Related papers: Remark on Continued Fractions, Moebius Transformat…
The aim of this note is to show the existence of a correspondance between certain algebraic continued fractions in fields of power series over a finite field and automatic sequences in the same finite field. this connection is illustrated…
In this note, we describe an interpretation of the (continuous) Fourier transform from the perspective of the Chinese Remainder Theorem. Some related issues, including a new derivation of Poisson summation formula, are discussed.
Let H be the real algebra of quaternions. The notion of regular function of a quaternionic variable recently presented by G. Gentili and D. C. Struppa developed into a quite rich theory. Several properties of regular quaternionic functions…
Given a positive integer $N$ and $x$ irrational between zero and one, an $N$-continued fraction expansion of $x$ is defined analogously to the classical continued fraction expansion, but with the numerators being all equal to $N$. Inspired…
We study orthogonal polynomials associated with a continued fraction due to Hirschhorn. Hirschhorn's continued fraction contains as special cases the famous Rogers--Ramanujan continued fraction and two of Ramanujan's generalizations. The…
In this paper, we apply results on number systems based on continued fraction expansions to modular arithmetic. We provide two new algorithms in order to compute modular multiplication and modular division. The presented algorithms are…
We describe a general method of arithmetic coding of geodesics on the modular surface based on a two parameter family of continued fraction transformations studied previously by the authors. The finite rectangular structure of the…
We propose to consider ensembles of cycles (quadrics), which are interconnected through conformal-invariant geometric relations (e.g. "to be orthogonal", "to be tangent", etc.), as new objects in an extended Moebius--Lie geometry. It was…
The presence of large partial quotients can invalidate many classical limit theorems in the metric theory of continued fractions. A commonly employed strategy to overcome this problem is to discard the largest partial quotient when…
Tractable Boolean and arithmetic circuits have been studied extensively in AI for over two decades now. These circuits were initially proposed as "compiled objects," meant to facilitate logical and probabilistic reasoning, as they permit…
We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also…
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…
We introduce a random dynamical system related to continued fraction expansions. It uses random combination of the Gauss map and the R\'enyi (or backwards) continued fraction map. We explore the continued fraction expansions that this…
Several conjectural continued fractions found with the help of various algorithms are published in this paper.
This note is supposed to answer some questions on deformation theory in derived algebraic geometry. We show that derived algebraic geometry allows for a geometrical interpretation of the full cotangent complex and gives a natural setting…
In this paper we study a continuity of the "values" of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics. Our main theorem reveals a more finer…
Gosper developed an algorithm for performing arithmetic operations on continued fractions (CFs), getting a CF as the result. Straightforward implementation of the algorithm leads to infinite loops on some inputs. Here we present a modified…
We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n graded commutative associative algebra. The applications include a new approach to the classical theory of matrices with coefficients in…
A permutation is said to be cycle-alternating if it has no cycle double rises, cycle double falls or fixed points; thus each index $i$ is either a cycle valley ($\sigma^{-1}(i)>i<\sigma(i)$) or a cycle peak ($\sigma^{-1}(i)<i>\sigma(i)$).…
We show how an object from the combinatorially geometric version of the analytical number theory, namely geometric continued fractions, appears in the classical smooth dynamics, namely in the problem on the topological classification of…