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Topic: Farey sequence

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	Sequence (Site not responding. Last check: 2007-11-06)
		Cotolerant sequence In mathematical logic, a cotolerant sequence is a sequence of cointerpretable in.
		Hubble sequence The Hubble sequence is a classification of 1936.
		Sequence motif A sequence motif is a biological significance.
	www.brainyencyclopedia.com /topics/sequence.html (880 words)

	Farey sequence - Wikipedia, the free encyclopedia
		In mathematics, a Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size.
		A Farey sequence is sometimes called a Farey series, which is not strictly correct, because the terms are not summed.
		Farey conjectured that each term in a Farey sequence is the mediant of its neighbours â” however, so far as is known, he did not prove this property.
	en.wikipedia.org /wiki/Farey_sequence (703 words)

	Online Encyclopedia and Dictionary - Sequence
		In mathematics, a sequence is a list of objects (or events) which have been arranged in a linear fashion; such that each member comes either before, or after, every other member, and the order of members is important.
		A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements.
		If the terms of the sequence are a subset of a ordered set, then a monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing.
	www.fact-archive.com /encyclopedia/Sequence (407 words)

	PlanetMath: Farey sequence
		Farey sequences are a singularly useful tool in understanding the convergents that appear in continued fractions.
		An alternate view of the ``dynamics'' of how Farey sequences develop is given by Stern-Brocot trees.
		This is version 4 of Farey sequence, born on 2002-06-13, modified 2002-06-24.
	planetmath.org /encyclopedia/FareySequence.html (166 words)

	Farey sequence (Site not responding. Last check: 2007-11-06)
		In mathematics, a Farey sequence of order n isthe sequence of completely reduced fractions between 0 and 1 which, when in lowestterms, have denominators less than or equal to n, arranged inorder of increasing size.
		A Farey sequence is sometimes called a Farey series,which is not strictly correct, because the terms are not summed.
		Farey sequences are named after the British geologist John Farey,whose letter about these sequences was published in the Philosophical Magazine in 1816.
	www.therfcc.org /farey-sequence-215759.html (843 words)

	Sequence - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-06)
		In mathematics, a sequence is a list of object s (or event s) which have been arranged in a linear fashion; such that each member comes either before, or after, every other member, and the order of members is important.
		A more formal definition of a finite sequence with terms in a set S is a function from {1,2,..., n} to S for some n ≥0.
		An infinite sequence in S is a function from {1,2,...} (the set of natural number s) to S.
	encyclopedia.worldsearch.com /sequence.htm (524 words)

	Farey (Site not responding. Last check: 2007-11-06)
		Farey attended a local school in Woburn until he was sixteen years of age when he went to a school in Halifax, Yorkshire, where he studied mathematics, drawing and surveying.
		Then the "curious property" is that each member of the sequence is equal to the rational whose numerator is the sum of the numerators of the fractions on either side, and whose denominator is the sum of the denominators of the fractions on either side.
		Farey is immortal because he failed to understand a theorem which Haros had proved perfectly fourteen years before...
	www-groups.dcs.st-and.ac.uk /%7Ehistory/Mathematicians/Farey.html (1105 words)

	Farey sequence (Site not responding. Last check: 2007-11-06)
		In mathematics, a Farey sequence of order n is the sequence of completely reduced fractionss between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size.
		Each Farey sequence starts with the value 0, denominated by the fraction 0/1, and ends with the value 1, denominated by the fraction 1/1 (although some authors omit these terms).
		The middle term of a Farey sequence is always 1/2.
	www.sciencedaily.com /encyclopedia/farey_sequence (773 words)

	Sequence - free-definition (Site not responding. Last check: 2007-11-06)
		In mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion; such that each member either comes before, or after, every other member.
		A sequence is a function with a domain equal to the set of positive integers.
		A subsequence of a sequence S is a sequence formed from S by deleting some of the elements without disturbing the relative positions of the remaining elements.
	www.free-definition.com /Sequence.html (301 words)

	Cut The Knot!
		Farey series is characterized by two wonderful equivalent properties.
		The sequence of denominators of terms in the Farey series is palindromic.
		The Farey series are embedded into the Stern-Brocot tree for which this property comes almost for free.
	www.maa.org /editorial/knot/Pick.html (791 words)

