"소모스 수열(Somos sequence)"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 22개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
| − | + | ==개요== | |
| − | * | + | * 점화식으로 정의되는 수열 |
| + | * 소모스 4,5,6,7 은 정수수열이며 소모스 8,9는 정수수열이 아니다 | ||
| + | * 점화식만으로는 정수수열이 되는가가 자명하지 않다 | ||
| + | * 정수수열이 되는가의 문제 (integrality) | ||
| + | * 합동식을 생각할 때의 주기성 문제 (periodicity modulo n) '''[Robinson1992]''' | ||
| + | * [[타원곡선]]론과 클러스터 대수(cluster algebra) 등의 이론에서 등장 | ||
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| − | + | ==소모스-4 수열== | |
| − | * | + | * <math>a_{n+4}a_{n} = a_{n+3} a_{n+1} + a_{n+2}^2</math> |
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| − | + | * [[소모스-4 수열]] | |
| − | + | ==소모스5- 수열== | |
| − | < | + | * <math>a_{n+5}a_{n} = a_{n+4} a_{n+1} + a_{n+3} a_{n+2}</math> |
| + | * 1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, 1249441, 9434290, 68570323, 1013908933 | ||
| − | + | # RecurrenceTable[{a[n] a[5 + n] == a[2 + n] a[3 + n] + a[1 + n] a[4 + n], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1}, a, {n, 20}] | |
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| − | # RecurrenceTable[{a[n] a[5 + n] == a[2 + n] a[3 + n] + a[1 + n] a[4 + n], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1}, a, | ||
* [http://www-groups.dcs.st-and.ac.uk/%7Ejohn/Zagier/Solution5.1.html http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Solution5.1.html] | * [http://www-groups.dcs.st-and.ac.uk/%7Ejohn/Zagier/Solution5.1.html http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Solution5.1.html] | ||
| − | * http://oeis.org/A006721 | + | * http://oeis.org/A006721 |
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| − | + | ==소모스-6 수열== | |
| − | * <math>a_{n+6}a_{n} = a_{n+5} a_{n+1} +a_{n+4}a_{n+2}+ a_{n+3}^2</math | + | * <math>a_{n+6}a_{n} = a_{n+5} a_{n+1} +a_{n+4}a_{n+2}+ a_{n+3}^2</math> |
| − | * 1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, 2534423, 14161887, 232663909, 3988834875[http://oeis.org/A006722 ] | + | * 1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, 2534423, 14161887, 232663909, 3988834875[http://oeis.org/A006722 ] |
| − | # RecurrenceTable[{a[n] a[n - 6] == a[n - 1] a[n - 5] + a[n - 2] a[n - 4] + a[n - 3]^2, a[1] == 1, | + | # RecurrenceTable[{a[n] a[n - 6] == a[n - 1] a[n - 5] + a[n - 2] a[n - 4] + a[n - 3]^2, a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1}, a, {n, 20}] |
| − | * [http://cis.csuohio.edu/%7Esomos/somos6.html http://cis.csuohio.edu/~somos/somos6.html] | + | * [http://cis.csuohio.edu/%7Esomos/somos6.html http://cis.csuohio.edu/~somos/somos6.html] |
| − | * http://oeis.org/A006722 | + | * http://oeis.org/A006722 |
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| − | + | ==소모스-8 수열== | |
| − | * <math>a_{n+8}a_{n} = a_{n+7} a_{n+1} +a_{n+6}a_{n+2}+a_{n+5}a_{n+3}+a_{n+4}^2</math | + | * <math>a_{n+8}a_{n} = a_{n+7} a_{n+1} +a_{n+6}a_{n+2}+a_{n+5}a_{n+3}+a_{n+4}^2</math> |
| − | * 1,1,1,1,1,1,1,1,4,7,13,25,61,187,775,5827,14815,420514/7,28670773/91,6905822101/2275 | + | * 1,1,1,1,1,1,1,1,4,7,13,25,61,187,775,5827,14815,420514/7,28670773/91,6905822101/2275 |
| − | # RecurrenceTable[{a[n] a[n - 8] == | + | # RecurrenceTable[{a[n] a[n - 8] == a[n - 1] a[n - 7] + a[n - 2] a[n - 6] + a[n - 3] a[n - 5] + a[n - 4]^2, a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1, a[7] == 1, a[8] == 1}, a, {n, 20}] |
