172 research outputs found
Partial monoids: associativity and confluence
A partial monoid is a set with a partial multiplication (and
total identity ) which satisfies some associativity axiom. The partial
monoid may be embedded in a free monoid and the product is
simulated by a string rewriting system on that consists in evaluating the
concatenation of two letters as a product in , when it is defined, and a
letter as the empty word . In this paper we study the profound
relations between confluence for such a system and associativity of the
multiplication. Moreover we develop a reduction strategy to ensure confluence
and which allows us to define a multiplication on normal forms associative up
to a given congruence of . Finally we show that this operation is
associative if, and only if, the rewriting system under consideration is
confluent
On Drinfel'd associators
In 1986, in order to study the linear representations of the braid group
coming from the monodromy of the Knizhnik-Zamolodchikov differential
equations,Drinfel'd introduced a class of formal power series on
noncommutative variables. These formal series can be considered as a class of
associators. We here give an interpretation of them as well as some new tools
over Noncommutative Evolution Equations. Asymptotic phenomena are also
discussed
M\"obius inversion formula for monoids with zero
The M\"obius inversion formula, introduced during the 19th century in number
theory, was generalized to a wide class of monoids called locally finite such
as the free partially commutative, plactic and hypoplactic monoids for
instance. In this contribution are developed and used some topological and
algebraic notions for monoids with zero, similar to ordinary objects such as
the (total) algebra of a monoid, the augmentation ideal or the star operation
on proper series. The main concern is to extend the study of the M\"obius
function to some monoids with zero, i.e., with an absorbing element, in
particular the so-called Rees quotients of locally finite monoids. Some
relations between the M\"obius functions of a monoid and its Rees quotient are
also provided.Comment: 12 pages, r\'esum\'e \'etendu soumis \`a FPSAC 201
Different goals in multiscale simulations and how to reach them
In this paper we sum up our works on multiscale programs, mainly simulations.
We first start with describing what multiscaling is about, how it helps
perceiving signal from a background noise in a ?ow of data for example, for a
direct perception by a user or for a further use by another program. We then
give three examples of multiscale techniques we used in the past, maintaining a
summary, using an environmental marker introducing an history in the data and
finally using a knowledge on the behavior of the different scales to really
handle them at the same time
Tables, Memorized Semirings and Applications
We define and construct a new data structure, the tables, this structure
generalizes the (finite) -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
A formal calculus on the Riordan near algebra
29 p.International audienceThe Riordan group is the semi-direct product of a multiplicative group of invertible series and a group, under substitution, of non units. The Riordan near algebra, as introduced in this paper, is the Cartesian product of the algebra of formal power series and its principal ideal of non units, equipped with a product that extends the multiplication of the Riordan group. The later is naturally embedded as a subgroup of units into the former. In this paper, we prove the existence of a formal calculus on the Riordan algebra. This formal calculus plays a role similar to those of holomorphic calculi in the Banach or Fréchet algebras setting, but without the constraint of a radius of convergence. Using this calculus, we define \emph{en passant} a notion of generalized powers in the Riordan group
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