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Journal articles on the topic 'Farey sequence'

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Published: 4 June 2021

Last updated: 30 January 2023

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1

Ripan Saha. "On Some Special Property of The Farey Sequence." Mathematical Journal of Interdisciplinary Sciences 7, no. 2 (March 6, 2019): 121–23. http://dx.doi.org/10.15415/mjis.2019.72016.

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In this paper, some special property of the Farey sequence is discussed. We prove in each term of the Farey sequence, the sum of elements in the denominator is two times of the sum of elements in the numerator. We also prove that the Farey sequence contains a palindrome structure.

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2

Setyati, Endang, and Raymond Sugiarto. "Pengenalan Tulisan Pada Iklan Pinggir Jalan yang Melengkung Menggunakan Shape Context." Journal of Intelligent System and Computation 3, no. 2 (October 1, 2021): 78–84. http://dx.doi.org/10.52985/insyst.v3i2.202.

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Membaca sebuah tulisan yang sama di bidang melengkung berbeda dengan di bidang datar, karena tulisan pada bidang melengkung bergantung pada permukaan bidang lengkungnya. Pada saat ini, banyak sekali tulisan pada iklan pinggir jalan yang ditempel pada bidang melengkung di sepanjang jalan. Tulisan yang digunakan berupa huruf dan angka, dengan berbagai macam background, bentuk dan warna yang diambil di pinggir jalan dengan menggunakan Farey Shape Context. Fitur Farey ini bergantung pada DSS (Digital Straight Line Segment) endpoint dan menggunakan pecahan Augmented Farey sequence. DSS endpoint ini dijadikan sebagai titik fitur atau feature point untuk menemukan shape context dari citra. DSS endpoint tersebut digunakan sebagai acuan bounding box yang akan digunakan sebagai object boundary yang dimana setiap sudutnya merupakan reference point. Untuk melakukan Binning Farey Rank, Augmented Farey Table (AFT) harus dibentuk terlebih dahulu berdasarkan Augmented Farey Sequence yang merupakan pengembangan dari Farey Sequence. Farey Sequence hanya meliputi pecahan dengan pembilang dan penyebut yang positif, sedangkan Augmented Farey Sequence meliputi pecahan dengan pembilang dan penyebut positif serta negatif. Pada penelitian ini digunakan 500 data iklan di pinggir jalan yang melengkung, dimana 70% digunakan sebagai data sample. Dari 70% data sample tersebut didapatkan ribuan karakter berupa huruf dan angka yang dijadikan data sample. Berdasarkan hasil uji coba penelitian yang dilakukan pada 500 Gambar dimana 30% sebagai data testing, maka hasil Farey Shape Context untuk mengenali tulisan berupa huruf dan angka pada iklan pinggir jalan yang melengkung mencapai akurasi benar 74.94% dan salah 25.06%.

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3

Shiu, Peter, and R. R. Hall. "The index of a Farey sequence." Michigan Mathematical Journal 51, no. 1 (April 2003): 209–23. http://dx.doi.org/10.1307/mmj/1049832901.

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4

Li, Junxian, Albert Tamazyan, and Alexandru Zaharescu. "Ducci iterates and similar ordering of visible points in convex regions." International Journal of Number Theory 16, no. 01 (July 25, 2019): 1–28. http://dx.doi.org/10.1142/s1793042120500013.

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Hardy et al. first introduced the notion of similar ordering of pairs of rationals, and Mayer proved that pairs of Farey fractions in [Formula: see text] are similarly ordered when [Formula: see text] is large enough. We generalize Mayer’s result to Ducci iterates of Farey sequence and visible points in certain regions in the plane. We also study the distribution of values of generalized indices of these sequences.

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5

Tou, Erik R. "The Farey Sequence: From Fractions to Fractals." Math Horizons 24, no. 3 (February 2017): 8–11. http://dx.doi.org/10.4169/mathhorizons.24.3.8.

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6

HEERSINK, BYRON. "Equidistribution of Farey sequences on horospheres in covers of and applications." Ergodic Theory and Dynamical Systems 41, no. 2 (October 7, 2019): 471–93. http://dx.doi.org/10.1017/etds.2019.71.

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We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$, where $\unicode[STIX]{x1D6E5}$ is a finite-index subgroup of $\text{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$, where for all $j$, $\mathbf{a}_{j}$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\text{SL}(n+1,\mathbb{R})$ developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.

