■初等関数研究所■

[0001 ガンマ関数 2019/02/09(土) 17:29:38.68 ID:xWYHJ5Wt]

初等関数（しょとうかんすう、英: Elementary function）とは、
実数または複素数の1変数関数で、代数関数、指数関数、対数関数、
三角関数、逆三角関数および、それらの合成関数を作ることを
有限回繰り返して得られる関数のことである

ガンマ関数、楕円関数、ベッセル関数、誤差関数などは初等関数でない
初等関数のうちで代数関数でないものを初等超越関数という
双曲線関数やその逆関数も初等関数である

初等関数の導関数はつねに初等関数になる

[0745 １３２人目の素数さん 2019/06/15(土) 19:01:18.18 ID:u9iyV9ev]

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[0746 １３２人目の素数さん 2019/06/15(土) 19:43:32.42 ID:IIziHdYA]

縦3マス、横4マスの12マスのうちランダムに選ばれた
2マスにそれぞれ宝が眠っている
AEIBFJ…の順で縦に宝を探していく方法をとるP君と、
ABCDEFGH…の順で横に宝を探していく方法をとるQ君が、
同時に地点Aから探索を開始した
どっちの方が有利？

ABCD
EFGH
I JK L

P1st Q1st even
[1,] 0 0 1
[2,] 4 5 6
[3,] 26 27 13
[4,] 84 83 23
[5,] 203 197 35
[6,] 413 398 50
[7,] 751 722 67
[8,] 1259 1210 87
[9,] 1986 1910 109
[10,] 2986 2875 134

完全追尾型多項式が完成しました

宝の個数は2

P1st={12n^4+28n^3-42n^2-52n-3(-1)^n+51}/48

Q1st={12n^4+20n^3-18n^2-20n-3(-1)^n+3}/48

even={10n^2+8n+(-1)^n-9}/8

■Wolframに入力すると既約分数表示になるので御注意

P1st／Q1st

=8(n-1){(n-2)n-6}/{2n(n+2)(6n^2-2n-5)-3(-1)^n+3}+1

[0747 １３２人目の素数さん 2019/06/15(土) 19:44:17.04 ID:IIziHdYA]

P1stとQ1stは、『宝一つの時の自陣当たり数』の二乗と
それぞれの差分を表す関数の和で求められる

■P1stを求める

宝一つの時の自陣当たり数

n(n+1)/2-1　……�@

P1stは�@^2と差分の和

差分は0 0 1 3 7 13 22 34 50 70 95 125 161 203
252 308 372 444 525 615……

それを表す関数

(4n^3-6n^2-4n-3(-1)^n+3)/48　……�A

計算知能で�@^2+�Aを入力すると

P1st={12n^4+28n^3-42n^2-52n-3(-1)^n+51}/48

■Q1stを求める

宝一つの時の自陣当たり数

n(n+1)/2-1　……�@

Q1stは�@^2と差分の和

差分は0 1 2 2 1 -2 -7 -15 -26 -41 -60 -84 -113
-148 -189……

それを表す関数は　

(-4n^3+18n^2+28n-3(-1)^n-45)/48　……�B

計算知能で�@^2+�Bを入力すると

Q1st={12n^4+20n^3-18n^2-20n-3(-1)^n+3}/48

■evenを求める

evenは、n(n+1)-1と同着数の和

同着数は1 2 4 6 9 12 16 20 25……

これを表す関数は　{2n^2-1+(-1)^(n)}/8　……�C

n(n+1)-1　……�D

計算知能で�C+�Dを入力すると

even={10n^2+8n+(-1)^n-9}/8

[0748 １３２人目の素数さん 2019/06/15(土) 19:44:43.84 ID:IIziHdYA]

P1st Q1st even
[1,] 0 0 1
[2,] 4 5 6
[3,] 26 27 13
[4,] 84 83 23
[5,] 203 197 35
[6,] 413 398 50
[7,] 751 722 67
[8,] 1259 1210 87
[9,] 1986 1910 109
[10,] 2986 2875 134
[11,] 4320 4165 161
[12,] 6054 5845 191
[13,] 8261 7987 223
[14,] 11019 10668 258
[15,] 14413 13972 295
[16,] 18533 17988 335
[17,] 23476 22812 377
[18,] 29344 28545 422
[19,] 36246 35295 469
[20,] 44296 43175 519

Table[(12n^4+28n^3-42n^2-52n-3(-1)^n+51)/48,{n,1,20}]

