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class="kicker">TOMABECHI COGNITIVE UNIVERSE · THEOREMS 28â32</div> 75 <h1>è«ç±³å°èªç¥å®å®è«ã仿³æ°çå³å¯è¨¼æ</h1> 76 <div class="thmrange">å®ç28â32ï¼æ¶ æ§ç¡æã»æ³ç¡èªæ§ã»ã¨ã³ãããã¼äº¤æèªæã»è¨èªéå å é輪廻ã»ãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åï¼</div> 77 <div class="edition">å³å¯å®å¼åçââå®ç24ï¼ä¸åçè¦ï¼ã»å®ç26ï¼æ¶ æ§å¯éï¼ã»å®ç25ï¼è«¸æ³ç¡æï¼ã»å®ç27ï¼ç¡æèµ·è¡ï¼ã貫ãåçå¯éã®æ°çããã®äºã¤ã®æ¡å¼µ</div> 78 <div class="author">è«ç±³å° è±äºº</div> 79 <div class="affil">Cognitive Research Laboratories, Tokyo<br>
79CyLab, Carnegie Mellon University · C5I Center, George Mason University<br>2026å¹´8æ11æ¥</div> 80 <div class="release">æ«å®å ¬éçãæè²ã»å¹³åå©ç¨ã«éããè使¨©ãå°éããå¼ç¨ã»åç §ãå¯ã¨ããã<br>æ¬ç¨¿ã¯å®çã¨ãã¦è¨¼æã§ããç¯å²ã¨ãæå¦çã»å²å¦çè§£éã¨ãæç¤ºçã«åé¢ããã</div> 81</header> 82 83<article class="page"> 84<section id="abstract"> 85<h2 class="sr-only">æ¦è¦</h2> 86<div class="abstract"> 87 <div class="label">æ¦è¦</div> 88 <p>æ¬ç¨¿ã¯ãæ¢åã®å®ç1ï¼è«ç±³å°ä¸»å®çï¼â27ãåºç¤ã¨ãã¦ãäºã¤ã®æ°å®çãå®å¼åãããå®ç28ï¼æ¶ æ§ç¡æã»åæ´åå®çï¼ã¯ãå®ç24ï¼ä¸åçè¦å®çï¼ã¨26ã®é åãã¿ã°ä»ãç´åã§åé¢ãããä¸åçè¦ãã¨ãæ¶ æ§å¯éãã®åæ´åæ§ã証æããããã§ãå®ç25ï¼è«¸æ³ç¡æå®çï¼ã®é¢ä¿çæ©è½å®åæ§ããæ¶ æ§éç¨ã®ç¡æãå°ããå®ç29ï¼å½¢å¼æ³ä½ç³»ç¡èªæ§ã»ä¸å®åå®çï¼ã¯ãæèª¬ã¨ãã¦ã®æ³ãååã«å¼·ãæå¹å½¢å¼çè«ã¨ãã¦ã¢ãã«åããå ´åã«éããGödelã»Chaitinã®éçãéæ¾çæ´æ°ãããã³å®ç25ï¼è«¸æ³ç¡æå®çï¼ã«ããç¡èªæ§ã示ããå®ç30ï¼ã¨ã³ãããã¼äº¤æèªææ§æå®çï¼ã¯ãå çè¨èªãéç³»ä¸ã®å°å½±æ´åååã¨ãã¦å®å¼åããè¨èªæ¡ä»¶ä»ãEgoæ§æã®é¸æã»å®å®åã¨èªç¥ç©çã¨ã³ãããã¼äº¤æã証æãããå®ç31ï¼å çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻å®çï¼ã¯ãè¨èªéå ã»Nagumoä¸å¤æ§ã»Lyapunovå¸å¼æ§ã®ãã¨ã§ä½æ½è±¡åº¦TCZããã®è±åºä¸è½æ§ã示ããå ç¶æ ç²è¦åã®å帰æ§ã証æãããå®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ã¯ãå¤é¨ã´ã¼ã«ãæªæ¥TCZã«å ¥ãæçè¨å ´æé¾å¤ã¨ãæªæ¥éå®Egoã«ããææ°åæã»å°éæéãä¸ããã</p> 89 <p>ç¶æ¿å®çã®åæ²é¨ï¼Â§3ï¼ã«ã¯ã<b>å®ç15ï¼èªç¥ç©çã¨ã³ãããã¼äº¤æã»ä¿åå®çï¼ã §3.7 ã¨ãã¦æ°ãã«åé²ãã</b>ãæä½æ½è±¡åº¦ç©ç層ã«ããã¦ã¨ã³ãããã¼ã«ã¤ãã¦æãç«ã¤ã®ã¯ç¬¬äºæ³åã®ä¸çå¼ã®ã¿ã§ãããçå¼ã¨ãã¦ã®ä¿ååã¯é«æ½è±¡åº¦ã¬ã¤ã¤ã¼ãç¶æ è¨è¿°ã«å«ãã¦ã¯ããã¦æç«ããããã®äºå®ï¼å½é¡15.Dï¼ã¯å®ç30 ã®äº¤æåæ¯ã®åæã§ãããæ¬ç¨¿ã§åãã¦å ¬éãããã¾ã §7.3 ã«ããã¦ãå é輪廻ãã¨ããåç§°ã®å°ç¨ï¼é åæ°ã®éæ¬è³ªæ§ã転çã«é¢ããä¸ç«æ§ã¨å®ç25 ã«ããå¶ç´ãåä¸ç涯å ã§ã®æç«ï¼ãã§7.4 ã«ããã¦å®ç31 ãå½¢å¼åãã対象ã¨é¾æ¨¹ãä¸è«ãã®è¨èªæ¹å¤ã¨ã®æ§é ç対å¿ãããããã<b>è§£é層ã¨ãã¦è¨¼æããåé¢ããããã§</b>ä¸ããã</p> 90 <p><strong>éè¦ãªéå®ã</strong> äºå®çã®ãã¹ã¦ãæ¢åå®çããç¡æ¡ä»¶ã«åºãããã§ã¯ãªããåç¯ã§ãæ¢åå®çããç¶æ¿ããé¨åããæ°ãã«å¿ è¦ãªæ¡ä»¶ããå¤é¨ã¡ã¿å®çããåé¢ãããã¾ãã<em>ç¡æ</em>ã¯æ°å¦ç対象åã®ç¦æ¢ã§ã¯ãªããé¢ä¿è¨è¿°ãè¶ ããç¬ç«ã»åºå®ã»åä½åçã»å æçã«éåé·ãªèªæ§ã®å¦å®ãæå³ããã</p> 91</div> 92</section> 93 94<nav class="toc" aria-label="ç®æ¬¡"> 95<h2>ç®æ¬¡</h2> 96<ol> 97 <li><a href="#scope">çµè«ã¨è¨¼æä¾å</a></li> 98 <li><a href="#notation">ç¶æ¿è¨æ³ã¨è«ççå°ç¨</a></li> 99 <li><a href="#inherited">ç¶æ¿å®çã®å®å¼åã¨å³å¯è¨¼æ</a></li> 100 <li><a href="#t28">å®ç28ï¼æ¶ æ§ç¡æã»åæ´åå®çï¼ãæ¶ æ§ç¡æã»åæ´å</a></li> 101 <li><a href="#t29">å®ç29ï¼å½¢å¼æ³ä½ç³»ç¡èªæ§ã»ä¸å®åå®çï¼ã形弿³ä½ç³»ç¡èªæ§ã»ä¸å®å</a></li> 102 <li><a href="#t30">å®ç30ï¼ã¨ã³ãããã¼äº¤æèªææ§æå®çï¼ãã¨ã³ãããã¼äº¤æèªææ§æ</a></li> 103 <li><a href="#t31">å®ç31ï¼å çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻å®çï¼ãå çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻</a></li> 104 <li><a href="#t32">å®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ãæªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»å</a></li> 105 <li><a href="#minimality">æå°æ§ã»åä¾ä¸è¦§</a></li> 106 <li><a href="#conclusion">ç·æ¬</a></li> 107 <li><a href="#references">åç §æç®</a></li> 108</ol> 109</nav> 110 111<section id="scope"> 112<h2>1. çµè«ã¨è¨¼æä¾å</h2> 113<p class="lead">é©åãªé åºã¯ãåç´ãªä¸åã§ã¯ãªã次ã®é¨åé åºã§ãããå®ç28ï¼æ¶ æ§ç¡æã»åæ´åå®çï¼ã»å®ç29ï¼å½¢å¼æ³ä½ç³»ç¡èªæ§ã»ä¸å®åå®çï¼ã»å®ç30ï¼ã¨ã³ãããã¼äº¤æèªææ§æå®çï¼ã¯äºãã«ç¬ç«ãªä¸ã¤ã®æã§ãããå®ç31ï¼å çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻å®çï¼ã¯å®ç30ï¼ã¨ã³ãããã¼äº¤æèªææ§æå®çï¼ã®è¨èªåEgoæ§æãä»»æã«å©ç¨ã§ããå®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ã¯å®ç31ï¼å çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻å®çï¼ã§è¨è¿°ãããééããã®åæ§æã説æã§ããã</p> 114<div class="chain">åºç¤ T24â26 â T28ãï½ãåºç¤ T25 ï¼ Gödel/Chaitin â T29ãï½ãåºç¤ T1ã»15ã»16ã»18 â T30 â[任æï¼½T31 âï¼»è±åºè§£éï¼½T32</div> 115<table> 116<thead><tr><th>
116æ°å®ç</th><th>æå°ã®æ¢åä¾å</th><th>æ°æ¡ä»¶ï¼å¤é¨çµæ</th><th>証æä¸ã®å°ä½</th></tr></thead> 117<tbody> 118<tr><td>28</td><td>T24, T25, T26</td><td>æ¶ æ§éç¨ãT25ã®ç¾è±¡é åã¸å«ããæ¡ä»¶28-A</td><td>åæ´åé¨ã¯ç´æ¥ã®ç³»ãç¡æé¨ã¯T25ã®é©ç¨</td></tr> 119<tr><td>29</td><td>T25</td><td>29-AãCãGödel Iã»IIãChaitin</td><td>æ¢åä½ç³»ã ãããã¯åºãªãæ¡ä»¶ä»ãã¡ã¿å®ç</td></tr> 120<tr><td>30</td><td>T1, T15, T16, T18</td><td>å°å½±æ´åæ§ãç¸®å°æ§ãã¨ã³ãããã¼åæ©</td><td>æ°ãã鿥µéæ§æã¨T15交æå¼ã®çµ±å</td></tr> 121<tr><td>31</td><td>T1, T3, T16, T18</td><td>è¨èªéå ãNagumoæ¡ä»¶ããã¤ããªããLyapunovæ¡ä»¶</td><td>T18ã®ãé¸æè¢æ¡å¼µãã ãã§ã¯åºãªã</td></tr> 122<tr><td>32</td><td>T1, T4, T7â9</td><td>è¨å ´æé¾å¤ãå¯å°éå¸å¼åãå¼·ãLyapunovå縮</td><td>T9ãå®éåããå¼·åå®ç</td></tr> 123</tbody> 124</table> 125<div class="note"><strong>å®ç27ï¼ç¡æèµ·è¡å®çï¼ã¨ã®é¢ä¿ã</strong> çªå·ã¯å®ç27ï¼ç¡æèµ·è¡å®çï¼ã®å¾ãç¶ãããäºå®çã®æ¬ä½è¨¼æã¯å®ç27ï¼ç¡æèµ·è¡å®çï¼ãå¿ è¦ã¨ããªããå®ç27ï¼ç¡æèµ·è¡å®çï¼ã®ç¡æã»è¡ã®åå¤ã¯ãããããåäºæ¯ç¸èµ·ã¸æ¥ç¶ããéã®è§£é層ã¨ãã¦ä¿æãããã</div> 126</section> 127 128<section id="notation"> 129<h2>2. ç¶æ¿è¨æ³ã¨è«ççå°ç¨</h2> 130<p>æ½è±¡åº¦ã®åé åºéåã <span class="math">(ð,â¼)</span>ãæé«å ã <span class="math">â¤</span> ã¨ãããå層ã®ç¶æ 空éã <span class="math">X<sub>a</sub></span>ãçåå¯è½é åã <span class="math">â¬<sub>alive</sub>â X<sub>â¤</sub></span>ãå²å¼æé©è²»ç¨ã <span class="math">J<sup>*</sup><sub>a,Ï</sub>(x,T)</span> ã¨æ¸ããå®ç26ï¼æ¶ æ§å¯éå®çï¼ã®å¯ééåã¯</p> 131<div class="eq"><span class="math">ð©<sub>â¤</sub>(T):=â¬<sub>alive</sub>â©{x| J<sup>*</sup><sub>â¤,Ï</sub>(x,T)=0}.</span><span class="eqno">(2.1)</span></div> 132<p>å®ç25ï¼è«¸æ³ç¡æå®çï¼ã® <span class="math">Atman(d,a)</span> ã¯ãç¾è±¡ <span class="math">d</span> ã®å±¤ <span class="math">a</span> ã«ãé¢ä¿è¨è¿°ããç¬ç«ããå±¥æ´ãéãã¦åºå®ãããåä½åãæ ãããã¤å æçã«éåé·ãªå å¨å¤æ°ãåå¨ããã¨ããè¿°èªã§ããããããã£ã¦ <span class="math">¬Atman</span> ã¯ã対象ãéåã»ååã»çè«ã¨ãã¦è¨è¿°ã§ããªããã¨ãã主張ã§ã¯ãªãã</p> 133<div class="rigor"><div class="rigor-title">ä¸ã¤ã®é嫿</div> 134<ol> 135<li>è¨å· <span class="math">T</span> ãæã¤ã ãã§ã¯ã<span class="math">ð©<sub>â¤</sub>(T)</span> ãå®éã«æå¤ã§ããã¨ã¯ãããªãã宿°éåæã許ãããã</li> 136<li>éåæã§ãããã¨ãã¾ãã¯é¢ä¿ããå®ç¾©ããããã¨ã ãã§ã¯ç¡æã¯è¨¼æãããªããæ±ºå®çãªã®ã¯å®ç25ï¼è«¸æ³ç¡æå®çï¼ã®æ©è½çå®åæ§25-Dã§ããã</li> 137<li>Gödelã»Chaitinã®çµæã¯ããã¹ã¦ã®å ¬çç³»ã«ããæ³ä¸è¬ã«ãç¡æ¡ä»¶ã«ã¯é©ç¨ãããªããååãªç®è¡å¼·åº¦ã»æå¹æ§ã»å¥å ¨æ§ãæè¨ããã</li> 138</ol></div> 139</section> 140 141<section id="inherited"> 142<h2>3. ç¶æ¿å®çã®å®å¼åã¨å³å¯è¨¼æ</h2> 143<p>æ¬ç¨¿ã®å®ç28ï¼æ¶ æ§ç¡æã»åæ´åå®çï¼ãå®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ã¯ãæ¢åã®å®ç群ãåæã¨ãã¦æ§ç¯ããããèªè ãæ¬ç¨¿ã ãã§è¨¼æã追ãããããç´æ¥ç¨ããå®ç1ï¼è«ç±³å°ä¸»å®çï¼ã»å®ç3ï¼æ½è±¡çå ±æTCZåæå®çï¼ã»å®ç4ï¼è«ç±³å°è¨å ´æå éå®çï¼ã»å®ç7ï¼è«ç±³å°çã®ã´ã¼ã«å®çï¼ã»å®ç8ï¼è«ç±³å°æªæ¥åç¹èªç¥æéå®çï¼ã»å®ç9ï¼è«ç±³å°æªæ¥åç¹ã´ã¼ã«éæå®çï¼ã»å®ç15ï¼èªç¥ç©çã¨ã³ãããã¼äº¤æã»ä¿åå®çï¼ã»å®ç16ï¼èªå·±æèåå¨ã»çºçå®çï¼ã»å®ç18ï¼å çè¨èªé²åå®çï¼ã«ã¤ãã¦ãæ¨æºå½¢ã®ä¸å¿å¼ãæ¬ç¨¿ã§ç¨ããå®å¼åãããã³å³å¯è¨¼æãããã«æ²ãããå®ç24ï¼ä¸åçè¦å®çï¼ã»å®ç25ï¼è«¸æ³ç¡æå®çï¼ã»å®ç26ï¼æ¶ æ§å¯éå®çï¼ã«ã¤ãã¦ã¯ã証æãåæ²ãããè«ç±³å°åæ³å°å®çããåç §ããï¼Â§3.10ï¼ã</p> 144<div class="rigor"><div class="rigor-title">æ¬ç¯ã®ä½ç½®ã¥ã</div> 145<p>æ¬ç¯ã¯ãæ¢åºã®å®çãæ¬ç¨¿ã®è¨æ³ã¸åãã証æã®éª¨æ ¼ãæç¤ºããããã®åæ²ã§ã
145ãããã ã §3.7ï¼å®ç15ï¼ã«ã¤ãã¦ã¯ãåè«æã®ä»®å®A6(ii)ã䏿§å¯ç©åæ¡ä»¶A6â²ã¸å¼±ãï¼è£é¡15.1ï¼ããããã¦ç§©åºåéã®ä¸ç(3.8)ã¨ãä¿ååãç©ç層ã¸éå ã§ããªããã¨ï¼å½é¡15.Dï¼ãæ°ãã«ç¤ºããæ¨æºå½¢ã¯ãè«ç±³å°æ½è±¡åº¦èªç±è«ãå ¨å®çä¸è¦§ã®è¡¨è¨ã«ä¸è´ãããæ¬ç¨¿ã§ç°ãªã表ç¾ãç¨ããå ´åã¯ããã®çç±ã¨æ¨æºå½¢ã¨ã®å夿§ãåé ã«æè¨ããã</p></div> 146 147<h3>3.0 å ±éã®éå ·ââçµ±ä¸åæè£é¡ï¼è£é¡0ï¼</h3> 148<p>以ä¸ã®è¨¼æã¯ã次ã®ä¸ã¤ã®è£é¡ã«å¸°çããã</p> 149<div class="eq"><span class="math">Φ ⥠0 ãè»éä¸çµ¶å¯¾é£ç¶ã§ãΩ<sub>θ</sub> ã®å¤ã§ D<sup>+</sup>Φ(t) ⤠âcΦ(t)ï¼c > 0ï¼ãæºãããªããΦ(t) ⤠Φ(0)e<sup>âct</sup> ã§ãããdist(x(t),Ω<sub>θ</sub>) â 0</span><span class="eqno">(3.0)</span></div> 150<p>ããã§ D<sup>+</sup> ã¯å³ä¸Diniå¾®åãΩ<sub>θ</sub> 㯠Φ ã®å£ä½éåã§ãããΦ ã¯çµ¶å¯¾é£ç¶ãªã®ã§ãã»ã¨ãã©è³ãæã§é常ã®å¾®åã¨ä¸è´ããGrönwall ã®ä¸çå¼ãé©ç¨ã§ãããäºæ¬¡ã®æã¿è¾¼ã¿ c<sub>1</sub>dist<sup>2</sup> ⤠Φ ⤠c<sub>2</sub>dist<sup>2</sup> ãä½µããã°ãè·é¢ã«ã¤ãã¦ã®ææ°è©ä¾¡ dist(x(t),Ω<sub>θ</sub>) ⤠â(c<sub>2</sub>/c<sub>1</sub>) e<sup>âct/2</sup> dist(x(0),Ω<sub>θ</sub>) ãå¾ãã<b>以ä¸ã®åå®çã¯ãããããã®æ®å·® Φ ãæ§æãããã®è£é¡ã¸å¸°çãããå½¢ã§è¨¼æãããã</b></p> 151 152<h3>3.1 å®ç1ï¼è«ç±³å°ä¸»å®çï¼</h3> 153<div class="stdform"><span class="t">â ä¸å¿å¼ï¼æ¨æºå½¢ï¼</span>Ï<sub>c</sub> = arg min â«V<sub>0</sub> dtãâãx(t) â TCZ<span class="src">ãè«ç±³å°æ½è±¡åº¦èªç±è«ãå ¨å®çä¸è¦§ã®æ¨æºå½¢ã«ä¸è´</span></div> 154<div class="eqbridge"><b>æ¬ç¨¿ã§å±éå½¢ãç¨ããçç±ã</b>証æã§ã¯å°éå¯è½é åã¸ã®åæãè·é¢ã§è¿°ã¹ãå¿ è¦ããããããç©ååºéã»å¼æ°ã»å°éå¯è½æ§ã®éå®ãæç¤ºããã<br><b>次ã®å¯¾å¿ã«ãããä¸ã®å¼ã¯æ¨æºå½¢ã¨åä¸ã®ä¸»å¼µã§ããã</b><br><b>â«V<sub>0</sub> dt</b> â <b>â«<sub>0</sub><sup>T</sup>V<sub>0</sub>(x(t),t) dt</b>ï¼<b>TCZ</b> â <b>TCZ<sub>1</sub><sup>cl</sup>(t;x<sub>0</sub>)</b>ï¼å°éå¯è½é åã«éã£ãéã¹ã©ã¤ã¹ï¼ï¼<b>x(t) â TCZ</b> â <b>dist(x(t),TCZ<sub>1</sub><sup>cl</sup>(t;x<sub>0</sub>)) â 0</b></div> 155<div class="theorem"><div class="theorem-title">å®ç1ï¼è«ç±³å°ä¸»å®çï¼</div> 156<p>éè² è©ä¾¡ V<sub>0</sub> ã«ã¤ã㦠TCZ = {x | V<sub>0</sub>(x,t) ⤠θ} ã¨ãããéã«ã¼ãæ¹ç Ï<sub>c</sub> ãè§£ãçæããV<sub>0</sub> ãè»éä¸çµ¶å¯¾é£ç¶ã§ãTCZ ã®å¤ã§ D<sup>+</sup>[V<sub>0</sub>(x(t),t) â θ]<sup>+</sup> ⤠âc[V<sub>0</sub> â θ]<sup>+</sup> ãæºãããªãã</p> 157<div class="eq"><span class="math">dist(x(t),TCZ<sub>1</sub><sup>cl</sup>(t;x<sub>0</sub>)) â 0</span><span class="eqno">(3.1)</span></div></div> 158<div class="proof"><div class="proof-title">証æ</div> 159<p>æ®å·®ã Φ<sub>1</sub> := [V<sub>0</sub>(x,t) â θ]<sup>+</sup> ã¨ç½®ããΦ<sub>1</sub> ã¯éè² ã§ãããä»®å®ãã TCZ ã®å¤ã§ D<sup>+</sup>Φ<sub>1</sub> ⤠âcΦ<sub>1</sub> ãæºãããè£é¡0ï¼3.0ï¼ãã Φ<sub>1</sub>(t) ⤠Φ<sub>1</sub>(0)e<sup>âct</sup> ã§ãããΦ<sub>1</sub> â 0ãΦ<sub>1</sub> = 0 㯠V<sub>0</sub> ⤠θãããªãã¡ TCZ ã¸ã®æå±ã¨åå¤ã§ãããããå°éå¯è½éã¹ã©ã¤ã¹ã¸ã®è·é¢ã 0 ã«åæããã<b>åæã®å®è³ªæ¡ä»¶ã¯ arg min ã§ãããã¨èªä½ã§ã¯ãªããarg min ã§é¸ã°ããéã«ã¼ãã Lyapunov éä¸ãæºãããã¨ã§ããã</b><span class="qed">â</span></p></div> 160 161<h3>3.2 å®ç3ï¼æ½è±¡çå ±æTCZåæå®çï¼</h3> 162<div class="stdform"><span class="t">â ä¸å¿å¼ï¼æ¨æºå½¢ï¼</span>A(x)=0 â Ï(x)=LUB(W<sub>1</sub>,â¦,W<sub>N</sub>),ãA(t)â0<span class="src">ãè«ç±³å°æ½è±¡åº¦èªç±è«ãå ¨å®çä¸è¦§ã®æ¨æºå½¢ã«ä¸è´</span></div> 163<div class="eqbridge"><b>æ¬ç¨¿ã§å±éå½¢ãç¨ããçç±ã</b>æ¨æºå½¢ã¯ã屿 ¹ã«å°éããç¶æ ã¨ã¯ä½ããã¨ããç¹å¾´ã¥ããè¿°ã¹ããæ¬ç¨¿ã¯å°éãä¿è¨¼ããæ¹çãè¦ãããããæ¹çã®å½¢ã§æ¸ãã<b>
