Integration of Paths, Geometric Invariants and a Generalized Baker-Hausdorff Formula

The Hopf Algebra of Rooted Trees, Free Lie Algebras, and Lie Series

“…(If, for each w ∈ A * , α, w = α w and β, w = β w , then α β, w = (αβ) w given in (6).) Consider the graded commutative K-algebra K[T ] (graded by the degree of forests, or, more generally, by the weight of forests if A is a weighted alphabet), where the unity element is represented by the empty forest, and each monomial t 1 · · · t m with t 1 , .…”

“…(6) In some applications, the alphabet A may be infinite, but then it typically makes sense assigning a weight (a positive integer) to each letter, and, when needed, truncating the series (4) according to the weight w of each word w ∈ A. Note that the set A * of words on the alphabet A with a prescribed weight is finite provided that the subsets A k ⊂ A of letters with weight k (k ≥ 1) are finite.…”

“…A well-known result due to Chen [6] implies that the series (4) can be formally rewritten as the exponential of a series of vector fields obtained as nested commutators of E 1 and E 2 . One of the goals of the present work (Subsection 4.2) is to obtain explicit formulas for such series of vector fields (the continuous BakerCampbell-Hausdorff (BCH) formula), expressed in appropriate basis of the Lie algebra generated by the vector fields E 1 and E 2 .…”

Some Algebraic Aspects of the Center Problem for Ordinary Differential Equations

“…The following results were obtained in [11]. Their proofs are based on [14,Thm. 4.2] and the results of [21].…”

Numerical Evaluation of the Evans Function by Magnus Integration

“…We can formally write the solution to (2.3) as the Neumann series (2.6) which converges provided t 0 A(τ ) dτ < ∞. (This method of successive approximation for integral equations is also known as the Peano-Baker series, matrizant, Feynman-Dyson path ordered exponential, or Chen-Fleiss series [34,3,13,10,37]). …”