Table[(12n^4+20n^3-18n^2-20n-3(-1)^n+3)/48,{n,1,20}]

Table[(10n^2+8n+(-1)^n-9)/8,{n,1,20}]

[0749 １３２人目の素数さん 2019/06/15(土) 19:46:33.87 ID:IIziHdYA]

2n x 2n の正方形を
1 x 2 のドミノで埋める場合の数を考えます

たとえば、2x2の正方形を1x2のドミノで埋める場合の数は、2通りです

4x4の正方形を1x2のドミノで埋める場合の数は、36通りです

一般に、n=0,1,2,3,,,,のとき、
1, 2, 36, 6728, 12988816, 258584046368,,,
となり、一般項は、

Π[j=1 to n]Π[k=1 to n]{4cos^2 πj/(2n+1)+4cos^2 πk/(2n+1)}

となるようなのですが、
どのようにその公式が導かれるのでしょうか?

[0750 １３２人目の素数さん 2019/06/15(土) 19:47:24.25 ID:IIziHdYA]

wikipedia
https://en.wikipedia.org/wiki/Domino_tiling
によると
Temperley & Fisher (1961) and Kasteleyn (1961)
によって独立に発見されたとある
多分元論文は
Temperley, H. N. V.; Fisher, Michael E. (1961),
"Dimer problem in statistical mechanics-an exact result",
Philosophical Magazine, 6 (68): 1061-1063, doi:10.1080/14786436108243366

Kasteleyn, P. W. (1961), "The statistics of dimers on a lattice. I.
The number of dimer arrangements on a quadratic lattice", Physica,
27 (12): 1209-1225, Bibcode:1961Phy....27.1209K, doi:10.1016/0031-8914(61)90063-5.

原論文読むのが早い

[0751 １３２人目の素数さん 2019/06/15(土) 19:48:11.89 ID:IIziHdYA]

これに証明載ってるかも
https://inis.iaea.org/collection/NCLCollectionStore/_Public/38/098/38098203.pdf?r=1&;r=1

Section2
A famous result of Kasteleyn [8] and Temperley and Fisher [18] counts the
number of domino tilings of a chessboard (or any other rectangular region).
In this section we explain Kasteleyn's proof.


「ドミノによるタイル張り」(京大･理)　36p.
http://www.ms.u-tokyo.ac.jp/~kazushi/proceedings/domino.pdf

「長方形領域のドミノタイル張りについて」(青学大･理工)　17p.
http://www.gem.aoyama.ac.jp/~kyo/sotsuken/2010/fujino_sotsuron_2010.pdf

[0752 １３２人目の素数さん 2019/06/15(土) 19:48:28.87 ID:IIziHdYA]

Table[{1-n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)(n-7)(n-8)(n-9)(n-10)(n-11)(n-12)/13!}/4,{n,0,13}]

Table[(1-C(0,n-13))/4,{n,0,13}]

[0753 １３２人目の素数さん 2019/06/15(土) 19:59:39.17 ID:j0UIFbY0]

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[0754 １３２人目の素数さん 2019/06/15(土) 20:00:38.15 ID:j0UIFbY0]

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[0755 １３２人目の素数さん 2019/06/15(土) 20:01:05.63 ID:j0UIFbY0]

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[0756 １３２人目の素数さん 2019/06/15(土) 20:01:56.99 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0757 １３２人目の素数さん 2019/06/15(土) 20:02:36.46 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0758 １３２人目の素数さん 2019/06/15(土) 20:03:17.62 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0759 １３２人目の素数さん 2019/06/15(土) 20:04:01.43 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0760 １３２人目の素数さん 2019/06/15(土) 20:04:41.30 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0761 １３２人目の素数さん 2019/06/15(土) 20:05:28.20 ID:j0UIFbY0]

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Kvr3idZa3OUlonClNyGyT3u2Xrtde47Cr6m4tG1j7AurlCjmUXvLPaQDQLlhymjaNIkWblKiKeVhk601XohEk7mNq3FyXjLGTJjx4csGI9MHt9vijbaaAQMFGIi8A28SQA1Ie3oELhvKeuLzK9ZYmGMEVqj4GtOgB719u1e1KHHqpfnGgwmMFMpRTjoTrEl4f9KFathh
kjasjfkajkljfklajjfjksdjksjalkjflkjasjfkjasjfkjajjs2333354994998989029929050295895028902802058299202095898582982092029209029029
sgssl;slg;ld;l

[0762 １３２人目の素数さん 2019/06/15(土) 20:06:05.50 ID:j0UIFbY0]