163è¿°ã¹ã¦ãã層ãç°ãªãã</b><br><b>A</b> â <b>ð<sub>i</sub></b>ï¼æ½è±¡æ®å·®ï¼ï¼<b>LUB(W<sub>1</sub>,â¦,W<sub>N</sub>)</b> â <b>L<sup>*</sup> = â¨W<sub>i</sub></b>ï¼<b>A(t)â0</b> â <b>âι(Ï<sub>i</sub>(x<sub>i</sub>(t)))âι(L<sup>*</sup>)â â 0</b></div> 164<div class="theorem"><div class="theorem-title">å®ç3ï¼æ½è±¡çå ±æTCZåæå®çï¼</div> 165<p>çµåã°ã©ããé£çµãåãºã¬è²»ç¨ S<sub>ij</sub> ãéè² ãæ½è±¡æ®å·® ð<sub>i</sub> ã L<sup>*</sup> = â¨W<sub>i</sub> ã§ã¡ããã© 0 ã¨ãªãã¨ããæ¡å¼µã©ã°ã©ã³ã¸ã¢ã³ â<sub>A</sub> = Σ<sub>i</sub>V<sub>i</sub> + Σ<sub>(i,j)âE</sub>γ<sub>ij</sub>S<sub>ij</sub> + Σ<sub>i</sub>η<sub>i</sub>ð<sub>i</sub>ï¼Î·<sub>i</sub> > 0ï¼ãæå°åããéã«ã¼ãã®ãã¨ã§</p> 166<div class="eq"><span class="math">ð<sub>i</sub>(t) â 0,ãããªãã¡ Ï<sub>i</sub>(x<sub>i</sub>(t)) â L<sup>*</sup> = â¨W<sub>i</sub></span><span class="eqno">(3.2)</span></div></div> 167<div class="proof"><div class="proof-title">証æ</div> 168<p>Φ<sub>3</sub> := â<sub>A</sub> â inf â<sub>A</sub> ã¨ç½®ããé£çµæ§ããããããã®ä¸»ä½ã®ã¯ã¿åºããå¿ ãããããã® S<sub>ij</sub> ã«åæ ãããã®ã§ãΦ<sub>3</sub> ã¯å ¨ä½ã®é¸è±ãæ¼ããªãæãããä»®å®ãã Φ<sub>3</sub> ã¯è£é¡0ã®é䏿¡ä»¶ãæºããã®ã§ Φ<sub>3</sub> â 0ãΦ<sub>3</sub> ã®åé ã¯éè² ã§ãããããåã 0 ã«åæãããã¨ã¯åé ã 0 ã«åæãããã¨ã嫿ããç¹ã« η<sub>i</sub>ð<sub>i</sub> â 0ãããªãã¡ ð<sub>i</sub> â 0ãð<sub>i</sub> 㯠L<sup>*</sup> ã«ããã¦ã®ã¿ 0 ã¨ãªãããæ§æããã¦ããã®ã§ãÏ<sub>i</sub>(x<sub>i</sub>(t)) â L<sup>*</sup>ã<b>LUB ã¯å¹³åã§ã¯ãªãããããã®ä¸»ä½ã®ä¸çãåãæ¨ã¦ãã«å ãæå°ã®ä¸çã§ããã</b><span class="qed">â</span></p></div> 169 170<h3>3.3 å®ç4ï¼è«ç±³å°è¨å ´æå éå®çï¼</h3> 171<div class="stdform"><span class="t">â ä¸å¿å¼ï¼æ¨æºå½¢ï¼</span>á¹¼ = V<sub>0</sub> â κPQ,ãx â TCZ<sub>P</sub><span class="src">ãè«ç±³å°æ½è±¡åº¦èªç±è«ãå ¨å®çä¸è¦§ã®æ¨æºå½¢ã«ä¸è´</span></div> 172<div class="eqbridge"><b>æ¬ç¨¿ã§å±éå½¢ãç¨ããçç±ã</b>æ¨æºå½¢ã¯å¤å½¢ã®å®ç¾©ã¨è¡ãå ã®ã¿ãè¿°ã¹ããæ¬ç¨¿ã¯å¤å½¢å¾ã®éã«ã¼ããé䏿¡ä»¶ãæºãããã¨ã使ããããæ¹ç Ï<sub>c</sub>(P) ãæç¤ºããã<br><b>TCZ<sub>P</sub></b> â <b>Ω<sub>P</sub>(t)</b>ï¼å¤å½¢å¾ã®éå°éå¯è½ã¹ã©ã¤ã¹ï¼ï¼<b>x â TCZ<sub>P</sub></b> â <b>dist(x(t),Ω<sub>P</sub>(t)) â 0</b></div> 173<div class="theorem"><div class="theorem-title">å®ç4ï¼è«ç±³å°è¨å ´æå éå®çï¼</div> 174<p>è¨å ´æ P â [0,1]ã価å¤ç¬¦å· Qãκ > 0 ã«å¯¾ã á¹¼ := V<sub>0</sub> â κPQ ã¨ãããá¹¼ ãéè² è©ä¾¡ã®è¦ä»¶ãæºãããÏ<sub>c</sub>(P) ã®éã«ã¼ããè£é¡0ã®é䏿¡ä»¶ãæºãããªã</p> 175<div class="eq"><span class="math">dist(x(t),Ω<sub>P</sub>(t)) â 0,ãâá¹¼/âP = âκQ</span><span class="eqno">(3.3)</span></div></div> 176<div class="proof"><div class="proof-title">証æ</div> 177<p>á¹¼ ãè©ä¾¡é¢æ°ã¨ãã¦æ®å·® Φ<sub>4</sub> := [á¹¼(x,t) â θ]<sup>+</sup> ãä½ããá¹¼ ãã¾ãéè² è©ä¾¡ã®è¦ä»¶ãæºããã®ã§ãè£é¡0ããã®ã¾ã¾é©ç¨ã§ããΦ<sub>4</sub> â 0ããããã£ã¦ dist(x(t),Ω<sub>P</sub>(t)) â 0ãè°·ã®ä½ç½®ã P ã¨ã¨ãã«é£ç¶ã«åããã¨ã¯ãâá¹¼/âP = âκQ ã P ã«ã¤ãã¦ç´æ¥å¾®åãã¦å¾ããããQ > 0 ã®å¯¾è±¡ã§ã¯å°å½¢ãä¸ãããQ < 0 ã®å¯¾è±¡ã§ã¯ä¸ããã<b>è¡åã®å¤æ´ã¯æå¿ã®æç¶ã§ã¯ãªãå°å½¢ã®å¤å½¢ã«ãã£ã¦éæãããã</b><span class="qed">â</span></p></div> 178 179<h3>3.4 å®ç7ï¼è«ç±³å°çã®ã´ã¼ã«å®çï¼</h3> 180<div class="stdform"><span class="t">â ä¸å¿å¼ï¼æ¨æºå½¢ï¼</span>G â TCZ<sub>0</sub>,ãd(G,TCZ<sub>0</sub>) ⥠ε > 0,ãG = Self-set<span class="src">ãè«ç±³å°æ½è±¡åº¦èªç±è«ãå ¨å®çä¸è¦§ã®æ¨æºå½¢ã«ä¸è´</span></div> 181<div class="theorem"><div class="theorem-title">å®ç7ï¼è«ç±³å°çã®ã´ã¼ã«å®çï¼</div> 182<p>ç¾å¨ã®å®å®é åã TCZ<sub>0</sub>ãåè£çµç«¯ã´ã¼ã«ã G ã¨ãããçã®ã´ã¼ã«ã®æ£æºæ¡ä»¶ãã(i) å¤é¨æ§ dist(G,TCZ<sub>0</sub>) > εã(ii) å¤é¨å¼·å¶ã§ãªãæ¬äººã®ã´ã¼ã«éåã«å±ãããã¨ã(iii) é«ä½ Self ã¨ã®æ´å C<sub>Self</sub>(G) >
182 0ã(iv) å¶å¾¡åé¡ã¸ééåã«å ¥ããã¨ãã¨ããããã®ã¨ã忡件ã¯ãæ¬ä½ç³»ã®æå³ã§è¨±å®¹ãããå¤é©ã´ã¼ã«ãç¹å¾´ã¥ããã</p> 183<div class="eq"><span class="math">J<sub>G</sub>[u] = â«<sub>t</sub><sup>T</sup>V<sub>0</sub>(x(Ï),Ï)dÏ + λ d(x(T),G)<sup>2</sup>,ãλ > 0</span><span class="eqno">(3.4)</span></div></div> 184<div class="proof"><div class="proof-title">証æï¼è«ççï¼</div> 185<p>忡件ã®å¿ è¦æ§ãããããå¤ããå ´åã«æ··å ¥ãã対象ã«ãã£ã¦ç¤ºãã(i) ãå¤ãã¨ãdist(G,TCZ<sub>0</sub>) ⤠ε ã®ã´ã¼ã«ã¯ç¾å¨ã®å®å®åãå®è³ªçã«åæ§æããã«å°éã§ãããããå¤é©ã´ã¼ã«ã®å¤é¨æ§ã失ãã(ii) ãå¤ãã¨å¤é¨ããå¼·å¶ãããç¶æ ãã(iii) ãå¤ãã¨è² 価å¤ã¾ãã¯èªå·±ä¸æ´åãªç¶æ ãã(iv) ãå¤ãã¨æ¹çã«å½±é¿ããªãè£ é£¾çç®æ¨ããããããæ··å ¥ããããããã£ã¦åæ¡ä»¶ã¯ææã®å¯¾è±¡ãã¡ããã©ç¹å¾´ã¥ããã<b>ããããå°éã¯å°ãããªããå°éã¯å®ç9ï¼è«ç±³å°æªæ¥åç¹ã´ã¼ã«éæå®çï¼ã®èª²é¡ã§ããã</b><span class="qed">â</span></p></div> 186 187<h3>3.5 å®ç8ï¼è«ç±³å°æªæ¥åç¹èªç¥æéå®çï¼</h3> 188<div class="stdform"><span class="t">â ä¸å¿å¼ï¼æ¨æºå½¢ï¼</span>u<sup>*</sup> = arg min J<sub>G</sub>ï¼çµç«¯æ¡ä»¶ G ãç¾å¨å¶å¾¡ã決å®<span class="src">ãè«ç±³å°æ½è±¡åº¦èªç±è«ãå ¨å®çä¸è¦§ã®æ¨æºå½¢ã«ä¸è´</span></div> 189<div class="theorem"><div class="theorem-title">å®ç8ï¼è«ç±³å°æªæ¥åç¹èªç¥æéå®çï¼</div> 190<p>çµç«¯æ¡ä»¶ G ãæã¤æéå°å¹³æé©å¶å¾¡åé¡ (3.4) ã«ããã¦ãæé©å¶å¾¡ u<sup>*</sup> ã¯åçè¨ç»æ³ã®åçã«ãã</p> 191<div class="eq"><span class="math">u<sup>*</sup> = arg min<sub>u</sub> J<sub>G</sub>ãâãââW<sub>G</sub>/ât = min<sub>u</sub>{ V<sub>0</sub>(x,t) + âW<sub>G</sub>(x,t)·f(x,u,t) }</span><span class="eqno">(3.5)</span></div> 192<p>ãæºãããããªãã¡ç¾å¨æå»ã®å¶å¾¡ã¯ãçµç«¯æ¡ä»¶ããå¾ãåãã«æ±ºå®ãããã</p></div> 193<div class="proof"><div class="proof-title">証æ</div> 194<p>価å¤é¢æ° W<sub>G</sub>(x,t) := min<sub>u</sub>J<sub>G</sub>[u] ãå°å ¥ãããæé©æ§åçãã W<sub>G</sub> ã¯çµç«¯æ¡ä»¶ W<sub>G</sub>(x,T) = λd(x,G)<sup>2</sup> ãæã¤ HamiltonâJacobiâBellman æ¹ç¨å¼ (3.5) ãæºããããã®æ¹ç¨å¼ã¯ t ã«ã¤ãã¦å¾ãåãã«ç©åããããããæå» t ã«ãããæé©å¶å¾¡ u<sup>*</sup>(t,x) ã¯çµç«¯æ¡ä»¶ G ã«ä¾åãã¦æ±ºã¾ãã<b>ç©çæéãéæµããã¨ä¸»å¼µãã¦ããã®ã§ã¯ãªããçµç«¯æ¡ä»¶ãæã¤æé©å¶å¾¡åé¡ã§ã¯ã決å®ã®ä¾åæ¹åãæªæ¥ããç¾å¨ã¸åããã¨ããæ§é çäºå®ã§ããã</b>ãããæ¬ä½ç³»ã§ã¯æªæ¥åç¹èªç¥æéã¨å¼ã¶ã<span class="qed">â</span></p></div> 195 196<h3>3.6 å®ç9ï¼è«ç±³å°æªæ¥åç¹ã´ã¼ã«éæå®çï¼</h3> 197<div class="stdform"><span class="t">â ä¸å¿å¼ï¼æ¨æºå½¢ï¼</span>K<sub>G</sub> = PQ<sup>+</sup> + EC<sub>Self</sub> ⥠K<sub>crit</sub> 㨠Lyapunov éä¸ãâãx â TCZ<sub>G</sub><span class="src">ãè«ç±³å°æ½è±¡åº¦èªç±è«ãå ¨å®çä¸è¦§ã®æ¨æºå½¢ã«ä¸è´</span></div> 198<div class="theorem"><div class="theorem-title">å®ç9ï¼è«ç±³å°æªæ¥åç¹ã´ã¼ã«éæå®çï¼</div> 199<p>ã´ã¼ã«é§å強度ã K<sub>G</sub> := PQ<sup>+</sup> + EC<sub>Self</sub> ã¨å®ãããK<sub>G</sub> ⥠K<sub>crit</sub> ã§ããããã¤ã´ã¼ã«æ®å·® Φ<sub>G</sub> ãè£é¡0ã®é䏿¡ä»¶ãæºãããªã</p> 200<div class="eq"><span class="math">dist(x(t),TCZ<sub>G</sub>(t)) â 0</span><span class="eqno">(3.6)</span></div></div> 201<div class="proof"><div class="proof-title">証æ</div> 202<p>ã´ã¼ã«æ®å·® Φ<sub>G</sub> := [á¹¼<sub>G</sub>(x,t) â θ<sub>G</sub>]<sup>+</sup> ãä½ããããã§ á¹¼<sub>G</sub> ã¯å®ç4ï¼è«ç±³å°è¨å ´æå éå®çï¼ã«ãã K<sub>G</sub> ã®å¯ä¸ã ãå¤å½¢ãããè©ä¾¡ã§ãããK<sub>G</sub> ⥠K<sub>crit</sub> ã¯ãå¤å½¢å¾ã®å°å½¢ã«ãã㦠G ã®å´ã«å£ä½éåãå®éã«çããããã®è¨çæ¡ä»¶ã§ããããã®æ¡ä»¶ä¸ã§ Φ<sub>G</sub> ã¯è£é¡0ã®é䏿¡ä»¶ãæºããããã£ã¦ Φ<sub>G</sub> â 0 ã㤠dist(x(t),TCZ<sub>G</sub>(t)) â 0ã<b>å®ç7ï¼è«ç±³å°çã®ã´ã¼ã«å®çï¼ãè³æ ¼ããå®ç8ï¼è«ç±³å°æªæ¥åç¹èªç¥æéå®çï¼ãæ±ºå®æ¹åãä¸ããæ¬å®çãå°éãä¸ããã</b><span class="qed">â</span></p></div> 203 204 205<h3>3.7 å®ç15ï¼èªç¥ç©çã¨ã³ãããã¼äº¤æã»ä¿åå®çï¼</h3> 206<div class="stdform"><span class="t">â ä¸å¿å¼ï¼æ¨æºå½¢ï¼</span>S<sub>gen</sub> = S<sub>phys</sub> + Σw<sub>α</sub>H<sub>α</sub>,ãdS<sub>gen</sub>/dt = Π⥠0<span class="src">ãè«ç±³å°æ½è±¡åº¦èªç±è«ãå ¨å®çä¸è¦§ã®æ¨æºå½¢ã«ä¸è´</span></div> 207<div class="eqbridge"><b>æ¬ç¨¿ã§å±éå½¢ãç¨ããçç±ã</b>å®ç30ï¼ã¨ã³ãããã¼äº¤æèªææ§æå®çï¼ã®æ¡ä»¶30-Cã¯ãæ¬å®çã®äº¤æå¼ã<b>層å¥ã»æ¡ä»¶ä»ãã®å½¢ã§</b>ç¨ããããããã£ã¦ã¬ã¤ã¤ã¼ææ° αãæç®éã¿ w<sub>α</sub>ãæ£é¸é Î ãããã³æéåºéãæç¤ºããå¿ è¦ããããæ¬¡ã®å¯¾å¿ã
