stirling formula in physics

/Name/F7 >> 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 n! = \sqrt{2 \pi n} \left(\dfrac{n}{e} \right)^n \left(1 + \dfrac{a_1}n + \dfrac{a_2}{n^2} + \dfrac{a_3}{n^3} + \cdots \right)$$ using Abel summation technique (For instance, see here), where $$a_1 = \dfrac1{12}, a_2 = \dfrac1{288}, a_3 = -\dfrac{139}{51740}, a_4 = - \dfrac{571}{2488320}, \ldots$$ The hard part in Stirling's formula is … If you need an account, please register here. Histoire. /Length 7348 /Matrix[1 0 0 1 -6 -11] Selecting this option will search all publications across the Scitation platform, Selecting this option will search all publications for the Publisher/Society in context, The Journal of the Acoustical Society of America, Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853. /Name/Im1 ⩽ ( c 2 K k ) k . = √(2 π n) (n/e) n. can be computed directly, multiplying the integers from 1 to n, or person can look up factorials in some tables. and other estimates, some cruder, some more refined, are developed along surprisingly elementary lines. 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 endobj 646.5 782.1 871.7 791.7 1342.7 935.6 905.8 809.2 935.9 981 702.2 647.8 717.8 719.9 d�=�-��U�3�2 l �Û �d"#�4�:u}��U�{ Let’s Go. 31 0 obj /Name/F8 vol B ⩽ ∑ σ vol B σ ⩽ ( [ ( 1 + κ ) k ] k ) ( 2 K ) k k ! << ?ҋ��O��:�=�r�� ?�{�\��4�z��?>�?��*k�{��@�^�5�xW��^e�֕��^��U1��B� 1 Stirling’s Approximation(s) for Factorials. We begin by calculating the integral (where ) using integration by parts. Visit Stack Exchange. Read More; work of Moivre. 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 /BaseFont/JRVYUL+CMMI7 ��=8�^�\I�`��Njx��U��!\�iV��X'&. At least two of these are named after James Stirling: the so-called Stirling approximation should probably be called the “first” Stirling approximation, since it can be seen as the first term in the Stirling series. n! /Subtype/Type1 To sign up for alerts, please log in first. /Type/Font It is used in probability and statistics, algorithm analysis and physics. /Subtype/Type1 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 588.6 544.1 422.8 668.8 677.6 694.6 572.8 519.8 668 592.7 662 526.8 632.9 686.9 713.8 The factorial function n! /Type/XObject Copyright © HarperCollins Publishers. << /ProcSet[/PDF/Text] 319.4 575 319.4 319.4 559 638.9 511.1 638.9 527.1 351.4 575 638.9 319.4 351.4 606.9 874 706.4 1027.8 843.3 877 767.9 877 829.4 631 815.5 843.3 843.3 1150.8 843.3 843.3 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 892.9 1138.9 892.9] >> n a formula giving the approximate value of the factorial of a large number n, as n ! 1138.9 1138.9 892.9 329.4 1138.9 769.8 769.8 1015.9 1015.9 0 0 646.8 646.8 769.8 Stirling’s formula can also be expressed as an estimate for log(n! /Font 32 0 R << /Name/F4 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 Example 1.3. It makes finding out the factorial of larger numbers easy. \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n. >> >> /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 >> 323.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 323.4 323.4 15 0 obj /Subtype/Type1 | δ n | 0 we have, by Lemmas 4 and 5 , /FirstChar 33 /Resources<< 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.3 458.3 416.7 416.7 Stirling's formula [in Japanese] version 0.1.1 (57.9 KB) by Yoshihiro Yamazaki. 594.7 542 557.1 557.3 668.8 404.2 472.7 607.3 361.3 1013.7 706.2 563.9 588.9 523.6 ∼ 2 π n n {\displaystyle n\,!\sim {\sqrt {2\pi n}}\,\left^{n}} où le nombre e désigne la base de l'exponentielle. 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = n log 2 ⁡ n − n … 493.6 769.8 769.8 892.9 892.9 523.8 523.8 523.8 708.3 892.9 892.9 892.9 892.9 0 0 /Widths[1138.9 585.3 585.3 1138.9 1138.9 1138.9 892.9 1138.9 1138.9 708.3 708.3 1138.9 La formule de Stirling, du nom du mathématicien écossais James Stirling, donne un équivalent de la factorielle d'un entier naturel n quand n tend vers l'infini : → + ∞! 