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7

Mihaila, Ioana. "Farey Sums and Understanding Ratios." Mathematics Teacher 98, no. 3 (October 2004): 158–62. http://dx.doi.org/10.5951/mt.98.3.0158.

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8

Ringland, John, Naoum Issa, and Mark Schell. "From U sequence to Farey sequence: A unification of one-parameter scenarios." Physical Review A 41, no. 8 (April 1, 1990): 4223–35. http://dx.doi.org/10.1103/physreva.41.4223.

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9

PRATIHAR, SANJOY, PARTHA BHOWMICK, SHAMIK SURAL, and JAYANTA MUKHOPADHYAY. "SKEW CORRECTION OF DOCUMENT IMAGES BY RANK ANALYSIS IN FAREY SEQUENCE." International Journal of Pattern Recognition and Artificial Intelligence 27, no. 07 (November 2013): 1353004. http://dx.doi.org/10.1142/s0218001413530042.

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Skew correction of a scanned document page is an important preprocessing step in document image analysis. We propose here a fast and robust skew estimation algorithm based on rank analysis in Farey sequence. Our target document class comprises two major Indian scripts with headlines, namely Devnagari and Bangla. At the beginning, straight edge segments from the edge map of the document page are detected by our algorithm using properties of digital straightness. Straight edges derived in this manner are binned by Farey ranks in correspondence with their slopes. The principal bin, identified from these bins using the strength of accumulated edge points, represents the principal direction along the direction of headlines, from which the gross skew angle is estimated. A fast refinement algorithm is then applied with a finer tuning of Farey ranks, to detect the skew up to the desired level of precision. The algorithm has been tested on a diverse set of document images, containing Bangla and Devnagari scripts. Experimental results are quite encouraging in terms of accuracy, sensitivity to non-textual objects, effectiveness in dealing with unrestricted layouts, and computational efficiency.

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10

Poornima, M., S. Jagannathan, R. Chandrasekhar, and S. T Kumara. "Farey Sequence for Error Correcting Codes and Medical Images." International Journal of Engineering & Technology 7, no. 3.1 (August 4, 2018): 118. http://dx.doi.org/10.14419/ijet.v7i3.1.16812.

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A new scheme for Error Correcting Codes and the coding of medical images through Farey Sequence is brought in the paper. It brings out the reduced real point representations and more of integer point representations. The test case and outcomes are explained in the section 4 and 5.

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11

Haynes, Alan. "The distribution of special subsets of the Farey sequence." Journal of Number Theory 107, no. 1 (July 2004): 95–104. http://dx.doi.org/10.1016/j.jnt.2004.03.003.

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12

SCHWEIGER, F. "A 2-DIMENSIONAL ALGORITHM RELATED TO THE FAREY–BROCOT SEQUENCE." International Journal of Number Theory 08, no. 01 (February 2012): 149–60. http://dx.doi.org/10.1142/s179304211250008x.

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Moshchevitin and Vielhaber gave an interesting generalization of the Farey–Brocot sequence for dimension d ≥ 2 (see [N. Moshchevitin and M. Vielhaber, Moments for generalized Farey–Brocot partitions, Funct. Approx. Comment. Math.38 (2008), part 2, 131–157]). For dimension d = 2 they investigate two special cases called algorithm [Formula: see text] and algorithm [Formula: see text]. Algorithm [Formula: see text] is related to a proposal of Mönkemeyer and to Selmer algorithm (see [G. Panti, Multidimensional continued fractions and a Minkowski function, Monatsh. Math.154 (2008) 247–264]). However, algorithm [Formula: see text] seems to be related to a new type of 2-dimensional continued fractions. The content of this paper is first to describe such an algorithm and to give some of its ergodic properties. In the second part the dual algorithm is considered which behaves similar to the Parry–Daniels map.

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13

Devaney, Robert L. "The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence." American Mathematical Monthly 106, no. 4 (April 1999): 289. http://dx.doi.org/10.2307/2589552.

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14

Zaharescu, Alexandru, Christian Cobeli, and Florian P. Boca. "On the distribution of the Farey sequence with odd denominators." Michigan Mathematical Journal 51, no. 3 (December 2003): 557–74. http://dx.doi.org/10.1307/mmj/1070919560.

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15

Devaney, Robert L. "The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence." American Mathematical Monthly 106, no. 4 (April 1999): 289–302. http://dx.doi.org/10.1080/00029890.1999.12005046.