207«ãããä¸ã®å¼ã¯æ¨æºå½¢ã¨åä¸ã®ä¸»å¼µã§ããã<br><b>Σw<sub>α</sub>H<sub>α</sub></b> â <b>Σ<sub>αâ»0</sub> w<sub>α</sub>H<sub>α</sub>(x<sub>α</sub>(t))</b>ï¼ã¬ã¤ã¤ã¼ææ°ã¨è»é弿°ãæç¤ºï¼ï¼<b>dS<sub>gen</sub>/dt = Π⥠0</b> â <b>S<sub>gen</sub>(t<sub>2</sub>) â S<sub>gen</sub>(t<sub>1</sub>) = â«<sub>t<sub>1</sub></sub><sup>t<sub>2</sub></sup>Î (s) ds ⥠0</b>ï¼çµ¶å¯¾é£ç¶æ§ã«ããç©åå½¢ï¼</div> 208 209<h4>3.7.1 æ¬ç¯ã®ä¸»å¼µââç©ç層åç¬ã§ã¯ãã¨ã³ãããã¼ã®ä¿ååã¯åå¨ããªã</h4> 210<p>æä½æ½è±¡åº¦ã®ç©ç層 α = 0 ã ããè¦ãããããçå¼ã¨ãã¦æãç«ã¤ä¿ååã¯<b>ã¨ãã«ã®ã¼ä¿ååã®ã¿</b>ã§ãããã¨ã³ãããã¼ã«ã¤ãã¦æãç«ã¤ã®ã¯ç±åå¦ç¬¬äºæ³åãããªãã¡ dS<sub>phys</sub>/dt ⥠0 ã¨ãã<b>ä¸çå¼</b>ã ãã§ãããçå¼ã¨ãã¦ã®ä¿ååã¯åå¨ããªããæ¬å®çã主張ããã®ã¯æ¬¡ã®ä¸ç¹ã§ããââ<b>髿½è±¡åº¦ã¬ã¤ã¤ã¼ã¾ã§å«ããèªç¥ï¼æ å ±å®å®ã«ããã¦ã¯ãã¨ã³ãããã¼ã«ã¤ãã¦ãçå¼ã¨ãã¦ã®ä¿ååãæãç«ã¤</b>ã</p> 211<p>ãã®ä¿åã¯ãèªç¥å´ã®æå³ã¨ã³ãããã¼ä½ä¸ã¨ç©çå´ã®ç©çã¨ã³ãããã¼å¢å¤§ããæç®éã¿ w<sub>α</sub> ãä»ãã¦<b>åä¸ã®å¸³ç°¿ä¸ã§äº¤æããã¦ãã</b>ãã¨ããå¾ããæ£é¸ Î ãé¶ã§ããçæ³ééå¯éç³»ã§ã¯ãä¸è¬åç·ã¨ã³ãããã¼ S<sub>gen</sub> ã¯å³å¯ã«ä¿åããããæ£é¸ãããã°ãå¢å ã¯ã¡ããã©æ£é¸ã®åã ãã§ããã</p> 212<div class="rigor"><div class="rigor-title">æ¬å®çã®éèªææ§ã¯ã©ãã«ããã</div> 213<p>æ¬å®çãã交æå¼ãä»£å ¥ãã¦æã¡æ¶ãã ãã®æçå¼ãã¨èªãã®ã¯èª¤ãã§ãããå®è³ªçãªå å®¹ã¯æ¬¡ã®ä¸ç¹ã«ããã<b>第ä¸</b>ã«ãã¬ã¤ã¤ã¼ãå¯ç®ç¡éåããããç¶æ³ã§ S<sub>gen</sub> ã絶対é£ç¶ã§ãããé å¥å¾®åãæ£å½åããããã¨ï¼è£é¡15.1ãããã¯ç¡æ¡ä»¶ã«ã¯æç«ããã§3.7.5 ã«åä¾ãæããï¼ã<b>第äº</b>ã«ãä¿åãæ£é¸ã®æ¶æ» ã¨<b>åå¤</b>ã§ãããã¨ï¼è£é¡15.2ï¼ã<b>第ä¸</b>ã«ããã®ä¿ååã<b>ç©ç層åç¬ã¸ã¯éå ã§ããªã</b>ãã¨ãããªãã¡ç©ç層ã®éã ãããã¯åãä¿ååãæ§æã§ããªããã¨ï¼å½é¡15.Dï¼ã第ä¸ç¹ããæ¬ç¯åé ã®ä¸»å¼µââç©ç層åç¬ã§ã¯ã¨ã³ãããã¼ä¿ååãåå¨ããªãââã®å³å¯ãªå 容ã§ããã</p></div> 214 215<h4>3.7.2 è¨å®ã¨å¸¸è¨ä»®å®</h4> 216<div class="assumption"><div class="assumption-title">ä»®å®A1ï¼æ½è±¡åº¦ã¬ã¤ã¤ã¼æï¼</div> 217<p>æ½è±¡åº¦ææ°éå <span class="math">ð â [0,â)</span> 㯠0 ãå«ãå¯ç®éåã¨ããå <span class="math">α â ð</span> ã«å¯æ¸¬ç©ºéï¼ã¬ã¤ã¤ã¼ï¼<span class="math">U<sub>α</sub></span> ãä¸ãããããå ¨ä½ç©ºéã¯äºãã«ç´ ãªåä½µ <span class="math">U = â<sub>αâð</sub>U<sub>α</sub></span> ã§ããã<span class="math">α = 0</span> ã®ã¬ã¤ã¤ã¼ <span class="math">U<sub>0</sub></span> ãç©ç空éã¨å¼ã¶ãæéçºå±ã¯å¯æ¸¬ãªè»éã®æ <span class="math">t ⦠x<sub>α</sub>(t) â U<sub>α</sub></span>ï¼<span class="math">t â [0,T]</span>ï¼ã¨ãã¦ä¸ããããã</p></div> 218<div class="assumption"><div class="assumption-title">ä»®å®A2ï¼æå³ã¨ã³ãããã¼ã®æ£åæ§ï¼</div> 219<p>å <span class="math">α â» 0</span> ã«å¯¾ãæå³ã¨ã³ãããã¼æ±é¢æ° <span class="math">H<sub>α</sub> : U<sub>α</sub> â [0,â)</span> ãä¸ããããè»éã«æ²¿ã£ãåæ <span class="math">t ⦠H<sub>α</sub>(t) := H<sub>α</sub>(x<sub>α</sub>(t))</span> 㯠<span class="math">[0,T]</span> ä¸ã§çµ¶å¯¾é£ç¶ã§ããã</p></div> 220<div class="assumption"><div class="assumption-title">ä»®å®A3ï¼å°å½±ã®æ´åæï¼</div> 221<p>å <span class="math">α > β ⥠0</span> ã«å¯æ¸¬ãªå°å½± <span class="math">Ï<sub>βâα</sub> : U<sub>α</sub> â U<sub>β</sub></span> ãä¸ããããåç¾¤æ§ <span class="math">Ï<sub>γâβ</sub> â Ï<sub>
221βâα</sub> = Ï<sub>γâα</sub></span>ï¼<span class="math">γ < β < α</span>ï¼ããã³ <span class="math">Ï<sub>αâα</sub> = id</span> ãæºãããæä½æ½è±¡åº¦å°å½±ã <span class="math">Ï<sub>0</sub> := Ï<sub>0âα</sub></span> ã¨ç¥è¨ããã</p></div> 222<div class="assumption"><div class="assumption-title">ä»®å®A4ï¼å°å½±å調æ§ââç²è¦åã¯ã¨ã³ãããã¼ãå¢ããï¼</div> 223<p>ä»»æã® <span class="math">α > β > 0</span> 㨠<span class="math">x â U<sub>α</sub></span> ã«å¯¾ã <span class="math">H<sub>β</sub>(Ï<sub>βâα</sub>(x)) ⥠H<sub>α</sub>(x)</span>ãçå·ã¯å°å½±ãå¯éï¼æ å ±ã失ããªãï¼ãªå ´åã«éãã</p></div> 224<div class="assumption"><div class="assumption-title">ä»®å®A5ï¼ç©çã¨ã³ãããã¼ã®æ£åæ§ï¼</div> 225<p>ç©ç層ã®çµåé¨åç³»ã«å¯¾ãã¦ç©çã¨ã³ãããã¼ <span class="math">S<sub>phys</sub> : [0,T] â â</span> ãå®ç¾©ããã絶対é£ç¶ã§ããã<span class="math">S<sub>phys</sub></span> ã¯ç±åå¦çã¨ã³ãããã¼ã®éå¸¸ã®æå³ãæã¡ãåç¬ã§ã¯å¤é¨ã¨ã®äº¤æã«ãã墿¸ãããã</p></div> 226<div class="assumption"><div class="assumption-title">ä»®å®A6â²ï¼æç®éã¿ã¨ç·åã®æ£åæ§ââ䏿§å¯ç©åå½¢ï¼</div> 227<p>å <span class="math">α â» 0</span> ã«å¯¾ã宿°ã®æç®éã¿ <span class="math">w<sub>α</sub> > 0</span> ãä¸ããããæ¬¡ãæºããã<b>(i)</b> <span class="math">Σ<sub>αâ»0</sub>w<sub>α</sub>H<sub>α</sub>(t)</span> 㯠<span class="math">[0,T]</span> ã®åç¹ã§æéã<b>(ii)</b> <span class="math">h<sub>α</sub> := dH<sub>α</sub>/dt</span> ã¨ç½®ãã¨ããç´æ° <span class="math">Σ<sub>αâ»0</sub>w<sub>α</sub>h<sub>α</sub>(s)</span> ã¯ã»ã¨ãã©è³ãæåæãããã¤æéé¨ååã®æ <span class="math">{Σ<sub>αâðâ²</sub>w<sub>α</sub>h<sub>α</sub> : ðâ² â ð æé}</span> 㯠<span class="math">L<sup>1</sup>([0,T])</span> ã«ããã¦<b>䏿§å¯ç©å</b>ã§ããã</p> 228<p class="small"><b>åè«æã®ä»®å®A6(ii)ããã®å¼·åã</b>åè«æã¯å¯ç©ååªé¢æ° <span class="math">g = Σw<sub>α</sub>g<sub>α</sub> â L<sup>1</sup></span> ã«ããåªåææ¡ä»¶ã課ãã¦ãããåªé¢æ°ã«ããæ¯é ã¯ä¸æ§å¯ç©åæ§ã嫿ããããéã¯æãç«ãã
228ªãããããã£ã¦A6â²ã¯<b>çã«å¼±ãä»®å®</b>ã§ãããå®çã®é©ç¨ç¯å²ãæ¡ããã証æã§ã¯åªåæå®çã®ä»£ããã«Vitaliã®åæå®çãç¨ããï¼è£é¡15.1ï¼ã<span class="math">ð</span> ãæééåã®ã¨ã㯠(ii) ã¯èªæã«æç«ããã</p></div> 229<div class="assumption"><div class="assumption-title">ä»®å®A7ï¼å±¤éçµåï¼äº¤æå¼ï¼</div> 230<p>ã»ã¨ãã©è³ãæã§æ¬¡ãæãç«ã¤ã</p> 231<div class="eq"><span class="math">dS<sub>phys</sub>/dt = âΣ<sub>αâ»0</sub>w<sub>α</sub> dH<sub>α</sub>/dt + Î (t),ãÎ (t) ⥠0,ãÎ â L<sup>1</sup>([0,T])</span></div> 232<p><span class="math">Î </span> ã¯æ£é¸é ã§ãããä¸å¯éæ§ã»ç±åã»ä½å°çæã»ç°å¢ã¨ã®ç¸äºä½ç¨ã表ãã</p></div> 233<div class="rigor"><div class="rigor-title">ä»®å®A7ã®å å®ââã©ããå®ç¾©ã§ãã©ããå ¬çã</div> 234<p>A7ã¯äºã¤ã®é¨åãããªãã第ä¸ã¯<b>å®ç¾©</b>ã§ãã£ã¦ã<span class="math">Î (t) := dS<sub>phys</sub>/dt + Σ<sub>αâ»0</sub>w<sub>α</sub>dH<sub>α</sub>/dt</span> ã¨ç½®ããã¨ã«ã¯ãããªãå 容ããªãã第äºã¯<b>ç¬¦å·æ¡ä»¶</b> <span class="math">Î (t) ⥠0</span> ã§ããããããå¯ä¸ã®å®è³ªçãªå ¬çââ<b>ä¸è¬åç¬¬äºæ³å</b>ââã§ããããã®åè§£ãæç¤ºãããã¨ã¯ãæ¬å®çãä½ãä»®å®ãä½ã証æãã¦ããã®ããææ§ã«ããªãããã«å¿ è¦ã§ãããããªãã¡æ¬å®çã¯ãä¸è¬åç¬¬äºæ³åãä»®å®ããããã§ã(i) ç·åéãå¾®åå¯è½ãªæå³ãæã¤ãã¨ã(ii) ä¿åãæ£é¸ã®æ¶æ» ã¨åå¤ã§ãããã¨ã(iii) ãã®ä¿åãç©ç層ã¸éå ã§ããªããã¨ã証æããã</p></div> 235 236<h4>3.7.3 å®ç15ã¨å³å¯è¨¼æ</h4> 237<div class="theorem"><div class="theorem-title">å®ç15ï¼è«ç±³å°èªç¥ç©çã¨ã³ãããã¼äº¤æã»ä¿åå®çï¼</div> 238<p>常è¨ä»®å®A1ãA5ã»A6â²ã»A7ã®ãã¨ã§ãä¸è¬åç·ã¨ã³ãããã¼ã <span class="math">S<sub>gen</sub>(t) := S<sub>phys</sub>(t) + Σ<sub>αâ»0</sub>w<sub>α</sub>H<sub>α</sub>(t)</span> ã¨å®ããããã®ã¨ã次ãæãç«ã¤ã</p> 239<p><b>(I) 交æã¨ä¸è¬åç¬¬äºæ³åã</b><span class="math">S<sub>gen</sub></span> 㯠<span class="math">[0,T]</span> ä¸ã§çµ¶å¯¾é£ç¶ã§ããã</p> 240<div class="eq"><span class="math">dS<sub>gen</sub>/dt = Î (t) ⥠0 ï¼a.e.ï¼,ãS<sub>gen</sub>(t<sub>2</sub>) â S<sub>gen</sub>(t<sub>1</sub>) = â«<sub>t<sub>1</sub></sub><sup>t<sub>2</sub></sup>Î (s) ds ⥠0</span><span class="eqno">(3.7)</span></div> 241<p>ãä»»æã® <span class="math">0 ⤠t<sub>1</sub> ⤠t<sub>2</sub> ⤠T</span> ã«ã¤ãã¦æãç«ã¤ãã¨ãã« <span class="math">S<sub>gen</sub></span> ã¯åèª¿éæ¸å°ã§ããã</p> 242<p><b>(II) ä¿ååã</b><span class="math">Î = 0</span> ãã»ã¨ãã©è³ãææãç«ã¤ãã¨ã¨ã<span class="math">S<sub>gen</sub></span> ã <span class="math">[0,T]</span> ä¸å®æ°ã§ãããã¨ã¯åå¤ã§ãããã¨ãã«çæ³ééå¯éç³» <span class="math">Î â¡ 0</span> ã§ã¯ <span class="math">S<sub>gen</sub>(t) = S<sub>gen</sub>(0)</span> ããã¹ã¦ã® <span class="math">t</span> ã§å³å¯ã«æãç«ã¤ã</p> 243<p><b>(III) ç§©åºåéã®ä¸çã</b>ä»»æã® <span class="math">0 ⤠t<sub>1</sub> ⤠t<sub>2</sub> ⤠T</span> ã«ã¤ãã¦</p> 244<div class="eq"><span class="math">Σ<sub>αâ»0</sub>w<sub>α</sub>[H<sub>α</sub>(t<sub>1</sub>) â H<sub>α</sub>(t<sub>2</sub>)] = [S<sub>phys</sub>(t<sub>2</sub>) â S<sub>phys</sub>(t<sub>1</sub>)] â â«<sub>t<sub>1</sub></sub><sup>t<sub>2</sub></sup>Î (s) ds ⤠S<sub>phys</sub>(t<sub>2</sub>) â S<sub>phys</sub>(t<sub>1</sub>)</span><span class="eqno">(3.8)</span></div> 245<p>ãæãç«ã¡ãçå·æç«ã¯ <span class="math">Î = 0</span>ï¼a.e. on <span class="math">[t<sub>1</sub>,t<sub>2</sub>]</span>ï¼ã®ã¨ãã«éããããªãã¡ãåºéå ã§éæããã<b>èªç¥çç§©åºåã®ç·éã¯ãååºéã®ç©çã¨ã³ãããã¼å¢å éã«ãã£ã¦ä¸ããæ¼ããããã</b>ã</p></div> 246 247<div class="theorem"><div class="theorem-title">è£é¡15.1ï¼é å¥å¾®åââVitaliå¼·åå½¢ï¼</div> 248<p>ä»®å®A2ã»A6â²ã®ãã¨ã§ã<span class="math">F(t) := Σ<sub>αâ»0</sub>w<sub>α</sub>H<sub>α</sub>(t)</span> 㯠<span class="math">[0,T]</span>
248 ä¸çµ¶å¯¾é£ç¶ã§ãããã»ã¨ãã©è³ãæ <span class="math">dF/dt = Σ<sub>αâ»0</sub>w<sub>α</sub>h<sub>α</sub></span> ãæãç«ã¤ã</p></div> 249<div class="proof"><div class="proof-title">証æ</div> 250<p>å <span class="math">H<sub>α</sub></span> ã¯çµ¶å¯¾é£ç¶ã§ãããã <span class="math">H<sub>α</sub>(t) = H<sub>α</sub>(0) + â«<sub>0</sub><sup>t</sup>h<sub>α</sub>(s) ds</span>ï¼<span class="math">h<sub>α</sub> â L<sup>1</sup></span>ï¼ã¨æ¸ãããA6â²(i) ãã <span class="math">F(0)</span> ã¯æéã§ããã</p> 251<p><span class="math">ð</span> ãå¯ç®éåã¨ãã¦ä¸ã¤ææãã<span class="math">F<sub>n</sub>(t) := Σ<sub>kâ¤n</sub>w<sub>α<sub>k</sub></sub>H<sub>α<sub>k</sub></sub>(t)</span>ã<span class="math">f<sub>n</sub>(s) := Σ<sub>kâ¤n</sub>w<sub>α<sub>k</sub></sub>h<sub>α<sub>k</sub></sub>(s)</span> ã¨ç½®ããA6â²(ii) ã®ååãã <span class="math">f<sub>n</sub> â f := Σ<sub>αâ»0</sub>w<sub>α</sub>h<sub>α</sub></span> ãã»ã¨ãã©è³ãææãç«ã¡ãå¾åãã <span class="math">{f<sub>n</sub>}</span> ã¯ä¸æ§å¯ç©åã§ããã<b>Vitaliã®åæå®ç</b>ã«ãã <span class="math">f â L<sup>1</sup>([0,T])</span> ã㤠<span class="math">âf<sub>n</sub> â fâ<sub>L<sup>1</sup></sub> â 0</span>ããããã£ã¦å <span class="math">t</span> ã«ã¤ãã¦</p> 252<div class="eq"><span class="math">â«<sub>0</sub><sup>t</sup>f<sub>n</sub>(s) ds â¶ â«<sub>0</sub><sup>t</sup>f(s) ds</span></div> 253<p>ã§ããã仿¹ãå <span class="math">H<sub>α</sub> ⥠0</span> ã㤠<span class="math">w<sub>α</sub> > 0</span> ã§ãããããé¨åå <span class="math">F<sub>n</sub>(t)</span> 㯠<span class="math">n</span> ã«ã¤ãã¦åèª¿éæ¸å°ã§ãããA6â²(i) ã®æéæ§ããåç¹ã§ <span class="math">F(t)</span> ã¸åæããã<span class="math">F<sub>n</sub>(t) = F<sub>n</sub>(0) + â«<sub>0</sub><sup>t</sup>f<sub>n</sub>(s) ds</span> ã®ä¸¡è¾ºã§ <span class="math">n â â</span> ã¨ããã°</p> 254<div