X. Viennot - Academia.edu (original) (raw)
Papers by X. Viennot
ACM SIGGRAPH Computer Graphics, 1989
Herein is presented a new procedural method for generating images of trees. Many other algorithms... more Herein is presented a new procedural method for generating images of trees. Many other algorithms have already been proposed in the last few years focusing on particle systems, fractals, graftals and L-systems or realistic botanical models. Usually the final visual aspect of the tree depends on the development process leading to this form. Our approach differs from all the previous ones. We begin by defining a certain "measure" of the form of a tree or a branching pattern. This is done by introducing the new concept of ramification matrix of a tree. Then we give an algorithm for generating a random tree having as ramification matrix a given arbitrary stochastic triangular matrix. The geometry of the tree is defined from the combinatorial parameters implied in the analysis of the forms of trees. We obtain a method with powerful control of the final form, simple enough to produce quick designs of trees without loosing in the variety and rendering of the images. We also intro...
Annales des Sciences Mathematiques du Quebec
Nous construisons une famille de series formelles en deux variables dont tout element f(y, t) adm... more Nous construisons une famille de series formelles en deux variables dont tout element f(y, t) admet un developpement U en T-fraction, par rapport a la variable t, tel que la T-fraction duale U * est un developpement de f(t, y -1 ), toujours par rapport a t. Ce resultat est etabli grâce a une interpretation combinatoire du couple (U, U * ). Nous donnons l'exemple de la serie generatrice des polyominos parallelogrammes, qui fait intervenir des q-analogues de fonctions de Bessel.
Nonlinear Control Systems Design 1992, 1993
Publisher Summary This chapter discusses computing iterated derivatives along trajectories of non... more Publisher Summary This chapter discusses computing iterated derivatives along trajectories of nonlinear systems. It describes the theory of species. This theory combines the usual language of formal power series with the classical enumeration formulas. It allows to compute specific coefficients in the development of the solution of nonlinear differential equations. It develops a convenient language for the description of systems. The theory of species shows the internal structure of the classical algebric manipulations and synthetizes a long development into the description of a certain class of arborescences. This approach is particularly efficient for the design of computer algebra algorithms. There are new developments in this direction. Classical developments on implicit polynomial differential equations are also possible.
Lecture Notes in Computer Science, 1990
Page 1. TREES EVERYWHERE Xavier G6rard VIENNOT1 LaBRI2, Universit6 de Bordeaux I 33405 TALENCE Ce... more Page 1. TREES EVERYWHERE Xavier G6rard VIENNOT1 LaBRI2, Universit6 de Bordeaux I 33405 TALENCE Cedex, France ... Page 2. 19 It was started in Hydrogeology by Horton [28] and Strahler [70] in some work about the morphological structure of river networks. ...
Lecture Notes in Mathematics, 1978
Bases des algebres de Lie libres et factorisations des monofdes libres.
Advances in Applied Mathematics, 1994
... Des résultats exacts en dimension d=2 (nombre d'animaux, largeur moyenne,:.) ont été démo... more ... Des résultats exacts en dimension d=2 (nombre d'animaux, largeur moyenne,:.) ont été démontrés par Derrida, Hakim, Nadal et Vannimenus [17] ,[ 21] , ainsi que par Dhar [12] ,[13] en dimension d=2 et 3 . Ce dernier montre l'équivalence avec le modèle de gaz des hexagones ...
Introduction In studying representability of matroids, Lindstrom [42] gave a combinatorial interp... more Introduction In studying representability of matroids, Lindstrom [42] gave a combinatorial interpretation to certain determinants in terms of disjoint paths in digraphs. In a previous paper [25], the authors applied this theorem to determinants of binomial coe#cients. Here we develop further applications. As in [25], the paths under consideration are lattice paths in the plane. Our applications may be divided into two classes: first are those in which a determinant is shown to count some objects of combinatorial interest, and second are those which give a combinatorial interpretation to some numbers which are of independent interest. In the first class are formulas for various types of plane partitions, and in the second class are combinatorial interpretations for Fibonomial coe#cients, Bernoulli numbers, and the less-known Salie and Faulhaber numbers (which arise in formulas for sums of powers, and are closely related to Genocchi and Bernoulli numbers). Other enumerative appl
Formal Power Series and Algebraic Combinatorics, 2000
During the last thirty years, a growing interest for Pade approximants appeared in many theoretic... more During the last thirty years, a growing interest for Pade approximants appeared in many theoretical and applied fields, such as numerical analysis, theoretical physics, chemistry, electronics, … as shown in the books Baker [1], Baker, Graves-Morris [2], Brezinski [3], Gilewicz [11]. Pade approximants are strongly connected with continued fractions (see for example Henrici [16], Jones, Thron [17], Wall [25]) and orthogonal polynomials (see for example Brezinski [4, 5], Draux [7], Van Rossum [22], Wynn [26]). The so-called quotient-difference algorithm, or qd-algorithm, plays an important role in these theories. It was originated in Steifel [21] and studied by Rutishauser [19], Henrici [16,15]. (See also Brezinski [5], Gragg [12]).
