Farey Sequence Research Articles

An approximating k-ary GCD algorithm

In our paper we elaborate a new version of the k-ary GCD algorithm. Our algorithm is based on the Farey Series and surpasses all existing realizations of the k-ary algorithm. It can have practical applications inMathematics and Cryptography.

A tale of two fractals: The Hofstadter butterfly and the integral Apollonian gaskets

This paper unveils a mapping between a quantum fractal that describes a physical phenomena, and an abstract geometrical fractal. The quantum fractal is the Hofstadter butterfly discovered in 1976 in an iconic condensed matter problem of electrons moving in a two-dimensional lattice in a transverse magnetic field. The geometric fractal is the integer Apollonian gasket characterized in terms of a 300 BC problem of mutually tangent circles. Both of these fractals are made up of integers. In the Hofstadter butterfly, these integers encode the topological quantum numbers of quantum Hall conductivity. In the Apollonian gaskets an infinite number of mutually tangent circles are nested inside each other, where each circle has integer curvature. The mapping between these two fractals reveals a hidden threefold symmetry embedded in the kaleidoscopic images that describe the asymptotic scaling properties of the butterfly. This paper also serves as a mini review of these fractals, emphasizing their hierarchical aspects in terms of Farey fractions.

Complicated quasiperiodic oscillations and chaos from driven piecewise-constant circuit: Chenciner bubbles do not necessarily occur via simple phase-locking

We analyze the complex quasiperiodic oscillations and chaos generated by two coupled piecewise-constant hysteresis oscillators driven by a rectangular wave force. Oscillations generate Arnol’d resonance webs wherein lower dimensional resonance tongues extend such as that of a web in numerous directions. We provide the fundamental tools for bifurcation analysis of nonautonomous piecewise-constant oscillators. To optimize the outstanding simplicity of piecewise-constant circuits, we formulate a generalized procedure for calculating the Lyapunov exponents in nonautonomous piecewise-constant dynamics. The Lyapunov exponents in these dynamics can be calculated with a precision approximately similar to that of maps. We observe two-parameter Lyapunov diagrams near the fundamental resonance region called Chenciner bubbles, wherein the oscillation frequencies of the two oscillators and the force are synchronized with a ratio of 1:1:1. Inevitably, the hysteresis considerably distorts the Chenciner bubbles. This result suggests that the Chenciner bubbles do not necessarily occur due to simple phase-locking of two-dimensional tori that can be explained by homeomorphism on the circle. Furthermore, we observe the Farey sequence in the experimental measurements.

Mathematical analysis and STEM observations of arrangement of structural units in 〈001〉 symmetrical tilt grain boundaries.

A collaborative work between mathematics and atom-resolved scanning transmission electron microscopy (STEM) has been conducted. The grain boundary in a bicrystal of a simple rock-salt oxide can show a complicated arrangement of structural units, which can be well predicted by an algorithm utilizing the Farey sequence. The estimated arrangements had a nice agreement with those observed by STEM in atomic-scale up to several tens of nanometers.

Products of Farey Fractions

ABSTRACTThe Farey fractions of order n consist of all fractions in lowest terms lying in the closed unit interval and having denominator at most n. This article considers the products Fn of all nonzero Farey fractions of order n. It studies their size, given by log(Fn), and their divisibility properties by powers of a fixed prime, given by ordp(Fn), as a function of n. It presents evidence suggesting that information related to the Riemann hypothesis may be encoded in functions related to ordp(Fn) for a single fixed prime p. This encoding makes use of a relation of these products to the products Gn of all reduced and unreduced Farey fractions of order n, which are connected by Möbius inversion. It introduces new arithmetic functions which mix the Möbius function with functions of radix expansions to a fixed prime base p.

