⓪ | 2次方程式比較 | |||||
-1 | x^2+x-1=0 | -φ | 1/φ | |||
1 | x^2-x-1=0 | φ | -1/φ | |||
p-4 | ||||||
1 | x^2-x+1=0 | 0.5±⒤0.866603 | ||||
3 | x^2-3x+1=0 | φ^2 | 1/φ^2 | |||
5 | x^2-5x+1=0 | |||||
7 | x^2-7x+1=0 | |||||
9 | x^2-9x+1=0 | |||||
2k+1 | x^2-x+1.25=0 | 0.5±⒤1.0 | 9.8√2k+1 | ⋍ | 文献値 | |
3 | x^2-x+3.25=0 | 0.5±⒤1.73205 *0.85⋍ | 14.42798 | : | 14.13472 | |
5 | x^2-x+5.25=0 | 0.5±⒤2.23607 | 21.91347 | ⋍ | 21.02203 | |
7 | x^2-x+7.25=0 | 0.5±⒤2.64575 | 25.92836 | ⋍ | 25.01085 | |
9 | x^2-x+9.25=0 | 0.5±⒤3.0 | 29.4 | ⋍ | 30.42487 | |
11 | x^2-x+11.25=0 | 0.5±⒤3.31662 | 32.50292 | ⋍ | 32.93506 | |
13 | x^2-x+13.25=0 | 0.5±⒤3.60555 | 35.3344 | ⋍ | 37.58617 |
(参考) | ||||
楕円曲線の不変量 | ||||
①y^2⋍x^3+Px+Qの不変量は | ||||
j⋍2^8・3^3・P^3/(4P^3+27Q^2) | ||||
②y^2⋍x^3+Q⇒j⋍0 | ||||
③y^2⋍(x-A)(x-B)(x-C)⇒ | ||||
j⋍2^8(AB+BC+CA-A^2-B^2-C^2)/(A-B)^2(B-C)^2(C-A)^2 | ||||
④y^2⋍x(x-1)(x-A)⇒ | ||||
j⋍-2^8[A(A+1)+1]/A^2(A-1)^2 | ||||
⑤y^2-y⋍x^3-x^2⇒ | ||||
()^2⋍()^3-1/3()+19/108 | ||||
∵Y⋍y-1/2、X⋍x-1/3 | ||||
j⋍-2^12/11 | ||||
①重力定数g⋍9.8 | ||||
②πa(半径、面積) | ||||
③計量テンソルg{μν} | ||||
④ファインマン(標準形)積分⋍π^2/2a^2 | ||||
⑤ウィック回転⋍1/4π^2 | ||||
⑥SchwingerModelの2つのネーターカレント | ||||
jμ⋍I | ψ⋆(γμ)ψ | |||
j5μ⋍I | ψ⋆(γμ)(γ5)ψ⋍(εμν)jν | |||
⑦δ^2π+O(δ^3) | ||||
Z3⋍1+O(α) | ||||
π(q^2)⋍π2(q^2)+O(α^2) | ||||
π(0)⋍π2(0)+O(α^2) |
⓪9.8√(2k+1) | ②p⋍1/2±⒤9.8√(2k+1)f{μν} | ||||||||
(∵11√(2k+1)-√δ補正) | ※ | -2,75√δ | {{1,0} {0,0}} | ||||||
11√ | δ | -√δ | ⋍ | 補正値 | 文献値 | /√δ⋍9.8 | ⋍9.8√2k+1 | ||
11√ | 3 | 19.053 | - | 1.732051 | ※ | 14.28942 | 14.13472 | 8.160684 | 14.42798 |
11√ | 5 | 24.597 | - | 2.236068 | ⋍ | 22.36068 | 21.02203 | 9.401338 | 21.91347 |
11√ | 7 | 29.103 | - | 2.645751 | ⋍ | 26.45751 | 25.01085 | 9.453213 | 25.92836 |
11√ | 9 | 33 | - | 3 | ⋍ | 30 | 30.42487 | 10.14162 | 29.4 |
11√ | 11 | 36.483 | - | 3.316625 | ⋍ | 33.16625 | 32.93506 | 9.930294 | 32.50292 |
11√ | 13 | 39.661 | - | 3.605551 | ⋍ | 36.05551 | 37.58617 | 10.42453 | 35.3344 |
11√ | 15 | 42.603 | - | 3.872983 | ⋍ | 38.72983 | 37.58617 | 9.704707 | 37.95524 |
