A NEW SYMMETRIC TWO-DIMENSIONAL ALGORITHM

2006 ◽  
Vol 02 (04) ◽  
pp. 489-498
Author(s):  
PEDRO FORTUNY AYUSO ◽  
FRITZ SCHWEIGER
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Dual Graph ◽  
Plane Curve ◽  
Singularity Theory ◽  
Main Topic ◽  
Two Dimensional ◽  
Dimension 2 ◽  
Multidimensional Continued Fraction ◽  
Fraction Algorithm

Continued fractions are deeply related to Singularity Theory, as the computation of the Puiseux exponents of a plane curve from its dual graph clearly shows. Another closely related topic is Euclid's Algorithm for computing the gcd of two integers (see [2] for a detailed overview). In the first section, we describe a subtractive algorithm for computing the gcd of n integers, related to singularities of curves in affine n-space. This gives rise to a multidimensional continued fraction algorithm whose version in dimension 2 is the main topic of the paper.

EXPONENTIALLY STRONG CONVERGENCE OF NON-CLASSICAL MULTIDIMENSIONAL CONTINUED FRACTION ALGORITHMS

2002 ◽  
Vol 02 (04) ◽  
pp. 563-586
Author(s):  
KENTARO NAKAISHI
Keyword(s):  
Strong Convergence ◽  
Continued Fraction ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Deterministic Case ◽  
Convergence Properties ◽  
Almost Everywhere ◽  
Multidimensional Continued Fraction

Convergence properties of multidimensional continued fraction algorithms introduced by V. Baladi and A. Nogueira are studied. The paper contains an arithmetic proof of almost everywhere exponentially strong convergence of some two-dimensional Markovian random algorithms and dynamically defined ones. A special three-dimensional deterministic case is also discussed.

scholarly journals A generalized family of multidimensional continued fractions: TRIP Maps

2014 ◽  
Vol 10 (08) ◽  
pp. 2151-2186 ◽  
Author(s):  
Krishna Dasaratha ◽  
Laure Flapan ◽  
Thomas Garrity ◽  
Chansoo Lee ◽  
Cornelia Mihaila ◽  
...  
Keyword(s):  
Number Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Real Numbers ◽  
Multidimensional Continued Fractions ◽  
The Family ◽  
Before And After ◽  
Multidimensional Continued Fraction

Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

Extended continued fractions, recurrence relations and two-dimensional Markov processes

1989 ◽  
Vol 21 (2) ◽  
pp. 357-375 ◽  
Author(s):  
C. E. M. Pearce
Keyword(s):  
Markov Processes ◽  
Continued Fractions ◽  
Recurrence Relations ◽  
Equilibrium Distribution ◽  
Linear Recurrence ◽  
Natural State ◽  
Two Dimensional ◽  
State Spaces ◽  
General Order ◽  
Dimension 2

Connections between Markov processes and continued fractions have long been known (see, for example, Good [8]). However the usefulness of extended continued fractions in such a context appears not to have been explored. In this paper a convergence theorem is established for a class of extended continued fractions and used to provide well-behaved solutions for some general order linear recurrence relations such as arise in connection with the equilibrium distribution of state for some Markov processes whose natural state spaces are of dimension 2. Specific application is made to a multiserver version of a queueing problem studied by Neuts and Ramalhoto [13] and to a model proposed by Cohen [5] for repeated call attempts in teletraffic.

scholarly journals On convergence of branched continued fraction expansions of Horn's hypergeometric function $H_3$ ratios

2021 ◽  
Vol 13 (3) ◽  
pp. 642-650
Author(s):  
T.M. Antonova
Keyword(s):  
Uniform Convergence ◽  
Hypergeometric Function ◽  
Compact Subset ◽  
Complex Variable ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Continued Fraction Expansions ◽  
Branched Continued Fraction