0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;l

[0763 １３２人目の素数さん 2019/06/15(土) 20:07:54.13 ID:IIziHdYA]

>>82
どのスートが出るのも同様に確からしい
ジョーカーを除くトランプのカード52枚から
一枚のカードを箱に入れる

Ωの部分集合を事象と言う
Ω自身は全事象と言う

Ω=｛スペード，ハート，クラブ，ダイヤ｝となる

各 i （1≦i≦4） が根元事象である

ハートが出るという事象A=｛ハート｝で確率P(A)は

P(A)=1/4　となる

最初に箱に入れた時を i
山札をシャッフルしてダイヤが三枚出た後を ｊ として

箱の中のカードがハートであるという事象Aを考える.

A=｛（i，j）|　i または j がハート｝

Ω=｛（i，j）|1≦i≦4，1≦j≦49｝となり

この196通りの各要素が根元事象

[0764 １３２人目の素数さん 2019/06/15(土) 20:08:45.72 ID:IIziHdYA]

シャッフル後にダイヤのカードをn枚引いた時に
箱の中にダイヤ以外のスートが出る確率空間は

Ω＝｛（i，j）|1≦i≦4，1≦j≦52-n｝から

#A=4(52-n)-3(51-n)

　=208-4n-153+3n

　=55-n

#Aは事象Aに含まれる要素の個数

スペード・ハート・クラブの各スートの出る確率は

P(A)=(55-n)/(208-4n)

スペード・ハート・クラブである確率は

P(X)=(165-3n)/(208-4n)

ダイヤである確率は

q=1-(165-3n)/(208-4n)

しかしこのままでは
点(0,1/4),(13,0) を通らない

[0765 １３２人目の素数さん 2019/06/15(土) 20:09:21.00 ID:IIziHdYA]

■点(0,1/4),(13,0) を通るように二次関数にする

1-(165-3n)/(208-4n)　から
1-(165n-3n^2+3b)/(208n-4n^2+4b)　とおくと

n=0,b≧1のとき、1/4が出力できる

さらにn=13のときに(165n-3n^2+3b)=(208n-4n^2+4b)　
となれば、0が出力できる

このためには、分母を分子よりも小さくして

1-(165n-3n^2+3b)/(208n-7n^2+4b)　

その差分をb=117で回収すると完成

∴1-(165n-3n^2+351)/(208n-7n^2+468)　

式変形すると

(4n+9)(n-13)/(7n^2-208n-468)

■Wolfram入力

Table[(4n+9)(n-13)/(7n^2-208n-468),{n,0,13}]

[0766 １３２人目の素数さん 2019/06/15(土) 20:09:36.79 ID:IIziHdYA]

Functional Analysis

[0767 １３２人目の素数さん 2019/06/15(土) 20:10:35.58 ID:IIziHdYA]

■平方完成

y=ax^2-(-a+2)x-a-a+2
=a(x^2-(-a+2)x/a)-a-a+2
=a{(x-(-a+2)/(2a))^2-(-a+2)^2/(4a^2)}-a-a+2
=a(x-(-a+2)/(2a))^2-(-a+2)^2/(4a)-a-a+2
=a(x-(-a+2)/(2a))^2-(-a+2)^2/(4a)-2a+2
=a(x-(-a+2)/(2a))^2-(a^2-4a+4)/(4a)-2a+2
=a(x-(-a+2)/(2a))^2-(a^2-4a+4)/(4a)-(8a^2)/(4a)+(8a)/(4a)
=a(x-(-a+2)/(2a))^2-(a^2-4a+4+8a^2-8a)/(4a)
=a(x-(-a+2)/(2a))^2-(9a^2-12a+4)/(4a)

[0768 １３２人目の素数さん 2019/06/15(土) 20:11:00.33 ID:IIziHdYA]

初等関数（しょとうかんすう、英: Elementary function）とは、
実数または複素数の1変数関数で、代数関数、指数関数、対数関数、
三角関数、逆三角関数および、それらの合成関数を作ることを
有限回繰り返して得られる関数のことである

ガンマ関数、楕円関数、ベッセル関数、誤差関数などは初等関数でない
初等関数のうちで代数関数でないものを初等超越関数という
双曲線関数やその逆関数も初等関数である

初等関数の導関数はつねに初等関数になる

[0769 １３２人目の素数さん 2019/06/15(土) 20:12:29.99 ID:IIziHdYA]

384=8!!　

53760=2(10!!)+12!!

8755200=8(12!!)+13(14!!)