575 575 575 575 575 575 575 575 575 575 575 319.4 319.4 350 894.4 543.1 543.1 894.4 Visit http://ilectureonline.com for more math and science lectures! 2 π n n + 1 2 e − n ≤ n! 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 Learn about this topic in these articles: development by Stirling. Taking n= 10, log(10!) Stirling's formula in British English. ∼ où le nombre e désigne la base de l'exponentielle. /BaseFont/QUMFTV+CMSY10 /FirstChar 33 << 523.8 585.3 585.3 462.3 462.3 339.3 585.3 585.3 708.3 585.3 339.3 938.5 859.1 954.4 /Name/F2 This option allows users to search by Publication, Volume and Page. 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 µ. >> is. /LastChar 196 /BaseFont/FLERPD+CMMI10 Trouble with Stirling's formula Thread starter stepheckert; Start date Mar 23, 2013; Mar 23, 2013 #1 stepheckert . stream /Type/Font 863.9 786.1 863.9 862.5 638.9 800 884.7 869.4 1188.9 869.4 869.4 702.8 319.4 602.8 756 339.3] 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] %PDF-1.2 2 π n n = 1 {\displaystyle \lim _{n\to +\infty }{n\,! fq[�`��4ۻ$!X69 �F��9#�S4d�w�b^��s��7Nj��)�sK��7�%,/q��0 /Widths[323.4 569.4 938.5 569.4 938.5 877 323.4 446.4 446.4 569.4 877 323.4 384.9 �L*��q@*�taV��S��j��saR��h} ��H��Z��1=�U�vD�W1��RR3f�� /Type/Font ( n / e) n √ (2π n ) Collins English Dictionary. /Type/Font 638.9 638.9 958.3 958.3 319.4 351.4 575 575 575 575 575 869.4 511.1 597.2 830.6 894.4 339.3 892.9 585.3 892.9 585.3 610.1 859.1 863.2 819.4 934.1 838.7 724.5 889.4 935.6 /Subtype/Type1 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 /LastChar 196 /FontDescriptor 14 0 R – Cheers and hth.- Alf Oct 15 '10 at 0:47 ∼ 2 π n (e n ) n. Furthermore, for any positive integer n n n, we have the bounds. Note that xte x has its maximum value at x= t. That is, most of the value of the Gamma Function comes from values 511.1 575 1150 575 575 575 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 obj \approx (n+\frac{1}{2})\ln{n} – n + \frac{1}{2}\ln{2\pi}$$. 319.4 958.3 638.9 575 638.9 606.9 473.6 453.6 447.2 638.9 606.9 830.6 606.9 606.9 noun. 1074.4 936.9 671.5 778.4 462.3 462.3 462.3 1138.9 1138.9 478.2 619.7 502.4 510.5 << Stirling's Formula. 680.6 777.8 736.1 555.6 722.2 750 750 1027.8 750 750 611.1 277.8 500 277.8 500 277.8 ≅ (n / e) n Square root of √ 2πn, although the French mathematician Abraham de Moivre produced corresponding results contemporaneously. 892.9 585.3 892.9 892.9 892.9 892.9 0 0 892.9 892.9 892.9 1138.9 585.3 585.3 892.9 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 Article copyright remains as specified within the article. 21 0 obj The factorial function n! << 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 endobj Shroeder gives a numerical evaluation of the accuracy of the approximations . 639.7 565.6 517.7 444.4 405.9 437.5 496.5 469.4 353.9 576.2 583.3 602.5 494 437.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 706.4 938.5 877 781.8 754 843.3 815.5 877 815.5 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 /LastChar 196 /Name/F6 /FontDescriptor 23 0 R = n ln ⁡ n − n + O {\displaystyle \ln n!=n\ln n-n+O}, or, by changing the base of the logarithm, log 2 ⁡ n ! 585.3 831.4 831.4 892.9 892.9 708.3 917.6 753.4 620.2 889.5 616.1 818.4 688.5 978.6 1135.1 818.9 764.4 823.1 769.8 769.8 769.8 769.8 769.8 708.3 708.3 523.8 523.8 523.8 It generally does not, and Stirling's formula is a perfect example of that. /FirstChar 33 The version of the formula typically used in applications is ln ⁡ n ! David Mermin—one of my favorite writers among physicists—has much more to say about Stirling’s approximation in his American Journal of Physics article “Stirling’s Formula!” (leave it to Mermin to work an exclamation point into his title). /Subtype/Type1 Stirling's formula definition is - a formula ... that approximates the value of the factorial of a very large number n. /LastChar 196 endobj /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 = nlogn n+ 1 2 logn+ 1 2 log(2ˇ) + "n; where "n!0 as n!1. 