Zimmermann Type Cancellation in the Free Faa di Bruno Algebra

Center problem, Abel equation and the Faa di Bruno Hopf algebra for output feedback

A combinatorial interpretation is given of Devlin's word problem underlying the classical center-focus problem of Poincare for non-autonomous differential equations. It turns out that the canonical polynomials of Devlin are from the point of view of connected graded Hopf algebras intimately related to the graded components of a Hopf algebra antipode applied to the formal power series of Ferfera. The link is made by passing through control theory since the Abel equation, which describes a center, is equivalent to an output feedback equation, and the Hopf algebra of output feedback is derived fr…

Four explicit formulas for the prolongations of an infinitesimal Lie symmetry and multivariate Faa di Bruno formulas

In 1979, building on S. Lie's theory of symmetries of (partial) differrential equations, P.J. Olver formulated inductive formulas which are appropriate for the computation of the prolongations of an infinitesimal Lie symmetry to jet spaces, for an arbitrary number n\geq 1 of independent variables (x^1, ..., x^n) and for an arbitrary number m \geq 1 of dependent variables (y^1, ..., y^m). This paper is devoted to elaborate a formalism based on multiple Kronecker symbols which enables one to handle these ``unmanageable'' prolongations and to discover the underlying complicated combinatorics. Pro…

Centralizers of the Riordan Group

In this paper, we discuss centralizers in the Riordan group. We will see that Faà di Bruno's formula is an application of the Fundamental Theorem of Riordan arrays. Then the composition group of formal power series in ${\cal F}_1$ is studied to construct the centralizers of Bell type and Lagrange type Riordan arrays. Our tools are the $A$-sequences of Riordan arrays and Faà di Bruno's formula. Some combinatorial explanation and discussion about related algebraic topics are also given.

SISO Output Affine Feedback Transformation Group and Its Faa di Bruno Hopf Algebra

The general goal of this paper is to identify a transformation group that can be used to describe a class of feedback interconnections involving subsystems which are modeled solely in terms of Chen-Fliess functional expansions or Fliess operators and are independent of the existence of any state space models. This interconnection, called an output affine feedback connection, is distinguished from conventional output feedback by the presence of a multiplier in an outer loop. Once this transformation group is established, three basic questions are addressed. How can this transformation group be …

Combinatorics of Partial Derivatives

The natural forms of the Leibniz rule for the $k$th derivative of a product and of Faà di Bruno's formula for the $k$th derivative of a composition involve the differential operator $\partial^k/\partial x_1 ... \partial x_k$ rather than $d^k/dx^k$, with no assumptions about whether the variables $x_1,...,x_k$ are all distinct, or all identical, or partitioned into several distinguishable classes of indistinguishable variables. Coefficients appearing in forms of these identities in which some variables are indistinguishable are just multiplicities of indistinguishable terms (in particular, if a…

Combinatorial Hopf algebras in quantum field theory I

This manuscript stands at the interface between combinatorial Hopf algebra theory and renormalization theory. Its plan is as follows: Section 1 is the introduction, and contains as well an elementary invitation to the subject. The rest of part I, comprising Sections 2-6, is devoted to the basics of Hopf algebra theory and examples, in ascending level of complexity. Part II turns around the all-important Faa di Bruno Hopf algebra. Section 7 contains a first, direct approach to it. Section 8 gives applications of the Faa di Bruno algebra to quantum field theory and Lagrange reversion. Section 9 …

Combinatorial Hopf algebras from PROs

We introduce a general construction that takes as input a so-called stiff PRO and that outputs a Hopf algebra. Stiff PROs are particular PROs that can be described by generators and relations with precise conditions. Our construction generalizes the classical construction from operads to Hopf algebras of van der Laan. We study some of its properties and review some examples of application. We get in particular Hopf algebras on heaps of pieces and retrieve some deformed versions of the noncommutative Faà di Bruno algebra introduced by Foissy.

A combinatorial Hopf algebra for nonlinear output feedback control systems

In this work a combinatorial description is provided of a Faa di Bruno type Hopf algebra which naturally appears in the context of Fliess operators in nonlinear feedback control theory. It is a connected graded commutative and non-cocommutative Hopf algebra defined on rooted circle trees. A cancellation free forest formula for its antipode is given.

Associahedra, cyclohedra and inversion of power series

We introduce the Hopf monoid of sets of cycles and paths, which contains the Faà di Bruno Hopf monoid as a submonoid. We give cancellation-free and grouping-free formulas for its antipode, one in terms of tubings and one in terms of \emph{pointed} noncrossing partitions. We provide an explicit description of the group of characters of this Hopf monoid in terms of pairs of power series. Using graph associahedra, we relate paths and cycles to associahedra and cyclohedra, respectively. We give formulas for inversion in the group of characters in terms of the faces of these polytopes.

方程 $y^{\prime}=f(y)$ 的解与Bell多项式

Solution of the equation $y^{\prime}=f(y)$ and Bell Polynomials

本文通过Faa di Bruno公式，用部分Bell多项式表示微分方程$y^{\prime}=f(y)$的解，并借此计算部分Bell多项式的特殊值。

Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras

Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally in the combinatorics of set partitions, but also when studying compositions of diffeomorphisms on vector spaces and manifolds, and in the study of cumulants and moments in probability theory. We construct commutative and noncommutative Bell polynomials and explain how they give rise to Faà di Bruno Hopf algebras. We use the language of incidence Hopf algebras, and along the way provide a new description of antipodes in noncommutative incidence Hopf algebras, involving quasideterminants…

Formulas for the Connes-Moscovici Hopf algebra

We give explicit formulas for the coproduct and the antipode in the Connes-Moscovici Hopf algebra $\mathcal{H}_{\tmop{CM}}$. To do so, we first restrict ourselves to a sub-Hopf algebra $\mathcal{H}^1_{\tmop{CM}}$ containing the nontrivial elements, namely those for which the coproduct and the antipode are nontrivial. There are two ways to obtain explicit formulas. On one hand, the algebra $\mathcal{H}^1_{\tmop{CM}}$ is isomorphic to the Faà di Bruno Hopf algebra of coordinates on the group of identity-tangent diffeomorphism and computations become easy using substitution automorphisms rather t…

非对称operad的自然Hopf代数的多项式实现

Polynomial realizations of natural Hopf algebras of nonsymmetric operads

本文为自然Hopf代数引入新基，通过变量字母表构造多项式实现，并揭示其与多个已知Hopf代数（如词拟对称函数、Connes-Kreimer装饰版本等）的联系。

由非对称 operad 构建的 Hopf 代数的多项式实现

Polynomial realizations of Hopf algebras built from nonsymmetric operads

本文利用带一元与二元关系的非交换变量字母表，为非对称 operad 的自然 Hopf 代数构造多项式实现，并将其与多种经典 Hopf 代数联系起来。