A Fast Computation of the Best $${k}$$ k -Digit Rational Approximation to a Real Number

Given a real number \({\alpha}\), we aim at computing the best rational approximation with at most \({k}\) digits and with exactly \({k}\) digits at the numerator (denominator). Our approach exploits Farey sequences. Our method turns out to be very fast in the sense that, once the development of \({\alpha}\) in continued fractions is available, the required operations are just a few and their number remains essentially constant for any \({k}\) (in double precision finite arithmetic). Estimations of error bounds are also provided.

A scaling property of Farey fractions. Part II: convergence at rational points

The Farey sequence of order n consists of all reduced fractions a / b between 0 and 1 with positive denominator b less or equal to n. The sums of the inverse denominators 1 / b of the Farey fractions in prescribed intervals with rational bounds have a simple main term, but the deviations are determined by an interesting sequence of polygonal functions \(f_n\). In a former paper we obtained a limit function for \(n \rightarrow \infty \), which describes a scaling behaviour of the functions \(f_n\) in the vicinity of any fixed rational number a / b and which is independent of a / b. In this paper we prove that \(f_n(a/b)\) tends to zero for \(n \rightarrow \infty \) by using elementary representation formulas for \(f_n(a/b)\) as well as a variant of the prime number theorem. An application of this result immediately gives a global version of the scaling behaviour of the functions \(f_n\) around the rational numbers.

Label-based routing for a family of small-world Farey graphs.

We introduce an informative labelling method for vertices in a family of Farey graphs, and deduce a routing algorithm on all the shortest paths between any two vertices in Farey graphs. The label of a vertex is composed of the precise locating position in graphs and the exact time linking to graphs. All the shortest paths routing between any pair of vertices, which number is exactly the product of two Fibonacci numbers, are determined only by their labels, and the time complexity of the algorithm is O(n). It is the first algorithm to figure out all the shortest paths between any pair of vertices in a kind of deterministic graphs. For Farey networks, the existence of an efficient routing protocol is of interest to design practical communication algorithms in relation to dynamical processes (including synchronization and structural controllability) and also to understand the underlying mechanisms that have shaped their particular structure.

Optimal Design of Linear Space Code for MIMO Optical Wireless Communications

In this paper, the design of linear full-diversity space code (FDSC) is considered for an intensity-modulated direct-detection multiple-input–multiple-output optical wireless communication (IM/DD MIMO-OWC) system, in which the channel suffers from log-normal fading and different links have different variances. Utilizing our recently established error performance criterion for the design of FDSC with a maximum-likelihood detector, we formulate the design problem of optimizing both large-scale diversity gain and small-scale diversity into a max–min optimization problem with continuous–discrete mixed design variables. We propose the use of the Farey sequence in number theory as a powerful tool to solve this kind of optimization problem. By taking advantage of some available properties and by developing some new interesting properties on the Farey sequence, a closed-form solution is attained for IM/DD MIMO-OWC equipped with two transmitters and multiple receivers. This optimal design shows that a repetition code with an optimal power allocation is optimal.

Łojasiewicz exponents and Farey sequences

\noindent Let $I$ be an ideal of the ring of formal power series $\bK[[x,y]]$ with coefficients in an algebraically closed field $\bK$ of arbitrary characteristic. Let $\Phi$ denote the set of all parametrizations $\varphi=(\varphi_1,\varphi_2)\in \bK[[t]]^2$, where $\varphi \neq (0,0)$ and $\varphi (0,0)=(0,0)$. The purpose of this paper is to investigate the invariant \[ \Lo(I)=\sup_{\varphi \in \Phi}\left(\inf_{f\in I} \frac{\ord f \circ \varphi}{\ord \varphi}\right) \] \noindent called the {\it \L ojasiewicz exponent} of $I$. Our main result states that for the ideals $I$ of finite codimension the \L ojasiewicz exponent $\Lo(I)$ is a Farey number i.e. an integer or a rational number of the form $N+\frac{b}{a}$, where $a,b,N$ are integers such that $0<b<a<N$.

