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O(0,0), A((â3-â2+1)/2,0), B(â3/2,(â6-â3)/2), C(â3/2,-(â6-â3)/2)
ãšããŠãOA, AB, AC ãããããç·åã§çµã³ãŸãã
ããããŠã§ãã Y ååããO ãäžå¿ã« 90 床ãã€åããŠè€è£œããå³åœ¢ã§ãã
å€åŽã®ééã§é·ã 1 ã確ä¿ã§ããªãããã« 45 床ã〠8 ç¹åãçºæ³ã§äœããŸãããã7 ç¹ã ãš cos(2Ï/7) ãšãç»å Žã㊠3 ä¹æ ¹ã«ãªãããªãŒããšã
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ã§ã¯ããããç§ãæãã€ããæ¡ãæžããŸãã
O(0,0),A(1,0)ãšããŠååšäžã6çåããããã«B,C,D,E,FããšããŸãã
B(1/2,â3/2),C(-1/2,â3/2),D(-1,0),E(-1/2,-â3/2),F(1/2,-â3/2)ã§ãã
Eãäžå¿ãšããååŸ1ã®å£åŒ§FOã«æ¥ãçŽç·OFãšå¹³è¡ãªçŽç·ãšã
Cãäžå¿ãšããååŸ1ã®å£åŒ§OBã®äº€ç¹ãIãšããŸãã
ãŸãå£åŒ§FOãšçŽç·ã®æ¥ç¹ãGãšããŸããFG=GOãšãªããŸãã
座æšã¯
G((â3-1)/2,-(â3-1)/2),
I((â(4â3-3)+2â3-5)/4,(2+â3-â(12â3-9))/4)
ãšãªããŸãã
y軞ã«é¢ããŠIãšå¯Ÿç§°ãªç¹ãJãGãšå¯Ÿç§°ãªç¹ãHãšããŸãã
H(-(â3-1)/2,-(â3-1)/2),
J(-(â(4â3-3)+2â3-5)/4,(2+â3-â(12â3-9))/4)
ã§ãã
ãããŠ
OI,IA,IB,OJ,JC,JD,OH,HE,OG,GFã®åç·åãæããŸãã
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â(30-16â3+2â(120â3-207))+â(22-4â3-6â(4â3-3))+2(â6-â2)â5.885
ãšãªããŸãã
åçŽç·IGïŒå£åŒ§FOã®æ¥ç·ïŒãšåäœåã®äº€ç¹ãKãšãããšã
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ç§ã®æ¡ã§ã®äžå¿éšåã® 90 床ã¯ãã¹ã«åãã¢ã€ãã¢äœ¿ã£ãããã£ãšçããªãããªïŒ
ãïŒã€ã®ïŒãã®ã«ãŒã«ã«æºããŠããïŒã€ã®ïŒãã§ïŒïŒãäœãããšãè©Šã¿ããšçºãããšæããŸãããããã解ã¯ãããŸãã
å€åãã£ãšãã解ããããšæããŸããããšãããã
11=(4!)!!!!!!!!!!!!!/(4!)
ïŒååã¯13ééä¹ïŒ
(è¿œèš)ã¬ãŠã¹èšå·ã䜿ããšãããã
11=[4!!*log4]
11=4-[tan(4!!)]
11=[4+exp(â4)]
11=[-exp(â4)/cos(4)]
11=[(4!)^(-sin(4))]
(4!)!!!!!!!!!!!!!=24*11ãªã®ã§ã(4!)!!!!!!!!!!!!!/(4!)=11ã§ããïŒããã¯ç¢ºãã«çºããŸããïŒïŒïŒã
ïŒã€ã®ïŒã§ïŒïŒãäœããšãããã¿ã®ããšã¯ããïŒã€ã®ïŒãã®èšäºã«å«ãŸããäžèšã®åŒããå°ãããã®ã§ãã
ïŒïŒïŒïŒÎïŒÎïŒïŒïŒïŒïŒÎïŒâïŒïŒïŒÎïŒâïŒïŒïŒÎïŒâïŒïŒ
ãã®åŒãã¿ãŠäžèšãå°ãã次第ã§ãã
ïŒïŒ = â(ïŒ+ïŒïŒ)= â(Î(â(ïŒ))+Î(Î(ïŒ)))
â»ïŒïŒééä¹ãçšããããããããã«ãã解ã«ã¯ããã¯ãªããŸããâŠâŠ 以äžã§ãã
ããããããããïŒã€ã®ïŒãã§ïŒïŒãäœã£ããã¯ããã¯ãå©çšãããšããïŒã€ã®ïŒãã§ãä»»æã®æçæ°ã
äœãããããªæ°ãããããŸãã äžäŸããããŸãã 22/7 ããïŒã€ã®ïŒãã§äœã£ãŠã¿ãŸããã
(((3!)!!)!!!!!!!!!!!!!!!!!!!!!!!!!!)/(((3!)!!)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!)
=(48!!!!!!!!!!!!!!!!!!!!!!!!!!)/(48!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!)
