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| {{mergefrom|AGM method|date=September 2012}}
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| In [[mathematics]], the '''arithmetic–geometric mean (AGM)''' of two positive [[real number]]s {{math|''x''}} and {{math|''y''}} is defined as follows:
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| First compute the [[arithmetic mean]] of {{math|''x''}} and {{math|''y''}} and call it {{math|''a''<sub>1</sub>}}. Next compute the [[geometric mean]] of {{math|''x''}} and {{math|''y''}} and call it {{math|''g''<sub>1</sub>}}; this is the [[square root]] of the product {{math|''xy''}}:
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| :<math>\begin{align}
| | "The supposition is that he was saying it in a good cause," Simons says.<br>Richard HalliburtonThe Glorious Adventrue.<br>28 23:56 CABAS VANESSA BRUNO Kuchyne Bednr Kuchyne Bednr UNREGISTERED VERSION WOMENS RAY BAN SUNGLASSES 2013.<br>On paper, "One" seems like a pretty standard post Robyn piece of robotic synth pop, but since it cuts out all the chaff, it's pretty much [http://tinyurl.com/lmk4zx5 cheap dre beats] an immediate, seemingly unending fun spiral.<br>私は彼は、彼のキャビネットに連れて行ってくれたことは幸運だったと思う.<br><br>Here is more info on [http://tinyurl.com/lmk4zx5 cheap dre beats] have a look at our web-page. |
| a_1 &= \frac{1}{2}(x + y)\\
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| g_1 &= \sqrt{xy}
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| \end{align}</math>
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| Then iterate this operation with {{math|''a''<sub>1</sub>}} taking the place of {{math|''x''}} and {{math|''g''<sub>1</sub>}} taking the place of {{math|''y''}}. In this way, two [[sequence]]s {{math|(''a''<sub>''n''</sub>)}} and {{math|(''g''<sub>''n''</sub>)}} are defined:
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| :<math>\begin{align}
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| a_{n+1} &= \frac{1}{2}(a_n + g_n)\\
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| g_{n+1} &= \sqrt{a_n g_n}
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| \end{align}</math>
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| These two sequences [[limit of a sequence|converge]] to the same number, which is the '''arithmetic–geometric mean''' of {{math|''x''}} and {{math|''y''}}; it is denoted by {{math|''M''(''x'', ''y'')}}, or sometimes by {{math|agm(''x'', ''y'')}}.
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| This can be used for algorithmic purposes as in the [[AGM method]].
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| ==Example==
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| To find the arithmetic–geometric mean of {{math|''a''<sub>0</sub> {{=}} 24}} and {{math|''g''<sub>0</sub> {{=}} 6}}, first calculate their arithmetic mean and geometric mean, thus:
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| :<math>\begin{align}
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| a_1 &= \frac{1}{2}(24 + 6) = 15\\
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| g_1 &= \sqrt{24 \times 6} = 12
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| \end{align}</math>
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| and then iterate as follows:
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| :<math>\begin{align}
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| a_2 &= \frac{1}{2}(15 + 12) = 13.5\\
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| g_2 &= \sqrt{15 \times 12} = 13.41640786500\dots\\
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| \dots
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| \end{align}</math>
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| The first five iterations give the following values:
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| :{| class="wikitable"
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| |-
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| ! {{math|''n''}}
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| ! {{math|''a''<sub>''n''</sub>}}
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| ! {{math|''g''<sub>''n''</sub>}}
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| |-
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| | 0
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| | 24
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| | 6
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| |-
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| | 1
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| | {{underline|1}}5
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| | {{underline|1}}2
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| |-
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| | 2
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| | {{underline|13}}.5
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| | {{underline|13}}.416407864998738178455042…
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| |-
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| | 3
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| | {{underline|13.458}}203932499369089227521…
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| | {{underline|13.458}}139030990984877207090…
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| |-
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| | 4
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| | {{underline|13.4581714817}}45176983217305…
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| | {{underline|13.4581714817}}06053858316334…
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| |-
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| | 5
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| | {{underline|13.4581714817256154207668}}20…
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| | {{underline|13.4581714817256154207668}}06…
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| |}
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| As can be seen, the number of digits in agreement (underlined) approximately doubles with each iteration. The arithmetic–geometric mean of 24 and 6 is the common limit of these two sequences, which is approximately 13.4581714817256154207668131569743992430538388544.<ref>[http://www.wolframalpha.com/input/?i=agm%2824%2C+6%29 agm(24, 6) at WolframAlpha]</ref>
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| == History ==
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| The first algorithm based on this sequence pair appeared in the works of [[Adrien-Marie Legendre|Legendre]]. Its properties were further analyzed by [[Gauss]].<ref name="BerggrenBorwein2004">{{cite book|editor=J.L. Berggren, Jonathan M. Borwein, Peter Borwein|title=Pi: A Source Book|url=http://books.google.com/books?id=QlbzjN_5pDoC&pg=PA481|year=2004|publisher=Springer|isbn=978-0-387-20571-7|page=481|chapter=The Arithmetic-Geometric Mean of Gauss|author=David A. Cox}} first published in ''[[L'Enseignement Mathématique]]'', t. 30 (1984), p. 275-330</ref>
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| ==Properties==
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| The geometric mean of two positive numbers is never bigger than the arithmetic mean (see [[inequality of arithmetic and geometric means]]); as a consequence, {{math|(''g<sub>n</sub>'')}} is an increasing sequence, {{math|(''a<sub>n</sub>'')}} is a decreasing sequence, and {{math|''g<sub>n</sub>'' ≤ ''M''(''x'', ''y'') ≤ ''a<sub>n</sub>''}}. These are strict inequalities if {{math|''x'' ≠ ''y''}}.