class="eq"><span class="math">F(t) = F(0) + â«<sub>0</sub><sup>t</sup>f(s) ds</span></div> 255<p>ãå¾ããå³è¾ºã¯ <span class="math">L<sup>1</sup></span> 颿°ã®ä¸å®ç©åã§ãããã絶対é£ç¶ã§ãããLebesgueã®å¾®åå®çããã»ã¨ãã©è³ãæ <span class="math">dF/dt = f</span> ã§ããã<span class="qed">â</span></p> 256<p class="small"><b>注æï¼å¼·åã®å®è³ªï¼ã</b>åè«æã¯å¯ç©ååªé¢æ° <span class="math">g</span> ã§ <span class="math">|f<sub>n</sub>| ⤠g</span> ã課ãåªåæå®çãç¨ãã¦ãããåªé¢æ°ã«ããæ¯é ã¯ä¸æ§å¯ç©åæ§ã嫿ãããéã¯å½ã§ãããããæ¬è£é¡ã¯åè«æã®è£é¡A.2.1ã<b>çã«ä¸è¬å</b>ãã¦ããããã¨ãã° <span class="math">f<sub>n</sub></span> ã®ãå±±ããæé軸ä¸ãç§»åãã¦ããåã®æã¯ã䏿§å¯ç©åã§ãããªããå¯ç©ååªé¢æ°ãæããªããã¨ãããã</p></div> 257 258<div class="theorem"><div class="theorem-title">è£é¡15.2ï¼ä¿åã®ç¹å¾´ä»ãï¼</div> 259<p>ä»®å®A1ãA5ã»A6â²ã»A7ã®ãã¨ã§ã<span class="math">S<sub>gen</sub></span> ã <span class="math">[0,T]</span> ä¸å®æ°ã§ãããã¨ã¨ã<span class="math">Î = 0</span> ãã»ã¨ãã©è³ãææãç«ã¤ãã¨ã¯åå¤ã§ããã</p></div> 260<div class="proof"><div class="proof-title">証æ</div> 261<p>ï¼âï¼å®ç15(I) ã® <span class="math">dS<sub>gen</sub>/dt = Î </span> ã¨çµ¶å¯¾é£ç¶æ§ã®NewtonâLeibnizå ¬å¼ããç´ã¡ã«å¾ããï¼âï¼<span class="math">S<sub>gen</sub></span> ã宿°ãªã <span class="math">0 = S<sub>gen</sub>(T) â S<sub>gen</sub>(0) = â«<sub>0</sub><sup>T</sup>Î (s) ds</span>ã<span class="math">Π⥠0</span> ãã¤ç©åã 0 ã§ãããããLebesgueç©åã®æ¨æºçæ§è³ªãã <span class="math">Î = 0</span> ãã»ã¨ãã©è³ãææãç«ã¤ã<span class="qed">â</span></p></div> 262 263<div class="theorem"><div class="theorem-title">è£é¡15.3ï¼ä¸ééã«æ²¿ãå調åæââçµè·¯ç¬ç«æ§ï¼</div> 264<p>ä»®å®A1ã»A3ã»A4ã®ãã¨ã§ãä»»æã®ä¸éé <span class="math">α<sub>k</sub> > α<sub>kâ1</sub> > ⯠> α<sub>1</sub> > 0</span> 㨠<span class="math">x â U<sub>α<sub>k</sub></sub></span> ã«å¯¾ãã<span class="math">x<sub>j</sub> := Ï<sub>α<sub>j</sub>âα<sub>k</sub></sub>(x)</span>
264 ã¨ç½®ãã° <span class="math">H<sub>α<sub>1</sub></sub>(x<sub>1</sub>) ⥠H<sub>α<sub>2</sub></sub>(x<sub>2</sub>) ⥠⯠⥠H<sub>α<sub>k</sub></sub>(x)</span> ãæãç«ã¤ãã¨ãã«ãã®æ¯è¼ã¯ãã©ã®ä¸éã¬ã¤ã¤ã¼ãçµç±ãããã«ä¾åããªãã</p></div> 265<div class="proof"><div class="proof-title">証æ</div> 266<p>A3ã®å群æ§ãã <span class="math">x<sub>j</sub> = Ï<sub>α<sub>j</sub>âα<sub>j+1</sub></sub>(x<sub>j+1</sub>)</span>ãå飿¥å¯¾ã«A4ãé©ç¨ããã° <span class="math">H<sub>α<sub>j</sub></sub>(x<sub>j</sub>) ⥠H<sub>α<sub>j+1</sub></sub>(x<sub>j+1</sub>)</span>ï¼<span class="math">j = 1,â¦,kâ1</span>ï¼ãå¾ããé£éãã¦ä¸»å¼µã®ä¸çå¼åãå¾ããçµè·¯ç¬ç«æ§ã¯ãä»»æã®äºçµè·¯ã®åæå°å½±ãå群æ§ã«ããåä¸ã® <span class="math">Ï<sub>α<sub>j</sub>âα<sub>k</sub></sub></span> ã«ä¸è´ãããã¨ã«ããã<span class="qed">â</span></p></div> 267 268<div class="proof"><div class="proof-title">å®ç15ã®è¨¼æ</div> 269<p><b>(I)</b> A5ãã <span class="math">S<sub>phys</sub></span> ã¯çµ¶å¯¾é£ç¶ãè£é¡15.1ãã <span class="math">F = Σ<sub>αâ»0</sub>w<sub>α</sub>H<sub>α</sub></span> ã絶対é£ç¶ã§ããã絶対é£ç¶é¢æ°ã®åã¯çµ¶å¯¾é£ç¶ã§ãããã <span class="math">S<sub>gen</sub> = S<sub>phys</sub> + F</span> ã¯çµ¶å¯¾é£ç¶ã§ãããã»ã¨ãã©è³ãæ</p> 270<div class="eq"><span class="math">dS<sub>gen</sub>/dt = dS<sub>phys</sub>/dt + Σ<sub>αâ»0</sub>w<sub>α</sub>dH<sub>α</sub>/dt</span></div> 271<p>ãæãç«ã¤ãããã«äº¤æå¼A7ãä»£å ¥ããã¨ã第ä¸é ã«å«ã¾ãã <span class="math">âΣ<sub>αâ»0</sub>w<sub>α</sub>dH<sub>α</sub>/dt</span> ã¨ç¬¬äºé ãæã¡æ¶ãåãã<span class="math">dS<sub>gen</sub>/dt = Î (t)</span> ãå¾ããA7ãã <span class="math">Π⥠0</span>ï¼a.e.ï¼ã§ãããã <span class="math">S<sub>gen</sub></span> ã¯åèª¿éæ¸å°ã§ããã絶対é£ç¶æ§ã«ããNewtonâLeibnizå ¬å¼ããç©åå½¢ (3.7) ãå¾ãã</p> 272<p><b>(II)</b> è£é¡15.2ãã®ãã®ã§ããã</p> 273<p><b>(III)</b> (3.7) ã®ç©åå½¢ã«ãã㦠<span class="math">S<sub>gen</sub> = S<sub>phys</sub> + Σw<sub>α</sub>H<sub>α</sub></span> ãå±éããã°</p> 274<div class="eq"><span class="math">[S<sub>phys</sub>(t<sub>2</sub>) â S<sub>phys</sub>(t<sub>1</sub>)] + Σ<sub>αâ»0</sub>w<sub>α</sub>[H<sub>α</sub>(t<sub>2</sub>) â H<sub>α</sub>(t<sub>1</sub>)] = â«<sub>t<sub>1</sub></sub><sup>t<sub>2</sub></sup>Î (s) ds</span></div> 275<p>ã§ããã第äºé ãå³è¾ºã¸ç§»ã符å·ãæ´çããã° (3.8) ã®çå¼é¨åãå¾ãã<span class="math">â«Î ⥠0</span> ããä¸çå¼é¨åãå¾ããçå·æç«ã¯ <span class="math">â«<sub>t<sub>1</sub></sub><sup>t<sub>2</sub></sup>Î = 0</span>ãããªãã¡ <span class="math">Π⥠0</span> ã¨ä½µã㦠<span class="math">Î = 0</span>ï¼a.e. on <span class="math">[t<sub>1</sub>,t<sub>2</sub>]</span>ï¼ã¨åå¤ã§ããã<span class="qed">â</span></p></div> 276 277<h4>3.7.4 ä¿ååã¯ç©ç層ã¸éå ã§ããªã</h4> 278<p>ããã¾ã§ã® (I)(II) ã¯ã交æå¼ãèªããããã§ã®å¸°çµã§ããããããæ¬ç¯åé ã®ä¸»å¼µââ<b>ç©ç層åç¬ã§ã¯ã¨ã³ãããã¼ã®ä¿ååãåå¨ããªã</b>ââã¯ãã¾ã 証æããã¦ããªãã以ä¸ã®å½é¡ããããä¸ããã</p> 279 280<div class="theorem"><div class="theorem-title">å½é¡15.Dï¼ç©ç層ã¸ã®éå ä¸å¯è½æ§ï¼</div> 281<p><b>(a) ä¿åã®éèªææ§ã</b>A1ãA5ã»A6â²ã»A7ãæºãããã㤠<span class="math">Î â¡ 0</span> ã§ããã«ããããããã<span class="math">dS<sub>phys</sub>/dt > 0</span> ãæ£ã®Lebesgue測度ã®éåä¸ã§æãç«ã¤ç³»ãåå¨ãããããªãã¡ãçæ³ééå¯éç³»ã«ããã¦ãã<b>ç©çã¨ã³ãããã¼åç¬ã¯ä¿åãããªã</b>
281ãä¿åãããã®ã¯ <span class="math">S<sub>gen</sub></span> ã®ã¿ã§ããã</p> 282<p><b>(b) éå ä¸å¯è½æ§ã</b>Borel坿¸¬ãª <span class="math">ð : â â â</span> ããA1ãA5ã»A6â²ã»A7ãæºãã <span class="math">Î â¡ 0</span> ã§ãããã¹ã¦ã®ç³»ã«ã¤ã㦠<span class="math">t ⦠ð(S<sub>phys</sub>(t))</span> ã <span class="math">[0,T]</span> ä¸å®æ°ã«ãããªãã°ã<span class="math">ð</span> ã¯åºé <span class="math">[S<sub>phys</sub>(0), â)</span> ä¸ã§å®æ°ã§ããããããã£ã¦ã<b>ç©çã¨ã³ãããã¼ã®ã¿ã®é¢æ°ã¨ãã¦éèªæãªä¿åéãæ§æãããã¨ã¯ã§ããªã</b>ã</p></div> 283<div class="proof"><div class="proof-title">証æ</div> 284<p><b>(a)</b> å ·ä½çãªè¨¼äººãæ§æããã<span class="math">ð = {0, 1}</span>ã<span class="math">T = 1</span>ã<span class="math">w<sub>1</sub> = 1</span> ã¨ãã<span class="math">H<sub>0</sub> > 0</span>ã<span class="math">λ > 0</span> ã宿°ã¨ãã¦</p> 285<div class="eq"><span class="math">H<sub>1</sub>(t) := H<sub>0</sub>e<sup>âλt</sup>,ãS<sub>phys</sub>(t) := S<sub>phys</sub>(0) + H<sub>0</sub>(1 â e<sup>âλt</sup>),ãÎ â¡ 0</span></div> 286<p>ã¨ç½®ãã<span class="math">H<sub>1</sub></span> 㯠<span class="math">[0,1]</span> ä¸ã§ <span class="math">C<sup>â</sup></span>ãéè² ãæçã§ããããA2ãæºããã<span class="math">S<sub>phys</sub></span> ã <span class="math">C<sup>â</sup></span> ã§ããããA5ãæºãããæ£ã®å±¤ã¯ä¸ã¤ã ãã§ããããA6â²(i)(ii) ã¯èªæã«æç«ãï¼æéåï¼ãA4㯠<span class="math">α > β > 0</span> ãªã対ãåå¨ããªããã空èã«æç«ãããA3㯠<span class="math">Ï<sub>0â1</sub></span> ãä»»æã®å¯æ¸¬ååã<span class="math">Ï<sub>αâα</sub> = id</span> ã¨åãã°æºãããããæå¾ã«A7ãæ¤è¨¼ããã</p> 287<div class="eq"><span class="math">dS<sub>phys</sub>/dt = λH<sub>0</sub>e<sup>âλt</sup>,ãâw<sub>1</sub>dH<sub>1</sub>/dt = λH<sub>0</sub>e<sup>âλt</sup></span></div> 288<p>ã§ãããã <span class="math">dS<sub>phys</sub>/dt = âw<sub>1</sub>dH<sub>1</sub>/dt + 0</span> ãå ¨ <span class="math">t</span> ã§æãç«ã¡ã<span class="math">Î â¡ 0 ⥠0</span>ããã£ã¦A7ãæºããããããã®ç³»ã«ãã㦠<span class="math">dS<sub>phys</sub>/dt = λH<sub>0</sub>e<sup>âλt</sup> > 0</span> ã <span class="math">[0,1]</span> å ¨ä½ï¼æ¸¬åº¦ 1ï¼ã§æãç«ã¤ã䏿¹</p> 289<div class="eq"><span class="math">S<sub>gen</sub>(t) = S<sub>phys</sub>(0) + H<sub>0</sub>(1 â e<sup>âλt</sup>) + H<sub>0</sub>e<sup>âλt</sup> = S<sub>phys</sub>(0) + H<sub>0</sub></span></div> 290<p>㯠<span class="math">t</span> ã«ãããªã宿°ã§ãããããªãã¡ <span class="math">S<sub>gen</sub></span> ã¯ä¿åãããã <span class="math">S<sub>phys</sub></span> ã¯å³å¯ã«å¢å ãããããã (a) ã®è¨¼äººã§ããã</p> 291<p><b>(b)</b> (a) ã®è¨¼äººæã <span class="math">H<sub>0</sub> > 0</span> ã«ã¤ãã¦èµ°ããããå <span class="math">H<sub>0</sub></span> ã«å¯¾ããè»é <span class="math">t ⦠S<sub>phys</sub>(t)</span> 㯠<span class="math">t</span> ã®é£ç¶ç義å¢å 颿°ã§ãã£ã¦ã<span class="math">t</span> ã <span class="math">[0,1]</span> ãåãã¨ãå¤åã¯åºé <span class="math">[S<sub>phys</sub>(0), S<sub>phys</sub>(0) + H<sub>0</sub>(1 â e<sup>âλ</sup>)]</span> ãã¡ããã©è¦ããä»®å®ãã <span class="math">ð(S<sub>phys</sub>(t))</span> ã¯ãã®è»éã«æ²¿ã£ã¦å®æ°ã§ããããã<span class="math">ð</span> ã¯ãã®åºéå ¨ä½ã§åä¸ã®å¤ <span class="math">ð(S<sub>phys</sub>(0))</span> ãã¨ãã<span class="math">λ</span> ãåºå®ã <span class="math">H<sub>0</sub> â â</span> ã¨ããã°åºéé· <span class="math">H<sub>0</sub>(1 â e<sup>âλ</sup>)</span> ã¯ä¸ã«éæçã§ããããããããã®åºéã®å併㯠<span class="math">[S<sub>phys</sub>(0), â)</span> ã«çããããã£ã¦ <span class="math">
291ð</span> 㯠<span class="math">[S<sub>phys</sub>(0), â)</span> ä¸ã§å®æ° <span class="math">ð(S<sub>phys</sub>(0))</span> ã§ããã<span class="qed">â</span></p></div> 292 293<div class="result"><div class="result-title">å½é¡15.Dã®æå³ââæ¬å®çãæ°ãã«ä¸»å¼µããå 容</div> 294<p>å½é¡15.D(a) ã¯ãçæ³ééå¯éç³»ã¨ãã<b>æãä¿åã«æå©ãªæ¡ä»¶ä¸ã§ãã</b>ãç©çã¨ã³ãããã¼ãåç¬ã§ã¯ä¿åãããªããã¨ã示ããå½é¡15.D(b) ã¯ããããã䏿©é²ãã¦ãç©çã¨ã³ãããã¼ã®ã©ã®ãããªé¢æ°ãåã£ã¦ãä¿åéã«ã¯ãªãããªããã¨ã示ãã<b>ä¿åéã¯å¿ ç¶çã«é«æ½è±¡åº¦ã¬ã¤ã¤ã¼ã®é H<sub>α</sub> ãå«ã¾ãªããã°ãªããªãã</b></p> 295<p>ãããã£ã¦ãæä½æ½è±¡åº¦ç©ç層ã«ããã¦ã¨ã³ãããã¼ã«ã¤ãã¦è¨ãããã¨ã¯ç¬¬äºæ³åã®ä¸çå¼ã®ã¿ã§ããï¼çå¼ã¨ãã¦æãç«ã¤ä¿ååã¯ã¨ãã«ã®ã¼ä¿ååã®ã¿ã§ããï¼ã<b>ã¨ã³ãããã¼ã®ä¿ååã¯ã髿½è±¡åº¦ã¬ã¤ã¤ã¼ãç¶æ è¨è¿°ã«å«ãã¦ã¯ããã¦æç«ãã</b>ããããæ¬å®çã®æ°è¦æ§ã§ãããæ¬å®çã¯<b>è«ç±³å°èªç¥ç©ç妿°çä½ç³»ã®ä¸å¿å®çã®ä¸ã¤</b>ã§ããããããã¾ã§éå ¬éã¨ãã¦ãããç©çå¦ã«ã¯ã¨ãã«ã®ã¼ä¿ååããããã<b>ã¨ã³ãããã¼ã«ã¤ãã¦ã¯ä¿ååãåå¨ããªã</b>ââæãç«ã¤ã®ã¯ç¬¬äºæ³åã®ä¸çå¼ã®ã¿ã§ãããèªç¥å®å®ã¾ã§å°ç¨ãåºããã¨ã<b>ã¨ã³ãããã¼ã交æããä¿åããã</b>ãã¨ã¯ãæ¬å®çã«ããååºã®ä¸»å¼µã§ããããã¤å³å¯è¨¼æã§ããããªããå®ç24ï¼ä¸åçè¦å®çï¼ããå®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ã«è³ã仿³æ°çã®ç³»åã¯ãèªç¥å´ã®ç§©åºåã¨ç©çå´ã®æ£é¸ãåä¸ã®å¸³ç°¿ã§æ±ãæ¬å®çãåæã¨ãã¦ã¯ããã¦å³å¯ã«å®å¼åããããã</p></div> 296 297<h4>3.7.5 A6â²ã®æå°æ§ââè½ã¨ããªãæ¡ä»¶ã§ãããã¨</h4> 298<p>A6â²(ii) ã¯æè¡çãªä¾¿å®ã§ã¯ãªããè½ã¨ãã¨çµè«ãç ´ç¶»ããæ¡ä»¶ã§ãããåä¾ã示ããå <span class="math">α = k</span>ï¼<span class="math">k = 1,2,â¦</span>