=(48*22)/(48*7)
=22/7
ããããããã«ããææ³ã¯ãšãŠã匷åã§ãããšæããŸãã
ãã¿ãŸããã
ã¿ã€ãã®æã¡ééãããã£ãŠãããäžèšã®åé¡ã§åæçš¿ãçŽããŠããŸãã
ã¿ã€ãã®éœåäž
sqrtn(x,2)=âx
sqrtn(x,3)=âx
sqrtn(x,4)=âx

ãªãèšå·ã§è¡šããšãããšã
f(x):=x*(sqrtn(x,2)*(sqrtn(x,3)*(sqrtn(x,4)*(sqrtn(x,5)*(sqrtn(x,6)*())))))
ã§å®çŸ©ããf(x)ã®äžå®ç©å
â«f(x)dx ã¯ïŒ
> f(x):=x*(sqrtn(x,2)*(sqrtn(x,3)*(sqrtn(x,4)*(sqrtn(x,5)*(sqrtn(x,6)*())))))
ããã¯å
¥ãåã«ãªã£ãŠããããã§ã¯ãªãã®ã§
f(x)=x*sqrtn(x,2)*sqrtn(x,3)*âŠ
=x^(1+1/2+1/3+âŠ)
=0 (0âŠxïŒ1), 1 (x=1), +â (xïŒ1)
ãšåãã§ã¯ïŒ
ïŒåéãããã£ããããããªããïŒ
2è¡ç®ã®æå³ãããããããªãã®ã§ãããã©ããã«ãåæçš¿ããããã®ã§ããïŒ
ãã®äžã®åŒã¯å€æŽãããŠããŸããããïŒ
æåæçš¿ãã圢ã
x*sqrtn(x,2)*sqrtn(x,3)*sqrtn(x,4)*sqrtn(x,5)*sqrtn(x,6)*
ã ã£ãã®ã§ãããã¯å
¥ãåã«ãªã£ãŠããªããšæãè¿ãçŽãã«èšæ£ããŠ
x*(sqrtn(x,2)*(sqrtn(x,3)*(sqrtn(x,4)*(sqrtn(x,5)*(sqrtn(x,6)*))))))
ã®æ§ã«ã¿ã€ããçŽããŠãããŸããã
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å
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sqrtn(sqrtn(x,4),3)
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æãã€ããåé¡ã§ãã
1ãnãŸã§ãã3ã€ã®ã°ã«ãŒãã«åããã(åã°ã«ãŒãã«ã¯å°ãªããšã1ã€ã®æ°ãå
¥ã)
ããããã®2ã°ã«ãŒãããããããã1ã€ãã€æ°ãéžã¶ããããã®æ°ãããéžã°ããªãã£ãã°ã«ãŒãã確å®ãããã
mod 3以å€ã®åãæ¹ã¯ããã§ããããïŒ
æãã€ããã®ã¯...
(2é²æ³ã§1ãå¶æ°æ¡ã ã),(2é²æ³ã§1ãå¥æ°æ¡ã ãã®å¥æ°),(ãã以å€ã®æ°)
ã§ãã...åã£ãŠãŸãããããïŒ
ãŸãããã以å€ã§ã®åãæ¹ã£ãŠãããŸãã§ããããïŒ
t=(1+sqrt(5))/2
f(n)=floor(n*t^2)
g(n)=floor(t*floor(n*t)
h(n)=floor(t*floor(n*t^2))
ã§
n=1ïœ50ã§f(n),g(n),h(n)ãèšç®ããããš
gp > for(n=1,50,print(n";"f(n) " VS "g(n) " VS " h(n)))
1;2 VS 1 VS 3
2;5 VS 4 VS 8
3;7 VS 6 VS 11
4;10 VS 9 VS 16
5;13 VS 12 VS 21
6;15 VS 14 VS 24
7;18 VS 17 VS 29
8;20 VS 19 VS 32
9;23 VS 22 VS 37
10;26 VS 25 VS 42
11;28 VS 27 VS 45
12;31 VS 30 VS 50
13;34 VS 33 VS 55
14;36 VS 35 VS 58
15;39 VS 38 VS 63
16;41 VS 40 VS 66
17;44 VS 43 VS 71
18;47 VS 46 VS 76
19;49 VS 48 VS 79
20;52 VS 51 VS 84
21;54 VS 53 VS 87
22;57 VS 56 VS 92
23;60 VS 59 VS 97
24;62 VS 61 VS 100
25;65 VS 64 VS 105
26;68 VS 67 VS 110
27;70 VS 69 VS 113
28;73 VS 72 VS 118
29;75 VS 74 VS 121
30;78 VS 77 VS 126
31;81 VS 80 VS 131
32;83 VS 82 VS 134
33;86 VS 85 VS 139
34;89 VS 88 VS 144
35;91 VS 90 VS 147
36;94 VS 93 VS 152
37;96 VS 95 VS 155
38;99 VS 98 VS 160
39;102 VS 101 VS 165
40;104 VS 103 VS 168
41;107 VS 106 VS 173
42;109 VS 108 VS 176
43;112 VS 111 VS 181
44;115 VS 114 VS 186
45;117 VS 116 VS 189
46;120 VS 119 VS 194
47;123 VS 122 VS 199
48;125 VS 124 VS 202
49;128 VS 127 VS 207
50;130 VS 129 VS 210
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> æãã€ããã®ã¯...
