Algebraic and Algorithmic Aspects of Differential and Integral Operators Session

ACA'09

École de Technologie Supérieure (ETS), Montréal, Canada, 25-28 June, 2009

Markus Rosenkranz (Austrian Academy of Sciences, RICAM, Linz, Austria)

In this session, we would like to examine various relations between differential and integral operators. To this end, we want to bring together the following topics and communities:

- Factorization of Differential/Integral Operators
- Constructive Methods for D-Modules
- Linear Initial/Boundary Value Problems and Green's Operators
- Rota-Baxter Algebras
- Symbolic Computation with Differential Polynomials
- Initial/Boundary Value Problems for Nonlinear Differential Equations

*
Abstract.
We consider the problem of systematic classification of integrable PDE initially known to possess a nonparametric zero curvature representation. The method is taken from M. Marvan, On the spectral parameter problem,
online on
Acta Appl. Math., DOI 10.1007/s10440-009-9450-4
or
http://arxiv.org/abs/0804.2031v2
.
Discussed will be an implementation of the method and results of classification of integrable Weingarten surfaces in E�.
*

*
Abstract. See the pdf.
*

**Slides.** For download here.

*
Abstract. The first purpose of this talk is to recall briefly constructive
results on Serre's reduction of determined / overdetermined / underdetermined
linear functional systems obtained recently by M. S. Boudellioua (Sultan
Qaboos University, Oman) and A. Quadrat (INRIA Sophia Antipolis, France).
Serre's reduction aims at finding an equivalent presentation of a
linear functional system which contains fewer equations and unknowns. We
shall explain why this problem can be reduced to the case where the
equivalent system contains only one equation.
Then, we will discuss our implementation of these results in an OreModules
package called Serre.
Finally, we will concentrate on the zero-dimensional case (D-finite or
holonomic linear functional systems) where we can go further in this
analysis and explain the links between these results and the Jacobson
normal form of a matrix with entries in a principal ideal domain (e.g.,
ODEs with rational coefficients).
The different results will be illustrated with explicit examples coming
from control theory and engineering sciences.
This is a work in progress in collaboration with A. Quadrat (INRIA Sophia
Antipolis, France).
*

**Slides.** For download here.

*
Abstract. See the pdf.
*

**Slides.** For download here.

*
Abstract. We use the algebra of difference operators (which we introduce as a
discrete analogue of differential operators) in order to prove the
enumeration formula for totally symmetric plane partitions (TSPP),
whose proof by hand is highly nontrivial (Stembridge 1995). Our
computer algebra (holonomic systems) approach is a computational
challenge that became feasible only by using new techniques. The
remarkable point in our new proof of TSPP is that the same technique
can be applied to the generalized problem q-TSPP which is still an
outstanding open problem in enumerative combinatorics.
*

**Slides.** For download here.

*
Abstract. Normal forms for matrix polynomials (such as Hermite, Popov etc) have been very useful in many areas of mathematics
(e.g. linear control theory). In this talk we will discuss normal forms for matrices of differential operators.
We show their usefulness in the context of systems of linear differential equations
and discuss the various computational challenges in computing the forms for arbitrary matrices of differential operators.
*

*
Abstract. Finite and infinite Lie symmetry pseudo-groups of differential equations are sets of continuous transformations leaving
invariant their solution spaces.
They are central to the modern exact approaches for nonlinear differential equations such as reduction of dimension,
determining invariant solutions and mappings.
In this talk we give an introduction to such methods and the algorithmic determination of the structure of such pseudo-groups;
and the related algebras of operators.
*

*
Abstract. See the pdf.
*

**Slides.** For download here.

*
Abstract.
Jacobson normal forms of matrices with entries in polynomial rings of ordinary differential
or difference operators with coefficients in a skew field play an important role in the study
of linear systems of ordinary differential or recurrence equations. Unfortunately, they generally
do not exist for matrices with entries in noncommutative polynomial rings in more
than one variable, i.e., they cannot be used to study linear systems of partial differential
equations, differential time-delay systems or multi-indexed recurrence equations.
*

*
The purpose of this talk is to show that every matrix over a noncommutative polynomial ring
in two independent variables admits an upper triangular reduction formed by three diagonal
blocks: the first diagonal block defines the torsion-free part of the linear system, the second one
defines the 1-dimensional part and the third one defines the 0-dimensional part of the system. Hence,
the corresponding linear system can be integrated in cascade by first solving the 0-dimensional
system, then the 1-dimensional one and finally the parametrizable one. This form for 2-dimensional
linear systems generalizes the Jacobson normal form for 1-dimensional linear systems. This normal form,
based on difficult results of module theory (e.g., pure submodules, purity filtration)
and homological algebra (e.g., extension functor, ext_D^i(ext_D^i(M, D), D), Baer's extensions),
can be computed by means of Gr�bner or Janet basis techniques.
*

**Slides.** For download here.

*
Abstract.
The notion of integro-differential algebra brings together the usual derivation structure with an algebraic
version of indefinite integration and evaluation. We construct the associated algebra of
integro-differential operators (used for modeling Green's operators for linear boundary problems)
directly in terms of normal forms. For polynomial coefficients, we can use skew polynomials,
defining the integro-differential Weyl algebra as a natural extension of the classical Weyl algebra in one variable.
Its algebraic properties and its relation to the localization of differential operators are studied.
Fixing the integration constant, we regain the integro-differential operators with polynomial coefficients.
*

**Slides.** For download here.

*
Abstract.
Using the methods of regularized moving frames of Fels and
Olver we obtained in our ISSAC'08 paper generating sets of invariants for
the second- and third-orders bivariate hyperbolic and non-hyperbolic
LPDOs.
Now we generalize the method to the case of arbitrary
order (hyperbolic and non-hyperbolic) LPDOs. A special
choice of the normalization equations was invented, so
that we were able prove some invariants form
a finite (small) generating set of invariants.
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*
Abstract. See the pdf.
*

*
Abstract. Burgers' equation is a fundamental equation in fluid mechanics. We know its discretization and ultra-discretization. The
ultra-discretization describes traffic flow. We talk about the discretization but over the prime field $F_2=Z/2Z$. The equation generates a beautiful pattern, which we call mandala.
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**Slides.** For download here.

*
Abstract. In this talk, we present two irreducibility criteria for the elements of a
large class of skew-polynomial rings. The proofs rely heavily on non-commutative valuations and extensions thereof. The results apply, in particular, to ordinary linear differential operators and linear difference operators having coefficients in
not-necessarily-commutative fields with valuations. This is a joint with R. Churchill.
*

If you are also interested in joining the session, please contact us.