Products of Farey graphs are totally geodesic in the pants graph

We show that for a surface [Formula: see text], the subgraph of the pants graph determined by fixing a collection of curves that cut [Formula: see text] into pairs of pants, once-punctured tori, and four-times-punctured spheres is totally geodesic. The main theorem resolves a special case of a conjecture made in [2] and has the implication that an embedded product of Farey graphs in any pants graph is totally geodesic. In addition, we show that a pants graph contains a convex [Formula: see text]-flat if and only if it contains an [Formula: see text]-quasi-flat.

Stable, fast computation of high-order Zernike moments using a recursive method

Zernike moments and Zernike polynomials have been widely applied in the fields of image processing and pattern recognition. When high-order Zernike moments are computed, both computing speed and numerical accuracy become inferior. The main purpose of this study is to propose a stable, fast method for computing high-order Zernike moments. Based on the recursive formulas for computing Zernike radial polynomials, this study develops stable, fast algorithms to compute Zernike moments. Symmetry under group action and Farey sequence are both applied to shorten the computing time. The experimental results show that the proposed method took 5.292 seconds to compute the top 500-order Zernike moments of an image with 512×512 pixels. The normalized mean square error is 0.00124846 if 450-order moments are used to reconstruct the image. When computing the high-order Zernike moments, the proposed method outperformed other compared methods in terms of speed and accuracy.

A scaling property of Farey fractions

The Farey sequence of order n consists of all reduced fractions a/b between 0 and 1 with positive denominator b less than or equal to n. The sums of the inverse denominators 1/b of the Farey fractions in prescribed intervals with rational bounds have a simple main term, but the deviations are determined by an interesting sequence of polygonal functions $$f_n$$ . For $$n \rightarrow \infty $$ we also obtain a certain limit function, which describes an asymptotic scaling property of functions $$f_n$$ in the vicinity of any fixed fraction a/b and which is independent of a/b. The result can be obtained by using only elementary methods. We also study this limit function and especially its decay behaviour by using the Mellin transform and analytical properties of the Riemann zeta function.

Products of binomial coefficients and unreduced Farey fractions

This paper studies the product [Formula: see text] of the binomial coefficients in the [Formula: see text]th row of Pascal’s triangle, which equals the reciprocal of the product of all the reduced and unreduced Farey fractions of order [Formula: see text]. It studies its size as a real number, measured by [Formula: see text], and its prime factorization, measured by the order of divisibility [Formula: see text] by a fixed prime [Formula: see text], each viewed as a function of [Formula: see text]. It derives three formulas for [Formula: see text], two of which relate it to base [Formula: see text] radix expansions of integers up to [Formula: see text], and which display different facets of its behavior. These formulas are used to determine the maximal growth rate of each [Formula: see text] and to explain structure of the fluctuations of these functions. It also defines analogous functions [Formula: see text] for all integer bases [Formula: see text] using base [Formula: see text] radix expansions replacing base [Formula: see text]-expansions. A final topic relates factorizations of [Formula: see text] to Chebyshev-type prime-counting estimates and the prime number theorem.

Poincaré sections for the horocycle flow in covers of $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$ and applications to Farey fraction statistics

For a given finite index subgroup $$H\subseteq \mathrm {SL}(2,\mathbb {Z})$$ , we use a process developed by Fisher and Schmidt to lift a Poincare section of the horocycle flow on $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$ found by Athreya and Cheung to the finite cover $$\mathrm {SL}(2,\mathbb {R})/H$$ of $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$ . We then use the properties of this section to prove the existence of the limiting gap distribution of various subsets of Farey fractions. Additionally, to each of these subsets of fractions, we extend solutions by Xiong and Zaharescu, and independently Boca, to a Diophantine approximation problem of Erdős, Szusz, and Turan.