> (2é²æ³ã§1ãå¶æ°æ¡ã ã),(2é²æ³ã§1ãå¥æ°æ¡ã ãã®å¥æ°),(ãã以å€ã®æ°)
>
> ã§ãã...åã£ãŠãŸãããããïŒ
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2é²æ³ã§1ãå¶æ°æ¡ã ã: 10100010(2)
2é²æ³ã§1ãå¥æ°æ¡ã ãã®å¥æ°: 1000001(2)
ãã以å€ã®æ°: 2é²æ³ã§1ãå¥æ°æ¡ã ãã®å¶æ°ãšå¶æ°æ¡å¥æ°æ¡ã®äž¡æ¹ã«1ãããæ°
ãšããæå³ã§ããããïŒ
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1001(2)=9(10)ãšããåããã£ããšãã«
第1ã°ã«ãŒããšç¬¬2ã°ã«ãŒãã®å: 1000(2)+1(2)=1001(2)
第1ã°ã«ãŒããšç¬¬3ã°ã«ãŒãã®å: 10(2)+111(2)=1001(2)
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å¶æ°æ¡ã ãã1ã®æ°ïŒ10, 1010,101010,...
å¥æ°æ¡ã ãã1ã®æ°ïŒ1,101,10101,1010101,...
ã®ã€ããã§ãã ^^;...
ã¡ãªã¿ã«ã
ããæ¹ããã{1,n,ãã®ä»ã®æ°}
ã®3ã°ã«ãŒãã«åããŠãå¯èœãšæããŠããã ããŸãã...
ãªãã»ã©ã
ããã ãšããŠã
1111(2)=15(10)ãšããåããã£ããšãã«
第1ã°ã«ãŒããšç¬¬2ã°ã«ãŒãã®å: 1010(2)+101(2)=1111(2)
第2ã°ã«ãŒããšç¬¬3ã°ã«ãŒãã®å: 1(2)+1110(2)=1111(2)
ã®ã©ã¡ããªã®ãåºå¥ãã€ããªããšæããŸãã
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ãã£ã...!!
æµ
ã¯ãã§ãã ^^;
ããããšãããããŸãã Orzãâ
1,2,3ãã©ã®ã°ã«ãŒãã«å
¥ããããå ŽååãããŠçŽ°ãã調ã¹ãããšã«ããã
æ¡ä»¶ãæºããåãæ¹ã¯
ãmod3ã§åããã
ã1ãšnãšãã®ä»ã«åããã
ã®2éããããªãããšã蚌æã§ããŸããã
蚌æã¯é·ããªããŸãã®ã§ãšããããçç¥ããŸãã
ããããæ§ãž
é¢çœãã§ãã ^^
ã©ã®ããã«èšŒæã§ããã®ãåãããŸããã ^^;
äžè¬ã«ãmod (m: 3以äžã®å¥æ°) ã§ã°ã«ãŒãåã(mã°ã«ãŒã)ããã°ããã¹ãŠã®ã°ã«ãŒãããã®å㯠mod mã§0ã«ãªãã®ã§ã
mã°ã«ãŒãã«åããŠãm-1ã°ã«ãŒãããåãåºããåâ¡r (mod m) ãªããåãåºããªãã£ãã°ã«ãŒãã¯m-rã®ã°ã«ãŒããšãããã
mod(m: 4以äžã®å¶æ°)ã§ã°ã«ãŒãåãããã°ãå
šãŠã®ã°ã«ãŒãããã®å㯠mod m㧠m/2 ã®ãªãã®ã§ãåãåºããªãã£ãã°ã«ãŒãã¯m/2-rã®ã°ã«ãŒããšãããã®ã§ãäžè¬åã§ããŸããã
ãŸãã1,n,ãã®ä»ã...
k,n,ãã®ä»ã§ããå
šéšã®åãn(n+1)/2 ãªã®ã§ã2ã°ã«ãŒãã®åãåŒãããã®ãk or n以å€ãªãããã®ä»ãšãããã®ã§å¯èœã§ããã
åæ§ã«ãmã°ã«ãŒãã®æã...
äŸãã°...
