Algebraic and Algorithmic Aspects of Differential and Integral Operators Session

ACA 2012

Sofia, Bulgaria, 25-28 June, 2012


Moulay Barkatou (University of Limoges, XLIM DMI, Limoges, France)
Thomas Cluzeau (University of Limoges, CNRS, XLIM UMR 7252, DMI, Limoges, France)
Georg Regensburger (INRIA Saclay - Île de France, Project Disco, Gif-sur-Yvette, France)
Markus Rosenkranz (University of Kent, SMSAS, Canterbury, United Kingdom)


The algebraic/symbolic treatment of differential equations is a flourishing field, branching out in a variety of subfields committed to different approaches. In this session, we want to give special emphasis to the operator perspective of both the underlying differential operators and various associated integral operators (e.g. as Green's operators for initial/boundary value problems).

In particular, we invite contributions in line with the following topics:

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

Please see also our previous ACA sessions AADIOS '11 , AADIOS '10 , AADIOS '09, and AADIOS '08.

See here for an MCS double Special Issue on previous AADIOS sessions.


Alexandre Benoit: Quasi-Optimal Multiplication of Linear Differential Operators

Abstract. The product of polynomials and the product of matrices are two of the most basic operations in mathematics; the study of their computational complexity is central in computer science. In this talk, we will be interested in the computational complexity of multiplying two linear differential operators. These algebraic objects encode linear differential equations, and form a non-commutative ring that shares many properties with the commutative ring of usual polynomials. Yet, the algorithmic study of linear differential operators is currently much less advanced than in the polynomial case: the complexity of multiplication has been addressed only recently, but not completely solved. The aim of the present work is to make a step towards filling this gap, and to solve an open question raised by van der Hoeven. This work is joint with Alin Bostan and Joris van der Hoeven.

Alin Bostan: Fast Computation of Common Left Multiples of Linear Ordinary Differential Operators

Abstract. We study tight bounds and fast algorithms for LCLMs of several linear differential operators with polynomial coefficients. We analyse the arithmetic complexity of existing algorithms for LCLMs, as well as the size of their outputs. We propose a new algorithm that recasts the LCLM computation in a linear algebra problem on a polynomial matrix. This algorithm yields sharp bounds on the coefficient degrees of the LCLM, improving by one order of magnitude the best bounds obtained using previous algorithms. The complexity of the new algorithm is almost optimal, in the sense that it nearly matches the arithmetic size of the output. Joint work with Frédéric Chyzak, Ziming Li and Bruno Salvy.

Slides. For download here.

François Boulier and Nicolas M. Thiéry: A Differential Algebra Package in Sage

Abstract. For download here.

Yongjae Cha: Homomorphism Between two Difference Operators

Abstract. For download here.

Slides. For download here.

Guillaume Cheze: An Efficient Algorithm for Computing Rational First Integrals of Polynomial Vector Fields

Abstract. For download here.

Thierry Combot: Algorithms for Non Integrabililty Proofs

Abstract. For download here.

Ivan Dimovski: Operational Calculi for Boundary Value Problems

Abstract. For download here.

Sette Diop: On a Differential Algebraic Approach of Control Observation Problems

Abstract. For download here.

Ruyong Feng: On the Structure of Compatible Rational Functions

Abstract. A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential and shift operators. In this talk, we present a theorem that describes the structure of compatible rational functions. The theorem enables us to decompose a solution of such a system as a product of a rational function, several symbolic powers, a hyperexponential function and a hypergeometric term. Some applications are also presented in this talk. This is joint work with Shaoshi Chen, Guofeng Fu and Ziming Li.

Xiao-Shan Gao: Sparse Differential Resultant for Laurent Differential Polynomials

Abstract. In this talk, we first introduce the concept of Laurent differentially essential systems and give a criterion for Laurent differentially essential systems in terms of their supports. Then the sparse differential resultant for a Laurent differentially essential system is defined and its basic properties are proved. In particular, order and degree bounds for the sparse differential resultant are given. Based on these bounds, an algorithm to compute the sparse differential resultant is proposed, which is single exponential in terms of the order, the number of variables, and the size of the Laurent differential system. Detailes of the results can be found in W. Li, C.M. Yuan, X.S. Gao, Sparse Differential Resultant for Laurent Differential Polynomials. arxiv-1111.1084v2, 62 pages, 2011.

Vladimir Gerdt: Computer Algebra Application to Numerical Solving of Nonlinear KdV-type Equations

Abstract. For download here.

