Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1507.06939

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

Kurusch Ebrahimi-Fard·W. Steven Gray
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 from the composition of iterated integrals rather than just the products of iterated integrals, which yields the shuffle algebra. This means that the primary algebraic structure at play in Devlin's approach is actually not the shuffle algebra, but a Faa di Bruno type Hopf algebra, which is defined in terms of the shuffle product but is a distinct algebraic structure.

Explore related subjects

math.RAmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kurusch Ebrahimi-Fard, W. Steven Gray. 2015-11-30. Center problem, Abel equation and the Faa di Bruno Hopf algebra for output feedback. https://doi.org/10.1093/imrn%2Frnw167

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On solutions of singular Sylvester equations in quaternions

The quaternionic equations ax-xb=0 and ax-xb=c are investigated, which are called homogeneous and inhomogeneous Sylvester equations, respectively. Conditions for the existence of solutions are provided. In addition, the general and nonzero solutions to these equations are derived applying quaternion square roots.

math.RA↗
arXiv2026

Positivity preservers over finite fields II

We say that a matrix over a finite field $\mathbb{F}_q$ is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in $\mathbb{F}_q$. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on $M_n(\mathbb{F}_q)$ were classified in every case except when $n=2$, $q\equiv1\pmod4$, and $q$ is not a square. We settle this remaining case, thereby completing the classification of entrywise positivity preservers over every finite field and in every dimension $n\ge2$. Our proof is based on a novel idempotent reduction that not only resolves the remaining case but also yields a self-contained proof of the complete classification, while avoiding several technical results used in the earlier arguments. As a further application of the same reduction, we classify the entrywise preservers of strongly nonsingular matrices, i.e., matrices whose leading principal minors are all nonzero. We also prove a more general theorem in odd characteristic: for every prescribed sign pattern of nonzero leading principal minors of matrices of a fixed dimension $n\ge2$, the entrywise preservers are precisely the positive scalar multiples of field automorphisms. Thus, in odd characteristic, preserving any nonzero leading-principal-minor sign pattern surprisingly forces the preservation of every such sign pattern.

math.RA↗
arXiv2026

Graphs of Moore-Penrose inverse of matrices possessing the treeangle property

It is known that the inverse of an invertible real square matrix satisfying the treeangle property, is a treediagonal matrix. A converse statement also holds. We show that the verbatim analogues are not true for the Moore-Penrose inverse, and obtain the precise structure of graphs corresponding to the Moore-Penrose inverse of matrices possessing the treeangle property.

math.RA↗