1,2,...,(m-2),n,ãã®ä»
ã«åããŠããã°ãå
šäœã®åãäžå®ãªã®ã§åãããšãèšããã®ã§ãäžè¬åã§ããŸããã
èªç¶æ°nã®åå²æ°ãšããŠ
n=6ãªã
[6]
[1, 5]
[2, 4]
[3, 3]
[1, 1, 4]
[1, 2, 3]
[2, 2, 2]
[1, 1, 1, 3]
[1, 1, 2, 2]
[1, 1, 1, 1, 2]
[1, 1, 1, 1, 1, 1]
以äž11éã
n=8ãªã
[8]
[1, 7]
[2, 6]
[3, 5]
[4, 4]
[1, 1, 6]
[1, 2, 5]
[1, 3, 4]
[2, 2, 4]
[2, 3, 3]
[1, 1, 1, 5]
[1, 1, 2, 4]
[1, 1, 3, 3]
[1, 2, 2, 3]
[2, 2, 2, 2]
[1, 1, 1, 1, 4]
[1, 1, 1, 2, 3]
[1, 1, 2, 2, 2]
[1, 1, 1, 1, 1, 3]
[1, 1, 1, 1, 2, 2]
[1, 1, 1, 1, 1, 1, 2]
[1, 1, 1, 1, 1, 1, 1, 1]
以äž22éããšnã«å¯ŸããŠãã®åå²æ°ã決ãŸãã®ã§ããããP(n)ã§è¡šãããšã«ããã
n=1,2,3,,20ã§ã¯
P(n);1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627
ãšèšãããšã«ãªãã
ããŠãã®äžèŠäžèŠåãªæ°ã®äžŠã³ã«äŸã®ã©ãããžã£ã³ãn=0,1,2,3,ã®ãã¹ãŠã«å¯Ÿã
P(5*n+4)==0 (mod 5)
P(7*n+5)==0 (mod 7)
P(11*n+6)==0 (mod 11)
ãçºèŠããã
P(13*n+7)==0 (mod 13)
ãšèª¿åã«ä¹ãããããããã¯å
šãæç«ããªãã
1960幎代ã§Atokinããã£ãš
P(11^3*13*n+237)==0 (mod 13)
ãçºèŠããã
ãã®åŸ
P(59^4*13*n+111247)==0 (mod 13)
ãèŠã€ããã
次ã®çŽ æ°17ã§ã¯
P(23^3*17*n+2623)==0 (mod 17)
P(41^4*17*n+1122838)==0 (mod 17)
çŽ æ°19ã§ã¯
P(101^4*19*n+815655)==0 (mod 19)
ä»ã«
P(999959^4*29*n+289956221336976431135321047)==0 (mod 29)
P(107^4*31*n+30064597)==0 (mod 31)
ãæç«ããŠãããšããã
ãŸãçŽ æ°5,7,11ã ããçµã¿åããããããªæ°ã«ã¯
If Ύ = 5^a*7^b*11^c and 24*λ ⡠1 (mod Ύ),
then P(Ύ*n + λ) ⡠0 (mod Ύ)
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ã«èª¿ã¹ãŠã¿ããš1000以äžã«ããæ¡ä»¶æ°ã§ã¯
25=>P(25*n + 24)==0 (mod 25)
35=>P(35*n + 19)==0 (mod 35)
49=>P(49*n + 47)==0 (mod 49)
55=>P(55*n + 39)==0 (mod 55)
77=>P(77*n + 61)==0 (mod 77)
121=>P(121*n + 116)==0 (mod 121)
125=>P(125*n + 99)==0 (mod 125)
175=>P(175*n + 124)==0 (mod 175)
245=>P(245*n + 194)==0 (mod 245)
275=>P(275*n + 149)==0 (mod 275)
343=>P(343*n + 243)==0 (mod 343)
385=>P(385*n + 369)==0 (mod 385)
539=>P(539*n + 292)==0 (mod 539)
605=>P(605*n + 479)==0 (mod 605)
625=>P(625*n + 599)==0 (mod 625)
847=>P(847*n + 600)==0 (mod 847)
875=>P(875*n + 474)==0 (mod 875)
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343=>P(343*n + 243)==0 (mod 343)
ã ãã¯n=0,1,2,,20ã«å¯Ÿã
245,294,0,196,0,0,0,196,98,0,98,0,0,0,98,98,0,196,0,0,0
ã䞊ã³ãäºæ³ã«åããã
ä»ã®ããå€ãã®å®äŸã芳å¯ããããšã§ãã®åå ã
343=7^3
ã§æ°ãæ§æããçŽ æ°7ã®éšåã®ææ°ã«ããããããå€æŽããŠ
äžè¬ã«7^b -> 7^(floor(b/2)+1)
ãšããŠåŠçããã°ãªããªãããšãå€æããã
å³ã¡b=3ãªã
floor(3/2)+1=1+1=2
ã€ãŸã(mod 343) ã§ã¯ãªã(mod 7^2)=(mod 49)ã§åŠçããã
343=>P(343*n + 243)==0 (mod 49)ãªã
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
ãšäºæ³ã«åèŽããã
ããã§æ¬¡ã®å€æŽãå ããããã
If Ύ = 5^a*7^b*11^c and 24*λ ⡠1 (mod Ύ),
then P(Ύ*n + λ) ⡠0 (mod 5^a*7^(floor(b/2)+1)*11^c)
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ãªåŸ®åŠãªäžå€èŠåãæœãã§ããããã§ããã