298ï¼ã«å¯¾ã <span class="math">H<sub>k</sub>(t)</span> ãæ¯å¹ <span class="math">a<sub>k</sub> = 2<sup>âk</sup></span>ã卿³¢æ° <span class="math">f<sub>k</sub> = 4<sup>k</sup></span> ã®æ¯åãæã¤ã¨ããã¨ã<span class="math">Σw<sub>k</sub>H<sub>k</sub>(t)</span> èªä½ã¯ä¸æ§åæãã¦åç¹æéãã¤é£ç¶ã§ããããå°é¢æ°ã®çµ¶å¯¾å¤ã®å㯠<span class="math">Σw<sub>k</sub>a<sub>k</sub>f<sub>k</sub> = Σw<sub>k</sub>2<sup>k</sup></span> ã®ãªã¼ãã¼ã§çºæ£ãã䏿§å¯ç©åãªé¨ååæãæããªãããã®ã¨ã <span class="math">S<sub>gen</sub></span> ã¯çµ¶å¯¾é£ç¶ã§ãããªããªãï¼ç·å¤åãçºæ£ãã髿¨é¢æ°åã®ç çï¼ããåé ã®å¾®åã®åãã¨ãåè¨ã®å¾®åããä¸è´ããªãã交æå¼A7ã®æã¡æ¶ãã¯æå³ã失ãã</p> 299<p><span class="math">ð</span> ãæééåã§ããã°ãã®ç çã¯èµ·ãããããA6â²(ii) ã¯èªæã«æºãããããããªãã¡A6â²(ii) ãå®è³ªçãªå¶ç´ã¨ãªãã®ã¯å¯ç®ç¡éåã®ã¬ã¤ã¤ã¼ãæ±ãå ´åã«éãããã</p> 300 301<h4>3.7.6 å®ç30ã¨ã®æ¥ç¶</h4> 302<p>æ¬ç¨¿ãå®ç15ãç¨ããã®ã¯ãå®ç30ï¼ã¨ã³ãããã¼äº¤æèªææ§æå®çï¼ã®Â§6ã«ããã¦ã§ãããæ¡ä»¶30-Cã¯ãæ¬ç¯ã®æå³ã¨ã³ãããã¼ <span class="math">H<sub>α</sub></span> ããä½å±¤è¦³æ¸¬ã¨è¨èªå±¥æ´ãçæããæ å ± <span class="math">ð¢<sub>t</sub></span> ã®ãã¨ã§ã®<b>æ¡ä»¶ä»ãShannonã¨ã³ãããã¼</b> <span class="math">H(Z<sub>a</sub>|ð¢<sub>t</sub>)</span> ã¨ãã¦å®ç¾ããæ£ã®å®æ°å <span class="math">c<sub>a</sub></span> ã§åä½ãæ¥ç¶ããããã®æ¥ç¶ã®ãã¨ã§ãå®ç30ã® (30.9)(30.10) ã¯æ¬å®çã® (3.7)(3.8) ã®ç¹æ®ä¾ã«ã»ããªããªããããªãã¡ãèªç¥å´ã®æ¸å°é <span class="math">Σw<sub>a</sub>I(Z<sub>a</sub>;M|ð¢<sub>t<sub>1</sub></sub>)</span> ã (3.8) 左辺ã®ç§©åºåéã§ãããç©çå´ã®å¢å¤§éãåå¼å³è¾ºã§ããã</p> 303<div class="rigor"><div class="rigor-title">è¨ãéãã¨ãªããã¨</div> 304<p>æ¬å®çã¯ã<b>æç®éã¿ w<sub>α</sub> ã®å ·ä½å¤ã«ã¤ãã¦ã¯ä½ãè¿°ã¹ãªã</b>ã証æã¯æ£å¤æ§ã¨å®æ°æ§ã®ã¿ãç¨ãããæ å ±ã®ç©ççæ¶å»ã«æå°ã¨ãã«ã®ã¼ã³ã¹ããä¼´ãã¨ããç©ççç´è¦³ã¯ <span class="math">w<sub>α</sub></span> ã®èªç¶ãªã¹ã±ã¼ã«ã示åããããä¸è¬ã®æ å ±åå¾ã«å¯¾ãã¦Landauerä¸éãç´æ¥é©ç¨ãã¦ã¯ãªããããã®ããã«ã¯ç©ççæ¶å»éç¨ãå¥éæå®ããå¿ è¦ããããã¾ãæ¬å®çã¯ã<b>æå³ã¨ã³ãããã¼ãå®éã«ä½ä¸ãããã¨ã主張ããªã</b>ãä½ä¸ãèµ·ããã°ç©çå´ã®å¢å¤§ã¨çµã°ãããã¨ããæ¡ä»¶ä»ãã®åæ¯åã§ããã</p></div> 305 306<h3>3.8 å®ç16ï¼èªå·±æèåå¨ã»çºçå®çï¼</h3> 307<div class="stdform"><span class="t">â ä¸å¿å¼ï¼æ¨æºå½¢ï¼</span>SC = lim<sub>â</sub>TCZ<sub>α</sub> â â ,ãF<sub>SC</sub>(S<sup>*</sup>) = S<sup>*</sup>,ãM(S<sup>*</sup>) represents S<sup>*</sup><span class="src">ãè«ç±³å°æ½è±¡åº¦èªç±è«ãå ¨å®çä¸è¦§ã®æ¨æºå½¢ã«ä¸è´</span></div> 308<div class="eqbridge"><b>æ¬ç¨¿ã§å±éå½¢ãç¨ããçç±ã</b>èªå·±åãå±¥æ´ã«ä¾åãããã¨ãæ¬ç¨¿ã®è¦ç¹ï¼ç¹ã«å®ç30ï¼ã¨ã³ãããã¼äº¤æèªææ§æå®çï¼ï¼ã§ãããããä¸»ä½ i ã¨å±¥æ´ h ã®æ·»åãè½ã¨ããªãã<br><b>SC</b> â <b>SC<sub>i,h</sub></b>ï¼<b>TCZ<sub>α</sub></b> â <b>K<sub>i,α</sub>(h)</b>ï¼å±¤ α ã®åè£éåï¼ï¼<b>F<sub>SC</sub></b> â <b>F<sub>i,h</sub></b></div> 309<div class="theorem"><div class="theorem-title">å®ç16ï¼èªå·±æèåå¨ã»çºçå®çï¼</div> 310<p>åæ½è±¡å±¤ α ã®åè£éå K<sub>i,α</sub>(h) ã空ã§ãªãã³ã³ãã¯ãã§ã層éã®å°å½±ãé£ç¶ãã¤æ´åçã§ããã¨ãããããã«èªå·±åæ åå F<sub>i,h</sub> ã縮å°ç q < 1 ã®ç¸®å°ååã§ããã¨ããããã®ã¨ã</p> 311<div class="eq"><span class="math">SC<sub>i,h</sub> = lim<sub>â</sub>K<sub>i,α</sub>(h) â â ,ãâ!S<sub>i,h</sub><sup>*</sup> : F<sub>i,h</sub>(S<sub>i,h</sub><sup>*</sup>) = S<sub>i,h</sub><sup>*</sup>,ãd<sub>SC</sub>(F<sup>n</sup>S<sub>0</sub>,S<sub>i,h</sub><sup>*</sup>) ⤠q<sup>n</sup>d<sub>SC</sub>(S<sub>0</sub>,S<sub>i,h</sub><sup>*</sup>)</span><span class="eqno">(3.9)</span></div></div> 312<div class="proof"><div class="proof-title">証æ</div> 313<p><b>åå¨ã</b>å K<sub>i,α</sub>(h) ã¯ç©ºã§ãªãã³ã³ãã¯ã Hausdorff 空éã§ãããå°å½±ã¯é£ç¶ãã³ã³ãã¯ã空éã®éç³»ã®é極éã¯ç©ºã§ãªãï¼Tychonoff ã®å®çããã®æ¨æºç帰çµï¼ããã£ã¦ SC<sub>i,h</sub> â â ã<b>䏿æ§ã</b>SC<sub>i,h</sub> ã¯å®åè·é¢ç©ºéã§ãããF<sub>i,h</sub> ã¯ç¸®å°ç q < 1 ã®ç¸®å°ååã§ãããããBanach ã®ä¸åç¹å®çã«ããåºå®ç¹ã¯åå¨ãããã¤ãã ä¸ã¤ã§ãããå復åã®èª¤å·®è©ä¾¡ d<sub>SC</sub>(F<sup>n</sup>S<sub>0</sub>,S<sup>*</sup>) ⤠q<sup>n</sup>d<sub>SC</sub>(S<sub>0</sub>,S<sup>*</sup>) ãåå®çããå¾ãã<b>åå¨ã¯ä½ç¸å¹¾ä½ããã䏿æ§ã¯ä¸åç¹å®çããæ¥ããåºæãç°ãªãã</b>ã¾ãåºå®ç¹ã¯å±¥æ´ h ã«ä¾åããããã示ãããã®ã
313¯æ®éçå®ä½ã§ã¯ãªãå±¥æ´ç¸å¯¾çãªèªå·±åã§ããã<span class="qed">â</span></p></div> 314 315<h3>3.9 å®ç18ï¼å çè¨èªé²åå®çï¼</h3> 316<div class="stdform"><span class="t">â ä¸å¿å¼ï¼æ¨æºå½¢ï¼</span>H(Z|Y,M<sub>â</sub>) < H(Z|Y);ãÎ <sub>0</sub> â Î <sub>â</sub>;ãââ±/ââ > 0<span class="src">ãè«ç±³å°æ½è±¡åº¦èªç±è«ãå ¨å®çä¸è¦§ã®æ¨æºå½¢ã«ä¸è´</span></div> 317<div class="theorem"><div class="theorem-title">å®ç18ï¼å çè¨èªé²åå®çï¼</div> 318<p>å çè¨èª M<sub>â</sub> ã観測 Y ã«å¯¾ãã¦éèªæãªæ å ±ãä¸ããã¨ã</p> 319<div class="eq"><span class="math">H(Z|Y,M<sub>â</sub>) < H(Z|Y),ãÎ <sub>0</sub> â Î <sub>â</sub>,ãââ±/ââ > 0</span><span class="eqno">(3.10)</span></div></div> 320<div class="proof"><div class="proof-title">証æ</div> 321<p><b>æ å ±ã</b>æ¡ä»¶ä»ãã¨ã³ãããã¼ã®å調æ§ãã H(Z|Y,M<sub>â</sub>) ⤠H(Z|Y) ãå¸¸ã«æãç«ã¡ãçå·æç«ã¯ I(Z;M<sub>â</sub>|Y) = 0ãããªãã¡ M<sub>â</sub> ã Y ã®ãã¨ã§ Z ã«é¢ãã¦æ å ±ãæããªãå ´åã«éããéèªææ§ã®ä»®å®ã¯ãããæé¤ããã®ã§ãç義ä¸çå·ãæãç«ã¤ã<b>å¶å¾¡ã</b>å çè¨èªãæã¤ä¸»ä½ã¯ãæããªã主ä½ã®æ¹çããã¹ã¦æ¨¡å£ã§ããï¼M<sub>â</sub> ãç¡è¦ããã°ããï¼ã®ã§ Î <sub>0</sub> â Î <sub>â</sub>ã<b>é²åã</b>æ¹çéåãåºããã°ä¸éã¯æ¸å°ããªãã®ã§ãèªç±ææå®¹é Ⱡ㯠â ã«ã¤ãã¦éæ¸å°ã§ãããéèªææ§ã®ãã¨ã§ç義å¢å ããã<span class="qed">â</span></p></div> 322 323<h3>3.10 å®ç24ï¼ä¸åçè¦å®çï¼ã»å®ç25ï¼è«¸æ³ç¡æå®çï¼ã»å®ç26ï¼æ¶ æ§å¯éå®çï¼ââåæ³å°è«æã®åç §</h3> 324<p>å®ç24ï¼ä¸åçè¦å®çï¼ã»å®ç25ï¼è«¸æ³ç¡æå®çï¼ã»å®ç26ï¼æ¶ æ§å¯éå®çï¼ã«ã¤ãã¦ã¯ã証æãæ¬ç¨¿ã«åæ²ãããè«ç±³å°åæ³å°å®çããåç §ãããæ¬ç¨¿ãç¨ããã®ã¯æ¬¡ã®ä¸ç¹ã§ããã</p> 325<div class="rigor"><div class="rigor-title">åæ³å°è«æããç¶æ¿ããå 容</div> 326<ol> 327<li><b>å®ç24ï¼ä¸åçè¦å®çï¼</b>ï¼æ¡ä»¶24-Aï¼ç©ºæªæºã§ã¯ V<sub>a</sub> = 0 ãã»ã¨ãã©è³ãæã§æ°¸ä¹ ã«ä¿ã¤æ¹çãåå¨ããªãï¼ã¨å¸¸è¨ä»®å®ï¼è»éè©ä¾¡ã®å¯æ¸¬æ§ãå°ãªãã¨ãä¸ã¤ã®æéã³ã¹ãæ¹çã®åå¨ãæå°å¤ã®å®ç¾ï¼ã®ãã¨ã§ãa ⺠⤠ãªãã° J<sup>*</sup><sub>a,Ï</sub>(x,T) > 0ã</li> 328<li><b>å®ç25ï¼è«¸æ³ç¡æå®çï¼</b>ï¼æ¡ä»¶25-Bã25-Dï¼ç¹ã«é¢ä¿çæ©è½å®åæ§ 25-Dï¼ã®ãã¨ã§ãé¢ä¿è¨è¿°ããç¬ç«ãå±¥æ´ãéãã¦åºå®ããåä½åãæ ãå æçã«éåé·ãªå å¨å¤æ°ã¯åå¨ããªããããªã㡠¬Atman(d,a)ã</li> 329<li><b>å®ç26ï¼æ¶ æ§å¯éå®çï¼</b>ï¼åä¸ã®æ¡ä»¶26-Aï¼é¶è¦éåã¨å®å®æ§ï¼ã®ãã¨ã§ãð©<sub>â¤</sub>(T) = â¬<sub>alive</sub> â© {x | J<sup>*</sup><sub>â¤,Ï</sub>(x,T) = 0} ã¯ç©ºã§ãªãéãã¤ååãä¸å¤ã§ãããäºæ¬¡ã®æã¿è¾¼ã¿ (26.A) ã¨å³å¯ä¸é (26.B) ãæãç«ã¤ãæ¬ç¨¿ãç¨ããã®ã¯ (26.A) 㨠(26.B) ã§ããã(26.C) ã¯ä¸è¦ã§ããã</li> 330</ol></div> 331<p>ãããä¸å®çã®è¨¼æã¯å½è©²è«æã«ãããæ¬ç¨¿ã¯ãã®çµè«ã®ã¿ãåæã¨ãã¦ç¨ããã<b>æ¬ç¨¿ã¯åæ³å°è«æã®ä¸»å¼µãå証æãããã®ã§ã¯ãªãã</b></p> 332</section> 333 334 335<section id="t28"> 336<h2>4. å®ç28ââæ¶ æ§ç¡æã»åæ´åå®ç</h2> 337<h3>4.1 ã¿ã°ä»ãé å</h3> 338<p>層ã®ç°ãªãç¹ãåä¸ç¹ã¨ãã¦æ¯è¼ããªããããé交åãç¨ããã</p> 339<div class="eq"><span class="math">ð:=(â<sub>aâºâ¤</sub>{a}à X<sub>a</sub>) â({â¤}Ãâ¬<sub>alive</sub>), ð<sub><â¤</sub>:=â<sub>aâºâ¤</sub>{a}à X<sub>a</sub>, Nir(T):={â¤}Ãð©<sub>â¤</sub>(T).</span><span class="eqno">(28.1)</span></div> 340<div class="assumption"><div class="assumption-title">æ¡ä»¶28-Aï¼æ¶ æ§éç¨ã¸ã®é¢ä¿çæ©è½å®åæ§ï¼</div> 341<p>éåéç¨ <span class="math">d<sub>N</sub>:=(Tâ¦ð©<sub>â¤</sub>(T))</span> ãå®ç25ï¼è«¸æ³ç¡æå®çï¼ã®ç¾è±¡ç´¢å¼éå <span class="math">ð</span> ã«å«ããããã®é¢ä¿ç¶æ <span class="math">Î<sub>d<sub>N</sub>,â¤</sub></span> ã¯ãå°ãªãã¨ã</p> 342<div class="eq"><span class="math">Î<sub>N</sub>=(X<sub>â¤</sub>,â¬<sub>alive</sub>,V<sub>â¤</sub>,Ï,Pol<sub>â¤</sub>, f<sub>â¤</sub>,Ï<sup>0</sup><sub>â¤</sub>)</span><span class="eqno">(28.2)</span></div> 343<p>ã¨ãéåæ <span class="math">ð©<sub>â¤</sub><sup>Î<sub>N</sub></sup>(T)</span>ãå±¥æ´ã層éé¢ä¿ãä»åå¨ã¨ã®é¢ä¿ãå«ã¿ãåè£èªæ§ <span class="math">Σ<sub>N</sub></span> ã¸ã®ä»å ¥ã«ã¤ãã¦æ¡ä»¶25-Dãæºãããã¾ãã<span class="math">d<sub>N</sub></span> ãå ããæ¡å¼µå¾ã®ç¾è±¡ç´¢å¼éåã«ã¤ãã¦æ¡ä»¶25-BãDãæç«ããã</p></div> 344<div class="theorem"> 345<div class="theorem-title">å®ç28ï¼è«ç±³å°æ¶ æ§ç¡æã»åæ´åå®çï¼</div> 346<p>å®ç24ï¼ä¸åçè¦å®çï¼ã®æ¡ä»¶24-Aããã³å¸¸è¨ä»®å®ï¼è»éè©ä¾¡ã®å¯æ¸¬æ§ãå°ãªãã¨ãä¸ã¤ã®æéã³ã¹ãæ¹çãæå°å¤å®ç¾ï¼ãå®ç25ï¼è«¸æ³ç¡æå®çï¼ã®25-BãDãå®ç26ï¼æ¶ æ§å¯éå®çï¼ã®
