⋆ This work has benefited from the support of the French ANR (decision number ANR-2010-BLAN-0109-... more ⋆ This work has benefited from the support of the French ANR (decision number ANR-2010-BLAN-0109-03).
The goal of the present paper is to propose an enhanced ordinary differential equations solver by... more The goal of the present paper is to propose an enhanced ordinary differential equations solver by exploitation of the powerful equivalence method of Élie Cartan. This solver returns a target equation equivalent to the equation to be solved and the transformation realizing the equivalence. The target ODE is a member of a dictionary of ODE, that are regarded as well-known, or at least well-studied. The dictionary considered in this article are ODE in a book of Kamke. The major advantage of our solver is that the equivalence transformation is obtained without integrating differential equations. We provide also a theoretical contribution revealing the relationship between the change of coordinates that maps two differential equations and their symmetry pseudo-groups.
We give an algorithm which represents the radical J of a finitely generated differential ideal as... more We give an algorithm which represents the radical J of a finitely generated differential ideal as an intersection of radical differential ideals. The computed representation provides an algorithm for testing membership in J. This algorithm works over either an ordinary or a partial differential polynomial ring of characteristic zero. It has been programmed. We also give a method to obtain a characteristic set of J , if the ideal is prime. Keywords. Differential Algebra. Radical differential ideals. Characteristic sets. * The authors would like to thank the participants of the Special Year in Differential Algebra and Algebraic Geometry for their help and their comments, in particular Pr. William Sit and Raymond T. Hoobler. † This research was partially supported by EC contract ESPRIT B.R.A. n • 6846 POSSO. 1 We make precise in the following sections some of the notations and definitions used in this introduction.
Abstract. This paper presents a symbolic algorithm for computing the ODE systems which describe t... more Abstract. This paper presents a symbolic algorithm for computing the ODE systems which describe the evolution of the moments associated to a chemical reaction system, considered from a stochastic point of view. The algorithm, which is formulated in the Weyl algebra, seems more efficient than the corresponding method, based on partial derivatives. In particular, an efficient method for handling conservation laws is presented. The output of the algorithm can be used for a further investigation of the system behaviour, by numerical methods. Relevant examples are carried out.
Le formalisme des systemes de reactions chimiques (de maniere equivalente des reseaux de Petri st... more Le formalisme des systemes de reactions chimiques (de maniere equivalente des reseaux de Petri stochastiques) est tres souvent utilise en bilogie, en surete de fonctionnement etc. La serie generatrice associee a la master-equation est solution d'une equation d'evolution du type equation de Schrodinger. On adopte ici l'approche classique par le calcul des fonctions propres en se concentrant, dans cette premiere partie, sur le calcul de la distribution stationnaire pour un systeme comportant une seule espece chimique. On montre que, generiquement, la serie generatrice stationnaire est une fonction holomorphe dans tout le plan complexe. Des exemples de calcul (symbolique-numerique) sur ordinateur sont developpes.
This paper presents a symbolic algorithm for computing the ODE systems which describe the evoluti... more This paper presents a symbolic algorithm for computing the ODE systems which describe the evolution of the moments associated to a chemical reaction system, considered from a stochastic point of view. The algorithm, which is formulated in the Weyl algebra, seems more ecient than the corresponding method, based on partial derivatives. In particular, an ecient method for handling conservation laws is pre- sented. The output of the algorithm can be used for a further investiga- tion of the system behaviour, by numerical methods. Relevant examples are carried out.
This paper considers an approach for analyzing the identifiability of nonlinear controlled or unc... more This paper considers an approach for analyzing the identifiability of nonlinear controlled or uncontrolled dynamical systems. The method is based on the computation of the ideal containing the differential algebraic relations between the input and the output of the model, a such ideal is called input-output ideal. Our contribution consists in developing the corresponding algorithm in a symbolic computing language. This algorithm is based on differential algebra.