1805690880=15(14!!)+12(16!!)+9(18!!)

471092428800=10(16!!)+15(18!!)+16(20!!)+5(22!!)

[0770 １３２人目の素数さん 2019/06/15(土) 20:13:23.44 ID:IIziHdYA]

■フィボナッチ数列（英: Fibonacci sequence）

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,
987, 1597, 2584, 4181, 6765, 10946, …

Fn=(1/sqrt(5))(((1+sqrt(5))/2)^n-((1-sqrt(5))/2)^n)

[0771 １３２人目の素数さん 2019/06/15(土) 20:16:41.04 ID:IIziHdYA]

□□■■■□□□□■■■□□
□■■■■■□□■■■■■□
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□■■■■■■■■■■■■□
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□□□■■■■■■■■□□□
□□□□■■■■■■□□□□
□□□□□■■■■□□□□□
□□□□□□■■□□□□□□

[0772 １３２人目の素数さん 2019/06/15(土) 20:18:12.22 ID:IIziHdYA]

ガンマ関数とベータ関数
https://lecture.ecc.u-tokyo.ac.jp/~nkiyono/2006/miya-gamma.pdf

[0773 １３２人目の素数さん 2019/06/15(土) 20:19:49.24 ID:IIziHdYA]

とにかく今はゼータζ(s)だろ
素数とは一体なんなのか？
どんな調和があるのか
物理も化学もそう宇宙も全てが分かる瞬間こそ
素数そしてゼータ関数の解明である

フェルマー解いたワイルズは世界一有名な数学者の一人だろう
ゼータζゼロ点を解明しワイルズを超えたい

[0774 １３２人目の素数さん 2019/06/15(土) 20:22:18.33 ID:IIziHdYA]

Table[((-2)^(1+n)E^2+2Gamma[1+n,-2]+n Gamma[1+n,-2])/(E^2 n!),{n,1,28}]

Table[(e^2(-2)^(n+1)+n Γ(n+1,-2)+2 Γ(n+1,-2))/(e^2 n!),{n,1,28}]

{1, 0, 1, 2/3, 1, 16/15, 11/9, 142/105, 67/45, 4604/2835,
2771/1575, 59086/31185, 86327/42525, 4389248/2027025,
7533469/3274425, 222205682/91216125, 109456873/42567525,
2670957188/986792625, 16332117629/5746615875,
614053057522/206239658625, 1520442379271/488462349375,
126606575859992/38979295480125, 345404844856129/102088631019375,
15773069242557338/4482618980214375, 23501345644011017/6431583754220625,
4671255121834288564/1232720219558953125,
7547413632563686237/1923043542511966875,
23846953668187649602/5873549281427953125}

[0775 １３２人目の素数さん 2019/06/15(土) 20:23:17.97 ID:IIziHdYA]

第一種の合流型超幾何関数（クンマー）

1Ｆ1[a; b; z] = 1+Σ[k=1, ∞] {a(a+1)････(a+k-1)/b(b+1)････(b+k-1)} z^k/k!

1Ｆ1[-n; -2n; z] = {n!/(2n)!} Σ[k=0, n] {(2n-k)!/(n-k)!k!} z^k

[0776 １３２人目の素数さん 2019/06/15(土) 20:28:19.28 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0777 １３２人目の素数さん 2019/06/15(土) 20:28:55.07 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0778 １３２人目の素数さん 2019/06/15(土) 20:29:29.37 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0779 １３２人目の素数さん 2019/06/15(土) 20:30:08.32 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0780 １３２人目の素数さん 2019/06/15(土) 20:30:51.23 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0781 １３２人目の素数さん 2019/06/15(土) 20:31:29.44 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0782 １３２人目の素数さん 2019/06/15(土) 20:32:04.83 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0783 １３２人目の素数さん 2019/06/15(土) 20:32:42.79 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0784 １３２人目の素数さん 2019/06/15(土) 20:33:16.65 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0785 １３２人目の素数さん 2019/06/15(土) 20:33:46.09 ID:j0UIFbY0]

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Kvr3idZa3OUlonClNyGyT3u2Xrtde47Cr6m4tG1j7AurlCjmUXvLPaQDQLlhymjaNIkWblKiKeVhk601XohEk7mNq3FyXjLGTJjx4csGI9MHt9vijbaaAQMFGIi8A28SQA1Ie3oELhvKeuLzK9ZYmGMEVqj4GtOgB719u1e1KHHqpfnGgwmMFMpRTjoTrEl4f9KFathh
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sgssl;slg;ld;l