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16

Yamamoto, Kota, Seijiro Tsutsui, and Yuji Yamamoto. "Constrained paths based on the Farey sequence in learning to juggle." Human Movement Science 44 (December 2015): 102–10. http://dx.doi.org/10.1016/j.humov.2015.08.008.

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17

Fujimoto, Y. "The Fibonacci partition on the connectedness locus and the Farey sequence." Chaos, Solitons & Fractals 7, no. 4 (April 1996): 555–64. http://dx.doi.org/10.1016/0960-0779(95)00074-7.

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18

Pratihar, Sanjoy, and Partha Bhowmick. "On the Farey sequence and its augmentation for applications to image analysis." International Journal of Applied Mathematics and Computer Science 27, no. 3 (September 1, 2017): 637–58. http://dx.doi.org/10.1515/amcs-2017-0045.

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AbstractWe introduce a novel concept of theaugmented Farey table(AFT). Its purpose is to store the ranks of fractions of aFarey sequencein an efficient manner so as to return therankof any query fraction in constant time. As a result, computations on the digital plane can be crafted down to simple integer operations; for example, the tasks like determining the extent of collinearity of integer points or of parallelism of straight lines—often required to solve many image-analytic problems—can be made fast and efficient through an appropriate AFT-based tool. We derive certain interesting characterizations of an AFT for its efficient generation. We also show how, for a fraction not present in a Farey sequence, the rank of thenearest fractionin that sequence can efficiently be obtained by theregula falsimethod from the AFT concerned. To assert its merit, we show its use in two applications—one in polygonal approximation of digital curves and the other in skew correction of engineering drawings in document images. Experimental results indicate the potential of the AFT in such image-analytic applications.

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19

Hauck, T., and F. W. Schneider. "Chaos in a Farey Sequence through Period Doubling in the Peroxidase-Oxidase Reaction." Journal of Physical Chemistry 98, no. 8 (February 1994): 2072–77. http://dx.doi.org/10.1021/j100059a015.

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20

Kingston, S. Leo, and K. Thamilmaran. "Bursting Oscillations and Mixed-Mode Oscillations in Driven Liénard System." International Journal of Bifurcation and Chaos 27, no. 07 (June 30, 2017): 1730025. http://dx.doi.org/10.1142/s0218127417300257.

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We report the existence of bursting oscillations and mixed-mode oscillations in a Liénard system when it is driven externally by a sinusoidal force. The bursting oscillations transit from a periodic phase to spiking trains through chaotic windows, as the control parameter is varied. The mixed-mode oscillations appear via an alternate sequence of periodic and chaotic states, as well as Farey sequences. The primary and their associated secondary mixed-mode oscillations are detected for the suitable choices of system parameters. Additionally, the system is also found to possess multistability nature. Our investigations involve numerical simulations as well as real time hardware experiments using a simple analog electronic circuit. The experimental observations are in conformation with numerical results.

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21

Dubuc, Serge, and Abdul Malik. "Convex hull of powers of a complex number, trinomial equations and the Farey sequence." Numerical Algorithms 2, no. 1 (February 1992): 1–32. http://dx.doi.org/10.1007/bf02142203.

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22

Siemens, Ansgar, and Peter Schmelcher. "Formation and crossover of multiple helical dipole chains." Journal of Physics A: Mathematical and Theoretical 55, no. 37 (August 19, 2022): 375205. http://dx.doi.org/10.1088/1751-8121/ac86af.

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Abstract We investigate the classical equilibrium properties and metamorphosis of the ground state of interacting dipoles with fixed locations on a helix. The dipoles are shown to align themselves along separate intertwined dipole chains forming single, double, and higher-order helical chains. The number of dipole chains, and their properties such as chirality and length scale on which the chains wind around each other, can be tuned by the geometrical parameters. We demonstrate that all possible configurations form a self-similar bifurcation diagram which can be linked to the Stern–Brocot tree and the underlying Farey sequence. We describe the mechanism responsible for this behavior and subsequently discuss corresponding implications and possible applications.

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23

Albahadily, F. N., John Ringland, and Mark Schell. "Mixed‐mode oscillations in an electrochemical system. I. A Farey sequence which does not occur on a torus." Journal of Chemical Physics 90, no. 2 (January 15, 1989): 813–21. http://dx.doi.org/10.1063/1.456106.