SL$_2 (\mathbb Z)$-tilings of the torus, Coxeter–Conway friezes and Farey triangulations

The notion of SL _2 -tiling is a generalization of that of classical Coxeter–Conway frieze pattern. We classify doubly antiperiodic SL _2 -tilings that contain a rectangular domain of positive integers. Every such SL _2 -tiling corresponds to a pair of frieze patterns and a unimodular 2 \times 2 -matrix with positive integer coefficients. We relate this notion to triangulated n -gons in the Farey graph.

The mediant method for fast mass/concentration detection in nanotechnologies

Modern nanotechnologies require methods for registration of variable physical parameters of the environment adjacent to the object. This situation is constrained by very small values, and their fast variation. This paper reports on a theoretical method for the fast parameter-to-frequency conversion (PFC). We introduce the basic formalism of number theory application for rational approximation of an unknown value, which can be naturally represented as the ratio of two independent calculable integer parameters. Mediants (a kind of Farey fractions), as well as its fundamental properties are shown, and their relation with measurement processes in PFC is presented. It is shown that the proposed mediant method is invariant, i.e., strongly independent of any known kinds of timing jitter, which is the most inconvenient kind of noise for any problem requiring the fast variable parameter registration. This method also gives the measurement uncertainty at the rate of time-standard reproducibility. Here, we present the construction of the experimental prototype. We also provide some screenshots of the experimental data obtained using this equipment. Finally, the results of numerical simulation of parameters typical for nano-applications are reported.

On the Amplitude of External Perturbation and Chaos via Devil's Staircase — Stability of Attractors

We designed a chaotic memristive circuit proposed by Chua and Muthuswamy, and analyzed the behavior of the voltage of the capacitor, electric current in the inductor and the voltage of the memristor by adding an external sinusoidal oscillation of a type γω cos ωt to [Formula: see text], when [Formula: see text] is expressed by y(t)/C, and studied Devil's staircase route to chaos. We compared the frequency of the driving oscillation fs and the frequency of the response fd in the window and assigned W = fs/fd to each window. When capacitor C = 1.0, we observe stable attractors of Farey sequences [Formula: see text], which can be interpreted as hidden attractors, while when C = 1.2, we observe disturbed attractors. Frequency dependent stability of chaotic circuits due to octonions is discussed.

Periodicity of non-expanding piecewise linear maps and effects of random noises

We consider non-expanding piecewise linear maps described as Sα, β(x) = αx + β(mod 1), where 0 < α, β < 1. This transformation is known as the Nagumo– Sato (NS) model. The NS model corresponds to a special case of Caianiello's model, and it describes simplified dynamics of a single neuron. In this thesis, we can describe regions explicitly in which Sα, β has a periodic point with period n for an arbitrary integer n, and clarify that these regions are associated with the Farey series, which are already observed experimentally in [5]. We shall explain a mathematical rigorous reason for a complexity of a periodicity of Sα, β that is observed by Aihara-Oku. Furthermore, we introduce an example that noises induce the asymptotic periodicity in the sense of Lasota-Machey, even if an original transformation has no periodicity.

Constrained paths based on the Farey sequence in learning to juggle

In this article we report the results of a study conducted to investigate the learning dynamics of three-ball juggling from the perspective of frequency locking. Based on the Farey sequence, we predicted that four stable coordination patterns, corresponding to dwell ratios of 0.83, 0.75, 0.67, and 0.50, would appear in the learning process. We examined the learning process in terms of task performance, taking into account individual differences in the amount of learning. We observed that the participants acquired individual-specific coordination patterns in a relatively early stage of learning, and that those coordination patterns were preserved in subsequent learning, even though performance in terms of number of successful consecutive throws increased substantially. This increase appeared to be related to a reduction in spatial variability of the juggling movements. Finally, the observed coordination patterns were in agreement with the predicted patterns, with the proviso that the pattern corresponding to a dwell ratio of 0.50 was not realized and only a hint of evidence was found for the dwell ratio of 0.67. This implies that the dwell ratios of 0.83 and 0.75 in particular exhibited a stable coordination structure due to strong frequency locking between the temporal variables of juggling.