In this paper we study heavy-weighted automata, a generalization of weighted automata in which the weights of the transitions can be formal power series. As for ordinary weighted automata, the behaviour of heavy-weighted automata is expressed in terms of formal power series. We propose several equivalent definitions for their semantics, including a system of behavioural differential equations (...

The theory of formal power series and derivation is developed from the point of view of the power matrix. A Loewner equation for formal power series is introduced. We then show that the matrix exponential is surjective onto the group of power matrices, and the coefficients are entire functions of finitely many coefficients of the infinitesimal generator. Furthermore coefficients of the solution...

The goal of this article is to define multivariate polynomials in arbitrary number of indeterminates and then to prove that they constitute a ring (over appropriate structure of coefficients). The introductory section includes quite a number of auxiliary lemmas related to many different parts of the MML. The second section characterizes the sequence flattening operation, introduced in [9], but ...

The goal of this article is to define multivariate polynomials in arbitrary number of indeterminates and then to prove that they constitute a ring (over appropriate structure of coefficients). The introductory section includes quite a number of auxiliary lemmas related to many different parts of the MML. The second section characterizes the sequence flattening operation, introduced in [7], but ...

For the quantum integer [n]q = 1+q+q + · · ·+q there is a natural polynomial multiplication such that [mn]q = [m]q ⊗q [n]q . This multiplication is described by the functional equation fmn(q) = fm(q)fn(q ), defined on a given sequence F = {fn(q)} ∞ n=1 of polynomials such that fn(0) = 1 for all n. If F = {fn(q)} ∞ n=1 is a solution of the functional equation, then there exists a formal power se...

Let $ k be a field, G totally ordered abelian group and \mathbb K = k((G)) the maximal field of generalised power series, endowed with canonical valuation v $. We study \mathrm{-Aut} preserving automorphisms subfield k(G)\subseteq K\subseteq $, where k(G) is fraction ring k[G] Under assumption that satisfies two lifting properties we are able to generalise refine Hofberger's decomposition \math...