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Papers by vincel hoang ngoc minh

HAL (Le Centre pour la Communication Scientifique Directe), Jun 26, 2011

In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0,... more In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0, x1} with coefficients in a Q-extension, A, subjected to some suitable conditions, there exists an unique algebra homomorphism ϕ from the Q-algebra generated by the convergent polyzêtas to A such that Φ is computed from ΦKZ Drinfel'd associator by applying ϕ to each coefficient. We prove ϕ exists and it is a free Lie exponential over X. Moreover, we give a complete description of the kernel of polyzêta and draw some consequences about a structure of the algebra of convergent polyzêtas and about the arithmetical nature of the Euler constant.

IFAC Proceedings Volumes, Jun 1, 1989

Abslract : Given a nonlmear analytical dynamic system (affine with respect to the I1lpul) , lis O... more Abslract : Given a nonlmear analytical dynamic system (affine with respect to the I1lpul) , lis Olllput funclion can be viewed as a signal paramelri:ed by the Iw/mltlVes of the mput functlOns, TIllS signal can be formally describcd by ils generating series. Hence we oblam (I symbolic transform that generalizes Laplace transforlll of signals depend only on the inne. We deve/o p here the basic lools of thal symbolzc calculus. We prove a correspondence theo/'em belu;een certam convolutlOns of Signals and Cauchy products of generatmg, serus. Finally the Taylor expansion of triangular Volterra kernels IS simply deduced.

HAL (Le Centre pour la Communication Scientifique Directe), Jun 26, 2011

In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0,... more In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0, x1} with coefficients in a Q-extension, A, subjected to some suitable conditions, there exists an unique algebra homomorphism ϕ from the Q-algebra generated by the convergent polyzêtas to A such that Φ is computed from ΦKZ Drinfel'd associator by applying ϕ to each coefficient. We prove ϕ exists and it is a free Lie exponential over X. Moreover, we give a complete description of the kernel of polyzêta and draw some consequences about a structure of the algebra of convergent polyzêtas and about the arithmetical nature of the Euler constant.

Publications mathématiques de l'Université de Franche-Comté Besançon, Jun 15, 2023

Extending the Eulerian functions, we study their relationship with zeta function of several varia... more Extending the Eulerian functions, we study their relationship with zeta function of several variables. In particular, starting with Weierstrass factorization theorem (and Newton-Girard identity) for the complex Gamma function, we are interested in the ratios of ζ(2k)/π 2k and their multiindexed generalization, we obtain an analogue situation and draw some consequences about a structure of the algebra of polyzetas values, by means of some combinatorics of words and noncommutative rational series. The same frameworks also allow to study the independence of a family of eulerian functions. Résumé.-(Familles de fonctions eulériennes impliquées dans la régularisation de polyzêtas divergents) En généralisant les fonctions euleriennes, nous étudions leurs relations avec la fonction zêta en plusieurs variables. En particulier, à partir du théorème de factorisation de Weierstrass (et l'identité de Newton-Girard) pour la fonction Gamma complexe, nous nous intéressons aux rapports ζ(2k)/π 2k et leurs généralisations. Nous obtenons une situation analogue et nous tirerons quelques conséquences sur une structure de l'algèbre des valeurs polyzêtas, au moyen de la combinatoire des mots et des séries rationnelles en variables non commutatifs. Le même cadre de travail permet également d'étudier l'indépendance d'une famille de fonctions euleriennes.

arXiv (Cornell University), Oct 10, 2009

In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0,... more In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0, x1} with coefficients in a Q-extension, A, subjected to some suitable conditions, there exists an unique algebra homomorphism ϕ from the Q-algebra generated by the convergent polyzêtas to A such that Φ is computed from ΦKZ Drinfel'd associator by applying ϕ to each coefficient. We prove ϕ exists and it is a free Lie exponential over X. Moreover, we give a complete description of the kernel of polyzêta and draw some consequences about a structure of the algebra of convergent polyzêtas and about the arithmetical nature of the Euler constant. Contents

HAL (Le Centre pour la Communication Scientifique Directe), Jul 20, 2015

In order to push the study of solutions of nonlinear differential equations involved in quantum e... more In order to push the study of solutions of nonlinear differential equations involved in quantum electrodynamics 1 , we focus here on combinatorial aspects of their renormalization at {0, 1, +∞}.

