Farey Sequence Research Articles

A property of the uniform distribution modulo 1

An interesting relationship between Farey fractions and the uniform distribution modulo 1 is discovered.

The group ring of ℚ/ℤ and an application of a divisor problem

First we prove some elementary but useful identities in the group ring ofQ/Z\mathbb {Q}/\mathbb {Z}. Our identities have potential applications to several unsolved problems which involve sums of Farey fractions. In this paper we use these identities, together with some analytic number theory and results about divisors in short intervals, to estimate the cardinality of a class of sets of fundamental interest.

AMBIGUITY IN THE DETERMINATION OF THE FREE ENERGY ASSOCIATED WITH THE CRITICAL CIRCLE MAP

AbstractWe consider a simple model to describe the widths of the mode-locked intervals for the critical circle map. By using two different partitions of the rational numbers based on Farey series and Farey tree levels, respectively, we calculate the free energy analytically at selected points for each partition. It emerges that the result of the calculation depends on the method of partition. An implication of this finding is that the generalized dimensions Dq are different for the two types of partition except when q=0; that is, only the Hausdorff dimension is the same in both cases.

Order Statistics in the Farey Sequences in Sublinear Time and Counting Primitive Lattice Points in Polygons

We present the first sublinear-time algorithms for computing order statistics in the Farey sequence and for the related problem of ranking. Our algorithms achieve a running times of nearly O(n 2/3), which is a significant improvement over the previous algorithms taking time O(n).We also initiate the study of a more general problem: counting primitive lattice points inside planar shapes. For rational polygons containing the origin, we obtain a running time proportional to D 6/7, where D is the diameter of the polygon.

Optimal precoder for amplify-and-forward half-duplex relay system

In this paper, an optimal unitary precoder is designed for an amplify-and-forward (AF) half-duplex relay system to obtain the maximum coding gain while the original ergodic channel capacity for the relay system is kept unchanged. A closed-form design is derived for quadrature amplitude modulation (QAM) signals by employing the properties of Farey sequence in number theory. Simulation results indicate that the proposed design greatly improves the bit error rate (BER) performance for the relay system.

Summing Euler’s Φ -function

Euler’s Φ(n) is the number of positive integers n with no factor in common with n. We wish to calculate This problem arises in connection with the Farey series Fn, i.e. all the fractions from 0/1 to 1/1 (inclusive) with denominator n in their lowest terms, in order of size, so that for example

Spectral analysis of transfer operators associated to Farey fractions

The spectrum of a one-parameter family of signed transfer operators associated to the Farey map is studied in detail. We show that when acting on a suitable Hilbert space of analytic functions they are self-adjoint and exhibit absolutely continuous spectrum and no non-zero point spectrum. Polynomial eigenfunctions when the parameter is a negative half-integer are also discussed.

Quotients of values of the Dedekind Eta function

Inspired by Riemann’s work on certain quotients of the Dedekind Eta function, in this paper we investigate the value distribution of quotients of values of the Dedekind Eta function in the complex plane, using the form $${\frac{\eta(A_jz)} {\eta(A_{j-1}z)}}$$ , where A j-1 and A j are matrices whose rows are the coordinates of consecutive visible lattice points in a dilation XΩ of a fixed region Ω in $${\mathbb{R}^2}$$ , and z is a fixed complex number in the upper half plane. In particular, we show that the limiting distribution of these quotients depends heavily on the index of Farey fractions which was first introduced and studied by Hall and Shiu. The distribution of Farey fractions with respect to the value of the index dictates the universal limiting behavior of these quotients. Motivated by chains of these quotients, we show how to obtain a generalization, due to Zagier, of an important formula of Hall and Shiu on the sum of the index of Farey fractions.