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24

THAMILMARAN, K., and M. LAKSHMANAN. "CLASSIFICATION OF BIFURCATIONS AND ROUTES TO CHAOS IN A VARIANT OF MURALI–LAKSHMANAN–CHUA CIRCUIT." International Journal of Bifurcation and Chaos 12, no. 04 (April 2002): 783–813. http://dx.doi.org/10.1142/s0218127402004681.

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We present a detailed investigation of the rich variety of bifurcations and chaos associated with a very simple nonlinear parallel nonautonomous LCR circuit with Chua's diode as its only nonlinear element as briefly reported recently [Thamilmaran et al., 2000]. It is proposed as a variant of the simplest nonlinear nonautonomous circuit introduced by Murali, Lakshmanan and Chua (MLC) [Murali et al., 1994]. In our study we have constructed two-parameter phase diagrams in the forcing amplitude-frequency plane, both numerically and experimentally. We point out that under the influence of a periodic excitation a rich variety of bifurcation phenomena, including the familiar period-doubling sequence, intermittent and quasiperiodic routes to chaos as well as period-adding sequences, occur. In addition, we have also observed that the periods of many windows satisfy the familiar Farey sequence. Further, reverse bifurcations, antimonotonicity, remerging chaotic band attractors, and so on, also occur in this system. Numerical simulation results using Poincaré section, Lyapunov exponents, bifurcation diagrams and phase trajectories are found to be in agreement with experimental observations. The chaotic dynamics of this circuit is observed experimentally and confirmed both by numerical and analytical studies as well PSPICE simulation results. The results are also compared with the dynamics of the original MLC circuit with reference to the two-parameter space to show the richness of the present circuit.

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25

Liu, Yun, and Xijuan Liu. "Bifurcations and Structures of the Parameter Space of a Discrete-Time SIS Epidemic Model." Journal of Mathematics 2022 (April 23, 2022): 1–14. http://dx.doi.org/10.1155/2022/2233452.

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The dynamics of a discrete-time SIS epidemic model are reported in this paper. Three types of codimension one bifurcation, namely, transcritical, flip, Neimark–Sacker (N-S) bifurcations, and their intersection codimension two bifurcations including 1 : 2, 1 : 3, and 1 : 4 resonances are discussed. The necessary and sufficient conditions for detecting these types of bifurcation are derived using algebraic criterion methods. Numerical simulations are conducted not only to illustrate analytical results but also to exhibit complex behaviors which include period-doubling bifurcation in period − 2 , − 4 , − 8 , − 16 orbits, invariant closed cycles, and attracting chaotic sets. Especially, here we investigate the parameter space of the discrete model. We also investigate the organization of typical periodic structures embedded in a quasiperiodic region. We identify period-adding, Farey sequence of periodic structures embedded in this quasiperiodic region.

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26

Geum, Young Hee, and Young Ik Kim. "On Locating and Counting Satellite Components Born along the Stability Circle in the Parameter Space for a Family of Jarratt-Like Iterative Methods." Mathematics 7, no. 9 (September 11, 2019): 839. http://dx.doi.org/10.3390/math7090839.

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This paper is devoted to an analysis on locating and counting satellite components born along the stability circle in the parameter space for a family of Jarratt-like iterative methods. An elementary theory of plane geometric curves is pursued to locate bifurcation points of such satellite components. In addition, the theory of Farey sequence is adopted to count the number of the satellite components as well as to characterize relationships between the bifurcation points. A linear stability theory on local bifurcations is developed based upon a small perturbation about the fixed point of the iterative map with a control parameter. Some properties of fixed and critical points under the Möbius conjugacy map are investigated. Theories and examples on locating and counting bifurcation points of satellite components in the parameter space are presented to analyze the bifurcation behavior underlying the dynamics behind the iterative map.

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27

OKU, MAKITO, and KAZUYUKI AIHARA. "NUMERICAL ANALYSIS OF TRANSIENT AND PERIODIC DYNAMICS IN SINGLE AND COUPLED NAGUMO–SATO MODELS." International Journal of Bifurcation and Chaos 22, no. 06 (June 2012): 1230021. http://dx.doi.org/10.1142/s0218127412300212.

