months (February-April 2003)
in order to collaborate on the
OreModules library (see ).
Within the framework of our PAI Alliance (Sect. ), Peter Clarkson and Elizabeth Mansfield (U. of Canterbury at Kent, UK) visited our project for one week in June 2003 to work with E. Hubert on moving frames and differential systems.
Ralf Hemmecke (RISC Linz, Austria) visited our project for one week
in November 2003 in order to start porting his libaldor and Algebra libraries
(see ).
A PAI Amadeus proposal for 2005 is being prepared.
Yuri Rappoport (CC RAS, Moscow) gave a talk in the Café seminar
in February 2003.
Andrzej Maciejewski (Zielona Góra, Poland) visited our project for three 1-week periods in March, October and December, to work with M. Bronstein, M. Przybylska and J-A. Weil on the integrability of dynamical systems.
As part of our Liapunov project (see ), S.A. Abramov and D.E. Khmelnov (CC RAS, Russia) visited our project for two 10-day periods in May and November 2003.
As part of our joint library development for the Aldor compiler
(see ), Stephen Watt (UWO, Canada)
visited our project for two 1-week periods in June and December 2003,
and Marc Moreno Maza (UWO, Canada) for one week in June 2003.
Lourdes Juan (Texas Tech University) visited our project for one week
in June 2003 and presented her results on the inverse problem in
differential Galois theory in the Café seminar.
Shiva Shankar (Chennai Mathematical Institute, India) visited our project for
1 week in July 2003 to work with A. Quadrat on behavioral control theory,
which he presented in the Café seminar.
Kathy Horadam (Melbourne, Australia) presented differential cryptanalysis
in the Café seminar in September 2003.
Evelyne Hubert was a member of the program committee of the ISSAC'2003 conference.
The goals of the association
femmes & mathématiques
Evelyne Hubert is member of the council of the association.
As such she participates to the monthly meetings of the
council to propose, discuss and
undertake the actions of the association. Her major contribution
this year is chairing the organization of the
forum des jeunes mathématiciennes
The forum is a biennial francophone conference where senior and junior female mathematicians can meet so as to favor mentoring, role model and identification. Beside the scientific talks, the forum stages debates on subject relating to science and education and talks given by researchers in humanities working on women studies. The present forum focuses on mathematics for (natural) sciences. It has received the financial support of both INRA and INRIA.
In collaboration with Sylvie Poupinel (ACI Grid),
Evelyne Hubert launched two
sessions of
Séminaires CroisésSéminaires Croisés provide opportunities
for students to present formally their work in a reasonably relaxed
atmosphere. It is an occasion for all to interact constructively with
other teams on the site.
In 2003, there has been 6 thematic days with a total of 24 talks, a
fair success.
Evelyne Hubert is a member of the Committee of Doctoral
Studies at INRIA Sophia (comité du suivi doctoral)
chaired by Thierry Vieville.
The committee evaluates the documents for new PhDs and
postdoctoral fellows and serves for advice in the course of
the doctoral studies.
The committee members act as moderators when required
by supervisors or students and possibly investigate
after decision of the committee.
The committee is also in charge of ranking the candidacies for doctoral and
postdoctoral grants.
Evelyne Hubert is a member of the Colors Committee chaired by
Rose Dieng. The committee is in charge of selecting project of
collaborations of INRIA teams with local industrial or academic actors.
In the framework of the Graduiertenkolleg of the University of Aachen
(Germany), A. Quadrat was invited to organize a summer school on the
effective algebraic aspects of linear control theory (theory,
applications and packages).
The
A. Quadrat organized the unique invited session at the Workshop on
Time-Delay Systems, IFAC, which has been held at INRIA Rocquencourt
in September 2003.
The
M. Bronstein is a member of the ILC (Industrial Liaison Committee),
advising body for the research center
M. Bronstein is a member of the editorial boards for the Journal
of Symbolic Computation and for the Algorithms and Computation
in Mathematics Springer monograph series.
M. Bronstein was the program chair for the ISSAC'2003 conference (Philadelphia, August 2003).
M. Bronstein has been elected vice-chair of the
J-A. Weil has organized the sessions "Equations differentielles I"
and "Equations differentielles II" at the conference
Evelyne Hubert presented a practical course of
computer algebra in the special week
Immersion Mathématique et Informatique
Evelyne Hubert has taught computer science and computer algebra in
the classes préparatoires scientifiques
at the Centre International de Valbonne (72 hours).