Lecture Notes in Biomathematics, 1985
Journal of Statistical Physics, 1989
Tree-like patterns appear in many domains of physics and the quantitative description of their mo... more Tree-like patterns appear in many domains of physics and the quantitative description of their morphology raises an interesting problem. To analyze their topological structure, we introduce combinatorial concepts, the bifurcation and length ratios and the ramification matrix, which generalize ideas originating in hydrogeology. Two-dimensional diffusion-limited aggregation (DLA) patterns are studied along these lines, and their statistical combinatorial properties are compared
Séminaire Lotharingien de Combinatoire, 2010
The aim of this note is to show how the introduction of certain tableaux, called Catalan alternat... more The aim of this note is to show how the introduction of certain tableaux, called Catalan alternative tableaux, provides a very simple and elegant description of the product in the Hopf algebra of binary trees defined by Loday and Ronco. Moreover, we use this description to introduce a new associative product on the space of binary trees.
Journal of Statistical Physics, 1989
Tree-like patterns appear in many domains of physics and the quantitative description of their mo... more Tree-like patterns appear in many domains of physics and the quantitative description of their morphology raises an interesting problem. To analyze their topological structure, we introduce combinatorial concepts, the bifurcation and length ratios and the ramification matrix, which generalize ideas originating in hydrogeology. Two-dimensional diffusion-limited aggregation (DLA) patterns are studied along these lines, and their statistical combinatorial properties are compared to those of random and growing binary trees and to experimental data for injection of water in clay.
Journal of Combinatorial Theory, Series A, 1992
Introduction In studying representability of matroids, Lindstrom [42] gave a combinatorial interp... more Introduction In studying representability of matroids, Lindstrom [42] gave a combinatorial interpretation to certain determinants in terms of disjoint paths in digraphs. In a previous paper [25], the authors applied this theorem to determinants of binomial coe#cients. Here we develop further applications. As in [25], the paths under consideration are lattice paths in the plane. Our applications may be divided into two classes: first are those in which a determinant is shown to count some objects of combinatorial interest, and second are those which give a combinatorial interpretation to some numbers which are of independent interest. In the first class are formulas for various types of plane partitions, and in the second class are combinatorial interpretations for Fibonomial coe#cients, Bernoulli numbers, and the less-known Salie and Faulhaber numbers (which arise in formulas for sums of powers, and are closely related to Genocchi and Bernoulli numbers). Other enumerative appl
ACM SIGGRAPH Computer Graphics, 1989
Herein is presented a new procedural method for generating images of trees. Many other algorithms... more Herein is presented a new procedural method for generating images of trees. Many other algorithms have already been proposed in the last few years focusing on particle systems, fractals, graftals and L-systems or realistic botanical models. Usually the final visual aspect of the tree depends on the development process leading to this form. Our approach differs from all the previous ones. We begin by defining a certain "measure" of the form of a tree or a branching pattern. This is done by introducing the new concept of ramification matrix of a tree. Then we give an algorithm for generating a random tree having as ramification matrix a given arbitrary stochastic triangular matrix. The geometry of the tree is defined from the combinatorial parameters implied in the analysis of the forms of trees. We obtain a method with powerful control of the final form, simple enough to produce quick designs of trees without loosing in the variety and rendering of the images. We also intro...
Annales des Sciences Mathematiques du Quebec
Nous construisons une famille de series formelles en deux variables dont tout element f(y, t) adm... more Nous construisons une famille de series formelles en deux variables dont tout element f(y, t) admet un developpement U en T-fraction, par rapport a la variable t, tel que la T-fraction duale U * est un developpement de f(t, y -1 ), toujours par rapport a t. Ce resultat est etabli grâce a une interpretation combinatoire du couple (U, U * ). Nous donnons l'exemple de la serie generatrice des polyominos parallelogrammes, qui fait intervenir des q-analogues de fonctions de Bessel.
Nonlinear Control Systems Design 1992, 1993
Publisher Summary This chapter discusses computing iterated derivatives along trajectories of non... more Publisher Summary This chapter discusses computing iterated derivatives along trajectories of nonlinear systems. It describes the theory of species. This theory combines the usual language of formal power series with the classical enumeration formulas. It allows to compute specific coefficients in the development of the solution of nonlinear differential equations. It develops a convenient language for the description of systems. The theory of species shows the internal structure of the classical algebric manipulations and synthetizes a long development into the description of a certain class of arborescences. This approach is particularly efficient for the design of computer algebra algorithms. There are new developments in this direction. Classical developments on implicit polynomial differential equations are also possible.
Lecture Notes in Computer Science, 1990
Page 1. TREES EVERYWHERE Xavier G6rard VIENNOT1 LaBRI2, Universit6 de Bordeaux I 33405 TALENCE Ce... more Page 1. TREES EVERYWHERE Xavier G6rard VIENNOT1 LaBRI2, Universit6 de Bordeaux I 33405 TALENCE Cedex, France ... Page 2. 19 It was started in Hydrogeology by Horton [28] and Strahler [70] in some work about the morphological structure of river networks. ...