11√ | 17 | 45.354 | - | 4.123106 | ⋍ | 41.23106 | 40.91871 | 9.924245 | 40.40644 |
11√ | 19 | 47.948 | - | 4.358899 | ⋍ | 43.58899 | 43.32707 | 9.939912 | 42.71721 |
11√ | 21 | 50.408 | - | 4.582576 | ⋍ | 45.82576 | 43.32707 | 9.454742 | 44.90924 |
11√ | 23 | 52.754 | - | 4.795832 | ⋍ | 47.95832 | 48.00515 | 10.00977 | 46.99915 |
11√ | 25 | 55 | - | 5 | ⋍ | 50 | 49.77383 | 9.954766 | 49 |
11√ | 27 | 57.158 | - | 5.196152 | ⋍ | 51.96152 | 49.77383 | 9.578978 | 50.92229 |
11√ | 29 | 59.237 | - | 5.385165 | ⋍ | 53.85165 | 52.97 | 9.836282 | 52.77462 |
11√ | 31 | 61.245 | - | 5.567764 | ⋍ | 55.67764 | 56.446 | 10.138 | 54.56409 |
11√ | 33 | 63.19 | - | 5.744563 | ⋍ | 57.44563 | 56.446 | 9.825987 | 56.29671 |
11√ | 35 | 65.077 | - | 5.91608 | ⋍ | 59.1608 | 59.347 | 10.03147 | 57.97758 |
11√ | 37 | 66.91 | - | 6.082763 | ⋍ | 60.82763 | 60.831 | 10.00055 | 59.61107 |
11√ | 39 | 68.695 | - | 6.244998 | ⋍ | 62.44998 | 60.831 | 9.740756 | 61.20098 |
11√ | 41 | 70.434 | - | 6.403124 | ⋍ | 64.03124 | 65.112 | 10.16879 | 62.75062 |
11√ | 43 | 72.132 | - | 6.557439 | ⋍ | 65.57439 | 65.112 | 9.929487 | 64.2629 |
11√ | 57 | 83.048 | - | 7.549834 | ⋍ | 75.49834 | 75.8 | 10.03996 | 73.98838 |
平均ヘイキン値チ | 9.80864 |
the minimizer f is 2Δf-|▽f|^2+R=const.(ペレルマン、p5) | ||||
the function f on a closed manfiled M (ペレルマン、p8) | ||||
u=(1,1),v=(-1/2,0)とおく | ||||
①f⋍u+⒤v | ∴⒤f⋍-v+⒤u | |||
複素共役、f*⋍u-⒤v | ∴⒤f*⋍-v-⒤u | |||
∴⒤[f+f*]⋍ー2v⋍1 (rank⋍1) | ||||
⒤f[μν]⋍{{1,0}{0,0}} (rank⋍1) | ||||
②γ={{1,1}、{0,1}} | [a,b,c] | |||
∴γ^2={{1,2},{0,1}} | [1,-1,3.25] | |||
③γ^2⒤v=(-1/2,0) | [1,-1,5.25] | |||
γ^2・⒤u=(3,1) | … | |||
γ^4・⒤u=(5,1) | [1,-1,2k+1.25] | |||
… | [1,-1,q.25] | |||
γ^2k・⒤u=(2k+1,1) | ∴x⋍1/2∓⒤√Q⋍√(2k+1) | |||
∴√γ^2k・⒤u=√(2k+1,1)⋍√(2k+1) | x⋍1/2∓⒤1.73205… | |||
④x^2+x+Q.25=(x+1/2)^2+Q=0 | φ(x)=e^∓√(2k+1)x | |||
□φ= | d^2φ=(2k+1)φ | |||
∴[□-(2k+1)]φ=0 |
・ファインマン経路積分 | |||||||||
1/AB⋍1/(A-B)^2・[1/k-1/k+1] | |||||||||
1/s(1-s)⋍-1/ℓ(ℓ+1)⋍1/(ℓ+ℓ+1)^2・[1/k-1/k+1] | |||||||||
⋍1/p^4・[1/k-1/k+1] | |||||||||
