The symbolic computation of series solutions to ordinary differential equations using trees (extended abstract)

Algorithms previously developed by the author give formulas which can be used for the efficient symbolic computation of series expansions to solutions of nonlinear systems of ordinary differential equations. As a by product of this analysis, formulas are derived which relate to trees to the coefficients of the series expansions, similar to the work of Leroux and Viennot, and Lamnabhi, Leroux and Viennot

Querying databases of trajectories of differential equations 2: Index functions

Suppose that a large number of parameterized trajectories (gamma) of a dynamical system evolving in R sup N are stored in a database. Let eta is contained R sup N denote a parameterized path in Euclidean space, and let parallel to center dot parallel to denote a norm on the space of paths. A data structures and indices for trajectories are defined and algorithms are given to answer queries of the following forms: Query 1. Given a path eta, determine whether eta occurs as a subtrajectory of any trajectory gamma from the database. If so, return the trajectory; otherwise, return null. Query 2. Given a path eta, return the trajectory gamma from the database which minimizes the norm parallel to eta - gamma parallel

Bialgebra deformations and algebras of trees

Let A denote a bialgebra over a field k and let A sub t = A((t)) denote the ring of formal power series with coefficients in A. Assume that A is also isomorphic to a free, associative algebra over k. A simple construction is given which makes A sub t a bialgebra deformation of A. In typical applications, A sub t is neither commutative nor cocommutative. In the terminology of Drinfeld, (1987), A sub t is a quantum group. This construction yields quantum groups associated with families of trees

Computer algebra and operators

The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions

The realization of input-output maps using bialgebras

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