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| {{math|''M''(''x'', ''y'')}} is thus a number between the geometric and arithmetic mean of {{math|''x''}} and {{math|''y''}}; in particular it is between {{math|''x''}} and {{math|''y''}}.
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| If {{math|''r'' ≥ 0}}, then {{math|''M''(''rx'',''ry'') {{=}} ''r M''(''x'',''y'')}}.
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| There is an integral-form expression for {{math|''M''(''x'',''y'')}}:
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| :<math>\begin{align}M(x,y) &= \frac\pi2\bigg/\int_0^{\pi/2}\frac{d\theta}{\sqrt{x^2\cos^2\theta+y^2\sin^2\theta}}\\
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| &=\frac{\pi}{4} (x + y) \bigg/ K\left( \frac{x - y}{x + y} \right)
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| \end{align}</math>
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| where {{math|''K''(''k'')}} is the [[elliptic integral|complete elliptic integral of the first kind]]:
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| :<math>K(k) = \int_0^{\pi/2}\frac{d\theta}{\sqrt{1 - k^2\sin^2(\theta)}} </math>
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| Indeed, since the arithmetic–geometric process converges so quickly, it provides an effective way to compute elliptic integrals via this formula. In engineering, it is used for instance in [[elliptic filter]] design.<ref name="Dimopoulos2011">{{cite book|author=Hercules G. Dimopoulos|title=Analog Electronic Filters: Theory, Design and Synthesis|url=http://books.google.com/books?id=6W1eX4QwtyYC&pg=PA147|year=2011|publisher=Springer|isbn=978-94-007-2189-0|pages=147–155}}</ref>
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| == Related concepts ==
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| The reciprocal of the arithmetic–geometric mean of 1 and the [[square root of 2]] is called [[Gauss's constant]], after [[Carl Friedrich Gauss]].
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| :<math>\frac{1}{M(1, \sqrt{2})} = G = 0.8346268\dots</math>
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| The [[geometric–harmonic mean]] can be calculated by an analogous method, using sequences of geometric and [[harmonic mean|harmonic]] means. The arithmetic–harmonic mean can be similarly defined, but takes the same value as the [[geometric mean]].
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| The arithmetic-geometric mean can be used to compute [[Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integrals of the first kind]]. A modified arithmetic-geometric mean can be used to efficiently compute [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integrals of the second kind]].<ref>{{Citation |last=Adlaj |first=Semjon |title=An eloquent formula for the perimeter of an ellipse |url=http://www.ams.org/notices/201208/rtx120801094p.pdf |journal=Notices of the AMS |volume=59 |issue=8 |pages=1094–1099 |date=September 2012 |doi=10.1090/noti879 |accessdate=2013-12-14}}</ref>
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| ==Proof of existence==
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| From [[inequality of arithmetic and geometric means]] we can conclude that:
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| :<math>g_n \leqslant a_n</math>
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| and thus
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| :<math>g_{n + 1} = \sqrt{g_n \cdot a_n} \geqslant \sqrt{g_n \cdot g_n} = g_n</math>
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| that is, the sequence {{math|''g<sub>n</sub>''}} is nondecreasing.