[0786 １３２人目の素数さん 2019/06/15(土) 21:37:26.03 ID:IIziHdYA]

cos π/5 +i sin π/5

[0787 １３２人目の素数さん 2019/06/15(土) 21:40:08.79 ID:yCJa48fV]

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kjasjfkajkljfklajjfjksdjksjalkjflkjasjfkjasjfkjajjs2333354994998989029929050295895028902802058299202095898582982092029209029029
sgssl;slg;ld;l

[0788 １３２人目の素数さん 2019/06/15(土) 21:40:56.29 ID:yCJa48fV]

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sgssl;slg;ld;l

[0789 １３２人目の素数さん 2019/06/15(土) 21:41:02.13 ID:IIziHdYA]

例えば、6なら、

6→3→10→5→16→8→4→2→1

のように、偶数なら2で割り、奇数なら3倍に1を足すのを
繰り返して、1になるまでの回数(∈Ｎ)

この数列がすべての自然数で定義されるかどうかの
証明は知りません

[0790 １３２人目の素数さん 2019/06/15(土) 21:41:35.31 ID:yCJa48fV]

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sgssl;slg;ld;l

[0791 １３２人目の素数さん 2019/06/15(土) 21:42:08.21 ID:yCJa48fV]

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sgssl;slg;ld;l

[0792 １３２人目の素数さん 2019/06/15(土) 21:42:18.62 ID:IIziHdYA]

Table[(1!/(1-k)!)/k!,{k,1,20}]

{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

[0793 １３２人目の素数さん 2019/06/15(土) 21:42:50.52 ID:yCJa48fV]

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sgssl;slg;ld;l

[0794 １３２人目の素数さん 2019/06/15(土) 21:43:01.79 ID:IIziHdYA]

2n x 2n の正方形を
1 x 2 のドミノで埋める場合の数を考えます

たとえば、2x2の正方形を1x2のドミノで埋める場合の数は、2通りです

4x4の正方形を1x2のドミノで埋める場合の数は、36通りです

一般に、n=0,1,2,3,,,,のとき、
1, 2, 36, 6728, 12988816, 258584046368,,,
となり、一般項は、

Π[j=1 to n]Π[k=1 to n]{4cos^2 πj/(2n+1)+4cos^2 πk/(2n+1)}

となるようなのですが、
どのようにその公式が導かれるのでしょうか?

[0795 １３２人目の素数さん 2019/06/15(土) 21:43:26.62 ID:yCJa48fV]

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sgssl;slg;ld;l

[0796 １３２人目の素数さん 2019/06/15(土) 21:43:59.51 ID:IIziHdYA]

FromDigits[{1,0,1,0,0,1,0,0}, 2]

164

[0797 １３２人目の素数さん 2019/06/15(土) 21:44:16.20 ID:yCJa48fV]

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sgssl;slg;ld;l

[0798 １３２人目の素数さん 2019/06/15(土) 21:44:58.71 ID:yCJa48fV]

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sgssl;slg;ld;l

[0799 １３２人目の素数さん 2019/06/15(土) 21:45:38.53 ID:yCJa48fV]

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sgssl;slg;ld;l

[0800 １３２人目の素数さん 2019/06/15(土) 21:46:24.22 ID:yCJa48fV]

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sgssl;slg;ld;l

[0801 １３２人目の素数さん 2019/06/15(土) 21:57:54.87 ID:yCJa48fV]

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sgssl;slg;ld;l

[0802 １３２人目の素数さん 2019/06/15(土) 21:58:33.79 ID:yCJa48fV]

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sgssl;slg;ld;l

[0803 １３２人目の素数さん 2019/06/15(土) 21:59:06.17 ID:yCJa48fV]

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sgssl;slg;ld;l

[0804 １３２人目の素数さん 2019/06/15(土) 21:59:42.55 ID:yCJa48fV]

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sgssl;slg;ld;l

[0805 １３２人目の素数さん 2019/06/15(土) 22:00:18.56 ID:yCJa48fV]

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sgssl;slg;ld;l

[0806 １３２人目の素数さん 2019/06/15(土) 22:00:52.43 ID:yCJa48fV]

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sgssl;slg;ld;l

[0807 １３２人目の素数さん 2019/06/15(土) 22:01:26.75 ID:yCJa48fV]

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sgssl;slg;ld;l

[0808 １３２人目の素数さん 2019/06/15(土) 22:02:06.57 ID:yCJa48fV]

