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Topic: Generalized continued fraction

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Continued fraction

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	Continued fraction - Wikipedia, the free encyclopedia
		Continued fractions are motivated by a desire to have a "mathematically pure" representation for the real numbers.
		Continued fractions also play a role in the study of chaos, where they tie together the Farey fractions which are seen in the Mandelbrot set with the Minkowski question mark function and the modular group Gamma.
		The distribution of the digits in continued fractions is given by the zero'th eigenvector of this operator, and is called the Gauss-Kuzmin distribution.
	en.wikipedia.org /wiki/Continued_fraction (2454 words)

	Continued fraction -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-29)
		While one cannot discern any pattern in the infinite continued fraction expansion of π, this is not true for e, the (additional info and facts about base of the natural logarithm) base of the natural logarithm: e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10,...].
		The numbers with periodic continued fraction expansion are precisely the solutions of (An equation in which the highest power of an unknown quantity is a square) quadratic equations with integer coefficients.
		The backwards (additional info and facts about shift operator) shift operator for continued fractions is the (A diagrammatic representation of the earth's surface (or part of it)) map called the Gauss map, which lops off digits of a continued fraction expansion:.
	www.absoluteastronomy.com /encyclopedia/c/co/continued_fraction.htm (2074 words)

	Generalized continued fraction - Wikipedia, the free encyclopedia
		In mathematics, a generalized continued fraction is a generalization of the concept of continued fraction in which the numerators are allowed to differ from unity.
		If the positive sign is chosen, then (as for ordinary continued fractions) all convergents of odd order are greater than x but uniformly decrease; and all convergents of even order are less than x but uniformly increase.
		For example, there is a close relationship between the continued fraction for the irrational real number α, and the way lattice points in two dimensions lie to either side of the line y = αx.
	en.wikipedia.org /wiki/Generalized_continued_fraction (368 words)

	Encyclopedia: Continued fraction
		In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers.
		One may also define infinite continued fractions as limits: In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity.
		While one cannot discern any pattern in the infinite continued fraction expansion of Ï, this is not true for e, the base of the natural logarithm: The mathematical constant e is the base of the natural logarithm function.
	www.nationmaster.com /encyclopedia/Continued-fraction (3637 words)

	Continued fraction - Encyclopedia, History, Geography and Biography
		The semiconvergents to the continued fraction expansion of a real number $x$ include all the rational approximations which are better than any approximation with a smaller denominator.
		While one cannot discern any pattern in the infinite continued fraction expansion of π, this is not true for e, the base of the natural logarithm: e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10,...].
		The backwards shift operator for continued fractions is the map $h(x)=1/x - \lfloor 1/x \rfloor,$ called the Gauss map, which lops off digits of a continued fraction expansion: $h([0;a_1,a_2,a_3,...]) = [0;a_2,a_3,...]$ .
	www.arikah.net /encyclopedia/Continued_fraction (2403 words)

	Continued fraction
		Truncating the continued fraction representation of a number x early yields a rational approximation for x which is in a certain sense the "best possible" rational approximation.
		Then the continued fraction representation of r is [0; i,...], where "..." is the continued fraction representation of f, which is also between 0 and 1.
		Every finite continued fraction is rational, and every rational number can be represented in precisely two different ways as a finite continued fraction (in one representation the final term in the continued fraction is 1; in the other, shorter, representation the final term is greater than 1).
	www.sciencedaily.com /encyclopedia/continued_fraction (1777 words)

	Continued fraction (Site not responding. Last check: 2007-09-29)
		The continued fraction representation of a rational number is almost unique: there are exactly two representations for everyrational number, which are exactly the same except that one ends with...
		Every finite continued fraction is rational, and every rationalnumber can be represented in precisely two different ways as a finite continued fraction (in one representation the final term inthe continued fraction is 1; in the other, shorter, representation the final term is greater than 1).
		The first term, as we see, is the first fraction; the first and second together give the second fraction, 22/7; the first, thesecond and the third give the third fraction 333/106, and so on with the rest; the result being that the series entire isequivalent to the original value.
	www.therfcc.org /continued-fraction-19431.html (1667 words)

	Numbers and Functions as Continued Fractions - Numericana
		An Introduction to Continued Fractions by Ron Knott of the University of Surrey.
		The ellipsis (...) indicates that the expression is to be continued indefinitely.
		Continued fractions for which the sequence of partial quotients is ultimately periodic are called periodic continued fractions and they correspond to quadratic irrationals [also called algebraic numbers of degree 2, these are irrational roots of polynomials of degree 2 with integral coefficients].
	home.att.net /~numericana/answer/fractions.htm (3614 words)

	Generalized continued fraction -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-29)
		If the positive sign is chosen, then (as for ordinary continued fractions) all convergents of odd order are greater than but uniformly decrease; and all convergents of even order are less than but uniformly increase.
		Then if and integer such that implies, then the limits are equal and the continued fraction has a definite value.
		For example, there is a close relationship between the continued fraction for the irrational real number α, and the way (additional info and facts about lattice point) lattice points in two dimensions lie to either side of the line y = αx.
	www.absoluteastronomy.com /encyclopedia/G/Ge/Generalized_continued_fraction.htm (348 words)

