arXiv subjects

Gérard Henry Edmond Duchamp

Publications and source records attributed to Gérard Henry Edmond Duchamp.

24 records · Page 2Linked to original sources

arXiv2006

Congruences Compatible with the Shuffle Product

This article is devoted to the study of monoids which can be endowed with a shuffle product with coefficients in a semiring. We show that, when the multiplicities do not belong to a ring with prime characteristic, such a monoid is a monoid of traces. When the characteristic is prime, we give a decomposition of the congruences $\equiv$ (or relators $R$) such that $A^*/\_\equiv=< A; R>$ admits a shuffle product. This decomposition involves only addition of primitive elements to the successive quotients. To end with, we study the compatibility with Magnus transformation and examine the case of congruences which are homogeneous for some weight function. The existence of such a weight function is also showed for congruences of depth one.

math.CO↗

arXiv2006

Transitive factorizations of free partially commutative monoids and Lie algebras

Let $\M(A,θ)$ be a free partially commutative monoid. We give here a necessary and sufficient condition on a subalphabet $B\subset A$ such that the right factor of a bisection $\M(A,θ)=\M(B,θ\_B).T$ be also partially commutative free. This extends strictly the (classical) elimination theory on partial commutations and allows to construct new factorizations of $\M(A,θ)$ and associated bases of $L\_K(A,θ)$.

math.CO↗

arXiv2005

Automata-based adaptive behavior for economic modeling using game theory

In this paper, we deal with some specific domains of applications to game theory. This is one of the major class of models in the new approaches of modelling in the economic domain. For that, we use genetic automata which allow to buid adaptive strategies for the players. We explain how the automata-based formalism proposed - matrix representation of automata with multiplicities - allows to define a semi-distance between the strategy behaviors. With that tools, we are able to generate an automatic processus to compute emergent systems of entities whose behaviors are represented by these genetic automata.

cs.MA↗

arXiv2005

Feynman graphs and related Hopf algebras

In a recent series of communications we have shown that the reordering problem of bosons leads to certain combinatorial structures. These structures may be associated with a certain graphical description. In this paper, we show that there is a Hopf Algebra structure associated with this problem which is, in a certain sense, unique.

cs.SC↗

arXiv2005

Transitive Hall sets

We give the definition of Lazard and Hall sets in the context of transitive factorizations of free monoids. The equivalence of the two properties is proved. This allows to build new effective bases of free partially commutative Lie algebras. The commutation graphs for which such sets exist are completely characterized and we explicit, in this context, the classical PBW rewriting process.

math.CO↗

arXiv2005

Tables, Memorized Semirings and Applications

We define and construct a new data structure, the tables, this structure generalizes the (finite) $k$-sets sets of Eilenberg \cite{Ei}, it is versatile (one can vary the letters, the words and the coefficients). We derive from this structure a new semiring (with several semiring structures) which can be applied to the needs of automatic processing multi-agents behaviour problems. The purpose of this account/paper is to present also the basic elements of this new structures from a combinatorial point of view. These structures present a bunch of properties. They will be endowed with several laws namely : Sum, Hadamard product, Cauchy product, Fuzzy operations (min, max, complemented product) Two groups of applications are presented. The first group is linked to the process of "forgetting" information in the tables. The second, linked to multi-agent systems, is announced by showing a methodology to manage emergent organization from individual behaviour models.

cs.MA↗