scholarly journals A CONTINUED FRACTION TITBIT

Fractals ◽  
1995 ◽  
Vol 03 (04) ◽  
pp. 641-650 ◽  
Author(s):  
MICHAEL KEANE
Keyword(s):  
Dynamical System ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Elementary Proof ◽  
Statistical Regularity

In 1812 Gauss, in a letter to Laplace, proposed without proof a formula explaining the statistical regularity of continued fractions. There has since been speculation concerning the manner in which Gauss arrived at this formula. In this article we present a plausible explanation, which at the same time gives an elementary proof of the full ergodic nature of the underlying dynamical system. The method seems to be of interest for other number-theoretic expansions.

THE DISTAL AND ASYMPTOTIC SETS FOR CONTINUED FRACTIONS

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950139
Author(s):  
WEIBIN LIU ◽  
SHUAILING WANG
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Underlying Space ◽  
And Topology

For continued fraction dynamical system [Formula: see text], we give a classification of the underlying space [Formula: see text] according to the orbit of a given point [Formula: see text]. The sizes of all classes are determined from the viewpoints of measure, Hausdorff dimension and topology. For instance, the Hausdorff dimension of the distal set of [Formula: see text] is one and the Hausdorff dimension of the asymptotic set is either zero or [Formula: see text] according to [Formula: see text] is rational or not.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals Extreme Value Theory for Hurwitz Complex Continued Fractions

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 840
Author(s):  
Maxim Sølund Kirsebom
Keyword(s):  
Invariant Measure ◽  
Extreme Value Theory ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Value Theory ◽  
Extreme Value ◽  
Poisson Law ◽  
Complex Continued Fractions

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we obtained several results concerning the extremes of nearest integer continued fractions as well.

scholarly journals Calculating astronomical refraction by means of continued fractions

1979 ◽  
Vol 89 ◽  
pp. 95-101
Author(s):  
S. Mikkola
Keyword(s):  
Asymptotic Expansion ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Astronomical Refraction

A continued fraction was derived for the summation of the asymptotic expansion of astronomical refraction. Using simple approximations for the last denominator of the fraction, accurate formulae, useful down to the horizon, were obtained. The method is not restricted to any model of the atmosphere and can thus be used in calculations based on actual aerological measurements.

scholarly journals Continued Fractions Related to $(t,q)$-Tangents and Variants

10.37236/2014 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
General Version ◽  
Continued Fraction Expansion ◽  
Special Cases ◽  
Tangent Function

For the $q$-tangent function introduced by Foata and Han (this volume) we provide the continued fraction expansion, by creative guessing and a routine verification. Then an even more recent $q$-tangent function due to Cieslinski is also expanded. Lastly, a general version is considered that contains both versions as special cases.

scholarly journals CONTINUED FRACTIONS, INTERMEDIATE FRACTIONS AND THEIR RELATION TO THE BEST APPROXIMATIONS

2020 ◽  
Vol 20 (3) ◽  
pp. 545-560
Author(s):  
LUKA MILINKOVIC ◽  
BRANKO MALESEVIC ◽  
BOJAN BANJAC
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Approximations ◽  
Best Approximations ◽  
Current State ◽  
The Subject ◽  
State Of Art

The subject of this paper is the current state of art in theory of continued fractions, intermediate fractions and their relation to the best rational approximations of the first and second kind. The paper provides an overview of the some well known and even some new properties of continued fractions, and the various terms associated with them. In addition to intermediate fractions, paper considers the fine intermediate fractions and gave some statements to position these fractions in the continued fraction representation of numbers.

scholarly journals On the convergence of multidimensional regular C-fractions with independent variables

2018 ◽  
Vol 26 (1) ◽  
pp. 18 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Open Disk ◽  
Efficient Tool ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Multiple Power ◽  
Multiple Power Series ◽  
Branched Continued Fraction

In this paper, we investigate the convergence of multidimensional regular С-fractions with independent variables, which are a multidimensional generalization of regular С-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. We have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional regular С-fraction with independent variables. And, in addition, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional regular С-fraction with independent variables.

scholarly journals A characterization of rational numbers by p-adic Ruban continued fractions

1985 ◽  
Vol 39 (3) ◽  
pp. 300-305 ◽  
Author(s):  
Vichian Laohakosol
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers

AbstractA type of p–adic continued fraction first considered by A. Ruban is described, and is used to give a characterization of rational numbers.

DESIGN OF -1/2n ORDER ANALOG FRACTANCE APPROXIMATION CIRCUIT USING CONTINUED FRACTIONS DECOMPOSITION

2012 ◽  
Vol 21 (04) ◽  
pp. 1250035 ◽  
Author(s):  
YAN LIU ◽  
YI-FEI PU ◽  
JI-LIU ZHOU ◽  
XIAO-DONG SHEN
Keyword(s):  
Fractional Calculus ◽  
Frequency Response ◽  
Experimental Evidence ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Analog Circuit ◽  
Synthesis Method ◽  
Network Synthesis ◽  
Network Function ◽  
Amplitude Frequency Response

In order to solve the most important problem of the fractional calculus (FC) application, the realization of analog circuit of fractance, continued fraction theory is applied to design the -1/2n order analog fractance approximation circuit. The author presents a network function of ideal fractance and decomposes it in continued fractions (CFs) form to obtain the corresponding analog fractance approximation circuit. The new circuit consists of ordinary passive RC component through network synthesis method. Simulations are performed for the verification of the new circuit. Experimental evidence has proved that the performance of novel -1/2n order analog fractance approximation circuit is good in both amplitude-frequency response and phase-frequency response.

scholarly journals Continued Fractions with Absolutely Convergent Even or Odd Part

1959 ◽  
Vol 11 ◽  
pp. 131-140 ◽  
Author(s):  
David F. Dawson
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Absolute Convergence ◽  
The Absolute

The purpose of this paper is to give conditions under which the absolute convergence of the subsequence of odd or of even approximants to a continued fraction implies convergence of the continued fraction. In § 2 we consider the problem in general, and in § 3 we impose a condition which gives absolute convergence of the odd or of the even part of the continued fraction and state conditions which imply convergence of the continued fraction.

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