ã©ãªããçŽ æ°23ã«å¯Ÿãã0ã«ç¹ããåååŒããåç¥ãªããç¥ããäžããã
(ããããæç®ããµã€ããæ¢ãåã£ãã®ã§ãããããã ãã¯èŠã€ããããªãããŸãã
ãŸãèªåã§æ¢ããŠã¿ãŠã¯ãããã§ãã
ã²ãããªäºãã次ã®åŒã¯(mod 23)ã§ã¯0ã«ãªããªãã®ã ããããšæã£ãã
æéã®é¢ä¿ã§äžéšãã確èªãããŠããªããïŒã§ããããªã«ãããããªãã ããã«ïŒïŒ
P(37^4*23*n+631052)
P(67^4*23*n+5476393)
P(95^4*23*n+18897974)
P(133^4*23*n+29309936)
P(179^4*23*n+60460032)
P(185^4*23*n+103152724)
GAIããã«ãã埡æçš¿ãèªç¶æ°ã®åå²æ°ãå匷ããŠã¿ãŠãã«é¢é£ãã話é¡ã«ã€ããŠèšèŒããŠãããµã€ããã¿ã€ããŸããã®ã§åŸ¡å ±åããããŸãã
â åå²æ°ãã°ã©ã³ãã«ããã«ã«ååžã§æ±ãã(http://zakii.la.coocan.jp/physics/75_partition_number.htm)ã§ãã
ã¬ãŠã¹èšå·ãå«ãé¢æ°ã®ãå®ç©åãã«ã€ããŠã¯ãªããšããªãã«ããŠã
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ãªãŒãã³ç©åã§ã¯è¢«ç©åé¢æ°ã®é£ç¶æ§ãèŠè«ãããŸãããã¬ãŠã¹èšå·å«ãé¢æ°ã®ãå®ç©åãã«ã€ããŠã¯
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ã¬ãŠã¹èšå·ãäœãšããªããšæããŸãã
äŸãã°[x]ã®äžå®ç©åã¯(2x-[x]-1)[x]/2+CãšæžããŸãã
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â«|6x^2-18x+12|dxãæ±ããã
é«æ ¡çã®ãšãã«ã以äžã®èšç®ãããèŠãããããŸãã
Abs(ïœ)ïŒïœïŒÂ±ïœ^2ããªã®ã§ã䞡蟺ã埮åãããšãïŒAbs(ïœ)ïŒïœ)âïŒÂ±ïŒïœïŒïŒAbs(ïœ)ããšæžããã
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解ç
â«|6x^2-18x+12|dx
=(2x^3-9x^2+12x)-|x-1|(2x^2-7x+5)+|x-2|(2x^2-5x+2)+C
ãšãªããŸãã
ãã®åé¡ã«ã€ããŠã¯æè¿èããã®ã§ãããäžè¬ã®çµ¶å¯Ÿå€ä»ãå€é
åŒã®äžå®ç©åã¯ä»¥äžã®ããã«ãªããŸãã
n次å€é
åŒf(x)ã«å¯Ÿããâ«|f(x)|dxã®è§£ã¯
F(x)=
â«f(x)dxãïŒn次ã®ä¿æ°ãæ£ã®å ŽåïŒ
-â«f(x)dxãïŒn次ã®ä¿æ°ãè² ã®å ŽåïŒ
ïŒç©åå®æ°ã¯äœã§ãå¯ïŒ
G(x,α)ã¯{F(x)-F(α)}÷(x-α)ã®å
# G(x,α)={F(x)-F(α)}/(x-α)ãšãããšx=αã§å®çŸ©ãããªãã®ã§NGã§ã
# G(x,α)ã¯F(x)-F(α)ã(x-α)ã§å²ã£ãåãšããå¿
èŠããããŸãã
ãããŠf(x)=0ã®å®æ°è§£ã®ãã¡x軞ã暪åã解
ïŒã€ãŸãf(x)=0,f(x+ε)f(x-ε)ïŒ0ã§ããxïŒãå°ããé ã«
a[1],a[2],âŠ,a[m]ãšãããš
mãå¶æ°ã®ãšã
â«|f(x)|dx = F(x)+Σ[k=1ïœm](-1)^kã»|x-a[k]|ã»G(x,a[k])
mãå¥æ°ã®ãšã
â«|f(x)|dx = Σ[k=1ïœm](-1)^(k-1)ã»|x-a[k]|ã»G(x,a[k])
ïŒããããç©åå®æ°çç¥ïŒ
ãšãªããŸãã
ïŒm=0ã®ãšãã¯â«|f(x)|dx=F(x)+Cã§ããïŒ
â«|6x^2-18x+12|dxïŒ|x^2-3x+2|(2ïœ-3)+2|x-1|+|x-2|+C
ã§ã©ãã§ããããïŒ
f(x)=|x^2-3x+2|(2x-3)+2|x-1|+|x-2|ãšãããš
f(3/4)=41/32
f(1)=1
ãšãªã£ãŠæžå°ããŠããŸãã®ã§ãã¡ãã£ãšéãããã§ãã
aãæ£ã®æ°ãšãããšã
â«[-a,a]|x^2+x-2|dx
ãèšç®ããã
ããã«å¯ŸããŠããã®äžå®ç©åå
¬åŒã§åŠçããã°ã©ã®æ§ã«ãªãã®ã§ããïŒ
f(x)=x^2+x-2=(x+2)(x-1)ãªã®ã§f(x)=0ã®è§£ã¯x=-2,1
ïŒã€ãŸãm=2,a[1]=-2,a[2]=1ïŒ
F(x)=â«f(x)dx=x^3/3+x^2/2-2xãâ»å®æ°é
ã¯0ãšãã
G(x,-2)={F(x)-F(-2)}/(x+2)
={(x^3/3+x^2/2-2x)-(-8/3+2+4)}/(x+2)
=(2x^3+3x^2-12x-20)/(x+2)=(2x^2-x-10)/6
G(x,1)={F(x)-F(1)}/(x-1)
={(x^3/3+x^2/2-2x)-(1/3+1/2-2)}/(x-1)
=(2x^3+3x^2-12x+7)/{6(x-1)}=(2x^2+5x-7)/6
mã¯å¶æ°ãªã®ã§
â«|f(x)|dx=F(x)+Σ[k=1ïœm](-1)^kã»|x-a[k]|ã»G(x,a[k])+C
=(x^3/3+x^2/2-2x)-|x+2|(2x^2-x-10)/6+|x-1|(2x^2+5x-7)/6+C