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| − | + | ==역사== | |
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| − | + | * http://www.google.com/search?hl=en&tbs=tl:1&q=Somos+sequence | |
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| − | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
* [http://jeff560.tripod.com/mathword.html Earliest Known Uses of Some of the Words of Mathematics] | * [http://jeff560.tripod.com/mathword.html Earliest Known Uses of Some of the Words of Mathematics] | ||
* [http://jeff560.tripod.com/mathsym.html Earliest Uses of Various Mathematical Symbols] | * [http://jeff560.tripod.com/mathsym.html Earliest Uses of Various Mathematical Symbols] | ||
| − | * [[ | + | * [[수학사 연표]] |
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| − | + | ==메모== | |
| + | * Math Overflow http://mathoverflow.net/search?q=somos | ||
* http://www.cut-the-knot.org/arithmetic/algebra/SimpleSomosSequence.shtml | * http://www.cut-the-knot.org/arithmetic/algebra/SimpleSomosSequence.shtml | ||
* http://faculty.uml.edu/jpropp/somos.html | * http://faculty.uml.edu/jpropp/somos.html | ||
* http://www.math.brown.edu/~jhs/Presentations/ICMSEDSLecture.pdf | * http://www.math.brown.edu/~jhs/Presentations/ICMSEDSLecture.pdf | ||
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| − | + | ==관련된 항목들== | |
| + | * [[소모스-4 수열]] | ||
| − | + | ==사전 형태의 자료== | |
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* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
| − | * | + | * http://en.wikipedia.org/wiki/Somos_sequence |
* http://en.wikipedia.org/wiki/Elliptic_divisibility_sequence | * http://en.wikipedia.org/wiki/Elliptic_divisibility_sequence | ||
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| − | + | ==관련논문== | |
| − | + | * Fedorov, Yuri N., and Andrew N. W. Hone. “Sigma-Function Solution to the General Somos-6 Recurrence via Hyperelliptic Prym Varieties.” arXiv:1512.00056 [nlin], November 30, 2015. http://arxiv.org/abs/1512.00056. | |
| − | * Hone, Andrew N. W. 2010. Analytic solutions and integrability for bilinear recurrences of order six. Applicable Analysis: An International Journal 89, no. 4: 473. doi:[http://dx.doi.org/10.1080/00036810903329977 10.1080/00036810903329977]. | + | * Gorman, Alexi Block, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons, and Jeremy Rouse. “The Density of Primes Dividing a Particular Non-Linear Recurrence Sequence.” arXiv:1508.02464 [math], August 10, 2015. http://arxiv.org/abs/1508.02464. |
| + | * Davis, Bryant, Rebecca Kotsonis, and Jeremy Rouse. “The Density of Primes Dividing a Term in the Somos-5 Sequence.” arXiv:1507.05896 [math], July 21, 2015. http://arxiv.org/abs/1507.05896. | ||
| + | * Hone, Andrew N. W. 2010. Analytic solutions and integrability for bilinear recurrences of order six. Applicable Analysis: An International Journal 89, no. 4: 473. doi:[http://dx.doi.org/10.1080/00036810903329977 10.1080/00036810903329977]. | ||
* Hone, A. N. W. 2007. Sigma function solution of the initial value problem for Somos 5 sequences doi:[http://dx.doi.org/10.1090/S0002-9947-07-04215-8 0.1090/S0002-9947-07-04215-8] | * Hone, A. N. W. 2007. Sigma function solution of the initial value problem for Somos 5 sequences doi:[http://dx.doi.org/10.1090/S0002-9947-07-04215-8 0.1090/S0002-9947-07-04215-8] | ||