In the framework of the Graduiertenkolleg ``Mathematics and Practice'' of the University of Kaiserslautern (Germany), A. Quadrat was invited to lecture (9 hours) on effective algebraic analysis and its applications in October 2003. 20 graduate students, lecturers and professors, working either on effective algebra and symbolic computation or on control theory, attended the lectures.
M. Bronstein participated as examiner on the doctoral panel of Anne Desidéri-Bracco (UNSA, December 2003).
Doctorates in progress in the project:
Thomas Cluzeau, University of Limoges:
Algorithmique modulaire des équations différentielles
linéaires.
Co-directed by Moulay Barkatou (University of Limoges)
and J-A. Weil.
Nicolas Le Roux, University of Limoges:
Local study of nonlinear differential systems.
Co-directed by Moulay Barkatou (University of Limoges)
and Evelyne Hubert.
Min Wu, UNSA and Academia Sinica (Beijing):
Factorization of systems of linear partial differential equations.
Co-directed by Ziming Li (Academia Sinica) and Manuel Bronstein.
Internships completed in 2003:
Damien Jamet, DEA internship from the University of Caen:
Simina Maris, DEA internship from the University of Limoges:
Daniel Robertz, University of Aachen, ``Computational Methods in Linear Control Theory'', directed by Alban Quadrat.
Rohit Shrivastava, IIT Delhi,
M. Bronstein presented his work at the following
conferences and workshops:
the
The
The first
In addition, he has attended the ISSAC'2003 conference
(Philadelphia, August 2003)
and the
S. Dalmas
participated in the W3C technical plenary meeting (Boston, March 2003).
S. Dalmas and M. Gaëtano attended an
OpenMath Thematic Network workshop (Bremen, November 2003).
M. Gaëtano attended the
E. Hubert
was invited by Michael Singer to the Mathematical Science Research Institute in Berkeley (USA) in March and April.
participated in March in the workshop
Computational Commutative Algebra in MSRI.
participated in the
CIMPA school on polynomial systems
presented her work on differential algebra for derivations that
satisfy non trivial commutation rules at the
First Latin American workshop on
polynomial systemsLaboratoire d'Arithmetique,
de Calcul Formel et d'Optimisation in Limoges
and at the Laboratoire Sciences et Technologies de
l'Information et de la Communication à Polytechnique.
in collaboration with Nicolas Le Roux presented their work on
the computation of power series solutions of nonlinear differential
systems at the International Symposium on Symbolic
and Algebraic Computation that was held in Philadelphia (USA) in
August.
visited twice the Laboratoire d'Arithmetique,
de Calcul Formel et d'Optimisation in Limoges for collaboration with
Nicolas Le Roux and Moulay Barkatou.
M. Przybylska presented her work at the following
conferences and workshops:
Young Researchers Marie Curie Meeting, Paris, France (March 2003).
The first
Workshop ``Structural Dynamical Systems in Linear Algebra and Control. Computational Aspects'', Capitolo-Monopoli, Bari, Italy (June 2003).
XIV International Congress on Mathematical Physics, Lisbon, Portugal (July 2003).
Geometry, Dynamical Systems and Celestial Mechanics. A tribute to Alain Chenciner, Paris, France (October 2003).
A. Quadrat presented his work at the following
conferences and workshops:
the
The European Control Conference (Cambridge, UK, September 2003).
A workshop of time-delay systems (INRIA Rocquencourt, September 2003).
A summer school (Otzenhausen, Germany, September 2003).
He was also invited to represent the working group
``Systèmes à Retards''of the GdR Automatique
at the ``Journées Nationales d'Automatique''
(Valenciennes, June 2003).
In the framework of our PAI Polonium (Sect. ),
A. Quadrat was invited for one week to the
university of Zielona Góra in October 2003,
where he gave seminar presentations at the
the Institute of Engineering Cybernetics (Wroclaw)
and the Institute of Control and Computational Engineering
(Zielona Góra).
He also gave a seminar presentation in the mathematics department
of the University of Kaiserslautern (Germany) in October 2003.
J.-A. Weil
has participated in:
the
The first
The
The ISSAC'2003 conference (Philadelphia, August 2003).
In addition, he has presented his work on differential Galois theory at the following conferences, workshops and seminars:
the
A Differential Galois Theory conference (Oberflockenbach, October 2003).