={2x^3+3x^2-12x-|x+2|(2x^2-x-10)+|x-1|(2x^2+5x-7)}/6+C
ãã£ãŠ
â«[-a,a]|x^2+x-2|dx
={2a^3+3a^2-12a-|a+2|(2a^2-a-10)+|a-1|(2a^2+5a-7)}/6
ã-{-2a^3+3a^2+12a-|-a+2|(2a^2+a-10)+|-a-1|(2a^2-5a-7)}/6
={4a^3-24a-|a+2|(2a^2-a-10)-|a+1|(2a^2-5a-7)+|a-1|(2a^2+5a-7)+|a-2|(2a^2+a-10)}/6
={4a^3-24a-(a+2)(2a^2-a-10)-(a+1)(2a^2-5a-7)+|a-1|(2a^2+5a-7)+|a-2|(2a^2+a-10)}/6ãïŒâµaïŒ0ïŒ
={27+|a-1|(2a^2+5a-7)+|a-2|(2a^2+a-10)}/6
ä»ãŸã§ã¯ã°ã©ãçãå©çšã
aã«ãã£ãŠå ŽååããããŠ
0<aâŠ1ãªã-2/3*a^3+4*a
1âŠaâŠ2ãªãa^2+7/3
2âŠaãªã2/3*a^3-4*a+9
ãšåå¥ã«çããŠãããšæããŸãã
ãã®å
¬åŒã«ããïŒã€ã®å Žåã«åããŠèšè¿°ããŠãããã®ã
{27+|a-1|*(2*a^2+5*a-7)+|a-2|*(2*a^2+a-10)}/6
ã®äžã€ã®åŒã ãã§æžãŸãããã®ãïŒ
(äžèšã®3ã€ã®åŒããããã®åŒãæãã€ãã®ã¯è³é£ã®æã ãã
äžã®çµ¶å¯Ÿå€ãå«ãåŒããäžèšã®3ã€ã®åŒãå°ãã®ã¯å®¹æããïŒ
ãã£ããããæ°ã®è©±é¡ãåºãã®ã§ãé¢é£ããŠ
x^nãx^2-x-1ã§å²ã£ãåãšäœãã«ã¯ãã£ããããæ°ãå¯æ¥ã«é¢ãã£ãŠããã
x^2 = (x^2 - x - 1)*(1) + (x + 1)
x^3 = (x^2 - x - 1)*(x + 1) + (2*x + 1)
x^4 = (x^2 - x - 1)*(x^2 + x + 2) + (3*x + 2)
x^5 = (x^2 - x - 1)*(x^3 + x^2 + 2*x + 3) + (5*x + 3)
x^6 = (x^2 - x - 1)*(x^4 + x^3 + 2*x^2 + 3*x + 5) + (8*x + 5)
x^7 = (x^2 - x - 1)*(x^5 + x^4 + 2*x^3 + 3*x^2 + 5*x + 8) + (13*x + 8)
x^8 = (x^2 - x - 1)*(x^6 + x^5 + 2*x^4 + 3*x^3 + 5*x^2 + 8*x + 13) + (21*x + 13)
x^9 = (x^2 - x - 1)*(x^7 + x^6 + 2*x^5 + 3*x^4 + 5*x^3 + 8*x^2 + 13*x + 21) + (34*x + 21)
x^10 = (x^2 - x - 1)*(x^8 + x^7 + 2*x^6 + 3*x^5 + 5*x^4 + 8*x^3 + 13*x^2 + 21*x + 34) + (55*x + 34)
x^11 = (x^2 - x - 1)*(x^9 + x^8 + 2*x^7 + 3*x^6 + 5*x^5 + 8*x^4 + 13*x^3 + 21*x^2 + 34*x + 55) + (89*x + 55)
x^12 = (x^2 - x - 1)*(x^10 + x^9 + 2*x^8 + 3*x^7 + 5*x^6 + 8*x^5 + 13*x^4 + 21*x^3 + 34*x^2 + 55*x + 89) + (144*x + 89)
x^13 = (x^2 - x - 1)*(x^11 + x^10 + 2*x^9 + 3*x^8 + 5*x^7 + 8*x^6 + 13*x^5 + 21*x^4 + 34*x^3 + 55*x^2 + 89*x + 144) + (233*x + 144)
x^14 = (x^2 - x - 1)*(x^12 + x^11 + 2*x^10 + 3*x^9 + 5*x^8 + 8*x^7 + 13*x^6 + 21*x^5 + 34*x^4 + 55*x^3 + 89*x^2 + 144*x + 233) + (377*x + 233)
x^15 = (x^2 - x - 1)*(x^13 + x^12 + 2*x^11 + 3*x^10 + 5*x^9 + 8*x^8 + 13*x^7 + 21*x^6 + 34*x^5 + 55*x^4 + 89*x^3 + 144*x^2 + 233*x + 377) + (610*x + 377)

>ãã£ããããæ° {F(n)}ïŒ0,1,1,2,3,5,8,13,21,35,ãã«é¢ããŠãå¿
ã瀺ããã挞ååŒã
>ããF(n+2)=F(n+1)+F(n)
>ãããã§ããã®ãã£ããããæ°ã®mä¹æ°:F(n)^mã«ã€ããŠèª¿ã¹ããšã
>F(n+3)^2=2*F(n+2)^2+2*F(n+1)^2-F(n)^2
>F(n+4)^3=3*F(n+3)^3+6*F(n+2)^3-3*F(n+1)^3-F(n)^3
>F(n+5)^4=5*F(n+4)^4+15*F(n+3)^4-15*F(n+2)^4-5*F(n+1)^4+F(n)^4
>ãã
>ãæç«ããŠããŸãã
>ãm=5ã6ããã«ææŠããŠã»ããã
(F(n+6))^5, (F(n+7))^6 ã¯æ¬¡ã®ããã«ãªããŸãã
(F(n+6))^5=8*(F(n+5))^5+40*(F(n+4))^5-60*(F(n+3))^5-40*(F(n+2))^5+8*(F(n+1))^5+(F(n))^5,
(F(n+7))^6=13*(F(n+6))^6+104*(F(n+5))^6-260*(F(n+4))^6-260*(F(n+3))^6+104*(F(n+2))^6+13*(F(n+1))^6-(F(n))^6.
