In this paper, we study linear control systems over Ore algebras. Within this mathematical framework, we can simultaneously deal with different classes of linear control systems such as time-varying systems of ordinary differential equations (ODEs), differential time-delay systems, underdetermined systems of partial differential equations (PDEs), multidimensional discrete systems, multidimensional convolutional codes, etc. We give effective algorithms which check whether or not a linear control system over some Ore algebra is controllable, parametrizable, flat or π-free.
In this paper, we consider systems of algebraic and non-linear partial differential equations and inequations. We decompose these systems into so-called simple subsystems and thereby partition the set of solutions. For algebraic systems, simplicity means triangularity, square-freeness and nonvanishing initials. Differential simplicity extends algebraic simplicity with involutivity. We build upon the constructive ideas of J. M. Thomas and develop them into a new algorithm for disjoint decomposition. The given paper is a revised version of Bächler et al. (2010) and includes the proofs of correctness and termination of our decomposition algorithm. In addition, we illustrate the algorithm with further instructive examples and describe its Maple implementation together with an experimental comparison to some other triangular decomposition algorithms.Keywords: disjoint triangular decomposition, simple systems, polynomial systems, differential systems, involutivity input by means of regular chains (if the input only consists of equations) or regular systems. However, the Thomas decomposition differs noticeably from this decomposition, since the Thomas decomposition is finer and demands disjointness of the solution set. For a detailed description of algorithms related to regular chains, we refer the reader to Moreno Maza (1999).The disjointness of the Thomas decomposition combined with the structural properties of simple systems provide a useful platform for counting solutions of polynomial systems. In fact, the Thomas decomposition is the only known method to compute the counting polynomial introduced by Plesken (2009a). We refer to §2.3 for details on this structure, counting and their applications.During his research on triangular decomposition, Thomas was motivated by the Riquier-Janet theory (cf. Riquier (1910); Janet (1929)), extending it to non-linear systems of partial differential equations. For this purpose he developed a theory of (Thomas) monomials, which generate an involutive monomial division nowadays called Thomas division (cf. Gerdt and Blinkov (1998a)). He gave a recipe for decomposing a non-linear differential system into algebraically simple and passive subsystems (cf. Thomas (1937)). A modified version of the differential Thomas decomposition was considered by Gerdt (2008) with its link to the theory of involutive bases (cf. Gerdt and Blinkov (1998a);Gerdt (2005Gerdt ( , 1999; Seiler (2010)). In this decomposition, the output systems are Janet-involutive in accordance to the involutivity criterion from Gerdt (2008) and hence they are coherent. For a linear differential system it is a Janet basis of the corresponding differential ideal, as computed by the Maple package Janet (cf. Blinkov et al. (2003)).The differential Thomas decomposition differs from that computed by the Rosenfeld-Gröbner algorithm (cf. Boulier et al. (2009, 1995)). The latter decomposition forms a basis of the diffalg, DifferentialAlgebra and BLAD packages (cf. Hubert (1996-2004); Boulier (2004Boulier ( -2009). Experimentally, we f...
Calcul effectif de bases de modules libres sur des algèbres de Weyl Résumé : Un résultat célèbre dû à J. T. Stafford montre qu'un module à gauche M stablement libre sur les algèbres de Weyl D = A n (k) ou B n (k) (k est un corps de caractéristique 0) vérifiant rank D (M) ≥ 2 est libre. Le but de ce papier est de donner une nouvelle preuve constructive de ce résultat ainsi qu'un algorithme effectif pour le calcul de bases de M. Cet algorithme, basé sur de nouvelles preuves constructives [11, 14] d'un résultat de J. T. Stafford sur le nombre de générateurs d'un idéal à gauche de D, est une sorte de méthode de pivot de Gauss appliquée à l'adjoint formel de la matrice de présentation de M. Nous montrons que le résultat de J. T. Stafford est un cas particulier d'un résultat plus général montrant qu'un D-module à gauche M stablement libre satisfaisant rank D (M) ≥ sr(D) est libre, où sr(D) désigne le rang stable de l'anneau D. Ce résultat est constructif dès lors que l'on peut tester la stabilité des vecteurs unimodulaires à coefficients dans D. Finalement, nous donnons un algorithme calculant la dimension projective d'un D-module à gauche défini par une résolution libre de type fini. Ce dernier résultat nous permet de vérifier si un D-module à gauche est stablement libre. Mots-clés : Modules stablement libres, modules libres, calcul effectif de bases, dimension projective, résultats de Stafford, algèbres de Weyl, systèmes linéaires multidimensionnels plats.
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