Slides. For download here.

Li Guo: Free Integro-Differential Algebras

Abstract. The concept of integro-differential algebra has been introduced recently in the study of boundary problems for differential equations. Its free objects can be obtained from those of the closely related differential Rota-Baxter algebra by taking quotients. We study their explicit construction in terms of integro-differential polynomials. Joint work with Georg Regensburger and Markus Rosenkranz.

Anja Korporal: Composition and Factorization of Generalized Inverses and Boundary Problems

Abstract. For download here.

Christoph Koutschan: Twisting q-holonomic Sequences by Complex Roots of Unity

Abstract. We present two new closure properties for q-holonomic sequences, namely twisting by complex roots of unity and raising q to a rational power. The proofs are constructive, work in the multivariate setting of d-finite sequences and are implemented in our Mathematica package HolonomicFunctions. The results are illustrated by twisting natural q-holonomic sequences which appear in quantum topology, namely the colored Jones polynomial of pretzel knots and twist knots. The recurrence of the twisted colored Jones polynomial can be used to compute the asymptotics of the Kashaev invariant of a knot at an arbitrary complex root of unity. This is joint work with Stavros Garoufalidis.

Slides. For download here.

Johannes Middeke: On the Computation of Pi-flat Outputs for Differential-delay Systems

Abstract. For download here.

Alban Quadrat and Daniel Robertz: Module Structure of Rings of Partial Differential Operators

Abstract. For download here.

Daniel Robertz: Implicitization of Parametrized Families of Analytic Functions

Abstract. The correspondence between solution sets of systems of algebraic equations and radical ideals of the affine coordinate ring is fundamental for algebraic geometry. This talk discusses aspects of an analogous correspondence between systems of polynomial differential equations and their analytic solutions. Implicitization problems for certain families of analytic functions are approached in different generality. While the linear case is understood to a large extent, the non-linear case requires new algorithmic methods, e.g., the use of differential inequations, as proposed by J. M. Thomas in the 1930s.

Markus Rosenkranz: Localization and the Mikusinski Calculus

Abstract. The basic idea that integration is "somehow" a division by the differential operator has haunted mathematicians ever since the time of Oliver Heaviside. Put on a firm algebraic basis by Jan Mikusinski, the operational calculus has now been vastly extended and generalized, notably by Ivan Dimovski and his schol. In this talk we sketch a new approach to algebraic operational calculi. Treating Green's operators on a par with the standard integral operator (Duhamel convolution), we build up a localization of a suitable operator ring that can be made to act on an algebraic space of hyperfunctions. Joint work with A. Korporal. Earlier work on this topic was pursued in collaboration with G. Regensburger.

Slides. For download here.

Margarita Spiridonova: Extended Heaviside Algorithm for Resonance Mean-Periodic Solutions of Nonlocal Cauchy Problems

Abstract. For download here.

Srinivasarao Thota and Shiv Datt Kumar: Boundary Problems for Linear Systems of Differential Equations over an Integro-Differential Algebra

Abstract. We present a new approach for solving boundary value problems for linear systems of differential equations allowing two-/multi-point as well as arbitrary Stieltjes conditions as in the scalar case. For expressing differential operators, boundary conditions, and Green's operators, we employ the algebra of integro-differential operators.

Slides. For download here.

Yulian Tsankov: Exact Solution of Local and Nonlocal BVPs for the Laplace Equation in a Rectangle

Abstract. For download here.

Franz Winkler: Algebraic Differential Equations - Rational Solutions and Classification

Abstract. For download here.

Chun-Ming Yuan: Differential Chow Forms

Abstract. In this talk, an intersection theory for generic differential polynomials is presented. The intersection of an irreducible differential variety of dimension d and order h with a generic differential hypersurface of order s is shown to be an irreducible variety of dimension d-1 and order h+s. As a consequence, the dimension conjecture for generic differential polynomials is proved. Based on the intersection theory, the Chow form for an irreducible differential variety is defined and most of the properties of the Chow form in the algebraic case are extended to its differential counterpart. Furthermore, the generalized differential Chow form is defined and its properties are proved. As an application of the generalized differential Chow form, the differential resultant of n + 1 generic differential polynomials in n variables is defined and properties similar to that of the Sylvester resultant of two univariate polynomials are proved.

Serguey V. Zemskov: On Finding a Complete Integral of Second-Order Hyperbolic PDEs with Constant Coefficients in Infinite Space

Abstract. For download here.