	Farey (Site not responding. Last check: 2007-11-06)
		Farey held this post for ten years and it was during this time that he was able to gain expertise in geology.
		Farey had already become interested in soils and rocks through carrying out his duties as land steward and he now took the opportunity to learn all that he could from Smith about stratification.
		Farey's article which is relevant to our history of mathematics archive was also published in the Philosophical Magazine and appeared in 1816.
	www-gap.dcs.st-and.ac.uk /~history/Mathematicians/Farey.html (1105 words)

	A Fresh Look at Number
		The moduli of the first 15 entries to the Farey sequence are shown in column 5 of Table 1.
		The sequence of moves is shown in column 8 of Table 1.
		The sequence of moves for the optimal transfer for TOH: n are again related to the exponent of the greatest power of n, referred to as the index, that divides evenly into a given integer.
	members.tripod.com /vismath4/kappraff1/index.html (5590 words)

	FAREY SEQUENCE - BINS (Site not responding. Last check: 2007-11-06)
		There is another sequence in addition to Stern Brocot, called the Farey sequence whose bin properties have been studied by the BCF group.
		SB: 0/1 1/3 1/2 2/3 1/1 Farey: 0/1 1/3 1/2 2/3 1/1
		SB: 0/1 1/4 1/3 2/5 1/2 3/5 2/3 3/4 1/1 Farey: 0/1 1/4 1/3 1/2 2/3 3/4 1/1
	www.ridgecrest.ca.us /~jebush/bcf/Farybin2.html (1014 words)

	Farey sequence - InformationBlast
		Farey sequences are named after the British geologist John Farey, whose letter about these sequences was published in the Philosophical Magazine in 1816.
		If p/q has neighbours a/b and c/d in some Farey sequence, with a/b < p/q < c/d then p/q is the mediant of a/b and c/d - in other words, p/q =( a + b)/( c + d).
		And if a/b and c/d are neighbours in a Farey sequence then the first term that appears between them as the order of the Farey sequence is increased is ( a + b)/( c + d), which first appears in the Farey sequence of order b + d.
	www.informationblast.com /Farey_sequence.html (845 words)

	Farey Series
		One C. Haros apparently anticipated Farey by fourteen years, but this fact was unknown to the mathematician Cauchy, who attributed the discovery to Farey, and others repeated Cauchy's statement.
		He also showed that if two fractions a/b and c/d satisfy bc - ad = 1, and x/y is another fraction between a/b and c/d that satisfies two similar conditions with regard to both a/b and c/d, then x/y is the mediant fraction of a/b and c/d.
		Farey did not prove that in his series any fraction is the mediant of its neighbors.
	www.cut-the-knot.com /blue/FareyHistory.shtml (1334 words)

	Sequence - Enpsychlopedia (Site not responding. Last check: 2007-11-06)
		In mathematics, a sequence is a list of objects (or events) arranged in a "linear" fashion, such that the order of the members is well defined and significant.
		Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2,4,6,...).
		The members of a sequence are also called its elements or terms, and the number of terms (possibly infinite) is called the length of the sequence.
	www.grohol.com /psypsych/Sequence (714 words)

	Farey Symbols and Fundamental domains (Site not responding. Last check: 2007-11-06)
		A generalized Farey sequence is a sequence of rationals (a_1 /b_1)< (a_2 /b_2)<...
		For a_i, a_(i + 1) in the Farey sequence, with corresponding label l_i not -3, the corresponding edge of the domain is a geodesic between a_i and a_(i + 1).
		Returns a sequence of pairs of cusps which are cusps of the Farey Symbol FS, and which are not adjacent in FS but which are images of 0 and infinity under some matrix in PSL_2(Z).
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text484.htm (515 words)

	Farey sequences
		to be the ascending sequence of fractions in the interval [0, 1] whose denominators are
		Farey sequences were discovered by a British geologist John Farey (1766 to 1826) who published a note in 1816 containing the result ( *) above.
		Farey has a biography of twenty lines in the Dictionary of National Biography in which he is described as a geologist.
	www-groups.dcs.st-and.ac.uk /~john/analysis/Lectures/A4.html (246 words)

	Farey sequence - Gurupedia
		In mathematics, a Farey sequence of order n is the sequence of completely reduced
		geologist John Farey, sr, whose letter about these sequences was published in the Philosophical Magazine in 1816.
		The Stern-Brocot tree is a data structure showing how the sequence is built up from 0 and 1, by taking successive mediants.
	www.gurupedia.com /f/fa/farey_sequence.htm (851 words)