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sgssl;slg;ld;l

[0809 １３２人目の素数さん 2019/06/15(土) 22:02:52.49 ID:yCJa48fV]

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sgssl;slg;ld;l

[0810 １３２人目の素数さん 2019/06/15(土) 22:03:31.73 ID:yCJa48fV]

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sgssl;slg;ld;l

[0811 １３２人目の素数さん 2019/06/15(土) 22:04:21.84 ID:yCJa48fV]

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sgssl;slg;ld;l

[0812 １３２人目の素数さん 2019/06/15(土) 22:04:56.35 ID:yCJa48fV]

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sgssl;slg;ld;l

[0813 １３２人目の素数さん 2019/06/15(土) 22:05:43.89 ID:yCJa48fV]

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sgssl;slg;ld;l

[0814 １３２人目の素数さん 2019/06/15(土) 22:06:33.22 ID:yCJa48fV]

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5f1wRmcZzXXh9xqvyHtHqtXG06b6iQ5OhfrAUZwU8Scwkh52X7iNRot4vwSfMrmjYoGlVIvhK9djdlkiGy03ly9O6SmKKfkBYZJCK8zLNCJux0nBGVJWVe90kIjRFBTCjOfe11bfeVXfLUM9mLp0zyFrfY4a1dC7nS9pShB2iDxRGp6Vn2SlReeXnc6mqJ6KfhY9L8gR
G1SMJQYyXNSnrSkV2JyMHPsH4umT9610YnFgDQnLn67betRIHPmdDewhiu5kGTYKxytAxhC1qJPuOVRw5q7ZpELCBwYEixZs6kmHkla4fdAZ5HQw7xnySrJ28cGOloZuerh0QEG6xb3JxuzHxFGdBWtgEy5decyx9iZpMlo5QVT14hFNLnp0zlcAGUfEdxAMjUP3lHkv
UHzFXQXLWlU2JyYfOkcUjg40eeXu8e77qS4rgvjHWwxRbA1IKsNmn3CR9ePAuumjHnrWXeVuNStOENI4pHd7a7EwvoL8D7oenfbfeVoMT4Jho808YluyiYSEkfV5E6qPA5SiIWgl3G2zIc1NGsEW0nQuGTP504IXiRebMTKIEZyBdIqgQg1tOUYg7LzaYKiKixwj59ph
cILXg6ToA2TskeIlZraUTbzQ5PkR9lGnJjBFN7aSfv8vkCEpe9hYmrfF47H0RcNX5k3Y3i5xgHKhiNu5T8GeXfcYWpG6eLzIDAFy8DF39cqoofzCDnk8Ogt5q2H4cQNTgsDQDYYmFl2kKkYeX6CZ9LQruT8LprERVMyn0lUCYqfO4sQqWCu8kiVnpdZgd9QqdVcOT639
SDK4t8ql63OVBPRjJe2DvhC0BHjXErI2RCeGdMeBPD539aNqdFVIPGHN1NIVVDyTM4gfJbnFDgC5slKjSO17coT9jUfpOvezlHg9lXM95eftZiKzTx36T6C88TnssEI0tM3SKlDidfP9neTR3feD1cDGkYAzaPDCjyD4a2OcqNChan0XweFrq0xqQYc6Oi6am5DWfurQ
Kvr3idZa3OUlonClNyGyT3u2Xrtde47Cr6m4tG1j7AurlCjmUXvLPaQDQLlhymjaNIkWblKiKeVhk601XohEk7mNq3FyXjLGTJjx4csGI9MHt9vijbaaAQMFGIi8A28SQA1Ie3oELhvKeuLzK9ZYmGMEVqj4GtOgB719u1e1KHHqpfnGgwmMFMpRTjoTrEl4f9KFathh
kjasjfkajkljfklajjfjksdjksjalkjflkjasjfkjasjfkjajjs2333354994998989029929050295895028902802058299202095898582982092029209029029
sgssl;slg;ld;l

[0815 １３２人目の素数さん 2019/06/15(土) 23:11:50.34 ID:IIziHdYA]

Table[sum[C(2n-1+α,k-1),{n,1,a}],{k,1,b}]

a=n(n+1)/2-1　
b=n(n+1)

を満たす差分追尾数列αを見つけてくれ〜(・ω・)ノ

[0816 １３２人目の素数さん 2019/06/16(日) 14:27:10.78 ID:EELeRVzV]

※次スレ

■初等関数研究村■
https://rio2016.5ch.net/test/read.cgi/math/1560604951/

[0817 １３２人目の素数さん 2019/06/16(日) 14:31:53.23 ID:EELeRVzV]

