About enveloping algebras of direct sums
We solve the PBW-like problem of normal ordering for enveloping algebras of direct sums.
cs.LO↗
arXiv subjects
Vu Nguyen Dinh
Publications and source records attributed to Vu Nguyen Dinh.
Towards a noncommutative Picard-Vessiot theory
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.
math.AG↗
Towards a Theory of Domains for Harmonic Functions and its Symbolic Counterpart
In this paper, we begin by reviewing the calculus induced by the framework of [10]. In there, we extended Polylogarithm functions over a subalgebra of noncommutative rational power series, recognizable by finite state (multiplicity) automata over the alphabet X = {x 0 , x 1 }. The stability of this calculus under shuffle products relies on the nuclearity of the target space [31]. We also concentrated on algebraic and analytic aspects of this extension allowing to index polylogarithms, at non positive multi-indices, by rational series and also allowing to regularize divergent polyzetas, at non positive multi-indices [10]. As a continuation of works in [10] and in order to understand the bridge between the extension of this "polylogarithmic calculus" and the world of harmonic sums, we propose a local theory, adapted to a full calculus on indices of Harmonic Sums based on the Taylor expansions, around zero, of polylogarithms with index x 1 on the rightmost end. This theory is not only compatible with Stuffle products but also with the Analytic Model. In this respect, it provides a stable and fully algorithmic model for Harmonic calculus. Examples by computer are also provided 6 .
cs.SC↗
Regions of the type C Catalan arrangement
In this paper, we give a bijection between rooted labeled ordered forests with a selected subset of their leaves and the regions of the type $C$ Catalan arrangement in $\R^n$. We thus obtain a bijective proof of the well-known enumeration formula of these regions ${2^n}n! \binom{2n}{n}$.
math.CO↗