2 The “algebra” K[[x]] ⋊ M of formal power series under multiplication and substitution 3 2.1 Basics on formal power series . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 “Algebra” of formal power series under substitution . . . . . . . . . . . . 4 2.2.1 Right-distributive algebras . . . . . . . . . . . . . . . . . . . . . . 4 2.2.2 Substitution of formal power series . . . . . . . . . ....

2 The Riordan skew algebra K[[x]] ⋊ M of formal power series under multiplication and substitution 3 2.1 Basics on formal power series . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Skew algebra of formal power series under substitution . . . . . . . . . . 4 2.2.1 Right-distributive algebras . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 Substitution of formal power series . . . . ....

The paper presents the abstract framework of hybrid formal power series. Hybrid formal power series are analogous to non-commutative formal power series. Formal power series are widely used in control systems theory. In particular, theory of formal power series is the main tool for solving the realization problem for linear and bilinear control systems. The theory of hybrid formal power series ...

In nonlinear control, it is helpful to choose a formalism well suited to computations involving solutions of controlled diierential equations, exponentials of vector elds, and Lie brackets. We show by means of an example |the computation of control variations that give rise to the Legendre-Clebsch condition| how a good choice of formalism , based on expanding diieomorphisms as products of expon...

On the basis of run semantics and breadth-first algebraic semantics, the algebraic characterizations for a classes of formal power series over complete strong bimonoids are investigated in this paper. As recognizers, weighted pushdown automata with final states (WPDAs for short) and empty stack (WPDAs[Formula: see text]) are shown to be equivalent based on run semantics. Moreover, it is demonst...

We study probability distributions over free algebras of trees. Probability distributions can be seen as particular (formal power) tree series [BR82; EK03], i.e. mappings from trees to a semiring K. A widely studied class of tree series is the class of rational (or recognizable) tree series which can be defined either in an algebraic way or by means of multiplicity tree automata. We argue that ...

We next want to construct a much larger ring in which infinite sums of multiples of elements of S are allowed. In order to insure that multiplication is well-defined, from now on we assume that S has the following additional property: (#) For all s ∈ S, {(s1, s2) ∈ S × S : s1s2 = s} is finite. Thus, each element of S has only finitely many factorizations as a product of two elements. For exampl...

In a recent paper we introduced Parikh slender languages and series as a generalization of slender languages de ned and studied by Andra siu, Dassow, P aun and Salomaa. Results concerning Parikh slender series can be applied in ambiguity proofs of context-free languages. In this paper an algorithm is presented for deciding whether or not a given N-algebraic series is Parikh slender. Category: F...

Gabrielov’s famous example for the failure of analytic Artin approximation in the presence of nested subring conditions is shown to be due to a growth phenomenon in standard basis computations for echelons, a generalization of the concept of ideals in power series rings. hh july 31, 2017 Introduction In the Séminaire Henri Cartan of 1960/61, Grothendieck posed the question whether analytically ...

A theorem of Christol states that a power series over a finite field is algebraic over the polynomial ring if and only if its coefficients can be generated by a finite automaton. Using Christol’s result, we prove that the same assertion holds for generalized power series (whose index sets may be arbitrary well-ordered sets of nonnegative rationals).