Presses Académiques Francophones eBooks, Oct 21, 2014

Il est bien etabli que le calcul formel permet de verifier des conjectures, des demonstrations, d... more Il est bien etabli que le calcul formel permet de verifier des conjectures, des demonstrations, des formules longues et penibles. L'innovation est qu'avec quelques operations elementaires sur des suites de symboles 0 et 1, on capte aussi, de facon exacte, de nouvelles formules, identites remarquables et conjectures insolites. Dans ce travail, nous illustrons cette derniere assertion en proposant des algorithmes via la combinatoire des mots de Lyndon. Nous montrons d'abord que l'algebre des polylogarithmes est isomorphe a celle du melange et nous en tirons des consequences concernant le calcul effectif de la monodromie, du comportement asymptotique, des relations algebriques et des equations fonctionnelles. Nous examinons ensuite le rapport entre les series generatrices commutatives des polylogarithmes et des fonctions hypergeometriques pour obtenir diverses sommations. La serie generatrice non commutative des polylogarithmes nous mene au calcul de l'associateur de Drinfel'd mis sous forme factorisee. Nous appliquons finalement ces etudes aux equations integro-differentielles et a l'obtention d'un systeme de reecriture des relations polynomiales entre les sommes d'Euler-Zagier.

• The output function of any control system can be viewed as a signal parametrized by the primiti... more • The output function of any control system can be viewed as a signal parametrized by the primitives of the input functions. This signal can be formally described by its generating series. Then the temporal behaviour of any system can be derived from this generating ...

Journal of Symbolic Computation, Nov 1, 2017

Extending Eulerian polynomials and Faulhaber's formula 1 , we study several combinatorial aspects... more Extending Eulerian polynomials and Faulhaber's formula 1 , we study several combinatorial aspects of harmonic sums and polylogarithms at non-positive multi-indices as well as their structure. Our techniques are based on the combinatorics of noncommutative generating series in the shuffle Hopf algebras giving a global process to renormalize the divergent polyzetas at non-positive multi-indices.

Springer eBooks, Oct 5, 2005

ABSTRACT

HAL (Le Centre pour la Communication Scientifique Directe), Oct 24, 2021

A Chen generating series, along a path and with respect to m differential forms, is a noncommutat... more A Chen generating series, along a path and with respect to m differential forms, is a noncommutative series on m letters and with coefficients which are holomorphic functions over a simply connected manifold in other words a series with variable (holomorphic) coefficients. Such a series satisfies a first order noncommutative differential equation which is considered, by some authors, as the universal differential equation, i.e., in this case, universality can be seen by replacing each letter by constant matrices (resp. holomorphic vector fields) and then solving a system of linear (resp. nonlinear) differential equations. Via rational series, on noncommutative indeterminates and with coefficients in rings, and their non-trivial combinatorial Hopf algebras, we give the first step of a noncommutative Picard-Vessiot theory and we illustrate it with the case of linear differential equations with singular regular singularities thanks to the universal equation previously mentioned. Contents 1 Introduction 2 2 Combinatorial framework 4 2.1 Factorization in bialgebras 4 2.2 Representative series 9 3 Triangularity, solvability and rationality 13 3.1 Syntactically exchangeable rational series 13 3.2 Exchangeable rational series and their linear representations 15 4 Towards a noncommutative Picard-Vessiot theory 20 4.1 Noncommutative differential equations 20 4.2 First step of a noncommutative Picard-Vessiot theory 23 5 Conclusion 25 References 25