Computing Farey Series

The Farey series of order n, Fn, consists of all the fractions between 0 and 1 (inclusive) with denominators less than or equal to n, arranged in order of magnitude and expressed in their lowest terms, so that for example The most obvious way of computing one is to assign a long series of stores and then insert the terms with denominator 1, with denominator 2, … , shifting the terms already there to make room for the new ones in their correct positions. This may require a large amount of storage and is fairly slow because of the large amount of shifting involved: for instance F1025 has 319765 terms, and occupies 400 printed pages (see Neville [1]). Neville also gives a table related to F100, which has 3045 terms. His procedures, a heroic application of pencil-and-paper methods, are described at the end of this paper.

Distribution of Farey fractions in residue classes and Lang–Trotter conjectures on average

We prove that the set of Farey fractions of order T T , that is, the set { α / β ∈ Q : gcd ⁡ ( α , β ) = 1 , 1 ≤ α , β ≤ T } \{\alpha /\beta \in \mathbb {Q}\ : \ \operatorname {gcd}(\alpha , \beta ) = 1, \ 1 \le \alpha , \beta \le T\} , is uniformly distributed in residue classes modulo a prime p p provided T ≥ p 1 / 2 + ε T \ge p^{1/2 +\varepsilon } for any fixed ε > 0 \varepsilon >0 . We apply this to obtain upper bounds for the Lang–Trotter conjectures on Frobenius traces and Frobenius fields “on average” over a one-parametric family of elliptic curves.

Farey fractions with prime denominators

Farey fractions with prime denominators

Period doubling in a periodically forced Belousov-Zhabotinsky reaction

Using an open-flow reactor periodically perturbed with light, we observe subharmonic frequency locking of the oscillatory Belousov-Zhabotinsky chemical reaction at one-sixth the forcing frequency (6:1) over a region of the parameter space of forcing intensity and forcing frequency where the Farey sequence dictates we should observe one-third the forcing frequency (3:1). In this parameter region, the spatial pattern also changes from slowly moving traveling waves to standing waves with a smaller wavelength. Numerical simulations of the FitzHugh-Nagumo equations show qualitative agreement with the experimental observations and indicate that the oscillations in the experiment are a result of period doubling.

Phase diagram and complex patterns in the modeling of the bromate-oxalic acid-Ce-acetone oscillating reaction

Simulations have been carried out on the bromate - oxalic acid - Ce(IV) - acetone oscillating reaction, under flow conditions, using Field and Boyd's model (J. Phys. Chem. 1985, 89, 3707). Many different complex dynamic behaviors were found, including simple periodic oscillations, complex periodic oscillations, quasiperiodicity and chaos. Some of these complex oscillations can be understood as belonging to a Farey sequence. The many different behaviors were systematized in a phase diagram which shows that some regions of complex patterns were nested with one inside the other. The existence of almost all known dynamic behavior for this system allows the suggestion that it can be used as a model for some very complex phenomena that occur in biological systems.

Arithmetic mean of differences of Dedekind sums

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums \(\frac{1}{\varphi(N)} \sum_{\mathop{\mathop{ 0 \le m< N}}\limits_{\gcd(m,N)=1}} \vert S(m,N)\vert\), as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form \(A_{h}(Q)=\frac{1}{\sum_{\frac{a}{q} \in {\cal F}_{Q}}h\left(\frac{a}{q}\right)} \times \sum_{\frac{a}{q} \in {\cal F}_{\!Q}}h\left(\frac{a}{q}\right) \vert s(a^{\prime},q^{\prime})-s(a,q)\vert\), where \(h:[0,1] \rightarrow {\Bbb C}\) is a continuous function with \(\int_0^1 h(t) \, {\rm d} t \ne 0\), \({\frac{a}{q}}\) runs over \({\cal F}_{\!Q}\), the set of Farey fractions of order Q in the unit interval [0,1] and \({\frac{a}{q}}<\frac{a^{\prime}}{q^{\prime}}\) are consecutive elements of \({\cal F}_{\!Q}\). We show that the limit limQ→∞Ah(Q) exists and is independent of h.

Relative blocking in posets

Poset-theoretic generalizations of set-theoretic committee constructions are presented. The structure of the corresponding subposets is described. Sequences of irreducible fractions associated to the principal order ideals of finite bounded posets are considered and those related to the Boolean lattices are explored; it is shown that such sequences inherit all the familiar properties of the Farey sequences.

