{"id":1128,"date":"2015-06-22T10:17:20","date_gmt":"2015-06-22T01:17:20","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1128"},"modified":"2021-09-24T17:32:26","modified_gmt":"2021-09-24T08:32:26","slug":"tok201403","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201403\/","title":{"rendered":"\u6771\u5de5\u59272014\uff1a\u7b2c3\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(1\\) \u500b\u306e\u3055\u3044\u3053\u308d\u3092\u6295\u3052\u3066, \u51fa\u305f\u76ee\u304c \\(1\\) \u304b \\(2\\) \u3067\u3042\u308c\u3070\u884c\u5217 \\(A = \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array} \\right)\\) \u3092, \u51fa\u305f\u76ee\u304c \\(3\\) \u304b \\(4\\) \u3067\u3042\u308c\u3070\u884c\u5217 \\(B = \\left( \\begin{array}{cc} 0 & -1 \\\\ 1 & 0 \\end{array} \\right)\\) \u3092, \u51fa\u305f\u76ee\u304c \\(5\\) \u304b \\(6\\) \u3067\u3042\u308c\u3070\u884c\u5217 \\(C = \\left( \\begin{array}{cc} -1 & 0 \\\\ 0 & 1 \\end{array} \\right)\\) \u3092\u9078\u3076.\r\n\u305d\u3057\u3066, \u9078\u3093\u3060\u884c\u5217\u306e\u8868\u3059 \\(1\\) \u6b21\u5909\u63db\u306b\u3088\u3063\u3066 \\(xy\\) \u5e73\u9762\u4e0a\u306e\u70b9 R \u3092\u79fb\u3059\u3068\u3044\u3046\u64cd\u4f5c\u3092\u884c\u3046. \u70b9 R \u306f\u6700\u521d\u306f\u70b9 \\((0,1)\\) \u306b\u3042\u308b\u3082\u306e\u3068\u3057, \u3055\u3044\u3053\u308d\u3092\u6295\u3052\u3066\u70b9 R \u3092\u79fb\u3059\u64cd\u4f5c\u3092 \\(n\\) \u56de\u7d9a\u3051\u3066\u884c\u3063\u305f\u3068\u304d\u306b\u70b9 R \u304c\u70b9 \\((0,1)\\) \u306b\u3042\u308b\u78ba\u7387\u3092 \\(p _ n\\) , \u70b9 \\((0,-1)\\) \u306b\u3042\u308b\u78ba\u7387\u3092 \\(q _ n\\) \u3068\u3059\u308b.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(p _ 1 , p _ 2\\) \u3068 \\(q _ 1 , q _ 2\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(p _ n + q _ n\\) \u3068 \\(p _ {n-1} + q _ {n-1}\\) \u306e\u95a2\u4fc2\u5f0f\u3092\u6c42\u3081\u3088. \u307e\u305f, \\(p _ n - q _ n\\) \u3068 \\(p _ {n-1} - q _ {n-1}\\) \u306e\u95a2\u4fc2\u5f0f\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\\(p _ n\\) \u3092 \\(n\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n<hr>\r\n<!--more-->\r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<p>A \\(( 0 , 1 )\\) , B \\(( 0 , -1 )\\) , C \\(( 1 , 0 )\\) , D \\(( -1 , 0 )\\) \u3068\u304a\u304f.\r\n\\[\\begin{align}\r\nA \\left( \\begin{array}{c} 0 \\\\ \\pm 1 \\end{array} \\right) & = \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array} \\right) \\left( \\begin{array}{c} 0 \\\\ \\pm 1 \\end{array} \\right) = \\left( \\begin{array}{c} \\pm 1 \\\\ 0 \\end{array} \\right) \\\\\r\nB \\left( \\begin{array}{c} 0 \\\\ \\pm 1 \\end{array} \\right) & = \\left( \\begin{array}{cc} 0 & -1 \\\\ 1 & 0 \\end{array} \\right) \\left( \\begin{array}{c} 0 \\\\ \\pm 1 \\end{array} \\right) = \\left( \\begin{array}{c} \\mp 1 \\\\ 0 \\end{array} \\right) \\\\\r\nC \\left( \\begin{array}{c} 0 \\\\ \\pm 1 \\end{array} \\right) & = \\left( \\begin{array}{cc} -1 & 0 \\\\ 0 & 1 \\end{array} \\right) \\left( \\begin{array}{c} 0 \\\\ \\pm 1 \\end{array} \\right) = \\left( \\begin{array}{c} 0 \\\\ \\pm 1 \\end{array} \\right)\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nA \\left( \\begin{array}{c} \\pm 1 \\\\ 0 \\end{array} \\right) & = \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array} \\right) \\left( \\begin{array}{c} \\pm 1 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} 0 \\\\ \\mp 1 \\end{array} \\right) \\\\\r\nB \\left( \\begin{array}{c} \\pm 1 \\\\ 0 \\end{array} \\right) & = \\left( \\begin{array}{cc} 0 & -1 \\\\ 1 & 0 \\end{array} \\right) \\left( \\begin{array}{c} \\pm 1 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} 0 \\\\ \\pm 1 \\end{array} \\right) \\\\\r\nC \\left( \\begin{array}{c} \\pm 1 \\\\ 0 \\end{array} \\right) & = \\left( \\begin{array}{cc} -1 & 0 \\\\ 0 & 1 \\end{array} \\right) \\left( \\begin{array}{c} \\pm 1 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} \\mp 1 \\\\ 0 \\end{array} \\right)\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u3055\u3044\u3053\u308d\u3092\u632f\u308b\u3068\u70b9 R \u306f, \u70b9 A \uff5e D \u9593\u3092\u79fb\u52d5\u3057, \u305d\u308c\u305e\u308c\u306e\u70b9\u304b\u3089\u79fb\u52d5\u3059\u308b\u78ba\u7387\u306f\u4e0b\u56f3\u306e\u901a\u308a\u3068\u306a\u308b.<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/tok20140301.svg\" alt=\"tok20140301\" class=\"aligncenter size-full\" \/>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\u306f\u3058\u3081, \u70b9 R \u306f\u70b9 A \u306b\u3042\u308b\u306e\u3067\r\n\\[\r\np _ 1 = \\underline{\\dfrac{1}{3}} , \\quad q _ 1 = \\underline{0}\r\n\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\np _ 2 & = 3 \\cdot \\dfrac{1}{3} \\cdot \\dfrac{1}{3} = \\underline{\\dfrac{1}{3}} \\\\\r\nq _ 2 & = 2 \\cdot \\dfrac{1}{3} \\cdot \\dfrac{1}{3} = \\underline{\\dfrac{2}{9}}\r\n\\end{align}\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\\(n\\) \u56de\u64cd\u4f5c\u5f8c\u306b, \u70b9 R \u304c\u70b9 C, D \u306b\u3042\u308b\u78ba\u7387\u3092\u305d\u308c\u305e\u308c \\(r _ n , s _ n\\) \u3068\u304a\u304f\u3068\r\n\\[\r\np _ n +q _ n +r _ n +s _ n = 1 \\quad ... [1]\r\n\\]\r\n\u4e0a\u306b\u66f8\u3044\u305f\u72b6\u614b\u9077\u79fb\u3088\u308a\r\n\\[\\begin{align}\r\np _ n & = \\dfrac{1}{3} p _ {n-1} +\\dfrac{1}{3} r _ {n-1} +\\dfrac{1}{3} s _ {n-1} \\\\\r\n& = \\dfrac{1}{3} p _ {n-1} +\\dfrac{1}{3} \\left( 1 -p _ {n-1} -q _ {n-1} \\right) \\quad ( \\ \\text{\u2235} \\ [1] \\ ) \\\\\r\n& = \\dfrac{1}{3} -\\dfrac{1}{3} q _ {n-1} \\quad ... [2] \\\\\r\nq _ n & = \\dfrac{1}{3} q _ {n-1} +\\dfrac{1}{3} r _ {n-1} +\\dfrac{1}{3} s _ {n-1} \\\\\r\n& = \\dfrac{1}{3} q _ {n-1} +\\dfrac{1}{3} \\left( 1 -p _ {n-1} -q _ {n-1} \\right) \\quad ( \\ \\text{\u2235} \\ [1] \\ ) \\\\\r\n& = \\dfrac{1}{3} -\\dfrac{1}{3} p _ {n-1} \\quad ... [3]\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, [2] [3] \u306e\u8fba\u3005\u3092\u52a0\u6e1b\u3059\u308c\u3070\r\n\\[\r\n\\underline{p _ n +q _ n = \\dfrac{2}{3} -\\dfrac{1}{3} \\left( p _ {n-1} +q _ {n-1} \\right)} \\quad ... [4] \\\\\r\n\\underline{p _ n -q _ n = \\dfrac{1}{3} \\left( p _ {n-1} -q _ {n-1} \\right)} \\quad ... [5]\r\n\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n[4] \u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\r\np _ n + q _ n -\\dfrac{1}{2} = -\\dfrac{1}{3} \\left( p _ {n-1} +q _ {n-1} -\\dfrac{1}{2} \\right)\r\n\\]\r\n\u306a\u306e\u3067, \u6570\u5217 \\(\\left\\{ p _ n +q _ n -\\dfrac{1}{2} \\right\\}\\) \u306f, \u521d\u9805 \\(p _ 0 +q _ 0 -\\dfrac{1}{2} = \\dfrac{1}{2}\\) , \u516c\u6bd4 \\(-\\dfrac{1}{3}\\) \u306e\u7b49\u6bd4\u6570\u5217, \u3059\u306a\u308f\u3061\r\n\\[\r\np _ n +q _ n -\\dfrac{1}{2} = \\dfrac{1}{2} \\left( -\\dfrac{1}{3} \\right)^n \\\\\r\n\\text{\u2234} \\quad p _ n +q _ n = \\dfrac{1}{2} \\left( -\\dfrac{1}{3} \\right)^n +\\dfrac{1}{2} \\quad ... [6]\r\n\\]\r\n\u307e\u305f, [5] \u3088\u308a, \u6570\u5217 \\(\\left\\{ p _ n -q _ n \\right\\}\\) \u306f, \u521d\u9805 \\(p _ 0 -q _ 0 = 1\\) , \u516c\u6bd4 \\(\\dfrac{1}{3}\\) \u306e\u7b49\u6bd4\u6570\u5217, \u3059\u306a\u308f\u3061\r\n\\[\r\np _ n -q _ n = \\left( \\dfrac{1}{3} \\right)^n \\quad ... [7]\r\n\\]\r\n\u3088\u3063\u3066, [6] [7] \u3092\u8fba\u3005\u52a0\u3048\u3066, \\(\\dfrac{1}{2}\\) \u500d\u3059\u308c\u3070\r\n\\[\r\np _ n = \\underline{\\dfrac{1}{2} \\left( \\dfrac{1}{3} \\right)^n +\\dfrac{1}{4} \\left( -\\dfrac{1}{3} \\right)^n +\\dfrac{1}{4}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(1\\) \u500b\u306e\u3055\u3044\u3053\u308d\u3092\u6295\u3052\u3066, \u51fa\u305f\u76ee\u304c \\(1\\) \u304b \\(2\\) \u3067\u3042\u308c\u3070\u884c\u5217 \\(A = \\left( \\begin{array}{cc} 0 &#038; 1 \\\\ -1 &#038; 0 \\end{array} \\right) &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/tok201403\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[116],"tags":[141,112],"class_list":["post-1128","post","type-post","status-publish","format-standard","hentry","category-toko_2014","tag-toko","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1128","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=1128"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1128\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1128"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1128"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1128"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}