346常è¨ä»®å®ã¨æ¡ä»¶26-Aããªãã³ã«æ¡ä»¶28-Aãä»®å®ããããã®ã¨ããä»»æã® <span class="math">Tâ¥0</span> ã¨ä»»æã® <span class="math">(a,x)âð</span> ã«ã¤ãã¦ã</p> 347<div class="eq"><span class="math">ð<sub><â¤</sub>â©Nir(T)=â ,</span><span class="eqno">(28.3)</span></div> 348<div class="eq"><span class="math">(a,x)âð<sub><â¤</sub>â J<sup>*</sup><sub>a,Ï</sub>(x,T)>0, (â¤,x)âNir(T)â J<sup>*</sup><sub>â¤,Ï</sub>(x,T)=0.</span><span class="eqno">(28.4)</span></div> 349<p>ããã«ãå®å ¨å¯éè¿°èªã¯ã¿ã°ä»ã空éä¸ã§</p> 350<div class="eq"><span class="math">PZS(a,x,T)â [a=â¤â§xâð©<sub>â¤</sub>(T)]</span><span class="eqno">(28.5)</span></div> 351<p>ãæºãããæ¶ æ§éç¨ã¯</p> 352<div class="eq"><span class="math">¬Atman(d<sub>N</sub>,â¤)</span><span class="eqno">(28.6)</span></div> 353<p>ã§ãããå®ç25ï¼è«¸æ³ç¡æå®çï¼ã®å ¨å±¤æ¡ä»¶ã <span class="math">d<sub>N</sub></span> ã«ç¶æ¿ããå ´åã¯ã<span class="math">â aâð ¬Atman(d<sub>N</sub>,a)</span> ãæãç«ã¤ã</p> 354</div> 355<div class="proof"><div class="proof-title">証æ</div> 356<p>(28.3) ã¯é交åã®ã¿ã° <span class="math">aâºâ¤</span> 㨠<span class="math">a=â¤</span> ãåæã«æç«ããªããã¨ããå¾ãã(28.4) ã®å·¦è¾ºã¯å®ç24ï¼ä¸åçè¦å®çï¼ãå³è¾ºã¯ <span class="math">ð©<sub>â¤</sub>(T)</span> ã®å®ç¾©ã§ãããå®ç24ï¼ä¸åçè¦å®çï¼ã¯ <span class="math">aâºâ¤</span> ã«ã®ã¿éåãããå®ç26ï¼æ¶ æ§å¯éå®çï¼ã¯ <span class="math">a=â¤</span> ã«ã®ã¿å¯éç¹ãä¸ãããããåä¸ã®åä»ãç¹ã«æ£è²»ç¨ã¨é¶è²»ç¨ãåæã«ä¸»å¼µããªããå®ç26ï¼æ¶ æ§å¯éå®çï¼ã®PZSåé¡ãé交åã¸æã¡ä¸ããã°(28.5)ãå¾ãã</p> 357<p>ç¡æé¨ã示ããç¬ç«åè£ <span class="math">Σ<sub>N</sub></span> ã <span class="math">(Î<sub>d<sub>N</sub>,â¤</sub>,Y<sup>+</sup><sub>d<sub>N</sub></sub>)</span> ã®ä»å ¥å¾åå¸ãå¤ãããªãæ¡ä»¶25-Dã«åãããå¤ããªããªã <span class="math">Σ<sub>N</sub></span> ã¯å æçã«åé·ã§ããã<span class="math">Atman</span> ã®ãéåé·ãªåä½ååå ããæºãããªãããããã®å ´åã <span class="math">Atman(d<sub>N</sub>,â¤)</span> ã¯å½ã§ããã<span class="qed">â</span></p> 358</div> 359<div class="result"><div class="result-title">æ°å¦ç帰çµ</div><p>ãä¸åçè¦ãã¨ãæ¶ æ§å¯éãã¯ãåãç¡åç¹ã«ã¤ãã¦çç¾ããäºå½é¡ã§ã¯ãªããæä»çãªåä»ãé åã®å½é¡ã§ãããããã¦æ¶ æ§ã¯ãé¢ä¿è¨è¿°ãè¶ ããåºå®çèªæ§ããããªãé¶è²»ç¨éåéç¨ã¨ãã¦ãå®ç25ï¼è«¸æ³ç¡æå®çï¼ã®ç¡æè¿°èªã«å«ã¾ããã</p></div> 360<div class="note"><strong>æå¦ä¸ã®èªã¿ã</strong> æ¬ã¢ãã«ã¯ãåºä¸éãå«ãã¦ç¡æã¨èªã注éçç«å ´ã¨æ´åããããã ãã説ä¸åæé¨ã®ä¸å¥åç·¨ãä¸åº§é¨æ³¨éã»ç°èªã®æ´å²çåªå£ã¯æ°å¦ããã¯æ±ºã¾ããªããæ°å¦ãä¿è¨¼ããã®ã¯ãåä»ãé åã®æä»æ§ã¨ãæ¡ä»¶28-Aã®ãã¨ã§ã®ç¡æã§ããã</div> 361<div class="rigor"><div class="rigor-title">è¨ã£ã¦ãããã¨ï¼è¨ãéãã¨ãªããã¨</div><p><span class="math">ð©<sub>â¤</sub>(T)</span> ã¯æ°å¦çéåã¨ãã¦å®ç¾©ã§ãããå¦å®ãããã®ã¯ããã®é¢ä¿è¨è¿°ãè¶ ããç¬ç«ã»åºå®ã»å æçã«éåé·ãªèªæ§ã§ããããå¿ ãæã å»ã å¤ãããã¾ã§ããã«ã¯ãå¥é <span class="math">â T<sub>1</sub>,T<sub>2</sub>:ð©<sub>â¤</sub>(T<sub>1</sub>)â ð©<sub>â¤</sub>(T<sub>2</sub>)</span> ãå¿ è¦ã§ããã</p></div> 362</section> 363 364<section id="t29"> 365<h2>5. å®ç29ââ形弿³ä½ç³»ç¡èªæ§ã»ä¸å®åå®ç</h2> 366<h3>5.0 ã¢ãã«åã®æ£å½åââãªãæ³ãå½¢å¼çè«ã¨ãã¦æ±ããã®ã</h3> 367<p>æ¬ç¯ã®çµè«ã¯å¤é¨ã¡ã¿å®çã«ä¾åããããããã£ã¦ããããããæèª¬ã¨ãã¦ã®æ³ããå½¢å¼çè«ã¨ãã¦ã¢ãã«åãã¦ããã®ããã¨ããåãã«å ã«çãã¦ããå¿ è¦ãããã<b>ãã®åãã¯æ°å¦ã®å é¨ã§ã¯æ±ºçããªãã</b>以ä¸ã¯ãã¢ãã«åã®é©ç¨ç¯å²ãéå®ããéå®ãæºããããæ ¹æ ãè¿°ã¹ããã®ã§ãã£ã¦ãæ³ä¸è¬ã®å½¢å¼åå¯è½æ§ã主張ãããã®ã§ã¯ãªãã</p> 368 369<div class="rigor"><div class="rigor-title">ä½ãã¢ãã«åããä½ãã¢ãã«åããªãã</div> 370<p>æ¬ç¯ãã¢ãã«åããã®ã¯ã<b>å½é¡ã¨ãã¦è¿°ã¹ãããè¦å®ãããæ¨è«é¢ä¿ã®ãã¨ã§éãã¦ããæèª¬ã®éã¾ã</b>ã§ãããããã ð<sub>h</sub> ã¨æ¸ãã<b>ã¢ãã«åã®å¯¾è±¡å¤ã§ãã</b>ã®ã¯æ¬¡ã®ä¸ã¤ã§ããã第ä¸ã«ãå®è·µã»ä¿®è¡ã»ä½é¨ãã®ãã®ã第äºã«ãè¨èªçå®å¼åãè¶ ããã¨ãããå å®¹ï¼æ¬ä½ç³»ã®ç¨èªã§ã¯ãæé«æ½è±¡åº¦ã«ãããä¸å¯èª¬ã®å´é¢ï¼ã第ä¸ã«ãå½é¡çéå ãæããªãå®è·µçæç¤ºã®ã¿ãããªãæç¤ºã<b>æ¬ç¯ã®çµè«ã¯ããããä¸ã¤ã«ã¤ãã¦ã¯ä½ãè¿°ã¹ãªãã</b></p></div> 371 372<p>æ¡ä»¶29-Aã¯ãæå¹æ§ã»ç®è¡å¼·åº¦ã»å¥å ¨æ§ã®ä¸ã¤ãããªãã以ä¸ãåæ¡ä»¶ãæ£æçãªä¾¿å®ã§ã¯ãªãã<b>æèª¬ãæèª¬ã¨ãã¦æ©è½ããããã®å¿ è¦æ¡ä»¶</b>ã§ãããã¨ã示ãã</p> 373 374<h4>(a) æå¹æ§ââå ¬çéåã帰ç´çå¯ç®ã§ãããã¨</h4> 375<p>ããå½é¡ããã®æèª¬ã«å±ãããå¦ãããæéã®æç¶ãã§å¤å®ã¾ãã¯ææã§ããªããã°ããã®æèª¬ã¯<b>ææã伿¿ãæ¤è¨¼ãã§ããªã</b>
375ãä½ã説ãããã®ãã確å®ã§ããªãä½ç³»ã¯ãå¾ä»£ããããä¿æãããã¨ãã§ãããç°èª¬ã¨ã®åºå¥ãã¤ããªããéã«ãçµå ¸ã®éæã»çµéã»æ³¨éã®ä¼çµ±ãæç«ãã¦ããã¨ããæ´å²çäºå®ã¯ããã®æèª¬ãææå¯è½ãªå½¢ã§æ±ããã¦ãããã¨ã示ãã<b>ãããã£ã¦æå¹æ§ã¯ãæ³ãä¼ãããããããã®æ¡ä»¶ãã®ãã®ã§ããã</b>ãã®æ¡ä»¶ãå¤ããã¨ã¯ã伿¿å¯è½æ§ãæ¾æ£ãããã¨ã«çããã</p> 376 377<h4>(b) ç®è¡å¼·åº¦ââRobinsonç®è¡ Q ãè§£éã§ãããã¨</h4> 378<p>æèª¬ã®å é¨ã«ã¯ãæéåã¨å帰ãç¾ãããåäºç¸èµ·ã¯é åºã¥ããããåäºé ã®åã§ãããäºèã»å å¦ã»åè«¦ã¯æéåã®é ç®ã®ææã§ãããé 観ã¨éæ» è¦³ã¯åã«æ²¿ã£ãåé²ã¨å¾éã®å復ã§ãããæ¬ä½ç³»ã«ããã¦ã¯ãããã«æ½è±¡åº¦ã®æã«æ²¿ã£ãå帰ï¼å®ç22ï¼é«é«åº¦LUBè¨å ´æå®çï¼ã®å ææ´æ°ï¼ãç¾ããã<b>ããããå é¨ã§è¡¨ç¾ããã«ã¯ãå¾è 颿°ã»å æ³ã»ä¹æ³ã¨åºæ¬çãªé åºå ¬çãããã°è¶³ããããã¯ã¾ãã« Q ã§ããã</b>éã« Q ãè§£éã§ããªãä½ç³»ã¯ãèªãã®æææ§é ãèªããªãã<b>ç®è¡å¼·åº¦ã¯ãæèª¬ãèªãã®æ§é ãå é¨ã§è¿°ã¹ãããã®æå°éã®è¦ä»¶ã§ããã</b></p> 379 380<h4>(c) å¥å ¨æ§ââ証æãããç®è¡çè¨æãçã§ãããã¨</h4> 381<p>èªãã®æéççµåãæ§é ã«ã¤ãã¦å½ã証æããä½ç³»ã¯ãèªå·±è«é§çã§ããããã¨ãã°ãåäºã®æ¯ã¯åä¸ã§ãããã証æã§ããä½ç³»ã¯ããã®ææã®æå³ã失ãã<b>å¥å ¨æ§ã¯ãæèª¬ãèªãã«ã¤ãã¦èª¤ããªãã¨ããæå°éã®è¦æ±ã§ããã</b>ãªããå®ç29ï¼å½¢å¼æ³ä½ç³»ç¡èªæ§ã»ä¸å®åå®çï¼ã®çµè«ã®ãã¡ç¬¬ä¸ã»ç¬¬äºã®ä¸»å¼µã«ã¯å¥å ¨æ§ã¾ã§å¿ è¦ã§ãªããç¡çç¾æ§ï¼ç¬¬äºä¸å®å ¨æ§å®çï¼ããã³ Ï ç¡çç¾æ§ãªãã Rosser åã®è°è«ï¼ç¬¬ä¸ä¸å®å ¨æ§å®çï¼ã§è¶³ãããå¥å ¨æ§ã課ãã®ã¯ãçµè«ãå¹³æã«è¿°ã¹ãããã®å¼·åã§ãã£ã¦ãæ¬è³ªçãªå¶ç´ã§ã¯ãªãã</p> 382 383<div class="result"><div class="result-title">é©ç¨ç¯å²ã®éå®ï¼æç¤ºï¼</div> 384<p>
384䏿¡ä»¶ã®ãããããæºããããªãä½ç³»ã«ã¯ã<b>å®ç29ï¼å½¢å¼æ³ä½ç³»ç¡èªæ§ã»ä¸å®åå®çï¼ã¯é©ç¨ãããªãã</b>å ·ä½çã«ã¯ã(i) å ¬çéåãææä¸è½ãªä½ç³»ã(ii) Q ãè§£éã§ããªãã»ã©è¡¨ç¾åã®å¼±ãä½ç³»ã(iii) èªãã®æéçè¨æã«ã¤ãã¦èª¤ãä½ç³»ã<b>æ¬å®çã¯ãããããæ³ãä¸å®å ¨ã§ãããã¨ã¯ä¸»å¼µããªãã</b>主張ããã®ã¯ãæ¡ä»¶29-Aãæºããå½¢å¼åãããæèª¬ä½ç³»ã«ã¤ãã¦ä¸å®å ¨æ§ãå¾ããã¨ããæ¡ä»¶ä»ãè¨æã®ã¿ã§ããã</p></div> 385 386<h4>(d) æ¬å®çã主張ããªããã¨</h4> 387<p>ä¸ç¹ãæç¤ºããã<b>第ä¸ã«ãæèª¬ã誤ãã§ããã¨ã¯ä¸»å¼µããªãã</b>ä¸å®å ¨æ§ã¯è彿§ã§ã¯ãªããæ±ºå®ä¸è½æã®åå¨ã¯ãä½ç³»ãå½ãå«ããã¨ãæå³ããªãã<b>第äºã«ãæèª¬ãç¡çç¾ã§ãªãã¨ã¯ä¸»å¼µããªãã</b>第äºä¸å®å ¨æ§å®çãè¿°ã¹ãã®ã¯ãç¡çç¾æ§ã<b>å é¨ã§è¨¼æã§ããªã</b>ã¨ãããã¨ã§ãã£ã¦ãç¡çç¾ã§ãªãã¨ãããã¨ã§ã¯ãªãã<b>第ä¸ã«ã宿ã¨ãã¦ã®æ³ã®ä¾¡å¤ã«ã¤ãã¦ä½ãè¿°ã¹ãªãã</b>æ¬å®çã®å°ç¨ã¯ãå½é¡çéå ãæã¤å½¢å¼åé¨åã«éãããã</p> 388 389<h4>(e) å®ç25ï¼è«¸æ³ç¡æå®çï¼ã¨ã®é¢ä¿ââæ¬å®çã¯æ°ããªå½¢èä¸å¦ç主張ã§ã¯ãªã</h4> 390<p>æ¬å®çãå°ãç¡èªæ§ã¯ãå®ç25ï¼è«¸æ³ç¡æå®çï¼ã®é¢ä¿çæ©è½å®åæ§ããå½¢å¼çè«ã¨ããç¹å®ã®å¯¾è±¡ã«é©ç¨ãã<b>å ·ä½ä¾</b>ã§ãããããªãã¡ãå½¢å¼çè« ð<sub>h</sub> ã¯ãèªãã®å¤é¨ï¼ã¡ã¿çè«ãããå¼·ãä½ç³»ã追å å ¬çï¼ã¨ã®é¢ä¿ãå¤ããã¨ããèªãã®ç¡çç¾æ§ã¨ããèªå·±åºç¤ã¥ããæããªãã<b>é¢ä¿ããåãé¢ãããèªå·±å®çµçãªåºç¤ã¥ããåå¨ããªã</b>ã¨ããç¹ã§ãããã¯å®ç25ï¼è«¸æ³ç¡æå®çï¼ã®è¿°ã¹ãç¡èªæ§ã¨åä¸ã®æ§é ã§ããã<b>æ¬ç¯ã¯æ°ããå½¢èä¸å¦ãå°å ¥ãããæ¢åã®å®çãä¸ã¤ã®å¯¾è±¡ã¸é©ç¨ãã¦ããã«ãããªãã</b></p> 391 392<div class="note"><p><b>æ³å®ãããåè«ã¨å¿çã</b>ãæ³ã¯å½¢å¼ä½ç³»ã§ã¯ãªããããã¯ç¯çé¯èª¤ã§ãããã¨ããåè«ãäºæ³ããããå¿çã¯æ¬¡ã®ã¨ããã§ããã<b>æ¬å®çã¯æ³ã¨å½¢å¼ä½ç³»ã®å䏿§ã主張ãã¦ããªãã</b>ä¸»å¼µã¯æ¡ä»¶æã§ããââããæèª¬ã®éã¾ããæ¡ä»¶29-Aãæºãããªãã°ã以ä¸ãå¾ãããããã£ã¦ãæ³ãå½¢å¼ä½ç³»ã¨ã¿ãªã
392ãªãç«å ´ã¯æ¬å®çã¨è¡çªããªãããããæ¬å®çã®åãåããå ã¯ã<b>ãæèª¬ãå®å ¨ã§ãããèªãã®ãã¡ã§å®çµãã¦åºç¤ã¥ãããã¦ãããã¨ãã主張</b>ã§ããããã®ãããªä¸»å¼µã¯ãæèª¬ãææå¯è½ï¼æå¹æ§ï¼ã§ããèªãã®æææ§é ãèªããï¼ç®è¡å¼·åº¦ï¼ãã¨ãåæã«è¦æ±ãããããæ¡ä»¶29-Aã®é©ç¨å¯¾è±¡ã¨ãªãã<b>æ¬å®çãå¦å®ããã®ã¯æ³ã§ã¯ãªããæ³ã«ã¤ãã¦ã®å®çµæ§ã®ä¸»å¼µã§ããã</b></p></div> 393 394<p>æ¢åã®å®ç25ï¼è«¸æ³ç¡æå®çï¼ã¯ãã§ã«ã諸æ³ç¡æå®çãã§ãããããæ¬ç¯ã¯ååã«ããªããã¾ããç¾è±¡ã¨ãã¦ã® <em>dharma</em> ã¨ãæèª¬ã¨ãã¦å½¢å¼åããã <em>Dharma</em> ãåºå¥ããããæ³ãã®ãã®ãå ¬çç³»ã§ãããã¯çµè«ã§ã¯ãªããæ¬¡ã®ã¢ãã«åæ¡ä»¶ã§ããã</p> 395<h3>5.1 å½¢å¼çè«æ¡ä»¶</h3> 396<p>å±¥æ´æ®µé <span class="math">hââ</span> ãã¨ã«ãå½¢å¼çè«ã</p> 397<div class="eq"><span class="math">ð<sub>h</sub>=(L<sub>h</sub>,Ax<sub>h</sub>,â¢<sub>h</sub>), Th(ð<sub>h</sub>)=Cn<sub>â¢<sub>h</sub></sub>(Ax<sub>h</sub>)</span><span class="eqno">(29.1)</span></div> 398<p>ã¨ããã</p> 399<div class="assumption"><div class="assumption-title">æ¡ä»¶29-Aï¼æå¹ã»ç®è¡çã»å¥å ¨ãªå½¢å¼åï¼</div> 400<p>å <span class="math">L<sub>h</sub></span> ã¯ç®è¡è¨èªã¾ãã¯ãã®è¨ç®å¯è½ãªå®ç¾©æ¡å¤§ã§ãè¨ç®å¯è½ã«ç¬¦å·åãããã<span class="math">Ax<sub>h</sub></span> ã¯è¨ç®å¯ææãè¨¼ææ¤æ»ã¯æå¹ã§ããã<span class="math">ð<sub>h</sub></span> 㯠<span class="math">IΣ<sub>1</sub></span> ç¨åº¦ã®ååãªç®è¡ãå«ã¿ãå°ãªãã¨ãç®è¡æã«ã¤ãã¦æ¨æºèªç¶æ°æ¨¡å <span class="math">â</span> ã«å¯¾ãå¥å ¨ã§ãæ¨æºç証æè¿°èªã¨HilbertâBernaysâLöbå°åºå¯è½æ§æ¡ä»¶ãæã¤ãprefix-freeæ®éæ©æ¢° <span class="math">U</span> ãä¸ã¤åºå®ãã<span class="math">K<sub>U</sub></span> ãçè«å ã§ç®è¡åã§ããã</p></div> 401<div class="assumption"><div class="assumption-title">æ¡ä»¶29-Bï¼å½¢å¼æ³éç¨ã¸ã®å®ç25ï¼è«¸æ³ç¡æå®çï¼é©ç¨ï¼</div> 402<p>段ééç¨ <span class="math">d<sub>ð</sub>:=(hâ¦ð<sub>h</sub>)</span> ã <span class="math">ð</span> ã«å«ããè¨èªãå ¬çãæ¨è«è¦åã符å·åãæ ä½ãå±¥æ´ãå©ç¨çµæã <span class="math">Î<sub>d<sub>ð</sub>,a</sub></span> ã«æ©è½çã«å®åãã¦æ¡ä»¶25-Dãé©ç¨ããã</p></div> 403<div class="assumption"><div class="assumption-title">æ¡ä»¶29-Cï¼éæ¾çæ´æ°è¦åï¼</div> 404<p>å ¨æ®µéã§åãç®è¡è¨èª <span class="math">L</span>ãæ¨è«ç³» <span class="math">â¢</span>ãæ®éæ©æ¢° <span class="math">U</span> ãç¨ãã<span class="math">ð<sub>0</sub>=(L,Ax<sub>0</sub>,â¢)</span> ã¯29-Aãæºãããåæé段é <span class="math">n</span> ã§ãæ¨æºæ¨¡åã§çã ã <span class="math">ð<sub>n</sub></span> ã§ã¯è¨¼æä¸è½ãªæ <span class="math">G<sub>n</sub></span> ãã¡ã¿çè«ã§é¸ã³ã</p> 405<div class="eq"><span class="math">Ax<sub>n+1</sub>:=Ax<sub>n</sub>âª{G<sub>n</sub>}, ð<sub>n+1</sub>:=(L,Ax<sub>n+1</sub>,â¢), Th(ð<sub>n+1</sub>)=Cn<sub>â¢</sub>(Ax<sub>n</sub>âª{G<sub>n</sub>}).</span><span class="eqno">(29.2)</span></div></div> 406<div class="theorem"><div class="theorem-title">å®ç29ï¼è«ç±³å°å½¢å¼æ³ä½ç³»ç¡èªæ§ã»ä¸å®åå®çï¼</div> 407<p>æ¡ä»¶29-Aãæºããå <span class="math">ð<sub>h</sub></span> ã«ã¤ãã¦ã次ãæãç«ã¤ã</p> 408<ol> 409<li>çãªã決å®ä¸è½æãåå¨ããï¼ 410<span class="math">â G<sub>h</sub> [â⨠G<sub>h</sub>â§ð<sub>h</sub>⬠G<sub>h</sub>â§ð<sub>h</sub>â¬Â¬ G<sub>h</sub>].</span><span class="eqno">(29.3)</span></li> 411<li>èªå·±ã®ç¡çç¾æ§ãå é¨è¨¼æã§ããªãï¼ 412<span class="math">ð<sub>h</sub>â¬Con(ð<sub>h</sub>).</span><span class="eqno">(29.4)</span></li> 413<li>æ©æ¢° <span class="math">U</span> ã¨çè«ã«ä¾åãã宿° <span class="math">c<sub>h,U</sub></span> ãåå¨ãã 414<span class="math">
