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Открыта: 05-02-2002

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Александр Борисович Коновалов

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Система компьютерной алгебры GAP: Summer School on Algebraic Analysis and Computer Algebra



AACA'09

Summer School on Algebraic Analysis and Computer Algebra
New Perspectives for Applications

July 13-17, 2009
RISC, Castle of Hagenberg, Austria
http://www.risc.uni-linz.ac.at/about/conferences/aaca09/


- Organizers
Markus Rosenkranz (Austrian Academy of Sciences, RICAM, Linz, Austria)
Franz Winkler (Research Institute for Symbolic Computation, Linz,
Austria)

- Lecturers
Jean-Francois Pommaret (Ecole Nationale des Ponts et Chaussees, France)
Alban Quadrat (INRIA, Sophia Antipolis, France)

- Schedule
The course contains two modules:
13-15 July: Theoretical Module (J.-F. Pommaret)
16-17 July: Practical Module (A. Quadrat)

Each day is divided into four blocks:
8:30-10:00 / 10:30-12:00 / 13:30-15:00 / 15:30-17:00

The event is collocated with the Fourth RISC/SCIEnce Training School:
http://www.risc.uni-linz.ac.at/projects/science/school/fourth/general.html

Possible funding via RISC/SCIEnce Transnational Access Programme:
http://www.risc.uni-linz.ac.at/projects/science/access/

In case of interest, please reply to this email.

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- Abstract of Theoretical Module

With only a slight abuse of language, one can say that the birth of
the " formal
theory" of systems of ordinary differential (OD) equations or partial
differential (PD) equations is coming from the work of M. Janet in
1920 along
algebraic ideas brought by D. Hilbert at the same time in his study of
sygyzies
for finitely generated modules over polynomial rings. The work of
Janet has then
been used (without any quotation !) by J.F. Ritt when he created
"differential
algebra" around 1930, namely when he became able to add the word
"differential"
in front of most of the classical concepts concerned with algebraic
equations,
successively passing from OD algebraic equations to PD algebraic
equations. In
1965 B. Buchberger invented Groebner bases, named in honor of his PhD
advisor W.
Groebner, whose earlier 1940 work on polynomial ideals and PD
equations with
constant coefficients provided a source of inspiration. However, Janet
and
Groebner approaches suffer from the same lack of "intrinsicness" as
they both
highly depend on the ordering of the n independent variables and
derivatives of
the m unknowns.

Meanwhile, "commutative algebra", namely the study of modules over
rings, was
facing a very subtle problem, the resolution of which led to the
modern but
difficult "homological algebra" with sequences and diagrams. Roughly,
one can
say that the problem was essentially to study properties of finitely
generated
modules not depending on the " presentation" of these modules by means
of
generators and relations. This very hard step is based on
homological/cohomological methods like the so-called "extension"
modules which
cannot therefore be avoided.

As before, using now rings of "differential operators" instead of
polynomial
rings led to "differential modules" and to the challenge of adding the
word
"differential" in front of concepts of commutative algebra.
Accordingly, not
only one needs properties not depending on the presentation as we just
explained
but also properties not depending on the coordinate system as it
becomes clear
from any application to mathematical or engineering physics where
tensors and
exterior forms are always to be met like in the space-time formulation
of
electromagnetism. Unhappily, no one of the previous techniques for OD
or PD
equations could work !.

By chance, the intrinsic study of systems of OD or PD equations has been
pioneered in a totally independent way by D. C. Spencer and
collaborators after
1960, using jet theory and diagram chasing in order to relate
differential
properties of the equations to algebraic properties of their "symbol", a
technique superseding the "leading term" approah of Janet or Groebner
but quite
poorly known by the mathematical community.

Accordingly, it was another challenge to unify the "purely differential"
approach of Spencer with the "purely algebraic" approach of
commutative algebra,
having in mind the necessity to use the previous homological algebraic
results
in this new framework. This sophisticated mixture of differential
geometry and
homological algebra, now called "algebraic analysis", has been
achieved after
1970 by V. P. Palamodov for the constant coefficient case, then by M.
Kashiwara
and B. Malgrange for the variable coefficient case.

The purpose of this intensive course held at RISC is to provide an
introduction
to "algebraic analysis" in a rather effective way as it is almost
impossible to
learn about this fashionable though quite difficult domain of pure
mathematics
today, through books or papers, and no such course is available
elsewhere.
Computer algbra packages like "OreModules" are very recent and a lot
of work is
left for the future.

Accordingly, the aim of the course will be to bring students in a self-
contained
way to a feeling of the general concepts and results that will be
illustrated by
many academic or engineering examples. By this way, any participant
will be able
to take a personal decision about a possible way to involve himself
into any
future use of computer algebra into such a new domain and be ready for
further
applications.

- Main References for Theoretical Module
J.-F. Pommaret, Partial Differential Control Theory, Kluwer, 2001, 2
vol, 1000
pp (See Zentralblatt review Zbl 1079.93001).
J.-F. Pommaret, Algebraic Analysis of Control Systems Defined by Partial
Differential Equations, in Advanced Topics in Control Systems Theory,
chapter 5,
Lecture Notes in Control and Information Sciences, LNCIS 311,
Springer, 2005,
155-223.
The second reference is an elementary introduction coming from a
series of
European courses.

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- Abstract of Practical Module
The purpose of the practical part of the lectures is to give deeper
insights
into constructive issues of algebraic analysis, present their
implementations in
the symbolic packages OreModules, OreMorphisms, Stafford, Quillen-
Suslin and
Serre, and illustrate them by means of different problems coming from
mathematical systems theory, control theory and mathematical physics.

In particular, we shall focus on different aspects of constructive
algebra,
module theory and homological algebra such as:

* Groebner basis computations over Ore algebras of functional
operators (e.g.,
differential/shift/time-delay/difference operators).
* Computation of finite free resolutions, dimensions, homomorphisms,
tensor
products, extension and torsion functors.
* Classification of module properties (e.g., torsion, torsion-free,
reflexive,
projective, stably free, free, decomposable, simple, pure modules) and
their
system-theoretic interpretations (e.g., autonomous elements,
minimal/successive/injective/Monge parametrizations, Bezout identities,
factorization/reduction and decomposition problems).

The different results and constructive algorithms will be illustrated by
examples coming from mathematical systems theory, control theory and
mathematical physics. Finally, the attendees will have to study explicit
problems by means of the packages OreModules, OreMorphisms, Stafford,
QuillenSuslin and Serre.

- Main Reference for Practical Module
A. Quadrat, Systems and Structures, An algebraic analysis approach to
mathematical systems theory, soon available
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