### A basis for Numerical Functionals

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In this note we investigate a relationship between the boundary behavior of power series and the composition of formal power series. In particular, we prove that the composition domain of a formal power series $g$ is convex and balanced which implies that the subset ${\overline{\mathbb{X}}}_{g}$ consisting of formal power series which can be composed by a formal power series $g$ possesses such properties. We also provide a necessary and sufficient condition for the superposition operator ${T}_{g}$ to map ${\overline{\mathbb{X}}}_{g}$ into itself or to map ${\mathbb{X}}_{g}$ into...

On this paper we compute the numerical function $\beta $ of the approximation theorem of M. Artin for the one-dimensional systems of formal equations.

In rings ${\Gamma}_{M}$ of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence $M={\left({M}_{l}\right)}_{l\ge 0}$ (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in ${\Gamma}_{M}$ such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to ${\Gamma}_{M}$, provided ${\Gamma}_{M}$ is stable under derivation. By a further development of the method, we obtain the main result of the paper, stating that...

One gives a formula for the calculation of the local intersection multiplicity index of analytic varieties, analogous to the Bézout formula in the case of algebraic varieties in the projective space, in the case of normal crossings. One obtains also a recurrent process for the calculation of the local intersection multiplicity index of plane analytic curves.

We give Lambek-Carlitz type characterization for completely multiplicative reduced incidence functions in Möbius categories of full binomial type. The $q$-analog of the Lambek-Carlitz type characterization of exponential series is also established.