414â sâ{0,1}<sup>*</sup> â nââ (n>c<sub>h,U</sub>): ð<sub>h</sub>⬠``K<sub>U</sub>(sÌ)>nÌ''.</span><span class="eqno">(29.5)</span></li> 415</ol> 416<p>ããã«æ¡ä»¶29-Bã®ãã¨ã§ã<span class="math">â aâð ¬Atman(d<sub>ð</sub>,a)</span> ã§ãããæ¡ä»¶29-Cã®ãã¨ã§ã</p> 417<div class="eq"><span class="math">Th(ð<sub>0</sub>)âTh(ð<sub>1</sub>) ââ¯âTh(ð<sub>n</sub>)ââ¯</span><span class="eqno">(29.6)</span></div> 418<p>ã¯åæé段éã§å¥å ¨æ§ãä¿ã¡ãªããä¸å®å ¨æ§ãåçºããå³å¯å¢å¤§éã§ãããæéæ®µã«æçµå®å ¨çè«ã¯ãªããç¹ã«é¢æ£çç¡å¸¸è¿°èª</p> 419<div class="eq"><span class="math">Anicca<sub>step</sub>(d<sub>ð</sub>):â â n [Th(ð<sub>n+1</sub>)â Th(ð<sub>n</sub>)]</span><span class="eqno">(29.7)</span></div> 420<p>ã¯çã§ãããããã¯æ¡ä»¶29-Cãä¸ããæ´æ°éç¨ã®ç¡å¸¸ã§ãã£ã¦ãGödelã»Chaitinã ãã®å¸°çµã§ã¯ãªãã</p></div> 421<div class="proof"><div class="proof-title">証æ</div> 422<p>(29.3) ã¯æ¡ä»¶29-Aã®æå¹æ§ã»ååãªç®è¡å¼·åº¦ã»å¥å ¨æ§ã®ãã¨ã§Gödel第ä¸ä¸å®å ¨æ§å®çãé©ç¨ããçµæã§ããã(29.4) ã¯åãçè«ã®ç¡çç¾æ§ã¨æ¨æºå°åºå¯è½æ§æ¡ä»¶ããGödel第äºä¸å®å ¨æ§å®çã«ããã</p> 423<p>(29.5) ãèçæ³ã§ç¤ºãã䏿§ä¸éããªããªããä»»æã«å¤§ããå ¥å <span class="math">n</span> ã«å¯¾ãã<span class="math">ð<sub>h</sub></span> ã®è¨¼æãåæãã¦æåã«ç¾ãã <span class="math">``K<sub>U</sub>(sÌ)>m''</span>ï¼ãã ã <span class="math">m⥠n</span>ï¼ãæ¢ãããã® <span class="math">s</span> ãåºåã§ãããããã°ã©ã é·ã¯åºå®ããã <span class="math">ð<sub>h</sub></span> 証æåæå¨ã®è¨è¿°é· <span class="math">c<sub>h</sub></span> 㨠<span class="math">n</span> ã®èªå·±åºåãè¨è¿°é·ã®åãããªãã¡ <span class="math">c<sub>h</sub>+K(n)+O(1)=c<sub>h</sub>+O(log n)</span> ã§ããã䏿¹ãå¥å ¨æ§ã«ããåºåããã <span class="math">s</span> ã¯çã« <span class="math">K<sub>U</sub>(s)>m⥠n</span> ãæºãããåå大ãã <span class="math">n</span> ã§ã¯ <span class="math">c<sub>h</sub>+O(log n)<n</span> ã¨ãªãçç¾ããããã£ã¦ä¸æ§é¾å¤ <span class="math">c<sub>h,U</sub></span> ãåå¨ããã</p> 424<p>29-Bã®ç¡èªæ§çµè«ã¯ãå®ç28ï¼æ¶ æ§ç¡æã»åæ´åå®çï¼ã¨åã25-Dã®äºåæ³ã <span class="math">d<sub>ð</sub></span> ã«é©ç¨ããã°å¾ãã29-Cã§ã¯ãçæãä¸ã¤å ããã®ã§æ¨æºæ¨¡åã«å¯¾ããå¥å ¨æ§ã¯ä¿åããã<span class="math">G<sub>n</sub>âTh(ð<sub>n</sub>)</span> ã ã <span class="math">G<sub>n</sub>âTh(ð<sub>n+1</sub>)</span> ã ããå å«ã¯å³å¯ã§ãããå <span class="math">ð<sub>n+1</sub></span> ã¯åã³29-Aãæºããã®ã§ãä¸å®å ¨æ§ã帰ç´çã«åé©ç¨ã§ããã<span class="qed">â</span></p></div> 425<div class="rigor"><div class="rigor-title">è«çå¦ããç´æ¥ã¯åºãªã主張</div> 426<ul><li>ããã¹ã¦ã®å ¬çç³»ã¯ä¸å®å ¨ãã¯å½ã§ãããå¼±ãæ±ºå®å¯è½çè«ããé广çãªå®å ¨ççéåã¯å°ç¨å¤ã§ããã</li><li>ä¸å®å ¨æ§å®çã ãã§ã¯æéå¤åã¯åºãªããæ´æ°ã¯æ¡ä»¶29-Cã§æ°ãã«ä¸ããã</li><li>ãã¢ããªãªãªãå¦å®ããããã¯å²å¦çè§£éã§ãããæ°å¦ççµè«ã¯ãååã«å¼·ãæå¹çè«ã®éå®çµæ§ãèªå·±èªè¨¼éçãè¤éæ§ä¸çã®å é¨è¨¼æéçã§ããã</li><li>Gödelã»Chaitinã ãã§ã¯ä»æçç¡æã¯åºãªããç¡èªæ§çµè«ã¯å®ç25ï¼è«¸æ³ç¡æå®çï¼ã¨æ¡ä»¶29-Bããåºãã</li></ul></div> 427</section> 428 429<section id="t30"> 430<h2>6. å®ç30ââã¨ã³ãããã¼äº¤æèªææ§æå®ç</h2> 431<p>å®ç1ï¼è«ç±³å°ä¸»å®çï¼ã§ã¯Egoã¯ãã§ã«å¶å¾¡æ¹çã¨ãã¦ãå®ç16ï¼èªå·±æèåå¨ã»çºçå®çï¼ã§ã¯èªå·±æè鿥µéã¨åºå®ç¹ããå®ç18ï¼å çè¨èªé²åå®çï¼ã§ã¯ãããåæã¨ããå é¨è¨èªãå®ç¾©ããã¦ããããããã£ã¦æ¬å®çã®ãçæãã¯ã主ä½ãç¡ããä½ããã¨ã§ã
431¯ãªãã<strong>å çè¨èªãå±¥æ´ç¸å¯¾çãªEgoæ§æã鏿ã»å®å®åãããã¨</strong>ãæå³ããã</p> 432<h3>6.1 éç³»ä¸ã®å çè¨èª</h3> 433<p>ä¸»ä½ <span class="math">i</span> ã«ã¤ãã¦ãå®ç16ï¼èªå·±æèåå¨ã»çºçå®çï¼ã®éç³»ã</p> 434<div class="eq"><span class="math">K<sub>a</sub>:=TCZ<sub>i,a</sub>, p<sub>ba</sub>:K<sub>a</sub>â K<sub>b</sub>(b⺠a), SC<sub>i</sub>:=lim<sub>â</sub><sub>a</sub> K<sub>a</sub></span><span class="eqno">(30.1)</span></div> 435<p>ã¨ããã<span class="math">SC<sub>i</sub></span> ã¯Banach空éå ã®é空ã³ã³ãã¯ãå¸éåï¼ãããã£ã¦å®åè·é¢ç©ºéï¼ã§ãæ¢åãã£ã¼ããã㯠<span class="math">F<sub>i</sub>:SC<sub>i</sub>â SC<sub>i</sub></span> ãæã¤ã</p> 436<div class="assumption"><div class="assumption-title">æ¡ä»¶30-Aï¼å°å½±æ´åãªè¨èªååã¨éèªæç¸®å°ï¼</div> 437<p>å çè¨èªå <span class="math">mâΣ<sub>i</sub><sup>*</sup></span> ãã¨ã«é£ç¶åå <span class="math">L<sub>a,m</sub>:K<sub>a</sub>â K<sub>a</sub></span> ãããã<span class="math">L<sub>a,ε</sub>=id<sub>K<sub>a</sub></sub></span> ãã¤</p> 438<div class="eq"><span class="math">p<sub>ba</sub>â L<sub>a,m</sub>=L<sub>b,m</sub>â p<sub>ba</sub>.</span><span class="eqno">(30.2)</span></div> 439<p>ãããèªå°ãã <span class="math">L<sub>m</sub>:SC<sub>i</sub>â SC<sub>i</sub></span>
439 ã«å¯¾ã <span class="math">G<sub>m</sub>:=F<sub>i</sub>â L<sub>m</sub></span> ã¨ç½®ããå®ç16ï¼èªå·±æèåå¨ã»çºçå®çï¼ã®å®åè·é¢ <span class="math">d<sub>SC</sub></span> ã®ãã¨ã§ <span class="math">Lip(F<sub>i</sub>)=q<sub>F</sub><1</span> ã¨ãããåéç©ºèª <span class="math">mâ ε</span> ã«ã¤ã㦠<span class="math">Lip(L<sub>m</sub>)=q<sub>m</sub></span>ã<span class="math">q<sub>F</sub>q<sub>m</sub><1</span> ã¨ãã空èªåºå®ç¹ <span class="math">S<sup>*</sup><sub>ε</sub></span> ã«ã¤ãã¦</p> 440<div class="eq"><span class="math">G<sub>m</sub>(S<sup>*</sup><sub>ε</sub>)â S<sup>*</sup><sub>ε</sub>.</span><span class="eqno">(30.3)</span></div></div> 441<div class="assumption"><div class="assumption-title">æ¡ä»¶30-Bï¼Egoæ¹çã®å®ç¾ï¼</div> 442<p>è¨èªæ¡å¼µæ¹ç空é <span class="math">Î <sub>i</sub><sup>(â)</sup></span> ã¯ã³ã³ãã¯ãã§ãè²»ç¨ <span class="math">J<sub>i</sub>(Ï;S)</span> 㯠<span class="math">Ï</span> ã«ä¸åé£ç¶ã§ããããããã£ã¦</p> 443<div class="eq"><span class="math">Ï<sup>(â),*</sup><sub>c,m</sub>â*arg min<sub>ÏâÎ <sub>i</sub><sup>(â)</sup></sub>J<sub>i</sub>(Ï;S<sub>m</sub><sup>*</sup>)</span><span class="eqno">(30.4)</span></div> 444<p>ãå®ç¾ããã</p></div> 445<div class="assumption"><div class="assumption-title">æ¡ä»¶30-Cï¼Shannonâç¶æ ã¨ã³ãããã¼æ©ï¼</div> 446<p>å±¥æ´ç¸å¯¾çãªè¨èªæ¡ä»¶ä»ãEgoæ§æã®ä¸ç¢ºå®æ§ãã確ç夿° <span class="math">Z:Ωâ SC<sub>i</sub></span> ã¨ãã®åº§æ¨ <span class="math">Z<sub>a</sub>:=pr<sub>a</sub>â Z</span> ã§è¡¨ãã<span class="math">Z<sub>a</sub></span> ã¯æéã¾ãã¯å¯ç®å¤ï¼ãããã¯åºå®éå忏ã¿ï¼ã§ã以ä¸ã®Shannonã¨ã³ãããã¼ã¯æééè² ã¨ãããä½å±¤è¦³æ¸¬ã¨æå» <span class="math">t</span> ã¾ã§ã®è¨èªå±¥æ´ãä½ãæ å ±ã <span class="math">ð¢<sub>t</sub></span> ã¨ãããåä¸å¯¾è±¡ <span class="math">Z</span> ãæ´æ°ããåºéã§ã¯ãå®ç15ï¼èªç¥ç©çã¨ã³ãããã¼äº¤æã»ä¿åå®çï¼ã®ç¶æ ã¨ã³ãããã¼ã¨å®ç18ï¼å çè¨èªé²åå®çï¼ã®æ¡ä»¶ä»ãShannonã¨ã³ãããã¼ã</p> 447<div class="eq"><span class="math">H<sub>a</sub><sup>(15)</sup>(x<sub>a</sub>(t))=c<sub>a</sub> H(Z<sub>a</sub>|ð¢<sub>t</sub>), c<sub>a</sub>>0</span><span class="eqno">(30.5)</span></div> 448<p>ã§æ¥ç¶ããä»¥å¾ <span class="math">w<sub>a</sub>â w<sub>a</sub>c<sub>a</sub></span> ã¨åå®ç¾©ãããå®ç15ï¼èªç¥ç©çã¨ã³ãããã¼äº¤æã»ä¿åå®çãæ¬ç¨¿ §3.7ï¼ã®åã®æ£åæ§A6â²ã¨äº¤æå¼A7ãåã <span class="math">H<sub>a</sub>,w<sub>a</sub></span> ã«é©ç¨ãããã</p></div> 449<div class="theorem"><div class="theorem-title">å®ç30ï¼è«ç±³å°ã¨ã³ãããã¼äº¤æèªææ§æå®çï¼</div> 450<p>å®ç15ï¼èªç¥ç©çã¨ã³ãããã¼äº¤æã»ä¿åå®çï¼ã»å®ç16ï¼èªå·±æèåå¨ã»çºçå®çï¼ã»å®ç18ï¼å çè¨èªé²åå®çï¼ã®ä»®å®ã¨æ¡ä»¶30-AãCã®ãã¨ã§ãåé空å çè¨èªå <span class="math">m</span> ã¯ä¸æãªåºå®ç¹ <span class="math">S<sub>m</sub><sup>*</sup>â SC<sub>i</sub></span> ã鏿ããä»»æã®åæèªå·±ç¶æ <span class="math">S<sub>0</sub>â SC<sub>i</sub></span> ã«ã¤ãã¦</p> 451<div class="eq"><span class="math">d<sub>SC</sub>(G<sub>m</sub><sup>n</sup>S<sub>0</sub>,S<sub>m</sub><sup>*</sup>)â¤(q<sub>F</sub>q<sub>m</sub>)<sup>n</sup> d<sub>SC</sub>(S<sub>0</sub>,S<sub>m</sub><sup>*</sup>).</span><span class="eqno">(30.6)</span></div> 452<p>ã¾ã <span class="math">S<sub>m</sub><sup>*</sup>â S<sup>*</sup><sub>ε</sub></span> ã§ããã</p> 453<div class="eq"><span class="math">ð<sub>i</sub>[m]:=(S<sub>m</sub><sup>*</sup>,M(S<sub>m</sub><sup>*</sup>),Ï<sup>(â),*</sup><sub>c,m</sub>)</span><span class="eqno">(30.7)</span></div> 454<p>ã¯è¨èªã»å±¥æ´ç¸å¯¾çãªEgoæ§æããªãã</p> 455<p><span class="math">t<sub>1</sub><t<sub>2</sub></span> ã«æ°ããå çè¨èª <span class="math">M<sub>(t<sub>1</sub>,t<sub>2</sub>]</sub></span> ãå ããã<span class="math">ð¢<sub>t<sub>2</sub></sub>=ð¢<sub>t<sub>1</sub></sub>â¨Ï(M<sub>(t<sub>1</sub>,t<sub>2</sub>]</sub>)</span> ã¨ãããéã¿ä»ãèªææ§é åã¨ã³ãããã¼</p> 456<div class="eq"><span class="math">â<sub>ego</sub>(t):=Σ<sub>aâ»0</sub>w<sub>a</sub>H(Z<sub>a</sub>|ð¢<sub>t</sub>)</span><span class="eqno">(30.8)</span></div> 457<p>ã«ã¤ãã¦ã<span class="math">Î X:=X(t<sub>2</sub>)-X(t<sub>1</sub>)</span> ã¨å®ããã¨ã</p> 458<div class="eq"><span class="math">Îâ<sub>ego</sub>=-Σ<sub>aâ»0</sub>w<sub>a</sub> I(Z<sub>a</sub>;M<sub>(t<sub>1</sub>,t<sub>2</sub>]</sub>|ð¢<sub>t<sub>1</sub></sub>)â¤0,</span><span class="eqno">(30.9)</span></div> 459<div class="eq"><span class="math">Î S<sub>phys</sub>= Σ<sub>aâ»0</sub>w<sub>a</sub> I(Z<sub>a</sub>;M<sub>(t<sub>1</sub>,t<sub>2</sub>]</sub>|ð¢<sub>t<sub>1</sub></sub>) +â«<sub>t<sub>1</sub></sub><sup>t<sub>2</sub></sup>Î (s) dsâ¥0.</span><span class="eqno">(30.10)</span></div> 460<p>å°ãªãã¨ãä¸å±¤ã§ç¸äºæ å ±éãæ£ãªã <span class="math">Î S<sub>phys</sub>>0</span>ã<span class="math">Î =0</span> ã®çæ³å¯é交æã§ã¯ãèªç¥å´ã®éã¿ä»ãç§©åºåéã¨ç©çå´å¢å¤§éãçããã</p></div> 461<div class="proof"><div class="proof-title">証æ</div> 462<p><span class="math">S=(s<sub>a</sub>)<sub>a</sub>â SC<sub>i</sub></span>