2 3 6 7 9
2 3 6 7 8 12 13 15 17
2 3 6 7 8 12 13 14 16 20 21 23 25 27
2 3 6 7 8 12 13 14 15 20 21 22 24 26 30 31 33 35 37 39
2 3 6 7 8 12 13 14 15 20 21 22 23 25 30 31 32 34 36 38 42 43 45 47 49 51 53

長軸choose数え上げ

[0818 １３２人目の素数さん 2019/06/16(日) 14:36:14.30 ID:EELeRVzV]

■高校生がDDoS攻撃で国内初検挙！ 青少年のサイバー犯罪の驚異
2014年10月01日 18時15分更新

[0819 １３２人目の素数さん 2019/06/16(日) 15:41:21.60 ID:EELeRVzV]

■連続投稿・重複

連続投稿・コピー＆ペースト
連続投稿で利用者の会話を害しているものは削除対象になります
個々の内容に違いがあっても、荒らしを目的としていると判断したものは同様です
コピー＆ペーストやテンプレートの存在するものは、アレンジが施してあれば
残しますが、全く変更されていない・一部のみの変更で内容の変わらないもの、
スレッドの趣旨と違うもの、不快感を与えるのが目的なもの、
などは荒らしの意図があると判断して削除対象になります


※お手数ですが削除依頼できる方お願いします<(_ _)>

[0820 １３２人目の素数さん 2019/06/16(日) 15:45:44.48 ID:EELeRVzV]

■書き込む前におやくそく

近頃、初心者さんが増えてきたので、お約束をつくってみました
基本的には、ユーザーの良識に任せたいのですが、
「明文化したルールがない」ということを逆手にとってなにをしてもいいと
誤解する人が多いので、、、
by　ひろゆき


【ロビーのお約束】 削除の要件（禁止されること）

荒らし依頼・ブラクラの張付け等第3者に迷惑がかかる行為
アダルト広告・勧誘・悪質な掲示板宣伝などのアドレス等張りつけ
煽り・煽りに対する返答・叩き・誹謗中傷等（差別発言等含む）
コピペ・アスキーアート等必要以上の張り付け または
第3者に迷惑が掛かる行為や発言であった場合は削除対象にします

[0821 １３２人目の素数さん 2019/06/16(日) 16:07:19.72 ID:EELeRVzV]

トランプの束がある
2〜10までの数字が描かれたカードが各スートに1枚ずつと、
ジョーカーのカードが24枚ある
全てを混ぜて無作為に切り直して12枚のカードを無作為に引いたとき
その12枚のカードのうちジョーカー以外にいずれも違う数字が
書かれている確率はいくらか

Sum[choose(24,k)*choose(9,12-k)*4^(12-k),{k,3,12}]/(choose(60,12))

Sum[C(24,k)C(9,12-k)4^(12-k),{k,3,12}]/(C(60,12))

出力　7371811052/66636135475

[0822 １３２人目の素数さん 2019/06/16(日) 16:42:14.88 ID:EELeRVzV]

>>532
Table[C(0,4mod n),{n,1,10}]

{1, 1, 0, 1, 0, 0, 0, 0, 0, 0}

[0823 １３２人目の素数さん 2019/06/16(日) 17:02:58.01 ID:EELeRVzV]

短軸有利☆

Table[C(9,k-1)+C(7,k-1)+C(5,k-1)+C(4,k-1)+C(1,k-1),{k,1,12}]

Table[sum[C(2n-1+C(0,n-2),k-1),{n,1,5}],{k,1,12}]

{5, 26, 73, 133, 167, 148, 91, 37, 9, 1, 0, 0}

k-1を一つにして式を短縮

[0824 １３２人目の素数さん 2019/06/16(日) 17:09:07.41 ID:EELeRVzV]

Table[C(0,n-1 mod7),{n,1,14}]

{1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0}

[0825 １３２人目の素数さん 2019/06/16(日) 17:17:50.84 ID:EELeRVzV]

Functional Analysis

[0826 １３２人目の素数さん 2019/06/16(日) 17:29:37.74 ID:EELeRVzV]

((-1)^n)(((-1)^n)n+n+4(-1)^n+2)/2

1 5 1 7 1 9 1 11 1 13 1 15 1 17 1

[0827 １３２人目の素数さん 2019/06/16(日) 17:30:15.12 ID:EELeRVzV]