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The Nagumo–Sato (NS) model is a one-dimensional piecewise linear map that describes simplified dynamics of a single neuron. The NS model and its network extension, coupled Nagumo–Sato models, exhibit complex behavior both in their transient dynamics and after converging to periodic orbits. However, the way the period and the transient length change against the parameters is not completely understood. In this study, we numerically investigate the transient and periodic dynamics in single and coupled NS models. Simulation results indicate the following observations. (1) The period of a single NS model shows layered structures associated with the Farey sequence. (2) Two coupled NS models show discontinuity in the transient length, even though the period does not change. (3) In the case of an associative memory model consisting of NS models, there exists a small parameter region where both the period and the transient length increase considerably. The dynamics within the region is much more complex than that outside the region.

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28

Windsor, Aaron J., and Candace S. Waddell. "FARE, a New Family of Foldback Transposons in Arabidopsis." Genetics 156, no. 4 (December 1, 2000): 1983–95. http://dx.doi.org/10.1093/genetics/156.4.1983.

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Abstract A new family of transposons, FARE, has been identified in Arabidopsis. The structure of these elements is typical of foldback transposons, a distinct subset of mobile DNA elements found in both plants and animals. The ends of FARE elements are long, conserved inverted repeat sequences typically 550 bp in length. These inverted repeats are modular in organization and are predicted to confer extensive secondary structure to the elements. FARE elements are present in high copy number, are heterogeneous in size, and can be divided into two subgroups. FARE1’s average 1.1 kb in length and are composed entirely of the long inverted repeats. FARE2’s are larger, up to 16.7 kb in length, and contain a large internal region in addition to the inverted repeat ends. The internal region is predicted to encode three proteins, one of which bears homology to a known transposase. FARE1.1 was isolated as an insertion polymorphism between the ecotypes Columbia and Nossen. This, coupled with the presence of 9-bp target-site duplications, strongly suggests that FARE elements have transposed recently. The termini of FARE elements and other foldback transposons are imperfect palindromic sequences, a unique organization that further distinguishes these elements from other mobile DNAs.

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29

Kawczyński, Andrzej L., and Peter E. Strizhak. "Period adding and broken Farey tree sequence of bifurcations for mixed-mode oscillations and chaos in the simplest three-variable nonlinear system." Journal of Chemical Physics 112, no. 14 (April 8, 2000): 6122–30. http://dx.doi.org/10.1063/1.481222.

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30

Hegedűs, Ferenc. "Topological analysis of the periodic structures in a harmonically driven bubble oscillator near Blake's critical threshold: Infinite sequence of two-sided Farey ordering trees." Physics Letters A 380, no. 9-10 (March 2016): 1012–22. http://dx.doi.org/10.1016/j.physleta.2016.01.022.

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31

Stechkin, S. B. "Farey sequences." Mathematical Notes 61, no. 1 (January 1997): 76–95. http://dx.doi.org/10.1007/bf02355009.

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32

Masè, Michela, Leon Glass, Marcello Disertori, and Flavia Ravelli. "Nodal recovery, dual pathway physiology, and concealed conduction determine complex AV dynamics in human atrial tachyarrhythmias." American Journal of Physiology-Heart and Circulatory Physiology 303, no. 10 (November 15, 2012): H1219—H1228. http://dx.doi.org/10.1152/ajpheart.00228.2012.

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The genesis of complex ventricular rhythms during atrial tachyarrhythmias in humans is not fully understood. To clarify the dynamics of atrioventricular (AV) conduction in response to a regular high-rate atrial activation, 29 episodes of spontaneous or pacing-induced atrial flutter (AFL), covering a wide range of atrial rates (cycle lengths from 145 to 270 ms), were analyzed in 10 patients. AV patterns were identified by applying firing sequence and surrogate data analysis to atrial and ventricular activation series, whereas modular simulation with a difference-equation AV node model was used to correlate the patterns with specific nodal properties. AV node response at high atrial rate was characterized by 1) AV patterns of decreasing conduction ratios at the shortening of atrial cycle length (from 236.3 ± 32.4 to 172.6 ± 17.8 ms) according to a Farey sequence ordering (conduction ratio from 0.34 ± 0.12 to 0.23 ± 0.06; P < 0.01); 2) the appearance of high-order alternating Wenckebach rhythms, such as 6:2, 10:2, and 12:2, associated with ventricular interval oscillations of large amplitude (407.7 ± 150.4 ms); and 3) the deterioration of pattern stability at advanced levels of block, with the percentage of stable patterns decreasing from 64.3 ± 35.2% to 28.3 ± 34.5% ( P < 0.01). Simulations suggested these patterns to originate from the combined effect of nodal recovery, dual pathway physiology, and concealed conduction. These results indicate that intrinsic nodal properties may account for the wide spectrum of AV block patterns occurring during regular atrial tachyarrhythmias. The characterization of AV nodal function during different AFL forms constitutes an intermediate step toward the understanding of complex ventricular rhythms during atrial fibrillation.

