Related papers: Remark on Continued Fractions, Moebius Transformat…
We provide a generalization of continued fractions to the Heisenberg group. We prove an explicit estimate on the rate of convergence of the infinite continued fraction and several surprising analogs of classical formulas about continued…
We introduce a $q$-analog of the higher continued fractions introduced by the last three authors in a previous work (together with Gregg Musiker), which are simultaneously a generalization of the $q$-rational numbers of Morier-Genoud and…
We statistically compare the relationships between frequencies of digits in continued fraction expansions of typical rational points in the unit interval and higher dimensional generalisations. This takes the form of a Large Deviation and…
We introduce and study in detail a special class of backward continued fractions that represents a generalization of R\'enyi continued fractions. We investigate the main metrical properties of the digits occurring in these expansions and we…
Continued fractions are linked to Stern's diatomic sequence 0,1,1,2,1,3,2,3,1,4,... (given by the recursion relation a_2n=a_n and a_{2n+1} = a_n + a_{n+1}, where a_0=0 and a_1=1), which has long been known. Using a particular…
In this notes we show how a problem regarding continued fractions of rational numbers, lead to several phenomena in number theory and dynamics, and eventually to the problem of shearing of divergent diagonal orbits in the space of adelic…
This paper presents a reinterpretation of a second-order linear recurrence sequence as a sequence of continuants derived from the convergents to a continued fraction. As a result, we are able to derive the generating function and Binet…
In 1993 Mahadevan and Keller used the Kirchhoff rod theory to predict the shape of a M\"obius band. Starting from the solution for a square cross-section (isotropic), they employ numerical continuation in the cross-sectional aspect ratio in…
A wardian calculus of sequences started almost seventy years ago constitutes the general scheme for extensions of the classical umbral operator calculus considered by many afterwards . At the same time this calculus is an example of the…
In this paper we recall some results and some criteria on the convergence of matrix continued fractions. The aim of this paper is to give some properties and results of continued fractions with matrix arguments. Then we give continued…
We define notions of differentiability for maps from and to the space of persistence barcodes. Inspired by the theory of diffeological spaces, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be…
We develop a theory of multi-stage degenerations of toric varieties over finite rank valuation rings, extending the Mumford--Gubler theory in rank one. Such degenerations are constructed from fan-like structures over totally ordered abelian…
$p$-adic continued fractions, as an extension of the classical concept of classical continued fractions to the realm of $p$-adic numbers, offering a novel perspective on number representation and approximation. While numerous $p$-adic…
We provide new similarities between regular continued fractions and L\"uroth series in terms of topological dynamics and Hausdorff dimension. In particular, we establish a complete analogue for the L\"uroth transformation of results by W.…
A generalization of the regular continued fractions was given by Chakraborty and Rao \cite{CR-2003}. For the transformation which generates this expansion and its invariant measure, the Perron-Frobenius operator is given and studied. For…
The theory of derivative of noninteger order goes back to Leibniz, Liouville and Riemann. Derivatives of fractional order have found many applications in recent studies in mechanics, physics, economics. In this paper we define the…
An infinite continued composition is an expression of the form \begin{equation*} \lim_{n\to\infty}t_0\circ t_1 \circ t_2 \circ \cdots \circ t_n(c)\;, \end{equation*} where the $t_i$ are maps from a set $D$ to itself, the initial value $c$…
We present several continued fraction algorithms, each of which gives an eventually periodic expansion for every quadratic element of ${\mathbb Q}_p$ over ${\mathbb Q}$ and gives a finite expansion for every rational number. We also give,…
Let $\Cc$ and $\Dd$ be two corings over a ring $A$ and $\Cc\stackrel{\lambda}{\longrightarrow}\Dd$ be a morphism of corings. We investigate the situation when the associated induced ("corestriction of scalars") functor…
In this paper we introduce the theory of ends and coends in the context of enriched bicategories. This will be an enriched version of the theory introduced in [Cor16], and a bicategorical version of the classical theory of enriched…