Proceedings of 1995 34th IEEE Conference on Decision and Control, 1995
ABSTRACT We introduce a new class of nonlinear systems called “finitely discretizable”. For a fin... more ABSTRACT We introduce a new class of nonlinear systems called “finitely discretizable”. For a finitely discretizable nonlinear system, any trajectory is completely determined by a finite number of the state derivatives. In the polynomial case, using the notion of dilations, we give sufficient conditions for a nonlinear system to be finitely discretizable and we relate this result to nilpotent Lie algebra of vector fields. We show that multirate sampling techniques can be an efficient tool to deal with the control problem of finitely discretizable systems. We conclude by some examples illustrating the relevance of finitely discretizable systems within the framework of control theory
ABSTRACT In this paper we give, an algorithm that computes the local minimal analytical realizati... more ABSTRACT In this paper we give, an algorithm that computes the local minimal analytical realization of finite generating power series. In the same time, this algorithm proves that any finite generating series has a polynomial realization: observation and vector fields components are commutative polynomials. That algorithm is implemented in the computer algebra system Scratchpad.
Proceedings of the 1995 international symposium on Symbolic and algebraic computation - ISSAC '95, 1995
We give an algorithm which represents the radical J of a finitely generated differential ideal as... more We give an algorithm which represents the radical J of a finitely generated differential ideal as an intersection of radical differential ideals. The computed representation provides an algorithm for testing membership in J. This algorithm works over either an ordinary or a partial differential polynomial ring of characteristic zero. It has been programmed. We also give a method to obtain a characteristic set of J , if the ideal is prime. Keywords. Differential Algebra. Radical differential ideals. Characteristic sets. * The authors would like to thank the participants of the Special Year in Differential Algebra and Algebraic Geometry for their help and their comments, in particular Pr. William Sit and Raymond T. Hoobler. † This research was partially supported by EC contract ESPRIT B.R.A. n • 6846 POSSO. 1 We make precise in the following sections some of the notations and definitions used in this introduction.
⋆ This work has benefited from the support of the French ANR (decision number ANR-2010-BLAN-0109-... more ⋆ This work has benefited from the support of the French ANR (decision number ANR-2010-BLAN-0109-03).
In this paper, we describe a new way to count isomor-phism classes of rooted triangular maps and ... more In this paper, we describe a new way to count isomor-phism classes of rooted triangular maps and unrooted triangular maps. We point out an explicit connection with the asymptotic expansion of the Airy function. The analysis presented here is used in a recent paper "Vidal (2007)" to present an algorithm that gen-erates in optimal amortized time an exhaustive list of triangular maps of a given size.
⋆ This work has benefited from the support of the French ANR (decision number ANR-2010-BLAN-0109-... more ⋆ This work has benefited from the support of the French ANR (decision number ANR-2010-BLAN-0109-03).
The goal of the present paper is to propose an enhanced ordinary differential equations solver by... more The goal of the present paper is to propose an enhanced ordinary differential equations solver by exploitation of the powerful equivalence method of Élie Cartan. This solver returns a target equation equivalent to the equation to be solved and the transformation realizing the equivalence. The target ODE is a member of a dictionary of ODE, that are regarded as well-known, or at least well-studied. The dictionary considered in this article are ODE in a book of Kamke. The major advantage of our solver is that the equivalence transformation is obtained without integrating differential equations. We provide also a theoretical contribution revealing the relationship between the change of coordinates that maps two differential equations and their symmetry pseudo-groups.
We give an algorithm which represents the radical J of a finitely generated differential ideal as... more We give an algorithm which represents the radical J of a finitely generated differential ideal as an intersection of radical differential ideals. The computed representation provides an algorithm for testing membership in J. This algorithm works over either an ordinary or a partial differential polynomial ring of characteristic zero. It has been programmed. We also give a method to obtain a characteristic set of J , if the ideal is prime. Keywords. Differential Algebra. Radical differential ideals. Characteristic sets. * The authors would like to thank the participants of the Special Year in Differential Algebra and Algebraic Geometry for their help and their comments, in particular Pr. William Sit and Raymond T. Hoobler. † This research was partially supported by EC contract ESPRIT B.R.A. n • 6846 POSSO. 1 We make precise in the following sections some of the notations and definitions used in this introduction.
Abstract. This paper presents a symbolic algorithm for computing the ODE systems which describe t... more Abstract. This paper presents a symbolic algorithm for computing the ODE systems which describe the evolution of the moments associated to a chemical reaction system, considered from a stochastic point of view. The algorithm, which is formulated in the Weyl algebra, seems more efficient than the corresponding method, based on partial derivatives. In particular, an efficient method for handling conservation laws is presented. The output of the algorithm can be used for a further investigation of the system behaviour, by numerical methods. Relevant examples are carried out.