| − | * Hone, A. N. W. 2005. Elliptic Curves and Quadratic Recurrence Sequences. Bulletin of the London Mathematical Society 37, no. 2 (April 1): 161 -171. doi:[http://dx.doi.org/10.1112/S0024609304004163 10.1112/S0024609304004163]. | + | * R.W. Gosper, R. Schroeppel, Somos sequence near-addition formulas and modular theta functions, arXiv:[http://arxiv.org/abs/math/0703470 math.NT/0703470v1], 15 March 2007. |
| + | * Hone, A. N. W. 2005. Elliptic Curves and Quadratic Recurrence Sequences. Bulletin of the London Mathematical Society 37, no. 2 (April 1): 161 -171. doi:[http://dx.doi.org/10.1112/S0024609304004163 10.1112/S0024609304004163]. | ||
* Swart, Christine, and Andrew Hone. 2005. Integrality and the Laurent phenomenon for Somos 4 sequences. math/0508094 (August 4). http://arxiv.org/abs/math/0508094 | * Swart, Christine, and Andrew Hone. 2005. Integrality and the Laurent phenomenon for Somos 4 sequences. math/0508094 (August 4). http://arxiv.org/abs/math/0508094 | ||
* van der Poorten, Alfred J. 2004. Elliptic curves and continued fractions. math/0403225 (March 14). [http://arxiv.org/abs/math/0403225. ]http://arxiv.org/abs/math/0403225. | * van der Poorten, Alfred J. 2004. Elliptic curves and continued fractions. math/0403225 (March 14). [http://arxiv.org/abs/math/0403225. ]http://arxiv.org/abs/math/0403225. | ||
| − | * Fomin, Sergey, and Andrei Zelevinsky. 2001. The Laurent phenomenon. math/0104241 (April 25). http://arxiv.org/abs/math/0104241. | + | * '''[FZ2001]'''Fomin, Sergey, and Andrei Zelevinsky. 2001. The Laurent phenomenon. math/0104241 (April 25). http://arxiv.org/abs/math/0104241. |
* '''[Robinson1992]'''R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619. doi:[http://dx.doi.org/10.1090/S0002-9939-1992-1140672-5 10.1090/S0002-9939-1992-1140672-5] | * '''[Robinson1992]'''R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619. doi:[http://dx.doi.org/10.1090/S0002-9939-1992-1140672-5 10.1090/S0002-9939-1992-1140672-5] | ||
* David Gale, Mathematical Entertainments: "The strange and surprising saga of the Somos sequences", Math. Intelligencer, 13(1) (1991), pp. 40-42. | * David Gale, Mathematical Entertainments: "The strange and surprising saga of the Somos sequences", Math. Intelligencer, 13(1) (1991), pp. 40-42. | ||
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| − | + | ==관련도서== | |
| + | * Gale, David. 1998. Tracking The Automatic Ant: And Other Mathematical Explorations. Springer, May 29. | ||
| + | [[분류:수열]] | ||
| − | + | ==메타데이터== | |
| − | + | ===위키데이터=== | |
| − | * | + | * ID : [https://www.wikidata.org/wiki/Q7560513 Q7560513] |
| − | + | ===Spacy 패턴 목록=== | |
| − | * [ | + | * [{'LOWER': 'somos'}, {'LEMMA': 'sequence'}] |
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2021년 2월 17일 (수) 04:48 기준 최신판
개요
- 점화식으로 정의되는 수열
- 소모스 4,5,6,7 은 정수수열이며 소모스 8,9는 정수수열이 아니다
- 점화식만으로는 정수수열이 되는가가 자명하지 않다
- 정수수열이 되는가의 문제 (integrality)
- 합동식을 생각할 때의 주기성 문제 (periodicity modulo n) [Robinson1992]
- 타원곡선론과 클러스터 대수(cluster algebra) 등의 이론에서 등장
소모스-4 수열
- \(a_{n+4}a_{n} = a_{n+3} a_{n+1} + a_{n+2}^2\)
소모스5- 수열
- \(a_{n+5}a_{n} = a_{n+4} a_{n+1} + a_{n+3} a_{n+2}\)
- 1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, 1249441, 9434290, 68570323, 1013908933
- RecurrenceTable[{a[n] a[5 + n] == a[2 + n] a[3 + n] + a[1 + n] a[4 + n], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1}, a, {n, 20}]
소모스-6 수열
- \(a_{n+6}a_{n} = a_{n+5} a_{n+1} +a_{n+4}a_{n+2}+ a_{n+3}^2\)