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33

Ruane, P. N. "Reviews - Motifs of mathematics: history and application of the mediant and the Farey sequence, by Scott B. Guthery. Pp. 243. £11.32. 2011. ISBN 978 1453810576. (Docent Press)." Mathematical Gazette 98, no. 541 (March 2014): 163–64. http://dx.doi.org/10.1017/s0025557200000991.

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34

Choi, Sang-Kyu, and Sherif T. Noah. "Mode-Locking and Chaos in a Jeffcott Rotor With Bearing Clearances." Journal of Applied Mechanics 61, no. 1 (March 1, 1994): 131–38. http://dx.doi.org/10.1115/1.2901387.

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A complex mode-locking (or entrainment) structure underlying the nonlinear whirling phenomenon of a horizontal Jeffcott rotor with a discontinuous nonlinearity (bearing clearance) was identified. A winding number is introduced as a measure of the ratio between two frequencies involved in the aperiodic whirling motions of the rotor system considered. Utilizing the winding number map, it was revealed that the alternating periodic and quasi-periodic responses take place according to the Farey number tree. The winding number varies in the form of the so-called “Devil’s staircase” as a certain system parameter varies. From the mode-locking pattern in the parameter space of the forcing amplitude and frequency, it was observed that as the forcing amplitude increases, the size of each locking interval increases so that its growth takes place in the form of “Arnol’d tongues,” where the winding number remains a rational number. Moreover, inside each locking zone, i.e., each “Arnol’d tongue,” there exist many smaller tongues similar to the main tongue, in which a sequence of period-doubling bifurcations leading to chaos occurred. The boundaries of each locking zone was obtained using a fixed-point algorithm along with the Floquet theory for checking the stability of the periodic solutions. The winding numbers were estimated utilizing a fixed-point algorithm modified to obtain quasi-periodic responses. A jump phenomenon was also observed by tracking multiple periodic solutions for several parameters of the rotor system.

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35

KHAN, SAMEEN AHMED. "Farey sequences and resistor networks." Proceedings - Mathematical Sciences 122, no. 2 (May 2012): 153–62. http://dx.doi.org/10.1007/s12044-012-0066-7.

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36

de Felipe, A. B., E. R. García Barroso, J. Gwoździewicz, and A. Płoski. "Łojasiewicz exponents and Farey sequences." Revista Matemática Complutense 29, no. 3 (March 31, 2016): 719–24. http://dx.doi.org/10.1007/s13163-016-0194-1.

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37

Boca, F. P. "On the index of Farey sequences." Quarterly Journal of Mathematics 53, no. 4 (December 1, 2002): 377–91. http://dx.doi.org/10.1093/qjmath/53.4.377.

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38

Singerman, David, and James Strudwick. "Petrie polygons, Fibonacci sequences and Farey maps." Ars Mathematica Contemporanea 10, no. 2 (February 5, 2016): 349–57. http://dx.doi.org/10.26493/1855-3974.864.e9b.

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39

Delmer, F., and J. M. Deshouillers. "On a Generalization of Farey Sequences, II." Journal of Number Theory 55, no. 1 (November 1995): 60–67. http://dx.doi.org/10.1006/jnth.1995.1127.

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40

Surányi, Gergő. "On sidon sequences of farey sequences, square roots and reciprocals." Mathematica Slovaca 70, no. 3 (June 25, 2020): 547–56. http://dx.doi.org/10.1515/ms-2017-0370.

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AbstractIn this paper, I will construct three families of Sidon sequences of certain subsets of ℝ, in particular I will study Farey sequences, square roots, and reciprocals. It will be shown that Sidon sequences over them have cardinality of between $\begin{array}{} \displaystyle c_1\frac{N^{3/4}} {\log{N}} \end{array}$ and c2N3/4, c3N, and $\begin{array}{} \displaystyle c_4 \frac{N\log{\log{N}}}{\log{N}}. \end{array}$

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ZHENG, WEI-MOU. "SYMBOLIC DYNAMICS FOR THE CIRCLE MAP." International Journal of Modern Physics B 05, no. 03 (February 10, 1991): 481–95. http://dx.doi.org/10.1142/s0217979291000298.

