Algebraic and Algorithmic Aspects of Differential and Integral Operators Session


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


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

Previous Session

The webpage for our session in 2008 can be found here.


The algebraic treatment of differential equations is a well-established field with close ties to the symbolic community (see also the ACA Session "Symbolic Symmetry Analysis and its Applications"). Algebraic methods often commence from an operator perspective on the underlying differential equations, e.g. in D-Module theory or in factoring linear differential operators (ODE/PDE, scalar/vector). On the other hand, integral operators have as yet received comparably little attention in an algebraic setting. In the context of linear differential equations, they arise naturally as Green's operators for initial/boundary value problems.

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:

Recent connections between these topics include algebraic structures combining derivations with integrals (e.g. differential Rota-Baxter algebras) and the correspondence between factorizations of differential and integral operators (e.g. by splitting their boundary value problems). We hope that we will have a stimulating discussion that may lead to further interrelations.


Hynek Baran: Classifying integrable systems via the spectral parameter problem

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�.

Moulay Barkatou: Local Reduced Forms of Systems of Linear Functional Equations and Applications

Abstract. See the pdf.

Slides. For download here.

Thomas Cluzeau: Serre's Reduction of Linear Functional Systems: Theory, Implementation and Applications

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.

Li Guo, William Sit and Ronghua Zhang: Rota-Baxter Type and Differential Type Algebras

Abstract. See the pdf.

Slides. For download here.

Christoph Koutschan: A Difference Operators Attack on Hard Combinatorial Problems

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.

George Labahn: Normal Forms of Matrices of Differential Polynomials

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.

Xuan Liu and Greg Reid: Symmetry Operators and Differential Equations

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.

Leon Pritchard and William Sit: Algebraic Constraints on Initial Values of Differential Equations

Abstract. See the pdf.

Slides. For download here.

Alban Quadrat: A Normal Form for 2-dimensional Linear Functional Systems

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.

Georg Regensburger, Markus Rosenkranz and Johannes Middeke: Integro-Differential Operators as an Ore Algebra

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.

Kate Shemyakova and Elizabeth Mansfield: Moving Frames for Laplace Invariants. General Case.

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.

Margarita Spiridonova: Duhamel-Type Representation of the Solution of a Linear Nonlocal Boundary Value Problem

Abstract. See the pdf.

Hiroshi Umemura: Discrete Burgers' Equation, Binomial Coefficients and Mandala

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.

Slides. For download here.

Yang Zhang: Irreducibility Criteria for Skew Polynomials

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.