HAL (Le Centre pour la Communication Scientifique Directe), Mar 9, 2020

This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations (3) and... more This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations (3) and our recent results on combinatorial aspects of zeta functions on several variables. In particular, we describe the action of the differential Galois group of 3 on the asymptotic expansions of its solutions leading to a group of associators which contains the unique Drinfel'd associator (or Drinfel'd series). Non trivial expressions of an associator with rational coefficients are also explicitly provided, based on the algebraic structure and the singularity analysis of the multi-indexed polylogarithms and harmonic sums. Contents 1. Knizhnik-Zamolodchikov equations and Drinfel'd series 2. Combinatorial framework 2.1. Shuffle and quasi-shuffle algebras 2.2. Diagonal series on bialgebras 2.3. Exchangeable and noncommutative rational series 3. Indexation by words and generating series 3.1. Indexation by words 3.2. Indexation by noncommutative rational series 3.3. Noncommutative generating series 4. Global asymptotic behaviors at singularities 4.1. The case of positive multi-indices 4.2. Structure of polyzetas 4.3. The case of negative multi-indices 5. A group of associators 5.1. The action of the Galois differential group 5.2. Associator Φ 5.3. Associators with rational coefficients 6. Conclusion Appendix A Appendix B Appendix C Appendix D References Math. classification: ??

arXiv (Cornell University), May 20, 2013

In this work, an effective construction, via Schützenberger's monoidal factorization, of dual bas... more In this work, an effective construction, via Schützenberger's monoidal factorization, of dual bases for the non commutative symmetric and quasi-symmetric functions is proposed.

Elsevier eBooks, 1990

Abslract : Given a nonlmear analytical dynamic system (affine with respect to the I1lpul) , lis O... more Abslract : Given a nonlmear analytical dynamic system (affine with respect to the I1lpul) , lis Olllput funclion can be viewed as a signal paramelri:ed by the Iw/mltlVes of the mput functlOns, TIllS signal can be formally describcd by ils generating series. Hence we oblam (I symbolic transform that generalizes Laplace transforlll of signals depend only on the inne. We deve/o p here the basic lools of thal symbolzc calculus. We prove a correspondence theo/'em belu;een certam convolutlOns of Signals and Cauchy products of generatmg, serus. Finally the Taylor expansion of triangular Volterra kernels IS simply deduced.

Springer eBooks, Jun 11, 2008

ABSTRACT Given a nonlinear control system, one can view its output function as a signal, parametr... more ABSTRACT Given a nonlinear control system, one can view its output function as a signal, parametrized by the primitives of the input functions. This signal can be formally described by its Fliess’ power series, that is a formal power series on noncommuting variables. The temporal behaviour of the system can be derived from this symbolic description by a transform, that we call “Evaluation transform” and that generalizes the inverse Laplace transform to the nonlinear area. We developpe here the basic tools of that symbolic calculus by introducing a “kernel” for our Evaluation transform. This kernel can be viewed as some “temporal memory” of the system in the Volterra’s meaning as well as in the programmation meaning.

Theoretical Computer Science, Feb 1, 1991

Given a nonlinear control system, one can view its output function as a signal, parametrized by t... more Given a nonlinear control system, one can view its output function as a signal, parametrized by the primitives of the input functions. This signal can be formally described by Fliess' power series, that is a formal power series on noncommuting variables. The temporal behaviour of the system can be derived from this symbolic description by a transform, called Evaluation transform, which generalizes the inverse Laplace transform to the nonlinear area. We develop here the basic tools of that symbolic calculus. We prove a correspondence theorem between certain convolutions of signals and Cauchy products of generating power series.

arXiv (Cornell University), Sep 18, 2022

This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations (KZ3) u... more This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations (KZ3) using our recent results on combinatorial aspects of zeta functions on several variables and software on noncommutative symbolic computations. In particular, we describe the actual solution of (KZ3) leading to the unique noncommutative series, ΦKZ , so-called Drinfel'd associator (or Drinfel'd series). Non-trivial expressions for series with rational coefficients, satisfying the same properties with ΦKZ , are also explicitly provided due to the algebraic structure and the singularity analysis of the polylogarithms and harmonic sums.