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The circle map is studied directly by means of symbolic sequences. By using the Farey representation of real numbers and the Farey transformations for symbols the oriented itineraries are analysed. The generalised Farey transformations for symbols are introduced for supercritical maps. Joints and the skeleton are described for the bifurcation structure.

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Essevaz-Roulet, B., P. Petitjeans, M. Rosen, and J. E. Wesfreid. "Farey sequences of spatiotemporal patterns in video feedback." Physical Review E 61, no. 4 (April 1, 2000): 3743–49. http://dx.doi.org/10.1103/physreve.61.3743.

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43

Svalbe, Imants, and Andrew Kingston. "Farey Sequences and Discrete Radon Transform Projection Angles." Electronic Notes in Discrete Mathematics 12 (March 2003): 154–65. http://dx.doi.org/10.1016/s1571-0653(04)00482-2.

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Nicholson, Thomas, and Marc Sabat. "FAREY SEQUENCES MAP PLAYABLE NODES ON A STRING." Tempo 74, no. 291 (December 19, 2019): 86–97. http://dx.doi.org/10.1017/s0040298219001001.

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AbstractNatural harmonics, i.e. partials and their harmonic series, may be isolated on a vibrating string by lightly touching specific points along its length. In addition to the two endpoints, stationary nodes for a given partial n present themselves at n − 1 locations along the string, dividing it into n parts of equal length. It is not the case, however, that touching any one of these nodes will necessarily isolate the nth partial and its integer multiples. The subset of nodes that will activate the nth partial (termed playable nodes by the authors) may be derived by following a mathematically predictable pattern described by so-called Farey sequences. The authors derive properties of these sequences and connect them to physical phenomena. This article describes various musical applications: locating single natural harmonics, forming melodies of neighbouring harmonics, sounding multiphonic aggregates, as well as predicting the relative tuneability of just intervals.

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Perez, G., Sudeshna Sinha, and H. A. Cerdeira. "Nonstandard Farey Sequences in a Realistic Diode Map." Europhysics Letters (EPL) 16, no. 7 (October 14, 1991): 635–41. http://dx.doi.org/10.1209/0295-5075/16/7/005.

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Gomes, Paulo, Nuno Franco, and Luís Silva. "Farey neighbors and hyperbolic Lorenz knots." Journal of Knot Theory and Its Ramifications 26, no. 09 (August 2017): 1743004. http://dx.doi.org/10.1142/s0218216517430040.

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Based on symbolic dynamics of Lorenz maps, we prove that, provided one conjecture due to Morton is true, then Lorenz knots associated to orbits of points in the renormalization intervals of Lorenz maps with reducible kneading invariant of type [Formula: see text], where the sequences [Formula: see text] and [Formula: see text] are Farey neighbors verifying some conditions, are hyperbolic.

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Goldman, Jay R. "Hurwitz sequences, the Farey process, and general continued fractions." Advances in Mathematics 72, no. 2 (December 1988): 239–60. http://dx.doi.org/10.1016/0001-8708(88)90029-1.

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Marszalek, Wieslaw. "Circuits with Oscillatory Hierarchical Farey Sequences and Fractal Properties." Circuits, Systems, and Signal Processing 31, no. 4 (February 10, 2012): 1279–96. http://dx.doi.org/10.1007/s00034-012-9392-3.

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Boca, Florin P. "An AF Algebra Associated with the Farey Tessellation." Canadian Journal of Mathematics 60, no. 5 (October 2008): 975–1000. http://dx.doi.org/10.4153/cjm-2008-043-1.

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AbstractWe associate with the Farey tessellation of the upper half-plane an AF algebra encoding the “cutting sequences” that define vertical geodesics. The Effros–Shen AF algebras arise as quotients of . Using the path algebra model for AF algebras we construct, for each τ ∈ ( 0, ¼], projections (En) in such that EnEn±1En ≤ τ En.

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Wannamaker, Robert. "Rhythmicon Relationships, Farey Sequences, and James Tenney’sSpectral CANON for CONLON Nancarrow(1974)." Music Theory Spectrum 34, no. 2 (October 2012): 48–70. http://dx.doi.org/10.1525/mts.2012.34.2.48.

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