The paper deals with the problem of convergence of the branched continued fractions with two branches of branching which are used to approximate the ratios of Horn's hypergeometric function $H_3(a,b;c;{\bf z})$. The case of real parameters $c\geq a\geq 0,$ $c\geq b\geq 0,$ $c\neq 0,$ and complex variable ${\bf z}=(z_1,z_2)$ is considered. First, it is proved the convergence of the branched continued fraction for ${\bf z}\in G_{\bf h}$, where $G_{\bf h}$ is two-dimensional disk. Using this result, sufficient conditions for the uniform convergence of the above mentioned branched continued fraction on every compact subset of the domain $\displaystyle H=\bigcup_{\varphi\in(-\pi/2,\pi/2)}G_\varphi,$ where \[\begin{split} G_{\varphi}=\big\{{\bf z}\in\mathbb{C}^{2}:&\;{\rm Re}(z_1e^{-i\varphi})<\lambda_1 \cos\varphi,\; |{\rm Re}(z_2e^{-i\varphi})|<\lambda_2 \cos\varphi, \\ &\;|z_k|+{\rm Re}(z_ke^{-2i\varphi})<\nu_k\cos^2\varphi,\;k=1,2;\; \\ &\; |z_1z_2|-{\rm Re}(z_1z_2e^{-2\varphi})<\nu_3\cos^{2}\varphi\big\}, \end{split}\] are established.

Extended continued fractions, recurrence relations and two-dimensional Markov processes

1989 ◽  
Vol 21 (02) ◽  
pp. 357-375 ◽  
Author(s):  
C. E. M. Pearce
Keyword(s):  
Markov Processes ◽  
Continued Fractions ◽  
Recurrence Relations ◽  
Equilibrium Distribution ◽  
Linear Recurrence ◽  
Natural State ◽  
Two Dimensional ◽  
State Spaces ◽  
General Order ◽  
Dimension 2

Connections between Markov processes and continued fractions have long been known (see, for example, Good [8]). However the usefulness of extended continued fractions in such a context appears not to have been explored. In this paper a convergence theorem is established for a class of extended continued fractions and used to provide well-behaved solutions for some general order linear recurrence relations such as arise in connection with the equilibrium distribution of state for some Markov processes whose natural state spaces are of dimension 2. Specific application is made to a multiserver version of a queueing problem studied by Neuts and Ramalhoto [13] and to a model proposed by Cohen [5] for repeated call attempts in teletraffic.

Reversibility, Continued Fractions, and Infinite Meander Permutations of Planar Homoclinic Orbits in Linear Hyperbolic Anosov Maps

2014 ◽  
Vol 24 (08) ◽  
pp. 1440008
Author(s):  
Bernold Fiedler
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Singularity Theory ◽  
Global Attractors ◽  
Homoclinic Orbits ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds ◽  
The Difference ◽  
Continued Fraction Expansions

Meander permutations have been encountered in the context of Gauss words, singularity theory, Sturm global attractors, plane Cartesian billiards, and Temperley–Lieb algebras, among others. In this spirit, we attempt to investigate the difference of orderings of homoclinic orbits on the stable and unstable manifolds of a planar saddle. As an example, we consider reversible linear Anosov maps on the 2-torus, and their relation to continued fraction expansions.

scholarly journals A new multidimensional continued fraction algorithm

2009 ◽  
Vol 78 (268) ◽  
pp. 2209-2222 ◽  
Author(s):  
Jun-ichi Tamura ◽  
Shin-ichi Yasutomi
Keyword(s):  
Continued Fraction ◽  
Multidimensional Continued Fraction ◽  
Fraction Algorithm

scholarly journals On the continued fraction algorithm

1970 ◽  
Vol 3 (3) ◽  
pp. 413-422 ◽  
Author(s):  
J. M. Mack
Keyword(s):  
Finite Number ◽  
Real Line ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Extreme Case ◽  
Rational Approximations ◽  
Real Numbers ◽  
Best Approximations ◽  
Rational Line ◽  
Fraction Algorithm

The fact that continued fractions can be described in terms of Farey sections is used to obtain a generalised continued fraction algorithm. Geometrically, the algorithm transfers the continued fraction process from the real line R to an arbitrary rational line l in Rn. Arithmetically, the algorithm provides a sequence of simultaneous rational approximations to a set of n real numbers θ1, …, θn in the extreme case where all of the numbers are rationally dependent on 1 and (say) θ1. All but a finite number of best approximations are given by the algorithm.

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