Phase locking control in the Circle Map

The phase-locking between two oscillators occurs when the ratio of their frequencies becomes locked in a ratio p/q of integer numbers over some finite domain of parameters values. Due to it, oscillators with some kind of nonlinear coupling may synchronize for certain set of parameters. This phenomenon can be better understood and studied with the use of a well-known paradigm, the Circle Map, and the definition of the winding number. Two diagrams related to this map are especially useful: the ‘Arnold tongues’ and the ‘devil’s staircase’. The synchronization that occurs in this map is described by the ‘Farey Series’. This property is the starting point for the development of control algorithms capable of locking the system under the action of an external excitation into a desired winding number. In this work, we discuss the main characteristics of the phase-locking phenomenon and consider three control algorithms designed to drive and keep the Circle Map into a desired winding number.

Oscillations, period doublings, and chaos in CO oxidation and catalytic mufflers

Early experimental observations of chaotic behavior arising via the period-doubling route for the CO catalytic oxidation both on Pt(110) and Ptgamma-Al(2)O(3) porous catalyst were reported more than 15 years ago. Recently, a detailed kinetic reaction scheme including over 20 reaction steps was proposed for the catalytic CO oxidation, NO(x) reduction, and hydrocarbon oxidation taking place in a three-way catalyst (TWC) converter, the most common reactor for detoxification of automobile exhaust gases. This reactor is typically operated with periodic variation of inlet oxygen concentration. For an unforced lumped model, we report results of the stoichiometric network analysis of a CO reaction subnetwork determining feedback loops, which cause the oscillations within certain regions of parameters in bifurcation diagrams constructed by numerical continuation techniques. For a forced system, numerical simulations of the CO oxidation reveal the existence of a period-doubling route to chaos. The dependence of the rotation number on the amplitude and period of forcing shows a typical bifurcation structure of Arnold tongues ordered according to Farey sequences, and positive Lyapunov exponents for sufficiently large forcing amplitudes indicate the presence of chaotic dynamics. Multiple periodic and aperiodic time courses of outlet concentrations were also found in simulations using the lumped model with the full TWC kinetics. Numerical solutions of the distributed model in two geometric coordinates with the CO oxidation subnetwork consisting of several tens of nonlinear partial differential equations show oscillations of the outlet reactor concentrations and, in the presence of forcing, multiple periodic and aperiodic oscillations. Spatiotemporal concentration patterns illustrate the complexity of processes within the reactor.

Multidimensional Farey partitions

The linear action of SL( n, ℤ +) induces lattice partitions on the ( n − 1)-dimensional simplex † n−1. The notion of Farey partition raises naturally from a matricial interpretation of the arithmetical Farey sequence of order r. Such sequence is unique and, consequently, the Farey partition of order r on A 1 is unique. In higher dimension no generalized Farey partition is unique. Nevertheless in dimension 3 the number of triangles in the various generalized Farey partitions is always the same which fails to be true in dimension n > 3. Concerning Diophantine approximations, it turns out that the vertices of an n-dimensional Farey partition of order r are the radial projections of the lattice points in ℤ + n ∩ [0, r] n whose coordinates are relatively prime. Moreover, we obtain sequences of multidimensional Farey partitions which converge pointwisely.

Euler products, Farey series, and the Riemann hypothesis. II.

Euler products, Farey series, and the Riemann hypothesis. II.

Cluster Approximation for the Farey Fraction Spin Chain

We consider the Farey fraction spin chain in an external field h. Utilising ideas from dynamical systems, the free energy of the model is derived by means of an effective cluster energy approximation. This approximation is valid for divergent cluster sizes, and hence appropriate for the discussion of the magnetizing transition. We calculate the phase boundaries and the scaling of the free energy. At h = 0 we reproduce the rigorously known asymptotic temperature dependence of the free energy. For h ≠ 0, our results are largely consistent with those found previously using mean field theory and renormalization group arguments.