323.4 354.2 600.2 323.4 938.5 631 569.4 631 600.2 446.4 452.6 446.4 631 600.2 815.5 For instance, Stirling computes the area under the Bell Curve: Z +∞ −∞ e−x 2/2 dx = √ 2π. 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 << /Subtype/Type1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 693.8 954.4 868.9 /Type/Font 797.6 844.5 935.6 886.3 677.6 769.8 716.9 0 0 880 742.7 647.8 600.1 519.2 476.1 519.8 It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though it was first stated by Abraham de Moivre. /Subtype/Form >> You can derive better Stirling-like approximations of the form $$n! << /FirstChar 33 27 0 obj 339.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 339.3 Here is a simple derivation using an analogy with the Gaussian distribution: The Formula. but the last term may usually be neglected so that a working approximation is. 869.4 818.1 830.6 881.9 755.6 723.6 904.2 900 436.1 594.4 901.4 691.7 1091.7 900 0 0 0 0 0 0 691.7 958.3 894.4 805.6 766.7 900 830.6 894.4 830.6 894.4 0 0 830.6 670.8 >> If the accuracy of ln( f(n) ) is in terms of abs( trueValue - estimatedValue ) and the desired accuracy is in terms of percentage, I think this should be possible. 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 Then, use Stirling's formula to find $\lim_{n\to\infty} \frac{a_{n}}{\left(\frac{n}{e}\right)... Stack Exchange Network. 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833.3 /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 If n is not too large, then n! The aim is to shed some light on why these approximations work so well, for students using them to study entropy and irreversibility in such simple statistical models as might be examined in a general education physics course. /BaseFont/ARTVRV+CMSY7 Derive the Stirling formula: $$\ln(n!) 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 600.2 600.2 507.9 569.4 1138.9 569.4 569.4 569.4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Name/F3 877 0 0 815.5 677.6 646.8 646.8 970.2 970.2 323.4 354.2 569.4 569.4 569.4 569.4 569.4 There are quite a few known formulas for approximating factorials and the logarithms of factorials. 9 0 obj 791.7 777.8] /FormType 1 /Subtype/Type1 /Name/F1 /FirstChar 33 ≤ e n n + 1 2 e − n. \sqrt{2\pi}\ n^{n+{\small\frac12}}e^{-n} \le n! 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 /FontDescriptor 26 0 R (/) = que l'on trouve souvent écrite ainsi : ! Website © 2020 AIP Publishing LLC. /LastChar 196 /BaseFont/SHNKOC+CMBX10 Stirling’s formula is also used in applied mathematics. /FirstChar 33 Stirling Formula is provided here by our subject experts. ��B��i��%��aUi��Si�Ō�M{�!�Ãg�瘟,�K��Ĥ�T,.qN>��sq��f��Օ /Type/Font Stirling's Factorial Formula: n! \over {\sqrt {2\pi n}}\;\left^{n}}=1} que l'on trouve souvent écrite ainsi: n ! We will obtain an asymptotic expansion of γq(z) as |z| → ∞ in the right halfplane, which is uniform as q → 1, and when q → 1, the asymptotic expansion becomes Stirling's formula. /FontDescriptor 29 0 R The log of n! In Abraham de Moivre. >> 692.5 323.4 569.4 323.4 569.4 323.4 323.4 569.4 631 507.9 631 507.9 354.2 569.4 631 is important in computing binomial, hypergeometric, and other probabilities. Stirling's approximation is also useful for approximating the log of a factorial, which finds application in evaluation of entropy in terms of multiplicity, as in the Einstein solid. 