This paper contributes to a theory of the behaviour of “finite-state” systems that is generic in the system type. We propose that such systems are modeled as coalgebras with a finitely generated carrier for an endofunctor on a locally finitely presentable category. Their behaviour gives rise to a new fixpoint of the coalgebraic type functor called locally finite fixpoint (LFF). We prove that if...

Given a formal power series g(x) = b0 +b1x+b2x2 +··· and a nonunit f(x) = a1x+ a2x2+··· , it is well known that the composition of g with f , g(f(x)), is a formal power series. If the formal power series f above is not a nonunit, that is, the constant term of f is not zero, the existence of the composition g(f(x)) has been an open problem for many years. The recent development investigated the ...

A famous result of Christol gives that a power series $$F(t)=\sum _{n\ge 0} f(n)t^n$$ with coefficients in finite field $$\mathbb {F}_q$$ characteristic p is algebraic over the rational functions t if and only there finite-state automaton accepting base-p digits n as input giving f(n) output for every $$n\ge 0$$ . An extension Christol’s theorem, complete description closure {F}_q(t)$$ , was la...

Pseudodifferential operators are formal Laurent series in the formal inverse ∂−1 of the derivative operator ∂ whose coefficients are holomorphic functions. Given a pseudodifferential operator, the corresponding formal power series can be obtained by using some constant multiples of its coefficients. The space of pseudodifferential operators is a noncommutative algebra over C and therefore has a...

In a recent paper,f André Gleyzal has constructed ordered fields consisting of certain "transfinite real numbers" and has established the interesting result that any ordered field can be considered as a subfield of one of these transfinite fields. These fields prove to be identical with fields of formal power series in which the exponents are allowed to range over a suitable ordered abelian gro...

Formal Laurent-Puiseux series of the form f(x) = ∑ k k0 akx k/n (1) are important in many branches of mathematics. Whereas Mathematica supports the calculation of truncated series with its Series command, and the Mathematica package SymbolicSum that is shipped with Mathematica version 2 is able to convert formal series of type (1) in some instances to their corresponding generating functions, i...

We investigate the theory of skew (formal) power series introduced by Droste, Kuske [5, 6], if the basic semiring is a Conway semiring. This yields Kleene Theorems for skew power series, whose supports contain finite and infinite words. We then develop a theory of convergence in semirings of skew power series based on the discrete convergence. As an application this yields a Kleene Theorem prov...