	Continued Fractions and Modular Forms
		However for continued fractions, two distinct cases have to be considered: the continuous and the discrete case.
		The main object of this lecture is the alternating sum of coefficients of a continued fraction.
		The alternating sum of coefficients of a continued fraction seems to be the first example where one needs not only upper bounds for sums of Kloosterman sums, but also their precise asymptotics.
	algo.inria.fr /seminars/sem99-00/vardi.html (1676 words)

	Continued Fractions - History
		The origin of continued fractions is traditionally placed at the time of the creation of Euclid's Algorithm.
		Continued fractions have also been utilized within computer algorithms for computing rational approximations to real numbers, as well as solving indeterminate equations.
		This brief sketch into the past of continued fractions is intended to provide an overview of the development of this field.
	archives.math.utk.edu /articles/atuyl/confrac/history.html (933 words)

	The Fifth Arithmetical Operation roots solving householder newton halley bernoulli mean arithmetic harmonic mean ratio ... (Site not responding. Last check: 2007-09-29)
		= -1, the generalized continued fraction expression for this root is:
		As we have seen in the above numerical examples, when trying to represent a cubic irrational by means of a second order continued fraction (traditional concept) then one get a disfigured image of the irrational.
		It is necessary to redefine the traditional representation of irrational numbers by means of continued fractions.
	mipagina.cantv.net /arithmetic/gencontfrac.htm (422 words)

	Continued Education (Site not responding. Last check: 2007-09-29)
		Continued fraction 1: In mathematics, a '''continued fraction''' is an expression such as 7: nity, the resulting expression is a generalized continued fraction.
		Generalized continued fraction 1: action''' is a generalization of the concept of continued fraction in which the numerators are allowed to 3: A generalized continued fraction is an expression such as: 14: nvergents are formed in a similar way to those of continued fractions.
		If all $\pm$ signs are pos 27: he positive sign is chosen, then (as for ordinary continued fractions) all convergents of odd order are great 29: The condition for the limits to be equal (and the continued fraction to have a definite value) is
	www.witchware.com /File/15001-Continued.Education.Html (518 words)

	This note's for you
		That is to say, this particular continued fraction does indeed converge to the irrational number it is supposed to represent.
		The continued fraction expansion for log[2](3) is [1,1,1,2,2,3,1,5,2,23,2,2,1,...] (see sequence A028507 of the On-Line Encyclopedia of Integer Sequences for more terms).
		The continued fraction expansion is [2, 3, 9, 2, 2, 4, 6, 2, 1, 1, 3, 1, 18] The convergents are: 7/3, 65/28, 137/59, 339/146, 1493/643,...
	www.research.att.com /~njas/sequences/DUNNE/TEMPERAMENT.HTML (3574 words)

	Citebase - Continued fractions and generalized patterns (Site not responding. Last check: 2007-09-29)
		Authors: Mansour, T. In [BS] Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation.
		Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation.
		We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of `forbidden' patterns, that seems to indicate that the enumerating sequence is always P-recursive.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0110037 (930 words)

	Generalized continued fraction (Site not responding. Last check: 2007-09-29)
		In mathematics, a generalized continued fraction is ageneralization of the concept of continued fraction in whichthe numerators are allowed to differ from unity.
		If the positive sign is chosen, then (as for ordinary continued fractions) all convergents of odd order are greater thanx but uniformly decrease; and all convergents of even order are less than x but uniformly increase.
		The condition for the limits to be equal (and the continued fraction to have adefinite value) is
	www.therfcc.org /generalized-continued-fraction-177417.html (167 words)

	Atlas: The proper generalization of the continued fraction by Alexander Bruno (Site not responding. Last check: 2007-09-29)
		Atlas: The proper generalization of the continued fraction by Alexander Bruno
		The algorithm of computation of the usual continued fraction of a number has several fine properties.
		The proposed generalization of the continued fraction is a motion along the surface of the convex hull [1, 3].
	atlas-conferences.com /c/a/l/z/41.htm (266 words)

	[No title]
		xciv / civ On a theorem of Legendre in the theory of continued fractions.
		#175 / m-01-pdf Worpitzky's theorem on continued fractions.
		xcv / lxii4 A generalization of Shiokawa?s rational approximation to the Rogers-Ramanujan continued fraction.
	www.geocities.com /furmend/Ba_Bz.txt (7631 words)

	Ae Ja Yee Research
		On the generalized Rogers-Ramanujan continued fraction (with B.C. Berndt), Ramanujan J., 7 (2003), 321-331, (dvi), (pdf).
		Continued fractions with three limit points (with G. Andrews, B. Berndt, J. Sohn, and A. Zaharescu), Adv.
		Overpartitions and generating functions for generalized Frobenius partitions (with S. Corteel and J. Lovejoy), Trends in Mathematics (Birkhauser), to appear, (pdf).
	www.math.psu.edu /yee/research.html (414 words)