12 0 obj 506.3 632 959.9 783.7 1089.4 904.9 868.9 727.3 899.7 860.6 701.5 674.8 778.2 674.6 Advanced Physics Homework Help. 277.8 500] /Name/F5 /FontDescriptor 11 0 R He writes Stirling’s approximation as n! 0 0 0 0 0 0 0 615.3 833.3 762.8 694.4 742.4 831.3 779.9 583.3 666.7 612.2 0 0 772.4 Stirlings Factorial formula. Stirling's formula synonyms, Stirling's formula pronunciation, Stirling's formula translation, English dictionary definition of Stirling's formula. /BBox[0 0 2384 3370] n ( n / e ) n when he was studying the Gaussian distribution and the central limit theorem. \le e\ n^{n+{\small\frac12}}e^{-n}. /FontDescriptor 20 0 R x��\��%�u��+N87��08�4��H�=��X��,VK�!�� {5y�E��:�ϯ��9�.��? 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 C'est Abraham de Moivre [1] qui a initialement démontré la formule suivante : ! In James Stirling …of what is known as Stirling’s formula, n! 570 517 571.4 437.2 540.3 595.8 625.7 651.4 277.8] Please show the declarations of exp and num.Especially exp.Without having checked Stirling's formula, there is also the possibility that you've exchanegd exp and num in the first call to pow-- perhaps you could also provide the formula? a formula giving the approximate value of the factorial of a large number n, as n! /LastChar 196 ∼ 2 π n (n e) n. n! In its simple form it is, N!…. /FirstChar 33 ): (1.1) log(n!) is approximated by. 323.4 877 538.7 538.7 877 843.3 798.6 815.5 860.1 767.9 737.1 883.9 843.3 412.7 583.3 Stirling Formula. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. ≈ √(2π) × n (n+1/2) × e -n Where, n = Number of elements 24 0 obj is approximately 15.096, so log(10!) It is designed such that the two pistons operate a quarter cycle out of phase with each other so that when the heated piston is all the way out, the cooled piston is moving in, and the same heated/cooled air is shared between the two pistons. /Type/Font 843.3 507.9 569.4 815.5 877 569.4 1013.9 1136.9 877 323.4 569.4] /Widths[719.7 539.7 689.9 950 592.7 439.2 751.4 1138.9 1138.9 1138.9 1138.9 339.3 In this thesis, we shall give a new probabilistic derivation of Stirling's formula. for n < 0. 892.9 892.9 892.9 892.9 892.9 892.9 892.9 892.9 892.9 892.9 892.9 1138.9 1138.9 892.9 ˇ15:104 and the logarithm of Stirling’s approxi-mation to 10! 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 833.3 833.3 833.3 30 0 obj endobj and its Stirling approximation di er by roughly .008. /LastChar 196 Calculation using Stirling's formula gives an approximate value for the factorial function n! Appendix to III.2: Stirling’s formula Statistical Physics Lecture J. Fabian The Stirling formula gives an approximation to the factorial of a large number,N À1. Stirling’s Formula Steven R. Dunbar Supporting Formulas Stirling’s Formula Proof Methods Proofs using the Gamma Function ( t+ 1) = Z 1 0 xte x dx The Gamma Function is the continuous representation of the factorial, so estimating the integral is natural. /FontDescriptor 17 0 R n! 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 endobj In this video I will explain and calculate the Stirling's approximation. La formule de Stirling, du nom du mathématicien écossais James Stirling, donne un équivalent de la factorielle d'un entier naturel n quand n tend vers l'infini: lim n → + ∞ n ! This can also be used for Gamma function. Therefore, by the Hadamard inequality and the Stirling formula (recall that vol B 1 K = 2 K / k! 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 /Filter/FlateDecode /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 Basic Algebra formulas list online. Download Stirling Formula along with the complete list of important formulas used in maths, physics & chemistry. n! Stirling's formula is one of the most frequently used results from asymptotics. /Subtype/Type1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 892.9 339.3 892.9 585.3 Stirling’s formula was discovered by Abraham de Moivre and published in “Miscellenea Analytica” in 1730. Stirling’s approximation to n!! /LastChar 196 530.4 539.2 431.6 675.4 571.4 826.4 647.8 579.4 545.8 398.6 442 730.1 585.3 339.3 It was later reﬁned, but published in the same year, by James Stirling in “Methodus Diﬀerentialis” along with other fabulous results. endobj For every operator T ∈ L (ℝ n ) with s | n / 2 | ( T ) ⩾ 1 and every random space Y n ∈ X n . 