Le formalisme des systemes de reactions chimiques (de maniere equivalente des reseaux de Petri st... more Le formalisme des systemes de reactions chimiques (de maniere equivalente des reseaux de Petri stochastiques) est tres souvent utilise en bilogie, en surete de fonctionnement etc. La serie generatrice associee a la master-equation est solution d'une equation d'evolution du type equation de Schrodinger. On adopte ici l'approche classique par le calcul des fonctions propres en se concentrant, dans cette premiere partie, sur le calcul de la distribution stationnaire pour un systeme comportant une seule espece chimique. On montre que, generiquement, la serie generatrice stationnaire est une fonction holomorphe dans tout le plan complexe. Des exemples de calcul (symbolique-numerique) sur ordinateur sont developpes.
This paper presents a symbolic algorithm for computing the ODE systems which describe the evoluti... more This paper presents a symbolic algorithm for computing the ODE systems which describe the evolution of the moments associated to a chemical reaction system, considered from a stochastic point of view. The algorithm, which is formulated in the Weyl algebra, seems more ecient than the corresponding method, based on partial derivatives. In particular, an ecient method for handling conservation laws is pre- sented. The output of the algorithm can be used for a further investiga- tion of the system behaviour, by numerical methods. Relevant examples are carried out.
This paper considers an approach for analyzing the identifiability of nonlinear controlled or unc... more This paper considers an approach for analyzing the identifiability of nonlinear controlled or uncontrolled dynamical systems. The method is based on the computation of the ideal containing the differential algebraic relations between the input and the output of the model, a such ideal is called input-output ideal. Our contribution consists in developing the corresponding algorithm in a symbolic computing language. This algorithm is based on differential algebra.
Proceedings of 1995 34th IEEE Conference on Decision and Control, 1995
ABSTRACT We introduce a new class of nonlinear systems called “finitely discretizable”. For a fin... more ABSTRACT We introduce a new class of nonlinear systems called “finitely discretizable”. For a finitely discretizable nonlinear system, any trajectory is completely determined by a finite number of the state derivatives. In the polynomial case, using the notion of dilations, we give sufficient conditions for a nonlinear system to be finitely discretizable and we relate this result to nilpotent Lie algebra of vector fields. We show that multirate sampling techniques can be an efficient tool to deal with the control problem of finitely discretizable systems. We conclude by some examples illustrating the relevance of finitely discretizable systems within the framework of control theory
ABSTRACT In this paper we give, an algorithm that computes the local minimal analytical realizati... more ABSTRACT In this paper we give, an algorithm that computes the local minimal analytical realization of finite generating power series. In the same time, this algorithm proves that any finite generating series has a polynomial realization: observation and vector fields components are commutative polynomials. That algorithm is implemented in the computer algebra system Scratchpad.
Proceedings of the 1995 international symposium on Symbolic and algebraic computation - ISSAC '95, 1995
We give an algorithm which represents the radical J of a finitely generated differential ideal as... more We give an algorithm which represents the radical J of a finitely generated differential ideal as an intersection of radical differential ideals. The computed representation provides an algorithm for testing membership in J. This algorithm works over either an ordinary or a partial differential polynomial ring of characteristic zero. It has been programmed. We also give a method to obtain a characteristic set of J , if the ideal is prime. Keywords. Differential Algebra. Radical differential ideals. Characteristic sets. * The authors would like to thank the participants of the Special Year in Differential Algebra and Algebraic Geometry for their help and their comments, in particular Pr. William Sit and Raymond T. Hoobler. † This research was partially supported by EC contract ESPRIT B.R.A. n • 6846 POSSO. 1 We make precise in the following sections some of the notations and definitions used in this introduction.
⋆ This work has benefited from the support of the French ANR (decision number ANR-2010-BLAN-0109-... more ⋆ This work has benefited from the support of the French ANR (decision number ANR-2010-BLAN-0109-03).
In this paper, we describe a new way to count isomor-phism classes of rooted triangular maps and ... more In this paper, we describe a new way to count isomor-phism classes of rooted triangular maps and unrooted triangular maps. We point out an explicit connection with the asymptotic expansion of the Airy function. The analysis presented here is used in a recent paper "Vidal (2007)" to present an algorithm that gen-erates in optimal amortized time an exhaustive list of triangular maps of a given size.
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