- 1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, 2534423, 14161887, 232663909, 3988834875[1]
- RecurrenceTable[{a[n] a[n - 6] == a[n - 1] a[n - 5] + a[n - 2] a[n - 4] + a[n - 3]^2, a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1}, a, {n, 20}]
소모스-8 수열
- \(a_{n+8}a_{n} = a_{n+7} a_{n+1} +a_{n+6}a_{n+2}+a_{n+5}a_{n+3}+a_{n+4}^2\)
- 1,1,1,1,1,1,1,1,4,7,13,25,61,187,775,5827,14815,420514/7,28670773/91,6905822101/2275
- RecurrenceTable[{a[n] a[n - 8] == a[n - 1] a[n - 7] + a[n - 2] a[n - 6] + a[n - 3] a[n - 5] + a[n - 4]^2, a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1, a[7] == 1, a[8] == 1}, a, {n, 20}]
역사
- http://www.google.com/search?hl=en&tbs=tl:1&q=Somos+sequence
- Earliest Known Uses of Some of the Words of Mathematics
- Earliest Uses of Various Mathematical Symbols
- 수학사 연표
메모
- Math Overflow http://mathoverflow.net/search?q=somos
- http://www.cut-the-knot.org/arithmetic/algebra/SimpleSomosSequence.shtml
- http://faculty.uml.edu/jpropp/somos.html
- http://www.math.brown.edu/~jhs/Presentations/ICMSEDSLecture.pdf
관련된 항목들
사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Somos_sequence
- http://en.wikipedia.org/wiki/Elliptic_divisibility_sequence
관련논문
- Fedorov, Yuri N., and Andrew N. W. Hone. “Sigma-Function Solution to the General Somos-6 Recurrence via Hyperelliptic Prym Varieties.” arXiv:1512.00056 [nlin], November 30, 2015. http://arxiv.org/abs/1512.00056.
- Gorman, Alexi Block, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons, and Jeremy Rouse. “The Density of Primes Dividing a Particular Non-Linear Recurrence Sequence.” arXiv:1508.02464 [math], August 10, 2015. http://arxiv.org/abs/1508.02464.
- Davis, Bryant, Rebecca Kotsonis, and Jeremy Rouse. “The Density of Primes Dividing a Term in the Somos-5 Sequence.” arXiv:1507.05896 [math], July 21, 2015. http://arxiv.org/abs/1507.05896.
- Hone, Andrew N. W. 2010. Analytic solutions and integrability for bilinear recurrences of order six. Applicable Analysis: An International Journal 89, no. 4: 473. doi:10.1080/00036810903329977.
- Hone, A. N. W. 2007. Sigma function solution of the initial value problem for Somos 5 sequences doi:0.1090/S0002-9947-07-04215-8
- R.W. Gosper, R. Schroeppel, Somos sequence near-addition formulas and modular theta functions, arXiv:math.NT/0703470v1, 15 March 2007.
- Hone, A. N. W. 2005. Elliptic Curves and Quadratic Recurrence Sequences. Bulletin of the London Mathematical Society 37, no. 2 (April 1): 161 -171. doi:10.1112/S0024609304004163.
- Swart, Christine, and Andrew Hone. 2005. Integrality and the Laurent phenomenon for Somos 4 sequences. math/0508094 (August 4). http://arxiv.org/abs/math/0508094
- van der Poorten, Alfred J. 2004. Elliptic curves and continued fractions. math/0403225 (March 14). [2]http://arxiv.org/abs/math/0403225.
- [FZ2001]Fomin, Sergey, and Andrei Zelevinsky. 2001. The Laurent phenomenon. math/0104241 (April 25). http://arxiv.org/abs/math/0104241.
- [Robinson1992]R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619. doi:10.1090/S0002-9939-1992-1140672-5
- David Gale, Mathematical Entertainments: "The strange and surprising saga of the Somos sequences", Math. Intelligencer, 13(1) (1991), pp. 40-42.
관련도서
- Gale, David. 1998. Tracking The Automatic Ant: And Other Mathematical Explorations. Springer, May 29.
메타데이터
위키데이터
- ID : Q7560513
Spacy 패턴 목록
- [{'LOWER': 'somos'}, {'LEMMA': 'sequence'}]