{"id":869,"date":"2013-08-07T01:04:39","date_gmt":"2013-08-06T16:04:39","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=869"},"modified":"2021-09-23T09:38:30","modified_gmt":"2021-09-23T00:38:30","slug":"thr201305","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr201305\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2013\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(2\\) \u6b21\u306e\u6b63\u65b9\u884c\u5217 \\(A\\) \u3092 \\(A = \\left( \\begin{array}{cc} -\\dfrac{1}{\\sqrt{2}} & -\\dfrac{1}{\\sqrt{2}} \\\\ \\dfrac{1}{\\sqrt{2}} & -\\dfrac{1}{\\sqrt{2}} \\end{array} \\right)\\) \u3067\u5b9a\u3081\u308b.\r\n\\(n = 1, 2, 3, \\cdots\\) \u306b\u5bfe\u3057\u3066, \u70b9 \\(\\text{P} {} _ n \\ \\left( x _ n , y _ n \\right)\\) \u3092\u95a2\u4fc2\u5f0f\r\n\\[\r\n\\left( \\begin{array}{c} x _ n \\\\ y _ n \\end{array} \\right) = A \\left( \\begin{array}{c} x _ {n-1} \\\\ y _ {n-1} \\end{array} \\right) + \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\quad ( n = 1, 2, 3, \\cdots ) \\ .\r\n\\]\r\n\u3067\u5b9a\u3081\u308b. \u305f\u3060\u3057, \\(x _ 0 = 1\\) , \\(y _ 0 = 0\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(A^4\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(n = 0, 1, 2, \\cdots\\) \u306b\u5bfe\u3057\u3066,\r\n\\[\r\n\\left( \\begin{array}{c} x _ n \\\\ y _ n \\end{array} \\right) = \\left( E-A^{n+1} \\right) (E-A)^{-1} \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\ .\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b. \u305f\u3060\u3057, \\(E\\) \u306f \\(2\\) \u6b21\u306e\u5358\u4f4d\u884c\u5217\u3068\u3059\u308b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u539f\u70b9 O \u304b\u3089 \\(\\text{P} {} _ n\\) \u307e\u3067\u306e\u8ddd\u96e2 \\(\\text{OP} {} _ n\\) \u304c\u6700\u5927\u3068\u306a\u308b \\(n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

    (1)<\/strong><\/p>\r\n

    \u539f\u70b9\u4e2d\u5fc3\u306b \\(\\theta\\) \u3060\u3051\u56de\u8ee2\u3059\u308b\u79fb\u52d5\u3092\u8868\u3059\u884c\u5217\u3092 \\(R( \\theta )\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nR( \\theta ) = \\left( \\begin{array}{cc} \\cos \\theta & -\\sin \\theta \\\\ \\sin \\theta & \\cos \\theta \\end{array} \\right) \\ .\r\n\\]\r\n\u6761\u4ef6\u3088\u308a\r\n\\[\r\nA = R \\left( \\dfrac{3 \\pi}{4} \\right) \\ .\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nA^4 = R( 3 \\pi ) = \\underline{\\left( \\begin{array}{cc} -1 & 0 \\\\ 0 & -1 \\end{array} \\right)} \\ .\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    \\[\\begin{align}\r\n\\left( E -A^{n+1} \\right) & (E-A)^{-1} \\\\\r\n& = \\left( E +A + \\cdots +A^n \\right) (E-A) (E-A)^{-1} \\\\\r\n& = \\textstyle\\sum\\limits _ {k=0}^n A^k \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(n = 0 , 1 , 2 , \\cdots\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\n\\left( \\begin{array}{c} x _ n \\\\ y _ n \\end{array} \\right) = \\textstyle\\sum\\limits _ {k=0}^n A^k \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\quad ... [ \\text{A} ] \\ .\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044.
    \r\n\u4ee5\u4e0b\u3067\u306f, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u3053\u308c\u3092\u793a\u3059.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(n = 0\\) \u306e\u3068\u304d, \u6761\u4ef6\u3088\u308a [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(n = \\ell \\ ( \\ell \\geqq 0 )\\) \u306e\u3068\u304d, [A] \u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\left( \\begin{array}{c} x _ {\\ell +1} \\\\ y _ {\\ell +1} \\end{array} \\right) & = A \\left( \\begin{array}{c} x _ {\\ell} \\\\ y _ {\\ell} \\end{array} \\right) +\\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\\\\r\n& = A \\textstyle\\sum\\limits _ {k=0}^{\\ell} A^k \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) +\\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\\\\r\n& = \\left( \\textstyle\\sum\\limits _ {k=1}^{\\ell +1} A^k +E \\right) \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\\\\r\n& = \\textstyle\\sum\\limits _ {k=0}^{\\ell +1} A^k \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\ .\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \\(n = \\ell +1\\) \u306e\u3068\u304d\u3082 [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