	Continued Fractions and 2D Hurwitz Polynomials -- from Mathematica Information Center
		A test based on continued fraction expansion for polynomials with complex coefficients decides whether the polynomial has all its roots in the left half-plane.
		The main result is a new test for polynomials in two variables and new algorithms testing necessary conditions of stability for these polynomials.
		They also show the strength of the continued fraction techniques and the role of positive functions in many areas of system theory.
	library.wolfram.com /infocenter/Articles/3634 (139 words)

	1991 Technical Reports \| SCS \| UW
		A generalized continued fraction algorithm associates with every real number x a sequence of integers; x is rational iff the sequence is finite.
		In this report I will give proofs of some simple theorems concerning continued fractions that are known to the cognoscenti, but for which proofs in the literature seem to be missing, incomplete, or hard to locate.
		Planning is the process of generating sequences of actions in order to provide a method for agents to modify the state of the world in which they exist.
	www.cs.uwaterloo.ca /research/tr/1991 (3476 words)

	Atlas: Empires in Sturmian Systems by Chris Hillman
		Empires were introduced by J. Conway in the context of Penrose tilings, but in fact this concept is quite general.
		In a subshift X, the cylinder Z(\alpha) of a word \alpha is the set of all sequences in which alpha appears.
		Here, a p-flat x+W is a translate of a p-dimensional subspace W and a digital approximation to x+W is a surface of p-dimensional facets with vertices in Z^(p+q), which remains "close" to x+W. The shifts X(v) arise as digital approximations to lines in R^2.
	atlas-conferences.com /cgi-bin/abstract/cabe-18 (377 words)

	Continued fraction transformations, Brjuno functions and BMO spaces - Marmi, Moussa, Yoccoz (ResearchIndex) (Site not responding. Last check: 2007-09-29)
		We analyse the relation between these functions and the various continued fraction transformations, and display the functional equation which is fulfilled by these highly singular functions.
		Marmi, P.Moussa, J.-C. Yoccoz, "Continued fraction transformations, Brjuno functions, and BMO spaces", Preprint SPh-T, C. Saclay, (1995).
		2 the invariant measures and the entropies for continued fract..
	citeseer.ist.psu.edu /marmi95continued.html (483 words)

	dynamical and control systems '03
		First, to general geodesic flows on manifolds, to show the existence of trajectories with unbounded energy when a generic quasi-periodic potential is added to the system.
		The diophantine condition to be used is formulated in terms of the growth rate of the continued fraction and is quite explicit and the loss of differentiability is optimal.
		In this talk we use Lyapunov s direct method to characterize the limit set of a nonlinear continuous semigroup in a Banach space.A sufficient condition for partial asymptotic stability of the equilibrium with respect to a continuous functional is proved.
	www.sissa.it /fa/workshop_old/DCS2003/Program.html (4178 words)

	Chapter IV (Site not responding. Last check: 2007-09-29)
		Break the continuity of the line by piercing it at the point 1/2, that is, remove the point 1/2 from the number line.
		The purpose of this sub-section is to clarify the principle of nested segments, to be used for the interpretation of non-terminating continued fractions.
		If the continued fraction is non-terminating, the broken line does not have the last segment the end of which lies on the dashed horizontal line.
	kr.cs.ait.ac.th /~radok/math/mat4/m44.htm (5798 words)

	Communications in the Analytic Theory of Continued Fractions (CATV)
		Pade approximants, orthogonal polynomials, recurrence relations, moments, linear fractional transformations, and generalizations of classical continued fractions are but a few examples of associated research areas within the scope of the publication.
		Abstracts, reviews, and papers that show connections with continued fraction theory are especially welcome, although this not a requirement for acceptance and publication.
		In general, it is appropriate to compose abstracts that do not exceed one tenth the length of the article, and contain no proofs.
	www.whitworth.edu /Academic/Department/MathComputerScience/research/CATCF.htm (2009 words)

	Category:Number theory - RSCI, The Science Classification Index (Site not responding. Last check: 2007-09-29)
		Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers and contains many open problems that are easily understood even by non-mathematicians.
		More generally, the field has come to be concerned with a wider class of problems that arose naturally from the study of integers.
		Number theory may be subdivided into several fields according to the methods used and the questions investigated.
	www.scindex.org /Category:Number_theory.html (136 words)

	continued fraction Resources
		The goal of this site is to provide a brief introduction to the field of continued fractions for...
		Using jigsaw puzzles to introduce the Continued Fraction, the simplest continued fraction is for Phi - the golden section; how continued fractions...
		the algorithm, one can derive the simple continued fraction of the rational p/q as opposed to...
	www.math-plans.com /directory/continued-fraction.html (555 words)

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