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| Furthermore, it is easy to see that it is also bounded above by the larger of {{math|''x''}} and {{math|''y''}} (which follows from the fact that both arithmetic and geometric means of two numbers both lie between them). Thus by the [[monotone convergence theorem]] the sequence is convergent, so there exists a {{math|''g''}} such that:
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| :<math>\lim_{n\to \infty}g_n = g</math>
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| However, we can also see that:
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| :<math>a_n = \frac{g_{n + 1}^2}{g_n}</math>
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| and so:
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| :<math>\lim_{n\to \infty}a_n = \lim_{n\to \infty}\frac{g_{n + 1}^2}{g_{n}} = \frac{g^2}{g} = g</math>
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| [[Q.E.D.]]
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| ==Proof of the integral-form expression==
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| This proof is given by Gauss.<ref name="BerggrenBorwein2004" />
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| Let
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| :<math>I(x,y) = \int_0^{\pi/2}\frac{d\theta}{\sqrt{x^2\cos^2\theta+y^2\sin^2\theta}},</math>
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| Changing the variable of integration to <math>\theta'</math>, where
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| :<math> \sin\theta = \frac{2x\sin\theta'}{(x+y)+(x-y)\sin^2\theta'}, </math>
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| gives
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| :<math>
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| \begin{align}
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| I(x,y) &= \int_0^{\pi/2}\frac{d\theta'}{\sqrt{\bigl(\frac12(x+y)\bigr)^2\cos^2\theta'+\bigl(\sqrt{xy}\bigr)^2\sin^2\theta'}}\\
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| &= I\bigl(\tfrac12(x+y),\sqrt{xy}\bigr).
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| \end{align}
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| </math>
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| Thus, we have
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| :<math>
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| \begin{align}
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| I(x,y) &= I(a_1, g_1) = I(a_2, g_2) = \cdots\\
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| &= I\bigl(M(x,y),M(x,y)\bigr) = \pi/\bigr(2M(x,y)\bigl).
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| \end{align}
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| </math>
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| The last equality comes from observing that <math>I(z,z) = \pi/(2z)</math>.
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| Finally, we obtain the desired result
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| :<math>M(x,y) = \pi/\bigl(2 I(x,y) \bigr). </math>
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| ==See also==
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| * [[Generalized mean]]
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| * [[Inequality of arithmetic and geometric means]]
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| * [[Gauss–Legendre algorithm]]
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| ==External links==
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| * [http://arithmeticgeometricmean.blogspot.de/ Arithmetic-Geometric Mean Calculator]
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| * [http://planetmath.org/convergenceofarithmeticgeometricmean/ Proof of convergence rate in PlanetMath]
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| ==References==
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| {{More footnotes|date=October 2008}}
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| *{{cite journal|last = Adlaj|first = Semjon|title = An eloquent formula for the perimeter of an ellipse|journal = Notices of the AMS|volume = 59|issue = 8|pages = 1094–1099|date = September 2012|url = http://www.ams.org/notices/201208/rtx120801094p.pdf}}
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| * [[Jonathan Borwein]], [[Peter Borwein]], ''Pi and the AGM. A study in analytic number theory and computational complexity.'' Reprint of the 1987 original. Canadian Mathematical Society Series of Monographs and Advanced Texts, 4. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1998. xvi+414 pp. ISBN 0-471-31515-X {{MR|1641658}}
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| * [[Zoltán Daróczy]], [[Zsolt Páles]], ''Gauss-composition of means and the solution of the Matkowski–Suto problem.'' Publ. Math. Debrecen 61/1-2 (2002), 157–218.
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| *{{SpringerEOM|author=M. Hazewinkel|title=Arithmetic–geometric mean process|urlname=a/a130280}}
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| *{{mathworld|urlname=Arithmetic-GeometricMean|title=Arithmetic–Geometric mean}}
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| <references />
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| {{DEFAULTSORT:Arithmetic-Geometric Mean}}
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| [[Category:Means]]
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| [[Category:Special functions]]
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| [[Category:Elliptic functions]]
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| [[Category:Articles containing proofs]]
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"The supposition is that he was saying it in a good cause," Simons says.
Richard HalliburtonThe Glorious Adventrue.
28 23:56 CABAS VANESSA BRUNO Kuchyne Bednr Kuchyne Bednr UNREGISTERED VERSION WOMENS RAY BAN SUNGLASSES 2013.
On paper, "One" seems like a pretty standard post Robyn piece of robotic synth pop, but since it cuts out all the chaff, it's pretty much cheap dre beats an immediate, seemingly unending fun spiral.
私は彼は、彼のキャビネットに連れて行ってくれたことは幸運だったと思う.
Here is more info on cheap dre beats have a look at our web-page.