äžè¬ã«ã¯ã次ã®ããã«ãªããŸãã
mãæ£æŽæ°ãs,tãå®æ°(ãã ããt^2+4*sâ 0,tâ 0)ãšãããšãã挞ååŒ
a(n+2)=s*a(n)+t*a(n+1)
ãæºããæ°å {a(n)} ã«å¯ŸããŠãçåŒ
(a(n+m+1))^m
=Σ[k=1ïœm+1](a(n+m+1-k))^m*((-1)^(k+1))*((-s)^(k*(k-1)/2))*(Î [j=1ïœk]A(m+2-j)/A(j))
ãæãç«ã¡ãŸãã
ããã§ã{A(n)}ã¯ä»¥äžã§å®ãŸãæ°åã§ãã
A(0)=0,
A(1)=1,
A(n+2)=s*A(n)+t*A(n+1) (nâ§0).
(蚌æ)
α=(t+â(t^2+4*s))/2,β=(t-â(t^2+4*s))/2 ãšããŸãã
a(n)=v*α^n+w*β^n (v,wã¯å®æ°) ãšè¡šããŸãã
G(z)=Σ[nâ§0](a(n))^m*z^n ãšãããšã
G(z)
=Σ[nâ§0](v*α^n+w*β^n)^m*z^n
=Σ[nâ§0]z^n*(Σ[jâ§0]comb(m,j)*(v*α^n)^j*(w*β^n)^(m-j))
=Σ[nâ§0]z^n*(Σ[jâ§0]comb(m,j)*(v^j*w^(m-j))*((α/β)^j*β^m)^n)
=Σ[nâ§0]Σ[jâ§0]comb(m,j)*(v^j*w^(m-j))*(z*(α/β)^j*β^m)^n
=Σ[jâ§0]comb(m,j)*(v^j*w^(m-j))*(Σ[nâ§0](z*(α/β)^j*β^m)^n)
=Σ[jâ§0]comb(m,j)*(v^j*w^(m-j))*(1/(1-z*(α/β)^j*β^m)).
ãã£ãŠã
G(z/(β^m))=Σ[jâ§0]comb(m,j)*(v^j*w^(m-j))*(1/(1-z*(α/β)^j)).
䞡蟺㫠Π[j=0ïœm](1-z*(α/β)^j) ãããããšã
G(z/(β^m))*Î [j=0ïœm](1-z*(α/β)^j)=(zã«ã€ããŠã®m次以äžã®å€é
åŒ)
ãšãªããŸãã䞡蟺㮠z^(n+m+1) ã®ä¿æ°ãæ¯èŒããŠã
[z^(n+m+1)](G(z/(β^m))*Î [j=0ïœm](1-z*(α/β)^j))=0.
[z^(n+m+1)](G(z/(β^m))*Î [j=0ïœm](1-z*(α/β)^j))
=Σ[k=0ïœm+1]([z^(n+m+1-k)]G(z/(β^m)))*([z^k]Î [j=0ïœm](1-z*(α/β)^j))).
ããã§ã[z^(n+m+1-k)]G(z/(β^m))=(a(n+m+1-k))^m*(1/β)^(m*(n+m+1-k)).
ãŸãã[z^k]Î [j=0ïœm](1-z*(α/β)^j)ã¯å°ã
åä»ã§ããã
[z^k]Î [j=0ïœm](1-z*(α/β)^j)
=((-1)^k)*((α/β)^(k*(k-1)/2))*Î [j=1ïœk](1-(α/β)^(m+2-j))/(1-(α/β)^j)
ãšãªããŸãã
(äžè¬ã«ã[z^k](Î [j=0ïœm](1-z*γ^j))
=((-1)^k)*(γ^(k*(k-1)/2))*Î [j=1ïœk](1-γ^(m+2-j))/(1-γ^j))
ãšãªããŸãããã®ããšã¯åŸã«èšŒæããŸãã)
ãã£ãŠã
[z^(n+m+1)](G(z/(β^m))*Î [j=0ïœm](1-z*(α/β)^j))
=Σ[k=0ïœm+1](a(n+m+1-k))^m*(1/β)^(m*(n+m+1-k))*((-1)^k)*((α/β)^(k*(k-1)/2))*Î [j=1ïœk](1-(α/β)^(m+2-j))/(1-(α/β)^j)
=Σ[k=0ïœm+1](a(n+m+1-k))^m*((-1)^k)*((α*β)^(k*(k-1)/2))*(Î [j=1ïœk](β^(m+2-j)-α^(m+2-j))/(β^j-α^j))*(1/β)^(m*(n+m+1))
=Σ[k=0ïœm+1](a(n+m+1-k))^m*((-1)^k)*((-s)^(k*(k-1)/2))*(Î [j=1ïœk]A(m+2-j)/A(j))*(1/β)^(m*(n+m+1)).
