Cyclic Vectors

Churchill, R. C.; Kovacic, Jerald J.

doi:10.1142/9789812778437_0007

Cited by 23 publications

(34 citation statements)

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“…For instance, apart from many other results, we prove that a linear system of PDEs defined by a full row rank matrix of PD operators (e.g., most of the classical linear systems of PDEs studied in mathematical physics and engineering science) is equivalent to a linear system defined by two PDEs in two unknown functions (see Example 9). This result can be seen as a generalization for linear systems of PDEs of the well-known cyclic vector theorem in the theory of ODEs (see, e.g., [10] and the references therein). Moreover, these results can be used to study nonlinear PD systems by means of their linearizations around a polynomial/rational/formal power series/locally convergent power series solution (e.g., shallow water waves, Poiseuille flow, flexible thread attached to one point in a vertical equilibrium position).…”

Section: If Rank D (M )mentioning

confidence: 80%

“…However, due to involved computations required in the corresponding algorithm, solutions of the quadratic equation (10) can in general only be computed in reasonable time for low order and low degree PD operators d 1 and d 2 .…”

Section: Theorem 5 ([9]mentioning

confidence: 99%

“…Corollary 2 can be understood as a generalization of the standard cyclic vector theorem for linear PD systems (see [10] and the references therein). 1 of Corollary 2 shows that every finitely generated torsion left D-module L can be generated by two elements.…”

Section: Rr N°8225mentioning

confidence: 99%

“…Theorem 11 is based on solving quadratic equations of the form (10). As already noticed in [36], for the computation of bases of finitely generated free modules, we only need to find α, β ∈ D such that…”

Section: Computation Of Basesmentioning

confidence: 99%

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A Constructive Study of the Module Structure of Rings of Partial Differential Operators

Quadrat

Robertz

2014

Acta Appl Math

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To cite this version:Alban Quadrat, Daniel Robertz. A constructive study of the module structure of rings of partial differential operators. [ Abstract: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 to 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. Key-words:Weyl algebras, Stafford's theorems, linear systems of partial differential equations, D-modules, mathematical systems theory, constructive algebra, symbolic computation. A constructive study of the module structure of rings of partial differential operators 4

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Section: If Rank D (M )mentioning

confidence: 80%

Section: Theorem 5 ([9]mentioning

confidence: 99%

Section: Rr N°8225mentioning

confidence: 99%

Section: Computation Of Basesmentioning

confidence: 99%

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A Constructive Study of the Module Structure of Rings of Partial Differential Operators

Quadrat

Robertz

2014

Acta Appl Math

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“…This has been extended to represent the zero set of zero dimensional polynomial ideals by means of the roots of a single polynomial [21,23,3,1,38,24]. Similarly, the cyclic vector construction shows that a linear differential system is equivalent to a single differential equation [4,16,33,12]. The resolvent representation is a generalization of both those 1 constructions to nonlinear differential systems.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic algorithms for computing resolvent representations of regular differential ideals

Cluzeau

Hubert²

2008

AAECC

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In a previous article [14], we proved the existence of resolvent representations for regular differential ideals. The present paper provides practical algorithms for computing such representations. We propose two different approaches. The first one uses differential characteristic decompositions whereas the second one proceeds by prolongation and algebraic elimination. Both constructions depend on the choice of a tuple over the differential base field and their success relies on the chosen tuple to be separating. The probabilistic aspect of the algorithms comes from this choice. To control it, we exhibit a family of tuples for which we can bound the probability that one of its element is separating.

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Background and Fundamental Concepts

Kauers

2023

Algorithms and Computation in Mathematics

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Cyclic Vectors

Cited by 23 publications

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A Constructive Study of the Module Structure of Rings of Partial Differential Operators

A Constructive Study of the Module Structure of Rings of Partial Differential Operators

Probabilistic algorithms for computing resolvent representations of regular differential ideals

Background and Fundamental Concepts

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