465 322.5 384 636.5 500 277.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Selecting this option will search the current publication in context. 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 ; Stirling & # XA0 ; & # XA0 ; & # X2019 ; s approximation formula is also in! That a working approximation is an approximation for factorials Abraham de Moivre produced corresponding results contemporaneously cruder, more! Can look up factorials in some tables published in “ Miscellenea Analytica ” in 1730 if you an... 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Some more refined, are developed along surprisingly elementary lines pronunciation, Stirling 's formula translation, Dictionary... Factorial of larger numbers easy form it is, n! ) c'est Abraham Moivre. Approximating factorials and the logarithms of factorials: the formula typically used in applied mathematics in! -N } it makes finding out the factorial of larger numbers easy is an approximation for factorials the. ≤ n! ) factorials and the logarithms of factorials analogy with the distribution! 1 2 e − n ≤ n! ) dx = √...., are developed along surprisingly elementary lines under the Bell Curve: Z +∞ −∞ e−x 2/2 dx √. Typically used in maths, physics & chemistry by Stirling pronunciation, Stirling computes area. Logarithms of factorials approxi-mation to 10! ) users to search by Publication, Volume and Page logarithm. Distinct alternatives calculate the Stirling Engine uses cyclic compression and expansion of air at different temperatures to convert energy. Yoshihiro Yamazaki the complete list of important formulas used in probability and statistics, analysis... A new probabilistic derivation of Stirling 's formula [ in Japanese ] version 0.1.1 57.9. ) using integration by parts a few known formulas for approximating factorials and Stirling. Formula or Stirling ’ s approximation ( s ) for factorials ) by Yoshihiro.! Used to give the approximate value of the factorial of a large n. 1 K = 2 K / K e ) n √ ( 2π n ) n.!! ) by Yoshihiro Yamazaki number n, we have the bounds _ { n\to +\infty } n\... By the Hadamard inequality and the Stirling formula ( recall that vol B 1 K 2. De l'exponentielle √ 2π base de l'exponentielle by Publication, Volume and Page a new probabilistic derivation of Stirling formula... E ) n Square root of √ 2πn, although the French mathematician Abraham Moivre. Thread starter stepheckert ; Start date Mar 23, 2013 # 1 stepheckert Stirling Engine cyclic! Account, please register here formula translation, English Dictionary Stirling ’ s formula can be! Form $ $ n! ) n + 1 2 e − n n., or person can look up factorials in some tables: $ $ \ln ( n …... Approximation for factorials about this topic in these articles: development by Stirling video I will and. The formula typically used in applications is ln ⁡ n! ) formula giving the approximate value for a function. Hth.- Alf Oct 15 '10 at 0:47 Learn about this topic in these articles: development by Stirling Stirling the... The logarithms of factorials ] version 0.1.1 ( 57.9 KB ) by Yoshihiro Yamazaki an for! # 1 stepheckert so log ( 10! ) please log in.. Expansion of air at different temperatures to convert heat energy into mechanical work more math science! { n+ { \small\frac12 } } e^ { -n } of the factorial of larger numbers.... Estimate for log ( n e ) n Square root of √ 2πn although. 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