Farey Sequence Research Articles

Velocity plateaus in traveling-wave electrophoresis

One-dimensional models are used to study traveling-wave electrophoresis, a tunable method for separating charged analytes. A traveling-electrode model reveals the mechanism for longitudinal oscillations. A stationary-electrode model explains the origin of mode-locked plateaus in the average velocity, predicts devil's staircases with nested Farey sequences, and reduces to a continuum sinusoidal model in the high electrode-density limit.

Distinct Clusterings and Characteristic Path Lengths in Dynamic Small-World Networks with Identical Limit Degree Distribution

Many real-world networks belong to a particular class of structures, known as small-world networks, that display short distance between pair of nodes. In this paper, we introduce a simple family of growing small-world networks where both addition and deletion of edges are possible. By tuning the deletion probability q t , the model undergoes a transition from large worlds to small worlds. By making use of analytical or numerical means we determine the degree distribution, clustering coefficient and average path length of our networks. Surprisingly, we find that two similar evolving mechanisms, which provide identical degree distribution under a reciprocal scaling as t goes to infinity, can lead to quite different clustering behaviors and characteristic path lengths. It is also worth noting that Farey graphs constitute the extreme case q t ≡0 of our random construction.

Nodal recovery, dual pathway physiology, and concealed conduction determine complex AV dynamics in human atrial tachyarrhythmias

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.

NUMERICAL ANALYSIS OF TRANSIENT AND PERIODIC DYNAMICS IN SINGLE AND COUPLED NAGUMO–SATO MODELS

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.

NEW INEQUALITIES ON THE HOMOGENEOUS FUNCTIONS

"Pdf File"DOI : http://dx.doi.org/10.22342/jims.15.1.43.49-59

Farey sequences and resistor networks

Farey sequences and resistor networks

How many equivalent resistances?

The order of the set of equivalent resistances, A(n) of n equal resistors combined in series and in parallel has been traditionally addressed computationally, for n up to 22. For larger n there have been constraints of computer memory. Here, we present an analytical approach using the Farey sequence with Fibonacci numbers as its argument. The approximate formula, A(n) ~ 2.55^n, obtained from the computational data up to n = 22 is consistent with the strict upper bound, A(n) ~ 2.618^n, presented here. It is further shown that the Farey sequence approach, developed for the A(n) is applicable to configurations other than the series and/or parallel, namely the bridge circuits and non-planar circuits. Expressions describing set theoretic relations among the sets A(n) are presented in detail. For completeness, programs to generate the various integer sequences occurring in this study, using the symbolic computer language MATHEMATCA, are also presented.

Hyperbolicity of the genus two Hatcher–Thurston complex

For the genus 1 surface with n punctures F 1,n , we show that the Hatcher–Thurston complex \({\fancyscript{HT}(F_{1,n})}\) is hyperbolic. For the genus 2 closed surface F 2, we show that the Hatcher–Thurston complex \({\fancyscript{HT}(F_{2})}\) is strongly relatively hyperbolic with respect to a family of its subspaces {X γ }, where γ ranges over all separating curves of F 2, and each X γ is isometric to the product of two Farey graphs.

Identities involving Farey fractions

The rational numbers a/q in [0, 1] can be counted by increasing height H(a/q) = max(a, q), or ordered as real numbers. Franel’s identity shows that the Riemann hypothesis is equivalent to a strong bound for a measure of the independence of these two orderings. We give a proof using Dedekind sums that allows weights w(q). Taking w(q) = χ(q) we find an extension to Dirichlet L-functions.

Geodesic continued fractions

There are infinitely many ways to express a rational number as a finite continued fraction with numerators 1 and integer denominators. We study the shortest continued fraction expansions for a given rational by interpreting these as geodesic paths in the Farey graph.

Circuits with Oscillatory Hierarchical Farey Sequences and Fractal Properties

We present two dual oscillating circuits having a wide spectrum of dynamical properties but relatively simple topologies. Each circuit has five bifurcating parameters, one nonlinear element of cubic current–voltage characteristics, one controlled element, LCR components and a constant biasing source. The circuits can be considered as two coupled oscillators (linear and nonlinear) that form dual jerk circuits. Bifurcation diagrams of the circuits show a rather surprising result that the bifurcation patterns are of the Farey sequence structure and the circuits’ dynamics is of a fractal type. The circuits’ fractal dimensions of the box counting (capacity) algorithm, Kaplan–Yorke (Lyapunov) type and its modified (improved) version are all estimated to be between 2.26 and 2.52. Our analysis is based on numerical calculations which confirm a close relationship of the circuits’ bifurcation patterns with those of the Ford circles and Stern–Brocot trees.