∴p⋍∓√(2ℓ+1)⋍∓√3,√5,√7…⋍∓√1-s | |||||||||
⓪9.8√(2k+1) | ①πa型√(2k+1){9.8±0.43} | ||||||||
(∵11√(2k+1)-δ補正) | ※ | -2,75δ | ※ | -2,75δ | |||||
11√ | δ | √(2k+1)⋍δ | 補正値 | 文献値 | /δ⋍9.8 | ||||
11√ | 3 | 19.0525589 | - | 1.73205081 | ※ | 14.2894192 | 14.13472 | ⋍ | 8.160684 |
11√ | 5 | 24.5967478 | - | 2.23606798 | ⋍ | 22.3606798 | 21.02203 | ⋍ | 9.401338 |
11√ | 7 | 29.1032644 | - | 2.64575131 | ⋍ | 26.4575131 | 25.01085 | ⋍ | 9.453213 |
11√ | 9 | 33 | - | 3 | ⋍ | 30 | 30.42487 | ⋍ | 10.14162 |
11√ | 11 | 36.4828727 | - | 3.31662479 | ⋍ | 33.1662479 | 32.93506 | ⋍ | 9.930294 |
11√ | 13 | 39.661064 | - | 3.60555128 | ⋍ | 36.0555128 | 37.58617 | ⋍ | 10.42453 |
11√ | 15 | 42.6028168 | - | 3.87298335 | ⋍ | 38.7298335 | 37.58617 | ⋍ | 9.704707 |
11√ | 17 | 45.3541619 | - | 4.12310563 | ⋍ | 41.2310563 | 40.91871 | ⋍ | 9.924245 |
11√ | 19 | 47.9478884 | - | 4.35889894 | ⋍ | 43.5889894 | 43.32707 | ⋍ | 9.939912 |
11√ | 21 | 50.4083326 | - | 4.58257569 | ⋍ | 45.8257569 | 43.32707 | ⋍ | 9.454742 |
11√ | 23 | 52.7541468 | - | 4.79583152 | ⋍ | 47.9583152 | 48.00515 | ⋍ | 10.00977 |
11√ | 25 | 55 | - | 5 | ⋍ | 50 | 49.77383 | ⋍ | 9.954766 |
11√ | 27 | 57.1576766 | - | 5.19615242 | ⋍ | 51.9615242 | 49.77383 | ⋍ | 9.578978 |
11√ | 29 | 59.2368129 | - | 5.38516481 | ⋍ | 53.8516481 | 52.97 | ⋍ | 9.836282 |
11√ | 31 | 61.245408 | - | 5.56776436 | ⋍ | 55.6776436 | 56.446 | ⋍ | 10.138 |
11√ | 33 | 63.1901891 | - | 5.74456265 | ⋍ | 57.4456265 | 56.446 | ⋍ | 9.825987 |
11√ | 35 | 65.0768776 | - | 5.91607978 | ⋍ | 59.1607978 | 59.347 | ⋍ | 10.03147 |
11√ | 37 | 66.9103878 | - | 6.08276253 | ⋍ | 60.8276253 | 60.831 | ⋍ | 10.00055 |
11√ | 39 | 68.694978 | - | 6.244998 | ⋍ | 62.44998 | 60.831 | ⋍ | 9.740756 |
11√ | 41 | 70.4343666 | - | 6.40312424 | ⋍ | 64.0312424 | 65.112 | ⋍ | 10.16879 |
11√ | 43 | 72.1318238 | - | 6.55743852 | ⋍ | 65.5743852 | 65.112 | ⋍ | 9.929487 |
11√ | 57 | 83.0481788 | - | 7.54983444 | ⋍ | 75.4983444 | 75.8 | ⋍ | 10.03996 |
平均値 | ⋍ | 9.887114 | |||||||
標偏差 | ⋍ | 0.436854 |