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We use the theory of bialgebras to provide the algebraic background for state space realization theorems for input-output maps of control systems. This allows us to consider from a common viewpoint classical results about formal state space realizations of nonlinear systems and more recent results involving analysis related to families of trees. If $H$ is a bialgebra, we say that $p \in H^*$ is differentially produced by the algebra $R$ with the augmentation $\epsilon$ if there is right $H$-module algebra structure on $R$ and there exists $f \in R$ satisfying $p(h) = \epsilon(f \cdot h)$. We characterize those $p \in H^*$ which are differentially produced., Comment: 16 pages

In this paper, we consider formal series associated with events, profiles derived from events, and statistical models that make predictions about events. We prove theorems about realizations for these formal series using the language and tools of Hopf algebras.

We provide an expository account of some of the Hopf algebras that can be defined using trees, labeled trees, ordered trees and heap ordered trees. We also describe some actions of these Hopf algebras on algebra of functions.

It is known that the vector space spanned by labeled rooted trees forms a Hopf algebra. Let k be a field and let R be a commutative k-algebra. Let H denote the Hopf algebra of rooted trees labeled using derivations D in Der(R). In this paper, we introduce a construction which gives R a H-module algebra structure and show this induces a differential algebra structure of H acting on R. The work here extends the notion of a R/k-bialgebra introduced by Nichols and Weisfeiler., Comment: 31 pages, 8 figures

A Hopf-algebraic structure on a vector space which has as basis a family of trees is described. Some applications of this structure to combinatorics and to differential operators are surveyed. Some possible future directions for this work are indicated.

The effective parallel symbolic computation of operators under composition is discussed. Examples include differential operators under composition and vector fields under the Lie bracket. Data structures consisting of formal linear combinations of rooted labeled trees are discussed. A multiplication on rooted labeled trees is defined, thereby making the set of these data structures into an associative algebra. An algebra homomorphism is defined from the original algebra of operators into this algebra of trees. An algebra homomorphism from the algebra of trees into the algebra of differential operators is then described. The cancellation which occurs when noncommuting operators are expressed in terms of commuting ones occurs naturally when the operators are represented using this data structure. This leads to an algorithm which, for operators which are derivations, speeds up the computation exponentially in the degree of the operator. It is shown that the algebra of trees leads naturally to a parallel version of the algorithm.

The theory of bialgebras is used to prove a state space realization theorem for input/output maps of dynamical systems. This approach allows for the consideration of the classical results of Fliess and more recent results on realizations involving families of trees. Two examples of applications of the theorum are given.