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Academic literature on the topic 'Arbre enraciné'
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Journal articles on the topic "Arbre enraciné"
Foissy, L. "Les algèbres de Hopf des arbres enracinés décorés, I." Bulletin des Sciences Mathématiques 126, no. 3 (March 2002): 193–239. http://dx.doi.org/10.1016/s0007-4497(02)01108-9.
Full textFoissy, L. "Les algèbres de Hopf des arbres enracinés décorés, II." Bulletin des Sciences Mathématiques 126, no. 4 (2002): 249–88. http://dx.doi.org/10.1016/s0007-4497(02)01113-2.
Full textFoissy, L. "Étude de l'algèbre de Lie double des arbres enracinés décorés." Advances in Mathematics 208, no. 2 (January 2007): 877–904. http://dx.doi.org/10.1016/j.aim.2006.04.002.
Full textSaïdi, Abdellatif, and Ridha Chatbouri. "Générateurs et Certaines Relations D'une Algèbre Pré-Lie sur les Arbres Enracinés." Communications in Algebra 41, no. 11 (November 2, 2013): 4033–45. http://dx.doi.org/10.1080/00927872.2012.699574.
Full textOger, Bérénice. "PreLie-decorated hypertrees." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AS,..., Proceedings (January 1, 2013). http://dx.doi.org/10.46298/dmtcs.12825.
Full textLevine, Lionel. "An Algebraic Analogue of a Formula of Knuth." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AN,..., Proceedings (January 1, 2010). http://dx.doi.org/10.46298/dmtcs.2867.
Full textBernardi, Olivier, and Eric Fusy. "A unified bijective method for maps: application to two classes with boundaries." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AN,..., Proceedings (January 1, 2010). http://dx.doi.org/10.46298/dmtcs.2869.
Full textLiu, Fu. "On bijections between monotone rooted trees and the comb basis." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 27th..., Proceedings (January 1, 2015). http://dx.doi.org/10.46298/dmtcs.2480.
Full textDissertations / Theses on the topic "Arbre enraciné"
Brieussel, Jérémie. "Croissance et moyennabilité de certains groupes d'automorphismes d'un arbre enraciné." Paris 7, 2008. http://www.theses.fr/2008PA077116.
Full textThe group of directed automorphisms of a rooted tree is shown to be amenable if the valencies of the tree are bounded. Using previous results of Wilson, amenable groups of non uniform exponential growth are constructed. Such groups are not elementary amenable by result of Osin. They are shown not even to be subexponentially amenable. These results generalize the construction of a family of groups of intermediate growth due to Grigorchuk. The growth function of the key exemple of such group is lowered by e^(n^a) with a=0,5207. Moreover, within this family are constructed groups with oscillating growth function, in particular groups the growth function of which is not equivalent to e^(n^a) for any real number a
Foissy, Loïc. "Les algèbres de Hopf des arbres enracinés décorés." Reims, 2002. http://www.theses.fr/2002REIMS010.
Full textConnes and Kreimer have introduced a Hopf algebra of (decorated) rooted trees Hr, in order to study Renormalization. We introduce here a Hopf algebra of planar decorated rooted trees Hpr, which construction generalizes the construction of Hr. This Hopf algebra satisfies a universal property in Hochschild cohomology. We show that it is self-dual. This property induces the existence of non-degenerate Hopf pairing between Hpr and itself. As a consequence, the dual basis of the basis of forests allows to find a basis of the space of the primitive elements of Hpr, and then to find all primitive elements of Hr, answering a question of Kreimer. Moreover, we study the Hr- and Hpr-comodules of finite dimension, and we establish the link between Hpr and several other Hopf algebras of trees, such as the Hopf algebras of Brouder and Frabetti, of Loday and Ronco, of Grossman and Larson, or the quantization of Hpr of Moerdijk and van der Laan
Ayadi, Mohamed. "Propriétés algébriques et combinatoires des espaces topologiques finis." Electronic Thesis or Diss., Université Clermont Auvergne (2021-...), 2022. http://www.theses.fr/2022UCFAC106.
Full textSaidi, Abdellatif. "Algèbres de Hopf d'arbres et structures pré-Lie." Phd thesis, Université Blaise Pascal - Clermont-Ferrand II, 2011. http://tel.archives-ouvertes.fr/tel-00720201.
