A constructive study of the module structure of rings of partial differential operators

The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series. © 2014 Springer Science+Business Media

Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be published in Lecture Notes in Computer Scienc

Shear Thickening Creep in Superplastic Silicon Nitride

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/65512/1/j.1151-2916.1992.tb05540.x.pd

Amoeba Techniques for Shape and Texture Analysis

Morphological amoebas are image-adaptive structuring elements for morphological and other local image filters introduced by Lerallut et al. Their construction is based on combining spatial distance with contrast information into an image-dependent metric. Amoeba filters show interesting parallels to image filtering methods based on partial differential equations (PDEs), which can be confirmed by asymptotic equivalence results. In computing amoebas, graph structures are generated that hold information about local image texture. This paper reviews and summarises the work of the author and his coauthors on morphological amoebas, particularly their relations to PDE filters and texture analysis. It presents some extensions and points out directions for future investigation on the subject.Comment: 38 pages, 19 figures v2: minor corrections and rephrasing, Section 5 (pre-smoothing) extende

Algebraic estimation in partial derivatives systems: parameters and differentiation problems

International audienceTwo goals are sought in this paper: namely, to provide a succinct overview on algebraic techniques for numerical differentiation and parameter estimation for linear systems and to present novel algebraic methods in the case of several variables. The state-of-art in the introduction is followed by a brief description of the methodology in the subsequent sections. Our new algebraic methods are illustrated by two examples in the multidimensional case. Some algebraic preliminaries are given in the appendix

Au Sujet des Approches Symboliques des Équations Intégro-Différentielles

International audienceRecent progress in computer algebra has opened new opportunities for the parameter estimation problem in nonlinear control theory, by means of integro-differential input-output equations. This paper recalls the origin of integro-differential equations. It presents new opportunities in nonlinear control theory. Finally, it reviews related recent theoretical approaches on integro-differential algebras, illustrating what an integro-differential elimination method might be and what benefits the parameter estimation problem would gain from it.Un résultat récent en calcul formel a ouvert de nouvelles opportunités pour l'estimation de paramètres en théorie du contrôle non linéaire, via des équations entrée-sortie intégro-différentielles. Ce chapitre rappelle les origines des équations intégro-différentielles. Il présente de nouvelles opportunités en théorie du contrôle non linéaire. Finalement, il passe en revue des approches théoriques récentes sur les algèbres intégro-différentielles, illustrant ce qu'une méthode d'élimination intégro-différentielle pourrait être et les bénéfices que le problème de l'estimation de paramètres pourrait en tirer