Farey

M.G. Farey, The Gas Chromatographic Determination of Isopropyl Nitrate in Slurry Explosives , Rept No ERDE-TN-41, Waltham Ab bey (Engl) (1971) 4)J. Diederichse n ... 

Lingappa, V.R. and Farey, K., Physiological Medicine A Clinical Approach to Basic Medical Physiology, McGraw-Hill, New York, 2000. 

The most convincing derivation of periodic structure, using the concepts of number theory, comes from a comparison with Farey sequences. The Farey scheme is a device to arrange rational fractions in enumerable order. Starting from the end members of the interval [0,1] an infinite tree structure is generated by separate addition of numerators and denominators to produce the Farey sequences ni of order n, where n limits the values of denominators... 

Each rational fraction, h/k, defines a Ford circle with a radius and y-coordinate of 1/(2k2), positioned at an -coordinate h/k. The Ford circles of any unimodular pair are tangent to each other and to the x-axis. The circles, numbered from 1 to 4 in the construction overleaf, represent the Farey sequence of order 4. This sequence has the remarkable property of one-to-one correspondence with the natural numbers ordered in sets of 2k2 and in the same geometrical relationship as the Ford circles of 4. 

The convergence follows the Fibonacci fractions which appear in the Farey tree structure that develops between the limits and . 

It is of interest to explore the possibility of an independent identification of magic numbers and the neutron spectrum by the relationship between neutron number and Farey sequences as mapped by Ford circles. Such an analysis presupposes the occurrence of periodic sequences of 32, 18, 8 and... 

All of the primary and secondary sequences can be traced back to tangent Ford circles. The two independent patterns have common points at the four most significant, generally accepted, magic numbers 2, 50, 82 and 126. The points at which the eleven hem lines intersect the golden ratio line are indicated by arrows. Ford circles from the Farey sequence (2k2 = 50) appear... 

The orbits from Venus to Ceres are represented by the unimodular series 4. In the outer system the Ford circles of only Uranus and Neptune are tangent, but the likeness to Farey sequences in atomic systems is sufficient to support the self-similarity conjecture. 

The principle that governs the periodic properties of atomic matter is the composition of atoms, made up of integral numbers of discrete sub-atomic units - protons, neutrons and electrons. Each nuclide is an atom with a unique ratio of protonsmeutrons, which defines a rational fraction. The numerical function that arranges rational fractions in enumerable order is known as a Farey sequence. A simple unimodular Farey sequence is obtained by arranging the fractions (n/n+1) as a function of n. The set of /c-modular sequences ... 

The equivalence between Sk, the infinite Farey tree structure and the nuclide mapping is shown graphically in Figure 8.4. The stability of a nuclide depends on its neutron imbalance which is defined, either by the ratio Z/N or the relative neutron excess, (N — Z) jZ. When these factors are in balance, Z2 + NZ — N2 = 0, with the solution Z = N(—1 /5)/2 = tN. The minimum (Z/N) = r and hence all stable nuclides are mapped by fractions larger than the golden mean. 

For Watt s description of the indicator see Robison, System of Mechanical Philosophy, vol. 2, pp. 156-7. On Farey and the indicator R. B. Prosser, Birmingham Inventors and Inventions (Birmingham The Journal Printing Works, New Street, 1881), p. 36 John Farey in Report of the Select Committee on the Law Relative to Patentsfor Inventions, 1829, p. 138 A. P. Woolrich, John Farey and his Treatise on the Steam Engine (1827) , History of Technology, 22 (2000), pp. 63-106. 

Cardwell, From Watt to Clausius, pp. 220-1 is convinced that Clapeyron had encountered the indicator diagram but is not sure how. John Farey, as previously noted, had seen an indicator diagram being taken in Russia in 1819. It is possible that the same thing happened to Clapeyron on whose presence and activities in Russia at this time see M. Bradley, Franco-Russian Engineering Links The Careers of Tame and Clapeyron, 1820-1830, Annals of Science, 38 (1981), pp. 291-312. 

Figure 2.9 Farey sequence of rational fractions. Starting with the first row and reading from right to left 3/1 is identified as the 12th rational fraction.

The stability problem is solved on noting that allowed fractions at small atomic number begin at unity and approach r with increasing Z. This trend should, by definition follow Farey fractions determined by Fibonacci numbers. The first few Fibonacci numbers are 0,1,1,2,3,5,8,13,21, etc. The ra-... 

All fractions beyond 3/4, at the centre of the Farey subset, are further seen to define points of intersection between the curves of constant A — 2Z and the straight line between coordinates of (14/19,0) and (2/3,87), at the intersection with the line 1 —> r. These lines provide the correct stability limits. They have the additional merit of automatically limiting the maximum allowed atomic number of a stable element to 83. 

Figure 2.11 Low-rank fractions in the Farey sequence between the Fibonacci fractions 1/1 and 2/3...

Conveniently, MMOs are characterized by a symbolic notation where L denotes the number of large and S the number of small oscillations during one period. Thus, the MMOs depicted in Fig. 27 are designated as F , P, and 1 P states. In the notation of the latter state, it is indicated that one period is built up from concatenated principal states. In fact, in the simulations, many such concatenated states were found for example, between the P and the P state, P(P) states with n going from 1 to 10 were observed. These sequences are called Farey sequences because a one-to-one correspondence of successive MMO states and the ordering of the rational numbers, which is conveniently represented in a Farey tree (see Fig. 31), can be established. In general, at low values of the resistance, the sequences of MMOs obey an incomplete Farey arithmetic. 

A Farey tree arises in number theory as a scheme for the generation of all the rational numbers between a given pair of rationals. This proceeds by the so-called Farey addition of two rationals piq and ris which is equal io p + q)l (r + s). 

Equation (16) was originally derived to model the reduction of In " from SCN solution on the HMDE. The bifurcation behavior of this system is summarized in the two-parameter bifurcation diagram in Fig. 29. Most remarkably, the two distinct MMO sequences of the model also show up in the experiment. Farey sequences were observed close to the Hopf bifurcation at low values of the series resistance, whereas at the high resistance end of the oscillatory regime, periodic-chaotic mixed-mode sequences were found. Owing to this good agreement of the bifurcation... 

Another most remarkable experimental study in which the two types of mixed-mode sequences were also observed was carried out by Albahadily et al, who studied the electrodissolution of copper in phosphoric acid from a rotating disk. Figure 30 shows a series of Farey states observed in this system, and in Fig. 31, the experimentally observed mixed-mode states are listed in the structure of a Farey tree. On the high rotation-rate end of the 1° state, alternating periodic and chaotic behavior appeared. The first period-doubled oscillation arising from the P parent state is reproduced in Fig. 32 together with the P parent state. In Fig. 33, a two-parameter bifurcation diagram is depicted in which the succession... 

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