Related papers: Simultaneous Convergent Continued Fraction Algorit…

We consider the geometric generalization of ordinary continued fraction to the multidimensional case introduced by F. Klein in 1895. A multidimensional periodic continued fraction is the union of sails with some special group acting freely…

In 1985, Robbins observed by computer the continued fraction expansion of certain algebraic power series over a finite field. Incidentally, he came across a particular equation of degree 4 in characteristic p=13. This equation has an…

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 introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the…

In this paper, we establish hybrid results on Diophantine approximation with primes from short intervals. In particular, we prove the following result in a slightly modified form: If $\alpha$ is an irrational number having a continued…

A continued fraction $v(\tau)$ of Ramanujan is evaluated at certain arguments in the field $K = \mathbb{Q}(\sqrt{-d})$, with $-d \equiv 1$ (mod $8$), in which the ideal $(2) = \wp_2 \wp_2'$ is a product of two prime ideals. These values of…

We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend, and use it…

We propose an SQP algorithm for mathematical programs with vanishing constraints which solves at each iteration a quadratic program with linear vanishing constraints. The algorithm is based on the newly developed concept of $\mathcal…

Power series are introduced that are simultaneously convergent for all real and p-adic numbers. Our expansions are in some aspects similar to those of exponential, trigonometric, and hyperbolic functions. Starting from these series and…

An $n$-ary integral quadratic form is a formal expression $Q(x_1,..,x_n)=\sum_{1\leq i,j\leq n}a_{ij}x_ix_j$ in $n$-variables $x_1,...,x_n$, where $a_{ij}=a_{ji} \in \mathbb{Z}$. We present a randomized polynomial time algorithm that given…

We give continued fraction expansions of the generating functions of Bernoulli numbers, Cauchy numbers, Euler numbers, harmonic numbers, and their generalized or related numbers. In particular, we focus on explicit forms of the convergents…

We introduce a family of maps generating continued fractions where the digit $1$ in the numerator is replaced cyclically by some given non-negative integers $(N_1,\ldots,N_m)$. We prove the convergence of the given algorithm, and study the…

The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These…

Let S be an infinite set of non-empty, finite subsets of the nonnegative integers. If p is an odd prime, let c(p) denote the cardinality of the set {T {\in} S : T {\subseteq} {1,...,p-1} and T is a set of quadratic residues (respectively,…

This paper presents a "two-dimensional Fourier Continuation" method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly…

We give continued fraction algorithms for a particular class of Fuchsian triangle groups. In particular, we give an explicit form of each such group that is a subgroup of the Hilbert modular group of its trace field and provide an interval…

We show a correspondence between simple continued fraction expansions of irrational numbers and irreducible permutative representations of the Cuntz algebra ${\cal O}_{\infty}$. With respect to the correspondence, it is shown that the…

Most well-known multidimensional continued fractions, including the M\"{o}nkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by…

In this paper we study the geodesic continued fraction in the case of the Shimura curve coming from the $(2,3,7)$-triangle group. We construct a certain continued fraction expansion of real numbers using the so-called coding of the…

We present an approach for regression problems that employs analytic continued fractions as a novel representation. Comparative computational results using a memetic algorithm are reported in this work. Our experiments included fifteen…