Full textZhao, Jinhua. "Maximum Bounded Rooted-Tree Problem : Algorithms and Polyhedra." Thesis, Université Clermont Auvergne (2017-2020), 2017. http://www.theses.fr/2017CLFAC044/document.
Full textGiven a simple undirected graph G = (V, E) with a so-called root node r in V, a rooted tree, or an r-tree, of G is either the empty graph, or a tree containing r. If a node-capacity vector c is given, then a subgraph of G is said to be bounded if the degree of each node in the subgraph does not exceed its capacity. Let w be an edge-weight vector and p a node-price vector. The Maximum Bounded r-Tree (MBrT) problem consists of finding a bounded r-tree T = (U, F) of G such that its weight is maximized. If the capacity constraint from the MBrT problem is relaxed, we then obtain the Maximum r-Tree (MrT) problem. This dissertation contributes to the study of the MBrT problem and the MrT problem.First we introduce the problems with their definitions and complexities. We define the associated polytopes along with a formulation for each of them. We present several polynomial-time combinatorial algorithms for both the MBrT problem (and thus the MrT problem) on trees, cycles and cactus graphs. Particularly, a dynamic-programming-based algorithm is used to solve the MBrT problem on trees, whereas on cycles we reduce it to some polynomially solvable problems in three different cases. For cactus graphs, we first show that the MBrT problem can be solved in polynomial time on a so-called cactus basis, then break down the problem on any cactus graph into a series of subproblems on trees and on cactus basis.The second part of this work investigates the polyhedral structure of three polytopes associated with the MBrT problem and the MrT problem, namely Bxy(G, r, c), Bx(G, r, c) and Rx(G, r). Bxy(G, r, c) and Bx(G, r, c) are polytopes associated with the MBrT problem, where Bxy(G, r, c) considers both edge- and node-indexed variables and Bx(G, r, c) considers only edge-indexed variables. Rx(G, r) is the polytope associated with the MrT problem that only considers edge-indexed variables. For each of the three polytopes, we study their dimensions, facets as well as possible ways of decomposition. We introduce some newly discovered constraints for each polytope, and show that these new constraints allow us to characterize them on several graph classes. Specifically, we provide characterization for Bxy (G, r, c) on cactus graphs with the help of a decomposition through 1-sum. On the other hand, a TDI-system that characterizes Bx(G,r,c) is given in each case of trees and cycles. The characterization of Rx(G,r) on trees and cycles then follows as an immediate result.Finally, we discuss the separation problems for all the inequalities we have found so far, and present algorithms or cut-generation heuristics accordingly. A couple of branch-and-cut frameworks are implemented to solve the MBrT problem together with a greedy-based matheuristic. We compare the performances of the enhanced formulations with the original formulations through intensive computational test, where the results demonstrate convincingly the strength of the enhanced formulations
Saïdi, Abdellatif. "Algèbres de Hopf d'arbres et structures pré-Lie." Thesis, Clermont-Ferrand 2, 2011. http://www.theses.fr/2011CLF22208/document.
Full textWe investigate in this thesis the Hopf algebra structure on the vector space H spanned by the rooted forests, associated with the pre-Lie operad. The space of primitive elements of the graded dual of this Hopf algebra is endowed with a left pre-Lie product denoted by ⊲, defined in terms of insertion of a tree inside another. In this thesis we retrieve the “derivation” relation between the pre-Lie structure ⊲ and the left pre-Lie product → on the space of primitive elements of the graded dual H0CK of the Connes-Kreimer Hopf algebra HCK, defined by grafting. We also exhibit a coproduct on the tensor product H⊗HCK, making it a Hopf algebra the graded dual of which is isomorphic to the enveloping algebra of the semidirect product of the two (pre-)Lie algebras considered. We prove that the span of the rooted trees with at least one edge endowed with the pre-Lie product ⊲ is generated by two elements. It is not free : we exhibit two families of relations. Moreover we prove a similar result for the pre-Lie algebra associated with the NAP operad. Finally, we introduce current preserving operads and prove that the pre-Lie operad can be obtained as a deformation of the NAP operad in this framework
Obradović, Jovana. "Cyclic operads : syntactic, algebraic and categorified aspects." Thesis, Sorbonne Paris Cité, 2017. http://www.theses.fr/2017USPCC191/document.
