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imported>Pythagoras0 |
imported>Pythagoras0 잔글 (찾아 바꾸기 – “<h5 (.*)">” 문자열을 “==” 문자열로) |
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| 7번째 줄: | 7번째 줄: | ||
| − | + | ==example 1== | |
* Diamond & Shurman 155p | * Diamond & Shurman 155p | ||
| 16번째 줄: | 16번째 줄: | ||
| − | + | ==example 2== | |
* Sums of sqaures of integers 126p | * Sums of sqaures of integers 126p | ||
| 25번째 줄: | 25번째 줄: | ||
| − | + | ==example 3== | |
* 1-2-3- of modular forms | * 1-2-3- of modular forms | ||
| 35번째 줄: | 35번째 줄: | ||
| − | + | ==history== | |
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
| 43번째 줄: | 43번째 줄: | ||
| − | + | ==related items== | |
* [[mathematics of x^3-x+1=0]]<br> | * [[mathematics of x^3-x+1=0]]<br> | ||
| 52번째 줄: | 52번째 줄: | ||
| − | + | ==encyclopedia== | |
* http://en.wikipedia.org/wiki/ | * http://en.wikipedia.org/wiki/ | ||
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| − | + | ==books== | |
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| − | + | ==articles== | |
| 92번째 줄: | 92번째 줄: | ||
| − | + | ==question and answers(Math Overflow)== | |
* http://mathoverflow.net/search?q= | * http://mathoverflow.net/search?q= | ||
| 101번째 줄: | 101번째 줄: | ||
| − | + | ==blogs== | |
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
| 111번째 줄: | 111번째 줄: | ||
| − | + | ==experts on the field== | |
* http://arxiv.org/ | * http://arxiv.org/ | ||
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| − | + | ==links== | |
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | * [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | ||
2012년 10월 28일 (일) 16:46 판
introduction
example 1
- Diamond & Shurman 155p
- \(x^3=d\)
example 2
- Sums of sqaures of integers 126p
- equation
number of solutions of \(x^4-2x^2+2=0\) in F_p = \(1+(\frac{-1}{p})+a_p\) where
\(q\prod_{n=1}^{\infty} (1-q^{2n})(1-q^{16n})=\sum_{n=1}^\infty a_nq^n\)
- Clear[g, p, M, a]
(*table of primes*)
Pr := Table[Prime[n], {n, 1, 20}]
(*equation*)
g[x_] := x^4 - 2 x^2 + 2
(*factorization of the discriminant & bad primes*)
FactorInteger[Discriminant[g[x], x]]
(* M[p] = number of solutions for the equation g[x]=0 modulo p*)
M[n_] := 0
Do[For[i = 0, i < p, i++,
M[p] = M[p] + If[Mod[PolynomialMod[g[i], p], p] == 0, 1, 0]], {p,
Pr}]
(*modification of the number of solutions *)
a[p_] := 1 + JacobiSymbol[-1, p] + M[p]
(*modular form*)
f[q_] := Series[
q*Product[(1 - q^(2 n))*(1 - q^(16 n)), {n, 1, 200}], {q, 0, 100}]
(*the coefficients of modular form f[q]*)
n[p_] := SeriesCoefficient[f[q], p]
(* output *)
title := {M_p, a_p, c_p};
TableForm[Table[{M[p], a[p], n[p]}, {p, Pr}] ,
TableHeadings -> {Pr, title}]
example 3
- 1-2-3- of modular forms
history
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- 2010년 books and articles
- http://gigapedia.info/1/squares
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
articles
- http://www.ams.org/mathscinet
- [1]http://www.zentralblatt-math.org/zmath/en/
- [2]http://arxiv.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
blogs
experts on the field