	Farey Sequences (Site not responding. Last check: 2007-11-06)
		Kopriva, J. "Contribution to the relation of the Farey series to the Riemann hypothesis on the zeros of the zeta function" (Czech), Casopis Pest.
		For a range of particle density including the half-filled case, it is shown that the minimum-energy states coincide with the large-U neutral ground state ionic configurations of the Falicov-Kimball model, thus providing a characterization of the latter as 'most homogeneous' ionic arrangements.
		It is found that the lines formed by periodic orbits, belonging to Farey sequences, are not smooth neither continuous.
	www.maths.ex.ac.uk /~mwatkins/zeta/farey.htm (3455 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		############################################################################ # # File: genrfncs.icn # # Subject: Procedures to generate sequences # # Author: Ralph E. Griswold # # Date: June 10, 2001 # ############################################################################ # # This file is in the public domain.
		# # friendseq(k) generate random friendly sequence from k values, 1 to k # (in a friendly sequence, successive terms differ by 1).
		# # powersofseq(i) sequence i ^ n, n = 1, 2, 3, 4,...n # # rabbitseq() rabbit sequence # # ratsseq(i) versumseq() with sort # # signaseq(r) signature sequence of r # # spectseq(r) spectral sequence integer(i * r), i - 1, 2, 3,...
	cvs.sourceforge.net /viewcvs.py/checkout/unicon/unicon/ipl/procs/genrfncs.icn (477 words)

	Definition of Farey sequence (Site not responding. Last check: 2007-11-06)
		Each Farey sequence starts with the value 0, denominated by the fraction
		The middle term of a Farey sequence is always
		If 0< p/q <1 then the Ford circles that are tangent to C[ p/q ] are precisely the Ford circles for fractions that are neighbours of p/q in some Farey sequence.
	www.wordiq.com /definition/Farey_sequence (715 words)

	next Farey fraction
		However quite about the only thing I know about Farey sequences is that if a'/b' is the "next Farey fraction", then ba'-ab'=1, so (1) bab'+1 and then a'=(ab'+1)/b.
		Then, next Farey fraction is =defined as the next element of the Farey sequence F_b.
		So you're assuming that the environment in which a/b lives is the Farey sequence of order b.
	www.forum-one.org /new-6026272-4346.html (2187 words)

	The Farey Room
		The term 'Farey Transform' is idiosyncratic on my part, I don't believe anyone else calls it that (I don't know what they might call it).
		The importance of Farey Trees to fractal phenomena was previously brought to my attention by Paul Cvitanovi\'c during some lectures in Paris in 1985.
		It in turn leads to the concept of a Rational Process, which defines a set of sequences of ratios that can be made to converge to...
	linas.org /art-gallery/farey/fthumb.html (1266 words)

	BCF-GROUP... Math, Physics, Discovery (Site not responding. Last check: 2007-11-06)
		We are willing to suffer the boredom of the professional mathematicians, and hope to let all you amateur mathematicians out there share our enjoyment of mathematics and joy of discovery.
		The Stern-Brocot sequence is formed by considering the unit interval: [0,1] and uses the mediant operator to insert fractions inside the interval.
		There is another sequence, called the Farey which has been around longer than Stern-Brocot which is very similar to, yet oh so different from the SB sequence.
	www.ridgecrest.ca.us /~jebush/bcf/bcf-group.html (302 words)

	Tuning-Math Archive Section 1: 850 - 874
		The first row of the Farey > sequence is [0/1, 1/1] and the nth row is obtained by inserting the > fraction (p1+q1)/(p2+q2) between p1/q1 and p2/q2 if it is in reduced > form (that is, if gcd(p1+q1, p2+q2)=1) and if q1+q2 <= n.
		For the Farey sequence, the rule is reduced form and q1 <= n and q2 <=n.
		For the Farey sequence, the rule is > > reduced form and q1 <= n and q2 <=n.
	www.robertinventor.com /tuning-math/s___1/msg__850-_874.html (3883 words)

	Citebase - On the distribution of the Farey sequence with odd denominators
		Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
		On the distribution of the Farey sequence with odd denominators
		F.P. Boca, R.N. Gologan, A. Zaharescu, On the index of Farey sequences, preprint arXiv math.NT/0201044, Quart.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0201046 (391 words)

	Farey sequence
		Farey conjectured that each term in a Farey sequence is the mediant of its neighbours - however, so far as is known, he did not prove this property.
		If a/b and c/d are neighbours in a Farey sequence, with a/b
		If p/q, which first appears in Farey sequence F
	www.brainyencyclopedia.com /encyclopedia/f/fa/farey_sequence.html (763 words)

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