Full textIn this thesis, we examine different frameworks for the general theory of cyclic operads of Getzler and Kapranov. As suggested by the title, we set up theoretical grounds of syntactic, algebraic and categorified nature for the notion of a cyclic operad.In the syntactic treatment, we propose a lambda-calculus-style formal language, called mu-syntax, as a lightweight representation of the entries-only cyclic operad structure. As opposed to the original exchangeable-output characterisation of cyclic operads, according to which the operations of a cyclic operad have inputs and an output that can be “exchanged” with one of the inputs, the entries-only cyclic operads have only entries (i.e. the output is put on the same level as the inputs). By employing the rewriting methods behind the formalism, we give a complete step-by-step proof of the equivalence between the unbiased and biased definitions of cyclic operads.Guided by the microcosm principle of Baez and Dolan and by the algebraic definitions of operads of Kelly and Fiore, in the algebraic approach we define cyclic operads internally to the category of Joyal’s species of structures. In this way, both the original exchangeable-output characterisation of Getzler and Kapranov, and the alternative entries-only characterisation of cyclic operads of Markl are epitomised as “monoid-like” objects in “monoidal-like” categories of species. Relying on a result of Lamarche on descent for species, we use these “monoid-like” definitions to prove the equivalence between the exchangeable-output and entries-only points of view on cyclic operads.Finally, we establish a notion of categorified cyclic operad for set-based cyclic operads with symmetries, defined in terms of generators and relations. The categorifications we introduce are obtained by replacing sets of operations of the same arity with categories, by relaxing certain defining axioms, like associativity and commutativity, to isomorphisms, while leaving the equivariance strict, and by formulating coherence conditions for these isomorphisms. The coherence theorem that we prove has the form “all diagrams of canonical isomorphisms commute”.For entries-only categorified cyclic operads, our proof is of syntactic nature and relies on the coherence of categorified operads established by Došen and Petrić. We prove the coherence of exchangeable-output categorified cyclic operads by “lifting to the categorified setting” theequivalence between entries-only and exchangeable-output cyclic operads, set up previously in the algebraic approach
Al-Kaabi, Mahdi Jasim Hasan. "Bases de monômes dans les algèbres pré-Lie libres et applications." Thesis, Clermont-Ferrand 2, 2015. http://www.theses.fr/2015CLF22599/document.
Full textIn this thesis, we study the concept of free pre-Lie algebra generated by a (non-empty) set. We review the construction by A. Agrachev and R. Gamkrelidze of monomial bases in free pre-Lie algebras. We describe the matrix of the monomial basis vectors in terms of the rooted trees basis exhibited by F. Chapoton and M. Livernet. Also, we show that this matrix is unipotent and we find an explicit expression for its coefficients, adapting a procedure implemented for the free magmatic algebra by K. Ebrahimi-Fard and D. Manchon. We construct a pre-Lie structure on the free Lie algebra $\mathcal{L}$(E) generated by a set E, giving an explicit presentation of $\mathcal{L}$(E) as the quotient of the free pre-Lie algebra $\mathcal{T}$^E, generated by the (non-planar) E-decorated rooted trees, by some ideal I. We study the Gröbner bases for free Lie algebras in tree version. We split the basis of E- decorated planar rooted trees into two parts O(J) and $\mathcal{T}$(J), where J is the ideal defining $\mathcal{L}$(E) as a quotient of the free magmatic algebra generated by E. Here $\mathcal{T}$(J) is the set of maximal terms of elements of J, and its complement O(J) then defines a basis of $\mathcal{L}$(E). We get one of the important results in this thesis (Theorem 3.12), on the description of the set O(J) in terms of trees. We describe monomial bases for the pre-Lie (respectively free Lie) algebra $\mathcal{L}$(E), using the procedure of Gröbner bases and the monomial basis for the free pre-Lie algebra obtained in Chapter 2. Finally, we study the so-called classical and pre-Lie Magnus expansions, discussing how we can find a recursion for the pre-Lie case which already incorporates the pre-Lie identity. We give a combinatorial vision of a numerical method proposed by S. Blanes, F. Casas, and J. Ros, on a writing of the classical Magnus expansion in $\mathcal{L}$(E), using the pre-Lie structure