      (3)<\/strong><\/p>\r\n

      \\(B _ n = \\textstyle\\sum\\limits _ {k=0}^n A^k\\) \u3068\u304a\u304f.
      \r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\nB _ 7 & = E+A+A^2+A^3-E-A-A^2-A^3 \\\\\r\n& = O \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nB _ {n+8} & = A^{n+1} B _ 7 +B _ n \\\\\r\n& = B _ n \\ .\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n\\left( \\begin{array}{c} x _ {n+8} \\\\ y _ {n+8} \\end{array} \\right) = \\left( \\begin{array}{c} x _ {n} \\\\ y _ {n} \\end{array} \\right) \\quad ... [1] \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6700\u5927\u5024\u3092\u3068\u308b \\(\\text{P} {} _ n\\) \u306f \\(\\text{P} {} _ 0 , \\text{P} {} _ 1 , \\cdots , \\text{P} {} _ 7\\) \u306e \\(8\\) \u500b\u306e\u3044\u305a\u308c\u304b\u3067\u3042\u308b.
      \r\n\u3055\u3089\u306b, \\(n = 1 , 2 , \\cdots\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nB _ n -B _ {n-1} & = A^n = R \\left( \\dfrac{3n \\pi}{4} \\right) \\\\\r\n\\text{\u2234} \\quad \\overrightarrow{\\text{P} {} _ {n-1} \\text{P} {} _ {n}} & = \\left( \\begin{array}{c} \\cos \\frac{3n \\pi}{4} \\\\ \\sin \\frac{3n \\pi}{4} \\end{array} \\right) \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(\\text{P} {} _ 0 , \\text{P} {} _ 1 , \\cdots , \\text{P} {} _ 7\\) \u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b, \u6b63 \\(8\\) \u89d2\u5f62\u3092\u306a\u3059.<\/p>\r\n\"tohoku_r_2013_05_01\"\r\n

      \u3053\u3053\u3067, \u539f\u70b9 O \u304b\u3089\u306e\u8ddd\u96e2\u304c\u6700\u5927\u3067\u3042\u308b\u306e\u306f \\(\\text{P} {} _ 3\\) \u306a\u306e\u3067
      \r\n[1] \u306b\u6ce8\u610f\u3059\u308c\u3070, \u6c42\u3081\u308b \\(n\\) \u306f\r\n\\[\r\nn =\\underline{8k+3} \\quad ( k = 0 , 1 , 2 , \\cdots ) \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(2\\) \u6b21\u306e\u6b63\u65b9\u884c\u5217 \\(A\\) \u3092 \\(A = \\left( \\begin{array}{cc} -\\dfrac{1}{\\sqrt{2}} & -\\dfrac{1}{\\sqrt{2}} \\\\ \\dfrac{1}{\\ … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[96],"tags":[148,111],"class_list":["post-869","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2013","tag-tohoku_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/869","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=869"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/869\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=869"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=869"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=869"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}