farey series
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2018 ◽  
Vol 156 (2) ◽  
pp. 465-480
Author(s):  
A. H. Ledoan
Keyword(s):  

2015 ◽  
Vol 31 (2) ◽  
pp. 165-172
Author(s):  
CRISTIAN COBELI ◽  
◽  
ALEXANDRU ZAHARESCU ◽  
◽  

We consider a certain linear recursive relation with integer parameters and study some of its algebraic and geometric properties, with the purpose of estimating the number of chains of valences in the Farey series.


2010 ◽  
Vol 4 (1) ◽  
pp. 57-74 ◽  
Author(s):  
Julyan H.E. Cartwright ◽  
Jack Douthett ◽  
Diego L. González ◽  
Richard Krantz ◽  
Oreste Piro

2008 ◽  
Vol 50 (2) ◽  
pp. 177-184
Author(s):  
BRIAN G. KENNY ◽  
TONY W. DIXON

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.


2008 ◽  
Vol 92 (524) ◽  
pp. 242-251
Author(s):  
Norman Routledge

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


2008 ◽  
Vol 92 (523) ◽  
pp. 55-62 ◽  
Author(s):  
Norman Routledge

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.


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