The Jacobian Conjecture 2n implies the Dixmier Problem n

The aim of the paper is to describe some ideas, approaches, comments, etc. regarding the Dixmier Conjecture, its generalizations and analogues

The prime spectrum and simple modules over the quantum spatial ageing algebra

For the algebra $A$ in the title, its prime, primitive and maximal spectra are classified. The group of automorphisms of $A$ is determined. The simple unfaithful $A$-modules and the simple weight $A$-modules are classified

Cluster Structure on Generalized Weyl Algebras

We introduce a class of non-commutative algebras that carry a non-commutative (geometric) cluster structure which are generated by identical copies of generalized Weyl algebras. Equivalent conditions for the finiteness of the set of the cluster variables of these cluster structures are provided. Some combinatorial data, called \textit{cluster strands,} arising from the cluster structure are used to construct irreducible representations of generalized Weyl algebras.Comment: in Algebras and Representation Theory, Volume 19, No1, Feb. 201

Symbolic-Numeric Methods for Nonlinear Integro-Differential Modeling

International audienceThis paper presents a proof of concept for symbolic and numeric methods dedicated to the parameter estimation problem for models formulated by means of nonlinear integro-differential equations (IDE). In particular, we address: the computation of the model input-output equation and the numerical integration of IDE systems

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

Quiver Generalized Weyl Algebras, Skew Category Algebras and Diskew Polynomial Rings

The aim of the paper is to introduce new large classes of algebras—quiver generalized Weyl algebras, skew category algebras, diskew polynomial rings and skew semi-Laurent polynomial rings

Computing polynomial solutions and annihilators of integro-differential operators with polynomial coefficients

International audienceIn this chapter, we study algorithmic aspects of the algebra of linear ordinary integro-differential operators with polynomial coefficients. Even though this algebra is not Noetherian and has zero divisors, Bavula recently proved that it is coherent, which allows one to develop an algebraic systems theory over this algebra. For an algorithmic approach to linear systems of integro-differential equations with boundary conditions, computing the kernel of matrices with entries in this algebra is a fundamental task. As a first step, we have to find annihilators of integro-differential operators, which, in turn, is related to the computation of polynomial solutions of such operators. For a class of linear operators including integro-differential operators, we present an algorithmic approach for computing polynomial solutions and the index. A generating set for right annihilators can be constructed in terms of such polynomial solutions. For initial value problems, an involution of the algebra of integro-differential operators then allows us to compute left annihilators, which can be interpreted as compatibility conditions of integro-differential equations with boundary conditions. We illustrate our approach using an implementation in the computer algebra system Maple