Lecture Notes in Mathematics, 1978
Bases des algebres de Lie libres et factorisations des monofdes libres.
Advances in Applied Mathematics, 1994
... Des résultats exacts en dimension d=2 (nombre d'animaux, largeur moyenne,:.) ont été démo... more ... Des résultats exacts en dimension d=2 (nombre d'animaux, largeur moyenne,:.) ont été démontrés par Derrida, Hakim, Nadal et Vannimenus [17] ,[ 21] , ainsi que par Dhar [12] ,[13] en dimension d=2 et 3 . Ce dernier montre l'équivalence avec le modèle de gaz des hexagones ...
Introduction In studying representability of matroids, Lindstrom [42] gave a combinatorial interp... more Introduction In studying representability of matroids, Lindstrom [42] gave a combinatorial interpretation to certain determinants in terms of disjoint paths in digraphs. In a previous paper [25], the authors applied this theorem to determinants of binomial coe#cients. Here we develop further applications. As in [25], the paths under consideration are lattice paths in the plane. Our applications may be divided into two classes: first are those in which a determinant is shown to count some objects of combinatorial interest, and second are those which give a combinatorial interpretation to some numbers which are of independent interest. In the first class are formulas for various types of plane partitions, and in the second class are combinatorial interpretations for Fibonomial coe#cients, Bernoulli numbers, and the less-known Salie and Faulhaber numbers (which arise in formulas for sums of powers, and are closely related to Genocchi and Bernoulli numbers). Other enumerative appl
Formal Power Series and Algebraic Combinatorics, 2000
During the last thirty years, a growing interest for Pade approximants appeared in many theoretic... more During the last thirty years, a growing interest for Pade approximants appeared in many theoretical and applied fields, such as numerical analysis, theoretical physics, chemistry, electronics, … as shown in the books Baker [1], Baker, Graves-Morris [2], Brezinski [3], Gilewicz [11]. Pade approximants are strongly connected with continued fractions (see for example Henrici [16], Jones, Thron [17], Wall [25]) and orthogonal polynomials (see for example Brezinski [4, 5], Draux [7], Van Rossum [22], Wynn [26]). The so-called quotient-difference algorithm, or qd-algorithm, plays an important role in these theories. It was originated in Steifel [21] and studied by Rutishauser [19], Henrici [16,15]. (See also Brezinski [5], Gragg [12]).
Lecture Notes in Biomathematics, 1985
Journal of Statistical Physics, 1989
Tree-like patterns appear in many domains of physics and the quantitative description of their mo... more Tree-like patterns appear in many domains of physics and the quantitative description of their morphology raises an interesting problem. To analyze their topological structure, we introduce combinatorial concepts, the bifurcation and length ratios and the ramification matrix, which generalize ideas originating in hydrogeology. Two-dimensional diffusion-limited aggregation (DLA) patterns are studied along these lines, and their statistical combinatorial properties are compared
Séminaire Lotharingien de Combinatoire, 2010
The aim of this note is to show how the introduction of certain tableaux, called Catalan alternat... more The aim of this note is to show how the introduction of certain tableaux, called Catalan alternative tableaux, provides a very simple and elegant description of the product in the Hopf algebra of binary trees defined by Loday and Ronco. Moreover, we use this description to introduce a new associative product on the space of binary trees.
Journal of Statistical Physics, 1989
Tree-like patterns appear in many domains of physics and the quantitative description of their mo... more Tree-like patterns appear in many domains of physics and the quantitative description of their morphology raises an interesting problem. To analyze their topological structure, we introduce combinatorial concepts, the bifurcation and length ratios and the ramification matrix, which generalize ideas originating in hydrogeology. Two-dimensional diffusion-limited aggregation (DLA) patterns are studied along these lines, and their statistical combinatorial properties are compared to those of random and growing binary trees and to experimental data for injection of water in clay.
Journal of Combinatorial Theory, Series A, 1992
Introduction In studying representability of matroids, Lindstrom [42] gave a combinatorial interp... more Introduction In studying representability of matroids, Lindstrom [42] gave a combinatorial interpretation to certain determinants in terms of disjoint paths in digraphs. In a previous paper [25], the authors applied this theorem to determinants of binomial coe#cients. Here we develop further applications. As in [25], the paths under consideration are lattice paths in the plane. Our applications may be divided into two classes: first are those in which a determinant is shown to count some objects of combinatorial interest, and second are those which give a combinatorial interpretation to some numbers which are of independent interest. In the first class are formulas for various types of plane partitions, and in the second class are combinatorial interpretations for Fibonomial coe#cients, Bernoulli numbers, and the less-known Salie and Faulhaber numbers (which arise in formulas for sums of powers, and are closely related to Genocchi and Bernoulli numbers). Other enumerative appl