arXiv (Cornell University), Sep 18, 2022

Basing on Picard-Vessiot theory of noncommutative differential equations and algebraic combinator... more Basing on Picard-Vessiot theory of noncommutative differential equations and algebraic combinatorics on noncommutative formal series with holomorphic coefficients, various recursive constructions of sequences of grouplike series converging to solutions of universal differential equation are proposed. These intensively use diagonal series in bialgebra and in a Loday's generalized bialgebra. As application, the unique solution, satisfying asymptotic conditions, of Knizhnik-Zamolodchikov equations is provided by dévissage. Contents 1. Introducing remarks 1 2. Combinatorial frameworks 9 2.1. Algebraic combinatorics on formal power series 9 2.2. More about diagonal series in concatenation-shuffle bialgebra and in a Loday's generalized bialgebra 15 3. Solutions of universal differential equation 18 3.1. Noncommutative differential equations 18 3.2. Knizhnik-Zamolodchikov equations 27 4. Conclusion 32 5. Appendices 34 5.1. Aprroximation solution for KZ 3 and identification the coefficients of log Φ KZ by Drinfel'd 34 5.2. KZ 3 , the simplest non-trial case 35 5.3. KZ 4 , other simplest non-trial case 36 References 38

The theory of noncommutative rational power series allows to express as iterated integrals some g... more The theory of noncommutative rational power series allows to express as iterated integrals some generating series associated to polylogarithms and polyzetas, also called MZV's (multiple zeta values: a generalization of the Riemann \zeta function). We introduce the Hurwitz polyzetas, as a multivalued generalization of the classical Hurwitz \zeta function. They are in fact generating series of the classical polyzetas in

Journal de Théorie des Nombres de Bordeaux, 2007

HAL (Le Centre pour la Communication Scientifique Directe), Jun 26, 2011

In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0,... more In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0, x1} with coefficients in a Q-extension, A, subjected to some suitable conditions, there exists an unique algebra homomorphism ϕ from the Q-algebra generated by the convergent polyzêtas to A such that Φ is computed from ΦKZ Drinfel'd associator by applying ϕ to each coefficient. We prove ϕ exists and it is a free Lie exponential over X. Moreover, we give a complete description of the kernel of polyzêta and draw some consequences about a structure of the algebra of convergent polyzêtas and about the arithmetical nature of the Euler constant.

IFAC Proceedings Volumes, Jun 1, 1989

Abslract : Given a nonlmear analytical dynamic system (affine with respect to the I1lpul) , lis O... more Abslract : Given a nonlmear analytical dynamic system (affine with respect to the I1lpul) , lis Olllput funclion can be viewed as a signal paramelri:ed by the Iw/mltlVes of the mput functlOns, TIllS signal can be formally describcd by ils generating series. Hence we oblam (I symbolic transform that generalizes Laplace transforlll of signals depend only on the inne. We deve/o p here the basic lools of thal symbolzc calculus. We prove a correspondence theo/'em belu;een certam convolutlOns of Signals and Cauchy products of generatmg, serus. Finally the Taylor expansion of triangular Volterra kernels IS simply deduced.

HAL (Le Centre pour la Communication Scientifique Directe), Jun 26, 2011

In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0,... more In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0, x1} with coefficients in a Q-extension, A, subjected to some suitable conditions, there exists an unique algebra homomorphism ϕ from the Q-algebra generated by the convergent polyzêtas to A such that Φ is computed from ΦKZ Drinfel'd associator by applying ϕ to each coefficient. We prove ϕ exists and it is a free Lie exponential over X. Moreover, we give a complete description of the kernel of polyzêta and draw some consequences about a structure of the algebra of convergent polyzêtas and about the arithmetical nature of the Euler constant.

