Farey sequence

Abstract: Our experiments on the manganese‐catalyzed Belousov–Zhabotinskii reaction in a stirred flow reactor reveal many sequences of distinct multipeaked periodic states. In the parameter ranges studied the waveform for each periodic state consists of an admixture of small and large amplitude oscillations. No chaos is discernible, and in many cases the transitions from one periodic state to another occur without any observable hysteresis. Two types of sequences were studied in detail, one with waveforms consisting of concatenations of two basic patterns and another with waveforms consisting of concatenations of three basic patterns. The sequences of states with two patterns are described well by Farey arithmetic, which provides rational approximations of irrational numbers. These states can be characterized by a firing number, the ratio of the number of small amplitude oscillations to the total number of oscillations per period. For our data this ratio is a monotone stepwise‐increasing function of flow rate, and the steps have a fractal dimension. The relationship between the observed sequence and the Farey arithmetic and the observation of a fractal dimension for the steps in the firing number suggest that the states formed by concatenating two patterns can be interpreted in terms of frequency locking on a 2 torus in phase space. The sequences of states with three basic patterns are described by a generalized Farey arithmetic that provides rational approximations for pairs of irrational numbers that are mutually irrational; this suggests that these states can be interpreted in terms of frequency locking on a 3 torus. A piecewise‐linear two‐dimensional map is shown to yield a phase diagram in qualitative accord with the measured phase diagram for these sequences.

Abstract: We develop the spacetime aspects of the computation of partition functions for string/M-theory on AdS(3) xM. Subleading corrections to the semi-classical result are included systematically, laying the groundwork for comparison with CFT partition functions via the AdS(3)/CFT(2) correspondence. This leads to a better understanding of the "Farey tail" expansion of Dijkgraaf et. al. from the point of view of bulk physics. Besides clarifying various issues, we also extend the analysis to the N=2 setting with higher derivative effects included.

Abstract: We analyze measurements of an oscillatory current in an electrochemical process in which copper dissolves into phosphoric acid from a rotating‐disk electrode. The focus is on a set of states in which each member consists of a different combination of large and small oscillations (mixed‐mode oscillations). This set of mixed‐mode oscillations is shown to constitute a Farey sequence, i.e., a periodic sequence for which a one‐to‐one correspondence exists with an ordered sequence of rational numbers. Plots of a measured quantity known as the ‘‘firing number’’ are presented which reveal a structure that is similar to a ‘‘devil’s staircase.’’ The states surrounding the mixed‐mode oscillations are analyzed by examining one‐dimensional maps, surfaces of section, and phase portraits constructed from experimental data. This analysis shows that the Farey sequence of these mixed‐mode oscillations is of a different nature than the Farey sequences associated with phase locking on a torus.

Abstract: The geometry of a flux lattice pinned by superconducting layers is studied. Under variation of magnetic field the lattice undergoes an infinite sequence of continuous transitions corresponding to different ways of selection of shortest distances. All possible lattices form a hierarchical structure identified as the hierarchy of Farey numbers. It is shown that dynamically accessible lattices are characterized by pairs of consecutive Fibonacci numbers.