Power Law:(23)John D. Barrow, Between Inner Space and Outer Space

(2008/11/13)
(DO図書館)
単純な法則に支配される宇宙が複雑な姿を見... ジョン・D. バロウ、John D. Barrow、 松浦 俊輔 (単行本 - 2002/11)
p.57,Fig5-1:質量と大きさの分布(原子から恒星まで)
p.57,Fig5-1:Distribution of mass and size(From the atom to the fixed star)

~~~
reading point in figure) as memo.
y:10/13.8->10^(10/13.8)->5.304*1e30
~~~
log
(x, y)=(1e-10, 1e-30),(1e10, 5.304e30)
->
(x, y/x)=(1e-10, 1e-30/1e-10),(1e10, 5.304e30/1e10)
=(1e-10, 1e-20),(1e10, 5.304e20)
Relative value)
=(1, 1e-20/5.304e20),(1e10/1e-10, 1)
=(1, 0.1885e-40),(1e20, 1)
A=2.036
B=0.1885e-40

~~~
A>0なので、更に
In addition, because A is 0 or more, it continues.
(x, y/x)=(1, 0.1885e-40),(1e20, 1/1e20)
=(1, 0.1885e-40),(1e20, 1e-20)
Relative value)
=(1, 0.1885e-40/1e-20),(1e20, 1)
=(1, 0.1885e-20),(1e20, 1)
A=1.036
B=0.1885e-20

~~~
A>0,then continues...)
(x, y/x)=(1, 0.1885e-20),(1e20, 1/1e20)
=(1, 0.1885e-20),(1e20, 1e-20)
Relative value)
=(1, 0.1885e-20/1e-20),(1e20, 1)
=(1, 0.1885),(1e20, 1)
A=0.036
B=0.1885

~~~
A>0,then continues...)
(x, y/x)=(1, 0.1885),(1e20, 1/1e20)
=(1, 0.1885),(1e20, 1e-20)
Relative value)
=(1, 0.1885/1e-20),(1e20, 1)
=(1, 0.1885e20),(1e20, 1)
A=-0.964
B=0.1885e20

~~~
LN(abs(A))-LN(B)のグラフで、LN(abs(A))=0の時のB値を求める。
B value at LN(abs(A)) =0 is requested in the graph of LN(abs(A)) - LN(B).

LN(y) = a*LN(abs(x)) + LN(b)
a = 5
LN(b) = LN(y) - a*LN(x)
= LN(0.1885e20) - 5*LN(abs(-0.964))
= 44.383 - 5*(-0.03666)
= 44.566
b = exp(44.566)
= 2.263e19

~~~
b = 2.263e19 に近い、Fn,Lnを求める。
Fn and Ln are obtained near b = 2.263e19.
http://humanbeing-etcman.blogspot.com/2008/11/power-law22-over-100-in-fn-and-ln.html
~~~
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibtable.html#100
Fn

83 : 99194853094755497
84 : 160500643816367088 = 24 x 32 x 13 x 29 x 83 x 211 x 281 x 421 x 1427
85 : 259695496911122585 = 5 x 1597 x 9521 x 3415914041
86 : 420196140727489673 = 6709 x 144481 x 433494437
87 : 679891637638612258 = 2 x 173 x 514229 x 3821263937
88 : 1100087778366101931 = 3 x 7 x 43 x 89 x 199 x 263 x 307 x 881 x 967
89 : 1779979416004714189 = 1069 x 1665088321800481
90 : 2880067194370816120 = 23 x 5 x 11 x 17 x 19 x 31 x 61 x 181 x 541 x 109441
91 : 4660046610375530309 = 132 x 233 x 741469 x 159607993
92 : 7540113804746346429 = 3 x 139 x 461 x 4969 x 28657 x 275449
93 : 12200160415121876738 = 2 x 557 x 2417 x 4531100550901
~~~:here
94 : 19740274219868223167 = 2971215073 x 6643838879
~~~:here
95 : 31940434634990099905 = 5 x 37 x 113 x 761 x 29641 x 67735001
96 : 51680708854858323072 = 27 x 32 x 7 x 23 x 47 x 769 x 1103 x 2207 x 3167
97 : 83621143489848422977 = 193 x 389 x 3084989 x 361040209
98 : 135301852344706746049 = 13 x 29 x 97 x 6168709 x 599786069
99 : 218922995834555169026 = 2 x 17 x 89 x 197 x 19801 x 18546805133
100 : 354224848179261915075 = 3 x 52 x 11 x 41 x 101 x 151 x 401 x 3001 x 570601

~~~
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/lucas200.html
Ln

85 : 580696784543856761 = 11 x 3571 x 1158551 x 12760031
86 : 939587134549734843 = 3 x 313195711516578281
87 : 1520283919093591604 = 22 x 59 x 349 x 19489 x 947104099
88 : 2459871053643326447 = 47 x 93058241 x 562418561
89 : 3980154972736918051 = 179 x 22235502640988369
90 : 6440026026380244498 = 2 x 33 x 41 x 107 x 2521 x 10783342081
91 : 10420180999117162549 = 29 x 521 x 689667151970161
92 : 16860207025497407047 = 7 x 253367 x 9506372193863
~~~:here
93 : 27280388024614569596 = 22 x 63799 x 3010349 x 35510749
~~~:here
94 : 44140595050111976643 = 3 x 563 x 5641 x 4632894751907
95 : 71420983074726546239 = 11 x 191 x 9349 x 41611 x 87382901
96 : 115561578124838522882 = 2 x 1087 x 4481 x 11862575248703
97 : 186982561199565069121 = 3299 x 56678557502141579
98 : 302544139324403592003 = 3 x 281 x 5881 x 61025309469041
99 : 489526700523968661124 = 22 x 19 x 199 x 991 x 2179 x 9901 x 1513909
100 : 792070839848372253127 = 7 x 2161 x 9125201 x 5738108801
~~~
誤差はあると思うが、Fnに近い？
A-Bの傾向線は、Lnと仮定していたが、Fnの可能性もあるのか、それとも混合値あるいは、
初期値の異なる別の数列なのか？TODO)
It is near Fn though it thinks the error margin to be?
Though the trend line of A-B was assumed to be Ln,...
Is there a possibility of Fn, too or a mixture value or an initial value another different progression?
~~~
end