Publications mathématiques de l'Université de Franche-Comté Besançon, Jun 15, 2023

Extending the Eulerian functions, we study their relationship with zeta function of several varia... more Extending the Eulerian functions, we study their relationship with zeta function of several variables. In particular, starting with Weierstrass factorization theorem (and Newton-Girard identity) for the complex Gamma function, we are interested in the ratios of ζ(2k)/π 2k and their multiindexed generalization, we obtain an analogue situation and draw some consequences about a structure of the algebra of polyzetas values, by means of some combinatorics of words and noncommutative rational series. The same frameworks also allow to study the independence of a family of eulerian functions. Résumé.-(Familles de fonctions eulériennes impliquées dans la régularisation de polyzêtas divergents) En généralisant les fonctions euleriennes, nous étudions leurs relations avec la fonction zêta en plusieurs variables. En particulier, à partir du théorème de factorisation de Weierstrass (et l'identité de Newton-Girard) pour la fonction Gamma complexe, nous nous intéressons aux rapports ζ(2k)/π 2k et leurs généralisations. Nous obtenons une situation analogue et nous tirerons quelques conséquences sur une structure de l'algèbre des valeurs polyzêtas, au moyen de la combinatoire des mots et des séries rationnelles en variables non commutatifs. Le même cadre de travail permet également d'étudier l'indépendance d'une famille de fonctions euleriennes.

arXiv (Cornell University), Oct 10, 2009

In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0,... more In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0, x1} with coefficients in a Q-extension, A, subjected to some suitable conditions, there exists an unique algebra homomorphism ϕ from the Q-algebra generated by the convergent polyzêtas to A such that Φ is computed from ΦKZ Drinfel'd associator by applying ϕ to each coefficient. We prove ϕ exists and it is a free Lie exponential over X. Moreover, we give a complete description of the kernel of polyzêta and draw some consequences about a structure of the algebra of convergent polyzêtas and about the arithmetical nature of the Euler constant. Contents

HAL (Le Centre pour la Communication Scientifique Directe), Jul 20, 2015

In order to push the study of solutions of nonlinear differential equations involved in quantum e... more In order to push the study of solutions of nonlinear differential equations involved in quantum electrodynamics 1 , we focus here on combinatorial aspects of their renormalization at {0, 1, +∞}.

Presses Académiques Francophones eBooks, Oct 21, 2014

Il est bien etabli que le calcul formel permet de verifier des conjectures, des demonstrations, d... more Il est bien etabli que le calcul formel permet de verifier des conjectures, des demonstrations, des formules longues et penibles. L'innovation est qu'avec quelques operations elementaires sur des suites de symboles 0 et 1, on capte aussi, de facon exacte, de nouvelles formules, identites remarquables et conjectures insolites. Dans ce travail, nous illustrons cette derniere assertion en proposant des algorithmes via la combinatoire des mots de Lyndon. Nous montrons d'abord que l'algebre des polylogarithmes est isomorphe a celle du melange et nous en tirons des consequences concernant le calcul effectif de la monodromie, du comportement asymptotique, des relations algebriques et des equations fonctionnelles. Nous examinons ensuite le rapport entre les series generatrices commutatives des polylogarithmes et des fonctions hypergeometriques pour obtenir diverses sommations. La serie generatrice non commutative des polylogarithmes nous mene au calcul de l'associateur de Drinfel'd mis sous forme factorisee. Nous appliquons finalement ces etudes aux equations integro-differentielles et a l'obtention d'un systeme de reecriture des relations polynomiales entre les sommes d'Euler-Zagier.

• The output function of any control system can be viewed as a signal parametrized by the primiti... more • The output function of any control system can be viewed as a signal parametrized by the primitives of the input functions. This signal can be formally described by its generating series. Then the temporal behaviour of any system can be derived from this generating ...