Counting spanning trees in a small-world Farey graph

The problem of spanning trees is closely related to various interesting problems in the area of statistical physics, but determining the number of spanning trees in general networks is computationally intractable. In this paper, we perform a study on the enumeration of spanning trees in a specific small-world network with an exponential distribution of vertex degrees, which is called a Farey graph since it is associated with the famous Farey sequence. According to the particular network structure, we provide some recursive relations governing the Laplacian characteristic polynomials of a Farey graph and its subgraphs. Then, making use of these relations obtained here, we derive the exact number of spanning trees in the Farey graph, as well as an approximate numerical solution for the asymptotic growth constant characterizing the network. Finally, we compare our results with those of different types of networks previously investigated.

Note on the prime divisors of Farey fractions

Let P1(n) ≥ P2(n) ≥· · · be the prime divisors of a natural number n arranged in the non-increasing order. The limit distribution of the sequences (log Pi(mn)/ log(mn), i ≥ 1) for m/n ꞓ 2 (lambda1; lambda2), n ≤ x, are considered. It is proved that under some conditions on lambdai the limit distribution of the sequences exists and is closely related to the Poisson–Dirichlet distribution.

The Stern–Brocot graph in Möbius number systems

We characterize interval Möbius number systems with sofic expansion subshifts and show that they can be obtained as factors of interval Möbius number systems with expansion subshifts of finite types. The endpoints of interval cylinders of such systems can be computed by a simple formula which generalizes the computation of Farey fractions in the Stern–Brocot graph. We treat in detail the bimodular number system which has many nice properties and could be used for exact real computer arithmetic.

Enumerating the Rationals from Left to Right

The rationals can be enumerated using Stern-Brocot sequences 풮ℬn, Calkin-Wilf sequences 풞풲n, or Farey sequences ℱn. We show that the rationals in 풮ℬn, 풞풲n, and ℱn can be computed using very similar second-order linear recurrence relations. (This, incidentally, obviates the need to compute previous sequences 풮ℬi, 풞풲i, ℱi with i < n.)

Distribution of Elements of Cosets of Small Subgroups and Applications

We obtain a series of estimates on the number of small integers and small order Farey fractions which belong to a given coset of a subgroup of order t of the group of units of the residue ring modulo a prime p, in the case when t is small compared to p. We give two applications of these results: to the simultaneous distribution of two high degree monomials x k1 and x k2 modulo p and to a question of J. Holden and P. Moree on fixed points of the discrete logarithm.

A transfer operator approach to random Fibonacci sequences

We consider two stochastic variants of a Fibonacci sequence generated by xn + 1 = xn ± xn − 1 and xn + 1 = |xn ± xn − 1|, respectively. Both the sequences begin with x0 = x1 = 1 and the ± signs in each iteration are independently chosen with probability p for +. We focus on the generalized Lyapunov exponent τ(q) of {xn}, which gives the growth (when q > 0) or decay (when q < 0) rate of the ensemble average of |xn|q when n → ∞. By using a technique that reduces the non-increasing segments in the sequences, we map the random sequences to a product of random 2 × 2 nonnegative matrices, which is closely related to the Farey fraction spin chain model. We show that there exists a critical value q* below which the ensemble average of |xn|q is dominated by the non-increasing or linearly increasing samples, and hence τ(q) is a constant. Both q* and τ(q) (and its derivatives) can be precisely calculated with the transfer operator method. Moreover, when q is a positive integer, τ(q) is exactly determined by a system of polynomial equations. The analysis can be readily generalized to the random sequence generated by xn + 1 = 2cos (π/k)xn ± xn − 1 (or xn + 1 = |2cos (π/k)xn ± xn − 1|) for any integer k ⩾ 3.

A note on Farey fractions with denominators in arithmetic progressions

As the conclusion of a line of investigation undertaken in two previous papers, we compute asymptotic frequencies for the values taken by numerators of differences of consecutive Farey fractions with denominators restricted to lie in arithmetic progression.

Farey graphs as models for complex networks

Farey sequences of irreducible fractions between 0 and 1 can be related to graph constructions known as Farey graphs. These graphs were first introduced by Matula and Kornerup in 1979 and further studied by Colbourn in 1982, and they have many interesting properties: they are minimally 3-colorable, uniquely Hamiltonian, maximally outerplanar and perfect. In this paper, we introduce a simple generation method for a Farey graph family, and we study analytically relevant topological properties: order, size, degree distribution and correlation, clustering, transitivity, diameter and average distance. We show that the graphs are a good model for networks associated with some complex systems.

Lightweight Encryption of Hierarchical Access Control Mechanism Key Using Tribes of Farey Fractions and its Analysis

In this paper we propose a mechanism to develop a lightweight encryption technique. We encrypt hierarchical single key-lock access control mechanism key by encoding key (key number). Key number is derived using Chinese reminder theorem and tribes of Gaussian or rational Farey fractions are used to encode the key number. The advantages of the proposed mechanism are i) no decoding overhead as access rights are derived without decoding the encoded the key number, ii) no need to derive the key number for already existing files. Hence this method secures access control system by encoding the key number and acts as a thin layer of security; however the performance of retrieval system remains unchanged. General Terms Access control mechanism, Cryptography, Network Security.