分数で表すと厳密な結果

小数で表すと近似になる

[0828 １３２人目の素数さん 2019/06/16(日) 17:45:47.52 ID:EELeRVzV]

短軸有利☆

Table[C(9,k-1)+C(7,k-1)+C(5,k-1)+C(4,k-1)+C(1,k-1),{k,1,12}]

Cの数は宝一つの時の当たり数の5
9+7+5+4+1=26は宝二個の時の当たり数になる

長軸有利☆

Table[C(9,k-1)+C(7,k-1)+C(6,k-1)+C(3,k-1)+C(2,k-1),{k,1,12}]

Cの数は宝一つの時の当たり数の5
9+7+6+3+2=27は宝二個の時の当たり数になる

[0829 １３２人目の素数さん 2019/06/16(日) 17:47:08.55 ID:EELeRVzV]

同様に20マスの場合は

短軸有利のCの数は宝一つの時の当たり数の9
17+15+13+11+10+8+5+4+1=84

長軸有利のCの数は宝一つの時の当たり数の9
17+15+13+12+8+7+6+3+2=83

[0830 １３２人目の素数さん 2019/06/16(日) 17:48:09.85 ID:EELeRVzV]

ヤン・ルカン、ジェフリーヒントン、Yoshua Bengioの

3人がチューリング賞受賞した

[0831 １３２人目の素数さん 2019/06/16(日) 17:50:56.14 ID:EELeRVzV]

100!中の二進数字の桁数を求める：

In[1]:=IntegerLength[100!, 2]

Out[1]=525

[0832 １３２人目の素数さん 2019/06/16(日) 17:53:19.34 ID:EELeRVzV]

q1..q2..q3..q4
q5..q6..q7..q8
q9q10q11q12

p1..p4..p7..p10
p2..p5..p8..p11
p3..p6..p9..p12

同じ座標なら数字の小さいほうが勝ち

［q2とq10］ & ［p4とp6］に宝が配置された時は

互いに数字の小さいほうを選んで勝負

q2 vs p4 で　q2の勝ちとなる

この後にq10とp6の探査をしても
情報としての価値はゼロ

[0833 １３２人目の素数さん 2019/06/16(日) 17:54:55.06 ID:EELeRVzV]

1 個のサイコロを 10 回投げたとき，1 または 2 の目が
ちょうど 4 回出る確率を求めよ

P(4)=C(10,4)P^4Q^(10-4)
　　　=C(10,4)(1/3)^4(2/3)^6

[0834 １３２人目の素数さん 2019/06/16(日) 17:56:16.64 ID:EELeRVzV]

シズカデス、メローネ

[0835 １３２人目の素数さん 2019/06/16(日) 17:56:49.74 ID:EELeRVzV]

(1 + (-1)^n)/2

[0836 １３２人目の素数さん 2019/06/16(日) 17:57:15.58 ID:EELeRVzV]

a_n = (-1/4 + i/4) ((-i)^n + i^(n + 1) + (-1 - i))

1 1 0 0 1 1 0 0

[0837 １３２人目の素数さん 2019/06/16(日) 18:07:02.39 ID:EELeRVzV]

e^(-2)

limit[ne^(-2)]

[0838 １３２人目の素数さん 2019/06/16(日) 18:07:36.10 ID:EELeRVzV]

確率空間においては, A ∈ F を事象 (event) と呼ぶ.

[0839 １３２人目の素数さん 2019/06/16(日) 18:08:23.59 ID:EELeRVzV]

■ランダムウォーク(random walk)

[0840 １３２人目の素数さん 2019/06/16(日) 18:09:04.24 ID:EELeRVzV]

a_n = 1/4 ((-1)^n - (1 + 2 i) (-i)^n - (1 - 2 i) i^n + 9)

1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2,

[0841 １３２人目の素数さん 2019/06/16(日) 18:09:45.38 ID:EELeRVzV]

合流型超幾何微分方程式
(confluent hypergeometric differential equation)

[0842 １３２人目の素数さん 2019/06/16(日) 18:10:37.54 ID:EELeRVzV]

(a-b-c)(a+b-c)(a-b+c)(a+b+c)

a^4-2a^2b^2-2a^2c^2+b^4-2b^2c^2+c^4

[0843 １３２人目の素数さん 2019/06/16(日) 18:15:42.96 ID:EELeRVzV]

(1+E^(IPi+InPi)+2n)/4

[0844 １３２人目の素数さん 2019/06/16(日) 18:16:08.82 ID:EELeRVzV]

ComplexExpand[(1+E^(I(1+n)Pi)+2n)/4]