Journal of Symbolic Computation, Nov 1, 2017

Extending Eulerian polynomials and Faulhaber's formula 1 , we study several combinatorial aspects... more Extending Eulerian polynomials and Faulhaber's formula 1 , we study several combinatorial aspects of harmonic sums and polylogarithms at non-positive multi-indices as well as their structure. Our techniques are based on the combinatorics of noncommutative generating series in the shuffle Hopf algebras giving a global process to renormalize the divergent polyzetas at non-positive multi-indices.

Springer eBooks, Oct 5, 2005

ABSTRACT

HAL (Le Centre pour la Communication Scientifique Directe), Oct 24, 2021

A Chen generating series, along a path and with respect to m differential forms, is a noncommutat... more A Chen generating series, along a path and with respect to m differential forms, is a noncommutative series on m letters and with coefficients which are holomorphic functions over a simply connected manifold in other words a series with variable (holomorphic) coefficients. Such a series satisfies a first order noncommutative differential equation which is considered, by some authors, as the universal differential equation, i.e., in this case, universality can be seen by replacing each letter by constant matrices (resp. holomorphic vector fields) and then solving a system of linear (resp. nonlinear) differential equations. Via rational series, on noncommutative indeterminates and with coefficients in rings, and their non-trivial combinatorial Hopf algebras, we give the first step of a noncommutative Picard-Vessiot theory and we illustrate it with the case of linear differential equations with singular regular singularities thanks to the universal equation previously mentioned. Contents 1 Introduction 2 2 Combinatorial framework 4 2.1 Factorization in bialgebras 4 2.2 Representative series 9 3 Triangularity, solvability and rationality 13 3.1 Syntactically exchangeable rational series 13 3.2 Exchangeable rational series and their linear representations 15 4 Towards a noncommutative Picard-Vessiot theory 20 4.1 Noncommutative differential equations 20 4.2 First step of a noncommutative Picard-Vessiot theory 23 5 Conclusion 25 References 25

HAL (Le Centre pour la Communication Scientifique Directe), Mar 9, 2020

This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations (3) and... more This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations (3) and our recent results on combinatorial aspects of zeta functions on several variables. In particular, we describe the action of the differential Galois group of 3 on the asymptotic expansions of its solutions leading to a group of associators which contains the unique Drinfel'd associator (or Drinfel'd series). Non trivial expressions of an associator with rational coefficients are also explicitly provided, based on the algebraic structure and the singularity analysis of the multi-indexed polylogarithms and harmonic sums. Contents 1. Knizhnik-Zamolodchikov equations and Drinfel'd series 2. Combinatorial framework 2.1. Shuffle and quasi-shuffle algebras 2.2. Diagonal series on bialgebras 2.3. Exchangeable and noncommutative rational series 3. Indexation by words and generating series 3.1. Indexation by words 3.2. Indexation by noncommutative rational series 3.3. Noncommutative generating series 4. Global asymptotic behaviors at singularities 4.1. The case of positive multi-indices 4.2. Structure of polyzetas 4.3. The case of negative multi-indices 5. A group of associators 5.1. The action of the Galois differential group 5.2. Associator Φ 5.3. Associators with rational coefficients 6. Conclusion Appendix A Appendix B Appendix C Appendix D References Math. classification: ??

arXiv (Cornell University), May 20, 2013

In this work, an effective construction, via Schützenberger's monoidal factorization, of dual bas... more In this work, an effective construction, via Schützenberger's monoidal factorization, of dual bases for the non commutative symmetric and quasi-symmetric functions is proposed.

Elsevier eBooks, 1990

Abslract : Given a nonlmear analytical dynamic system (affine with respect to the I1lpul) , lis O... more Abslract : Given a nonlmear analytical dynamic system (affine with respect to the I1lpul) , lis Olllput funclion can be viewed as a signal paramelri:ed by the Iw/mltlVes of the mput functlOns, TIllS signal can be formally describcd by ils generating series. Hence we oblam (I symbolic transform that generalizes Laplace transforlll of signals depend only on the inne. We deve/o p here the basic lools of thal symbolzc calculus. We prove a correspondence theo/'em belu;een certam convolutlOns of Signals and Cauchy products of generatmg, serus. Finally the Taylor expansion of triangular Volterra kernels IS simply deduced.