ããã 0 ã«çããã®ã§ã
(a(n+m+1))^m = Σ[k=1ïœm+1](a(n+m+1-k))^m*((-1)^(k+1))*((-s)^(k*(k-1)/2))*(Î [j=1ïœk]A(m+2-j)/A(j)).
------------------------------------------------------------------
[z^k](Î [j=0ïœm](1-z*γ^j))
=((-1)^k)*(γ^(k*(k-1)/2))*Î [j=1ïœk](1-γ^(m+2-j))/(1-γ^j)
ã§ããããšã®èšŒæïŒ
G(γ,z)=Î [j=0ïœm](1-z*γ^j)ãšãããG(γ,z)ãå±éãããšãã® z^k
ã®ä¿æ°ã U(γ,k) ãšããŸãã
ãããããšãG(γ,z)=Σ[k=0ïœm+1]U(γ,k)*z^k.
çåŒ (1-z*γ^(m+1))*G(γ,z)=(1-z)*G(γ,z*γ) ãæãç«ã¡ãŸãã
ãã®çåŒã®äž¡èŸºã®z^kã®ä¿æ°ãæ¯èŒããŠã
U(γ,k)-U(γ,k-1)*γ^(m+1)=U(γ,k)*γ^k-U(γ,k-1)*γ^(k-1).
ãã£ãŠã
U(γ,k)
=((γ^(m+1)-γ^(k-1))/(1-γ^k))*U(γ,k-1)
=((γ^(m+1)-γ^(k-1))*(γ^(m+1)-γ^(k-2))/((1-γ^k)*(1-γ^(k-1))))*U(γ,k-2)
=âŠ
=(Î [j=1ïœk](γ^(m+1)-γ^(k-j))/(1-γ^j))*U(γ,0)
=Î [j=1ïœk](γ^(m+1)-γ^(k-j))/(1-γ^j)
=((-1)^k)*(γ^(k*(k-1)/2))*Î [j=1ïœk](1-γ^(m+2-j))/(1-γ^j).
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ã¡ãªã¿ã«10ä¹ãŸã§ã®åŒã眮ããŠãããŸãã
F(n+8)^7=21*(F(n+7))^7+273*(F(n+6))^7-1092*(F(n+5))^7-1820*(F(n+4))^7 +1092*(F(n+3))^7+273*(F(n+2))^7-21*(F(n+1))^7-(F(n))^7
F(n+9)^8=34*(F(n+8))^8+714*(F(n+7))^8-4641*(F(n+6))^8-12376*(F(n+5))^8 +12376*(F(n+4))^8+4641*(F(n+3))^8-714*(F(n+2))^8-34*F(n+1))^8+(F(n))^8
F(n+10)^9=55*(F(n+9))^9+1870*(F(n+8))^9-19635*(F(n+7))^9-85085*(F(n+6))^9+136136*(F(n+5))^9+85085*(F(n+4))^9-19635*(F(n+3))^9-1870*(F(n+2))^9+55*(F(n+1))^9+(F(n))^9
F(n+11)^10=89*(F(n+10))^10+4895*(F(n+9))^10-83215*(F(n+8))^10-582505*(F(n+7))^10+1514513*(F(n+6))^10+1514513*(F(n+5))^10
-582505*(F(n+4))^10-83215*(F(n+3))^10+4895*(F(n+2))^10 +89*(F(n+1))^10-(F(n))^10
ãªã10ä¹ã®åŒãæ§æããã«ã¯ãããã°ã©ã çã«
gp > A(n)=matrix(n,n,i,j,binomial(i-1,n-j));
gp > charpoly(A(11),x)
%150 =
x^11 - 89*x^10 - 4895*x^9 + 83215*x^8 + 582505*x^7 - 1514513*x^6
- 1514513*x^5 + 582505*x^4 + 83215*x^3 - 4895*x^2 - 89*x + 1
ãªã
gp > A(11)
%151 =
[0 0 0 0 0 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 1 2 1]
[0 0 0 0 0 0 0 1 3 3 1]
[0 0 0 0 0 0 1 4 6 4 1]
[0 0 0 0 0 1 5 10 10 5 1]
[0 0 0 0 1 6 15 20 15 6 1]
[0 0 0 1 7 21 35 35 21 7 1]
[0 0 1 8 28 56 70 56 28 8 1]
[0 1 9 36 84 126 126 84 36 9 1]
[1 10 45 120 210 252 210 120 45 10 1]
ã®è¡åãæå³ããã(ãã¹ã«ã«ã®äžè§åœ¢ãå³è©°ãã§äœãã)
ãã®è¡åã®ç¹æ§æ¹çšåŒãå°ãã®ãcharpolyã³ãã³ãã§ãã
ãã®åŒãã%150=0ãšçœ®ããŠ
äžæ°ã«
x^11=89*x^10 + 4895*x^9 - 83215*x^8 - 582505*x^7 + 1514513*x^6
+ 1514513*x^5 - 582505*x^4 - 83215*x^3 + 4895*x^2 + 89*x - 1
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