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A relationship between continued fractions and Weyl groupoids of Cartan schemes of rank two is found. This allows to decide easily if a given Cartan scheme of rank two admits a finite root system. We obtain obstructions and sharp bounds for the entries of the Cartan matrices. Key words: Cartan matrix, continued fraction, Nichols algebra, Weyl groupoid
We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural extension
We present an unexpected connection between two map enumeration problems. The first one consists in counting planar maps with a boundary of prescribed length. The second one consists in counting planar maps with two points at a prescribed distance. W
It is widely believed that the continued fraction expansion of every irrational algebraic number $alpha$ either is eventually periodic (and we know that this is the case if and only if $alpha$ is a quadratic irrational), or it contains arbitrarily la
This is a translation of Eulers Latin paper De fractionibus continuis observationes into English. In this paper Euler describes his theory of continued fractions. He teaches, how to transform series into continued fractions, solves the Riccati-Differ
We exhibit a method to use continued fractions in function fields to find new families of hyperelliptic curves over the rationals with given torsion order in their Jacobians. To show the utility of the method, we exhibit a new infinite family of curv