Springer eBooks, Jun 11, 2008

ABSTRACT Given a nonlinear control system, one can view its output function as a signal, parametr... more ABSTRACT Given a nonlinear control system, one can view its output function as a signal, parametrized by the primitives of the input functions. This signal can be formally described by its Fliess’ power series, that is a formal power series on noncommuting variables. The temporal behaviour of the system can be derived from this symbolic description by a transform, that we call “Evaluation transform” and that generalizes the inverse Laplace transform to the nonlinear area. We developpe here the basic tools of that symbolic calculus by introducing a “kernel” for our Evaluation transform. This kernel can be viewed as some “temporal memory” of the system in the Volterra’s meaning as well as in the programmation meaning.

Theoretical Computer Science, Feb 1, 1991

Given a nonlinear control system, one can view its output function as a signal, parametrized by t... more Given a nonlinear control system, one can view its output function as a signal, parametrized by the primitives of the input functions. This signal can be formally described by Fliess' power series, that is a formal power series on noncommuting variables. The temporal behaviour of the system can be derived from this symbolic description by a transform, called Evaluation transform, which generalizes the inverse Laplace transform to the nonlinear area. We develop here the basic tools of that symbolic calculus. We prove a correspondence theorem between certain convolutions of signals and Cauchy products of generating power series.

arXiv (Cornell University), Sep 18, 2022

This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations (KZ3) u... more This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations (KZ3) using our recent results on combinatorial aspects of zeta functions on several variables and software on noncommutative symbolic computations. In particular, we describe the actual solution of (KZ3) leading to the unique noncommutative series, ΦKZ , so-called Drinfel'd associator (or Drinfel'd series). Non-trivial expressions for series with rational coefficients, satisfying the same properties with ΦKZ , are also explicitly provided due to the algebraic structure and the singularity analysis of the polylogarithms and harmonic sums.

arXiv (Cornell University), Sep 18, 2022

Basing on Picard-Vessiot theory of noncommutative differential equations and algebraic combinator... more Basing on Picard-Vessiot theory of noncommutative differential equations and algebraic combinatorics on noncommutative formal series with holomorphic coefficients, various recursive constructions of sequences of grouplike series converging to solutions of universal differential equation are proposed. These intensively use diagonal series in bialgebra and in a Loday's generalized bialgebra. As application, the unique solution, satisfying asymptotic conditions, of Knizhnik-Zamolodchikov equations is provided by dévissage. Contents 1. Introducing remarks 1 2. Combinatorial frameworks 9 2.1. Algebraic combinatorics on formal power series 9 2.2. More about diagonal series in concatenation-shuffle bialgebra and in a Loday's generalized bialgebra 15 3. Solutions of universal differential equation 18 3.1. Noncommutative differential equations 18 3.2. Knizhnik-Zamolodchikov equations 27 4. Conclusion 32 5. Appendices 34 5.1. Aprroximation solution for KZ 3 and identification the coefficients of log Φ KZ by Drinfel'd 34 5.2. KZ 3 , the simplest non-trial case 35 5.3. KZ 4 , other simplest non-trial case 36 References 38

The theory of noncommutative rational power series allows to express as iterated integrals some g... more The theory of noncommutative rational power series allows to express as iterated integrals some generating series associated to polylogarithms and polyzetas, also called MZV's (multiple zeta values: a generalization of the Riemann \zeta function). We introduce the Hurwitz polyzetas, as a multivalued generalization of the classical Hurwitz \zeta function. They are in fact generating series of the classical polyzetas in

Journal de Théorie des Nombres de Bordeaux, 2007