arXiv subjects
Publications and source records attributed to Pierre Simonnet.
In this short note, we reprove in a very elementary way some known facts about Pisano periods as well as some considerations about the link between Pisano periods and the order of roots of the characteristic equation. The technics only requires a small background in ring theory (merely the definition of a commutative ring). The tools set here can be reused for all linear recurrences with quadratic non-constant characteristic equation.
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 functionsover 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, bysome authors, as the universal differential equation, (i.e.) universality can beseen by replacing each letter by constant matrices (resp. analytic vector fields)and then solving a system of linear (resp. nonlinear) differential equations.Via rational series, on noncommutative indeterminates and with coefficients in rings, andtheir non-trivial combinatorial Hopf algebras, we give the first step of a noncommutativePicard-Vessiot theory and we illustrate it with the case of linear differential equationswith singular regular singularities thanks to the universal equation previously mentioned.
In this short note, we give a localized version of the basic triangle theorem, first published in 2011 (see [4]) in order to prove the independence of hyperlogarithms over various function fields. This version provides direct access to rings of scalars and avoids the recourse to fraction fields as that of meromorphic functions for instance.
In this work, we focus on on the approach by noncommutative formal power series to study the combinatorial aspects of the renormalization at the singularities in $\{0,1,+\infty\}$ of the solutions of nonlinear differential equations involved in quantum electrodynamics.
The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $\Game (D\_n({\bf\Sigma}^0\_2))$ for some natural number $n\geq 1$, where $\Game$ is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space $2^\omega$ into the class ${\bf\Delta}^1\_2$, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ${\bf\Delta}^1\_2$.
In this paper we propose a numerical method to solve the Cauchy problem based on the viscous shallow water equations in an horizontally moving domain. More precisely, we are interested in a flooding and drying model, used to modelize the overflow of a river or the intrusion of a tsunami on ground. We use a non conservative form of the two-dimensional shallow water equations, in eight velocity formulation and we build a numerical approximation, based on the Arbitrary Lagrangian Eulerian formulation, in order to compute the solution in the moving domain.
We investigate the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets. We show that, for each integer $n \geq 1$, there is a $D_{ω^n}({\bf Σ}^1_1)$-complete tree language L_n accepted by a (non deterministic) Muller tree automaton. On the other hand, we prove that a tree language accepted by an unambiguous Büchi tree automaton must be Borel. Then we consider the game tree languages $W_{(i,k)}$, for Mostowski-Rabin indices $(i, k)$. We prove that the $D_{ω^n}({\bf Σ}^1_1)$-complete tree languages L_n are Wadge reducible to the game tree language $W_{(i, k)}$ for $k-i \geq 2$. In particular these languages $W_{(i, k)}$ are not in any class $D_α({\bf Σ}^1_1)$ for $α< ω^ω$.
In this paper, we study the continuity of rational functions realized by Büchi finite state transducers. It has been shown by Prieur that it can be decided whether such a function is continuous. We prove here that surprisingly, it cannot be decided whether such a function F has at least one point of continuity and that its continuity set C(F) cannot be computed. In the case of a synchronous rational function, we show that its continuity set is rational and that it can be computed. Furthermore we prove that any rational Pi^0_2-subset of X^omega for some alphabet X is the continuity set C(F) of an omega-rational synchronous function F defined on X^omega.
We study the links between the topological complexity of an omega context free language and its degree of ambiguity. In particular, using known facts from classical descriptive set theory, we prove that non Borel omega context free languages which are recognized by Büchi pushdown automata have a maximum degree of ambiguity. This result implies that degrees of ambiguity are really not preserved by the operation of taking the omega power of a finitary context free language. We prove also that taking the adherence or the delta-limit of a finitary language preserves neither unambiguity nor inherent ambiguity. On the other side we show that methods used in the study of omega context free languages can also be applied to study the notion of ambiguity in infinitary rational relations accepted by Büchi 2-tape automata and we get first results in that direction.
We consider the set of infinite real traces, over a dependence alphabet (Gamma, D) with no isolated letter, equipped with the topology induced by the prefix metric. We then prove that all rational languages of infinite real traces are analytic sets and that there exist some rational languages of infinite real traces which are analytic but non Borel sets, and even Sigma^1_1-complete, hence of maximum possible topological complexity.