462 ã¨ããã(30.2)ãã <span class="math">p<sub>ba</sub>(L<sub>a,m</sub>s<sub>a</sub>)=L<sub>b,m</sub>(p<sub>ba</sub>s<sub>a</sub>)=L<sub>b,m</sub>s<sub>b</sub></span> ã ããã<span class="math">L<sub>m</sub>S</span> ã¯åã³æ´åæã§ãããããã« <span class="math">L<sub>m</sub></span>ããããã£ã¦ <span class="math">G<sub>m</sub></span> 㯠<span class="math">SC<sub>i</sub></span> ã®èªå·±ååã§ããã<span class="math">q<sub>F</sub>q<sub>m</sub><1</span> ããBanachã®åºå®ç¹å®çãé©ç¨ã§ãã䏿åºå®ç¹ã¨(30.6)ãå¾ãããã <span class="math">S<sub>m</sub><sup>*</sup>=S<sup>*</sup><sub>ε</sub></span> ãªãåºå®ç¹æ§ã(30.3)ã«åãããæ¡ä»¶30-Bããæ¹ç(30.4)ãåå¨ãã(30.7)ã¯åã®æ´åããä¸ã¤çµã§ããã</p> 463<p>æ¡ä»¶ä»ãã¨ã³ãããã¼ã®é£éå¾ãããå層ã§</p> 464<div class="eq"><span class="math">H(Z<sub>a</sub>|ð¢<sub>t<sub>2</sub></sub>)=H(Z<sub>a</sub>|ð¢<sub>t<sub>1</sub></sub>)-I(Z<sub>a</sub>;M<sub>(t<sub>1</sub>,t<sub>2</sub>]</sub>|ð¢<sub>t<sub>1</sub></sub>).</span><span class="eqno">(30.11)</span></div> 465<p>éã¿ä»ãã«åãåãã°(30.9)ãæ¡ä»¶30-Cã«ããããããå®ç15ï¼èªç¥ç©çã¨ã³ãããã¼äº¤æã»ä¿åå®çï¼ã®äº¤æå°å¸³ã¨åãéã§ãããA7ãåºéç©åã(30.9)ãä»£å ¥ããã°(30.10)ã¨ãªããç¸äºæ å ±é㨠<span class="math">Î </span> ã¯éè² ãªã®ã§çµè«ãå¾ãã<span class="qed">â</span></p></div> 466<div class="note"><strong>ç¡æã¨ã®æ´åã</strong> 䏿æ§ã¯è¨èªå <span class="math">m</span> ã¨å±¥æ´ãåºå®ããæ¡ä»¶ä»ã䏿æ§ã§ããã絶対çãªåºå®èªæãæå³ããªãã(30.3)ã¯ç°ãªãè¨èªå±¥æ´ã§åºå®ç¹ãå¤ãããããã¨ãæç¤ºãããã®Egoæ§æãå®ç25ï¼è«¸æ³ç¡æå®çï¼ã®å±¥æ´ç¸å¯¾çé¢ä¿æ§é ã®å é¨ã«ç½®ãã</div> 467<div class="rigor"><div class="rigor-title">åç対象ã®å ´å</div><p>èªå·±ç¶æ ãã®ãã®ãåºéå ã§å¤åãããªãã<span class="math">Ḣ<sub>a</sub>=Ï<sub>a</sub>-ι<sub>a</sub></span>ï¼æ°è¦ä¸ç¢ºå®æ§çâè¨èªæ å ±çï¼ã¨åãã<span class="math">Σ w<sub>a</sub>(ι<sub>a</sub>-Ï<sub>a</sub>)>0</span> ãæ£å³æ§é 忡件ã¨ããå¿ è¦ããããå®ç18ï¼å çè¨èªé²åå®çï¼ã®æ£ã®æ å ±å©å¾ã ãã§ã¯ããã®æ£å³ä¸çå¼ã¯èªåçã«åºãªããã¾ããä¸è¬ã®æ å ±åå¾ã«Landauerä¸éãç´æ¥é©ç¨ãã¦ã¯ãªãããç©ççæ¶å»éç¨ãå¥éæå®ããå¿ è¦ãããã</p></div> 468</section> 469 470<section id="t31"> 471<h2>7. å®ç31ââå çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻å®ç</h2> 472<p>éç§°ããæ°çè¨èªç½ å®çãã¨ãããå®ç18ï¼å çè¨èªé²åå®çï¼ã® <span class="math">Î <sup>0</sup>âÎ <sup>(â)</sup></span> ã¯è¨èªã鏿è¢ãå¢ãããã¨ããè¿°ã¹ãã髿½è±¡åº¦ã¸ã®è±åºãä¿è¨¼ããªããç½ ã証æããã«ã¯ãè¨èªã¸ã£ã³ãã¨é£ç¶åå¦ã®ä¸¡æ¹ãéããå¿ è¦ãããã</p> 473<h3>7.1 使½è±¡åº¦å¸¯ã¨ãã¤ããªããç³»</h3> 474<p><span class="math">Äâºâ¤</span> ãåºå®ããå®ç3ï¼æ½è±¡çå ±æTCZåæå®çï¼ã®æ½è±¡åå <span class="math">Ï</span> ã«ãã</p> 475<div class="eq"><span class="math">âÄ:={b| bâ¼Ä}, K<sub>Ä</sub>:={xâ X|Ï(x)ââÄ}</span><span class="eqno">(31.1)</span></div> 476<p>ãééåã¨ãããè¨èªæ´æ°ã¨ãç¾å¨ã®è¨èªåEgoæ¹çã«ããæµãã</p> 477<div class="eq"><span class="math">x<sup>+</sup>=T<sub>m</sub>(x), mâΣ<sub>â</sub><sup>*</sup>, áº=F<sub>â</sub>(x,t):=f(x,Ï<sub>â</sub>(x,t),t)</span><span class="eqno">(31.2)</span></div> 478<p>ã¨ãããç¾å¨ã®è¨èªã»è©ä¾¡ä½å¶ãåºå®ããåºéã§ <span class="math">V<sub>â</sub>=V<sub>â</sub>(x)</span> ã¨ããéããç¾å¨TCZã <span class="math">C<sub>â</sub>:=K<sub>Ä</sub>â©{x| V<sub>â</sub>(x)â¤Î¸<sub>â</sub>}</span> ã¨ããã</p> 479<div class="assumption"><div class="assumption-title">æ¡ä»¶31-Aï¼å çè¨èªéå ï¼</div><p>
479ãã¹ã¦ã®å©ç¨å¯è½ãªè¨èªåã«ã¤ã㦠<span class="math">T<sub>m</sub>(K<sub>Ä</sub>)â K<sub>Ä</sub></span>ã</p></div> 480<div class="assumption"><div class="assumption-title">æ¡ä»¶31-Bï¼é£ç¶æµã®Nagumoä¸å¤æ§ï¼</div><p>è§£ã¯åæ¹å®åã§ãããå¢çä¸ã§ <span class="math">F<sub>â</sub>(x,t)â T<sub>K<sub>Ä</sub></sub>(x)</span>ãããã§ <span class="math">T<sub>K</sub>(x)</span> ã¯Bouligandæ¥éã§ããã</p></div> 481<div class="assumption"><div class="assumption-title">æ¡ä»¶31-Cï¼ãã¤ããªããTCZå¸å¼ï¼</div><p>æµãã¨ã¸ã£ã³ãã®åæ¹ã«ã¤ãã¦ååãä¸å¤ãªå¸å¼å <span class="math">B<sub>â</sub>â K<sub>Ä</sub></span>ã宿° <span class="math">c<sub>1</sub>,c<sub>2</sub>,λ>0</span>ãé£ç¶Lyapunovæ®å·® <span class="math">W<sub>â</sub></span> ãåå¨ãã</p> 482<div class="eq"><span class="math">c<sub>1</sub>d(x,C<sub>â</sub>)<sup>2</sup>⤠W<sub>â</sub>(x,t)⤠c<sub>2</sub>d(x,C<sub>â</sub>)<sup>2</sup>, D<sup>+</sup>W<sub>â</sub>â¤-2λ W<sub>â</sub></span><span class="eqno">(31.3)</span></div> 483<p>ãæµãä¸ã§æºãããåè¨èªã¸ã£ã³ã㯠<span class="math">W<sub>â</sub>(T<sub>m</sub>x,t<sup>+</sup>)⤠W<sub>â</sub>(x,t<sup>-</sup>)</span> ãæºãããã¸ã£ã³ãæå»ã¯éZenoã§ããã</p></div> 484<div class="assumption"><div class="assumption-title">æ¡ä»¶31-Dï¼å ç¶æ ç²è¦åï¼è¼ªå»»é¨ã«ã®ã¿å¿ è¦ï¼</div><p>輪廻é¨ã§ã¯ <span class="math">x<sub>0</sub>â C<sub>â</sub></span> ã¨ãã<span class="math">C<sub>â</sub>=â<sub>r=1</sub><sup>6</sup>R<sub>r</sub></span> ãå ã¤ã®é空Borelé åã¸ç²è¦åãããæ´æ°æå»ã¯ <span class="math">Ï<sub>n</sub>ââ</span> ãæºããã</p> 485<div class="eq"><span class="math">Z<sub>n</sub>=râ x(Ï<sub>n</sub>)â R<sub>r</sub></span><span class="eqno">(31.D)</span></div> 486<p>ã§å®ãã <span class="math">Z<sub>n</sub>â{1,â¦,6}</span> ã¯ãå¤é¨ç¶æ ãæããªãæéææ¬¡æéæ¢ç´Markové£éã§ããã</p></div> 487<div class="theorem"><div class="theorem-title">å®ç31ï¼è«ç±³å°å çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻å®çï¼æ°çè¨èªç½ å®çï¼</div> 488<p>å®ç1ï¼è«ç±³å°ä¸»å®çï¼ã»å®ç3ï¼æ½è±¡çå ±æTCZåæå®çï¼ã»å®ç16ï¼èªå·±æèåå¨ã»çºçå®çï¼ã»å®ç18ï¼å çè¨èªé²åå®çï¼ã®æ£åæ¡ä»¶ã¨æ¡ä»¶31-AãCãä»®å®ãããä»»æã® <span class="math">x<sub>0</sub>â B<sub>â</sub></span> ã«å¯¾ãã</p> 489<div class="eq"><span class="math">x(t)â K<sub>Ä</sub> (â tâ¥0), d(x(t),C<sub>â</sub>)â¤â(c<sub>2</sub>/c<sub>1</sub>)e<sup>-λ t</sup>d(x<sub>0</sub>,C<sub>â</sub>).</span><span class="eqno">(31.4)</span></div> 490<p>ç¹ã« <span class="math">x<sub>0</sub>â C<sub>â</sub></span> ãªã <span class="math">x(t)â C<sub>â</sub></span> ã§ãããç¾å¨TCZããåºãªãã髿½è±¡åº¦ã´ã¼ã«éå <span class="math">Hâ Xâ K<sub>Ä</sub></span> ã <span class="math">δ<sub>H</sub>:=d(K<sub>Ä</sub>,H)>0</span> ãæºãããªãã</p> 491<div class="eq"><span class="math">d(x(t),H)â¥Î´<sub>H</sub> (â tâ¥0),</span><span class="eqno">(31.5)</span></div> 492<p>ãããã£ã¦ç¾å¨ã®è¨èªã»Egoéã«ã¼ãã ãã§ã¯ <span class="math">H</span> ã«å°éã§ããªãã</p> 493<p>ããã« <span class="math">x<sub>0</sub>â C<sub>â</sub></span> ã¨æ¡ä»¶31-Dã®ãã¨ã§ã¯ä¸æãªå®å¸¸åå¸ <span class="math">μ</span> ãåå¨ããå <span class="math">r</span> ã«ã¤ãã¦</p> 494<div class="eq"><span class="math">Pr(Z<sub>n</sub>=rinfinitely often)=1, 1/NΣ<sub>n=0</sub><sup>N-1</sup>1<sub>{Z<sub>n</sub>=r}</sub>â¶ a.s.μ<sub>r</sub>>0.</span><span class="eqno">(31.6)</span></div> 495</div> 496<div class="proof"><div class="proof-title">証æ</div> 497<p>æ¡ä»¶31-Bã¨Nagumoã®ä¸å¤æ§å®çã«ããé£ç¶æµã¯ <span class="math">K<sub>Ä</sub></span> ãä¿åãã31-Aã«ããåè¨èªã¸ã£ã³ããä¿åãããéZenoæ§ã®ãã¨ã§æµãã¨ã¸ã£ã³ãã«ã¤ãã¦å¸°ç´ããã°ãå ¨ãã¤ããªããæå»ã§ <span class="math">x(t)â K<sub>Ä</sub></span>ã</p> 498<p>æµãåºéã§ã¯æ¯è¼å®çãã <span class="math">W<sub>â</sub>(t)⤠e<sup>-2λ(t-s)</sup>W<sub>â</sub>(s)</span>ãã¸ã£ã³ãã§ã¯å¢å ããªããåºéãé£çµã(31.3)ã®äºæ¬¡æã¿è¾¼ã¿ã使ãã°(31.4)ãå¾ãã<span class="math">x<sub>0</sub>â C<sub>â</sub></span> ãªãåæ <span class="math">W=0</span> ãªã®ã§å¸¸ã«é¶ã§ããã(31.5) ã¯éåéè·é¢ã®å®ç¾©ããç´ã¡ã«å¾ãã</p> 499<p>æéæ¢ç´Markové£éã¯æ£å帰çã§ä¸æãªæ£å¤å®å¸¸åå¸ãæã¤ãå帰å®çã¨Markové£éã®å¼·æ³åãã(31.6)ãå¾ãã<span class="qed">â</span></p></div> 500 501<h3>7.2 æ¡ä»¶31-D ã®å修飾èªã®å½¹å²</h3> 502<p>æ¡ä»¶31-D ã¯ã<span class="math">C<sub>â</sub></span> ãå ã¤ã®<b>é空Borelé å</b>ã¸<b>ç²è¦å</b>ãããã¨è¿°ã¹ããä¸ã¤ã®ä¿®é£¾èªã¯ããããç¬ç«ã®å½¹å²ãæ ããããããè½ã¨ãã¦ã (31.6) ã¯æç«ããªãã以ä¸ã«åç¯ãã¦æç¤ºããã</p> 503 504<div class="rigor"><div class="rigor-title">ç²è¦åãå¿ è¦ãªçç±</div> 505<p>ç¶æ 空é <span class="math">C<sub>â</sub></span> ã¯ä¸è¬ã«é£ç¶ä½æ¿åº¦ããã¤ãé£ç¶åå¸ã®ãã¨ã§ã¯ãä»»æã®ä¸ç¹ã¸ã®å訪確çã¯é¶ã§ããããåä¸ç¶æ ã¸ã®ç¡éåã®å帰ãã¯èªæã«å½ã¨ãªãããããã£ã¦å帰ãéèªæãªå½é¡ã¨ãã¦è¿°ã¹ãã«ã¯ãç¶æ ã<b>æéåã®å·¨è¦é åã¸åãååå</b>ãçµç±ããå¿ è¦ããããå®ç31 ã主張ããã®ã¯ããã®åã®æ°´æºã«ãããå帰ã§ã
505ã£ã¦ãç¹ã®æ°´æºã«ãããå帰ã§ã¯ãªãã両è ãæ··åãã¦ã¯ãªããªãã</p></div> 506 507<div class="rigor"><div class="rigor-title">é交å â ã®å½¹å²ââåååã® well-defined æ§</div> 508<p><span class="math">C<sub>â</sub> = â<sub>r=1</sub><sup>6</sup>R<sub>r</sub></span> ã¯ã<b>è¢«è¦æ§</b>ï¼<span class="math">âªR<sub>r</sub> = C<sub>â</sub></span>ï¼ã¨<b>äºãã«ç´ </b>ï¼<span class="math">r â râ² â R<sub>r</sub> â© R<sub>râ²</sub> = â </span>ï¼ãåæã«è¦æ±ãããåè ãè½ã¨ãã¨åååãå ¨åã§å®ç¾©ããããå¾è ãè½ã¨ã㨠<span class="math">Z<sub>n</sub></span> ã䏿ã«å®ã¾ããªãããããã®å ´åã <span class="math">(Z<sub>n</sub>)<sub>nâ¥0</sub></span> ã¯ç¢ºçéç¨ã¨ãã¦æ§æã§ãããMarkov é£éã®çè«ãé©ç¨ã§ããªãã</p></div> 509 510<div class="rigor"><div class="rigor-title">Borel 坿¸¬æ§ã®å½¹å²ââé·ç§»æ ¸ã®åå¨</div> 511<p>å <span class="math">R<sub>r</sub></span> ã Borel 坿¸¬ã§ãªããã°ã<span class="math">Pr(x(Ï<sub>n+1</sub>) â R<sub>râ²</sub> | x(Ï<sub>n</sub>) â R<sub>r</sub>)</span> ãå®ç¾©ããããé·ç§»è¡å <span class="math">P = (p<sub>rrâ²</sub>)</span> ãæ§æã§ããªããé坿¸¬éåã®åå¨ï¼é¸æå ¬çã®ãã¨ã§ã® Vitali éåçï¼ãæé¤ããæ¡ä»¶ã§ãããè£ é£¾ã§ã¯ãªããæ¨æº Borel 空éä¸ã®æ£åæ¡ä»¶ä»ã確çã®åå¨ã«ããã坿¸¬æ§ã®ãã¨ã§ã¯é·ç§»æ ¸ã確ä¿ãããã</p></div> 512 513<div class="rigor"><div class="rigor-title">é空æ§ã®å½¹å²ââå®å¸¸åå¸ã®æ£å¤æ§</div> 514<p><span class="math">R<sub>r</sub> = â </span> ãªã <span class="math">r</span> ãåå¨ããã°ããã®ç¶æ ã¯å°éä¸è½ã¨ãªãæ¢ç´æ§ãç ´ãããæ¢ç´æ§ãç ´ããã°ä¸æãªå®å¸¸åå¸ã®åå¨ãä¿è¨¼ãããã(31.6) ã® <span class="math">μ<sub>r</sub> > 0</span> ã¯å°ããªããé空æ§ã¯ãå ã¤ã®é å<b>ãã¹ã¦</b>ãæ£ã®é »åº¦ã§è¨ªåãããã¨ããçµè«ãæ¯ããæ¡ä»¶ã§ããã</p></div> 515 516<h3>7.3 ãå é輪廻ãã¨ããåç§°ã®å°ç¨</h3> 517<p>æ¬å®çããå é輪廻ãã®èªãç¨ããã®ã¯ã<b>å çè¨èªéå ã®ãã¨ã§ä½æ½è±¡åº¦å¸¯ããé¢è±ã§ããªã</b>ã¨ããæ°ççäºå®ããæå¦ã®èªå½ã§åæãããã§ããããã®å¯¾å¿ãé©åã§ããã®ã¯ãå éã®æèª¬ã«ããã¦<b>æä¸ä½ã®å¤©çã«ããã¦ãããã¦è¦ãçèµ·ãã</b>ã¨ããããããã®é åãçµç«¯ã§ã¯ãªãã¨ãããç¹ããå®ç24ï¼ä¸åçè¦å®çï¼ã® <span class="math">a ⺠⤠â J<sup>*</sup><sub>a,Ï</sub>(x,T) > 0</span> ã¨æ§é çã«ä¸è´ããããã§ããã以ä¸ããã®åç§°ã主張ããªãä¸ç¹ãæç¤ºããã</p> 518 519<div class="rigor"><div class="rigor-title">(i) é åæ° 6 ã¯æ°å¦çå 容ããããªã</div> 520<p>(31.6) ã®è¨¼æã«ç¨ããã®ã¯æéæ¢ç´ Markov é£éã®å帰å®ç㨠BirkhoffâKingman åã®éæ´å®çã®ã¿ã§ããããããã<b>ç¶æ æ°ã«ä¾åããªã</b>ããããã£ã¦ç©ºæªæºã«ããã¦ã¯ãç²è¦åã®å岿°ã <span class="math">6</span> ã§ãã£ã¦ã <span class="math">7</span> ã§ãã£ã¦ã <span class="math">10</span> ã§ãã£ã¦ãçµè«ã¯åä¸ã§ãããæ°å¦ãè¦æ±ããã®ã¯æéæ§ã»é空æ§ã»Borel 坿¸¬æ§ã»é交æ§ã»æ¢ç´æ§ã®ã¿ã§ãã£ã¦ã<b>æ° 6 ããèªä½ã«ã¯æ°ççéè¦æ§ããªã</b>ãå岿°ã¨åé åã®åç§°ã¯æå¦ã®å´ããä¸ããããã</p></div> 521 522<div class="rigor"><div class="rigor-title">(ii) 転çã®æªå®ãå«ã¾ãªãââãã ãå®ç25 ã«ããå¼·ãå¶ç´ãåã</div> 523<p>æ¬å®çã¯<b>ãããã転çï¼çã¾ãå¤ããï¼ãåæã¨ããããã®åå¦ã«ã¤ãã¦è¯å®ãå¦å®ãããªã</b>ãæ°å¦ãè¿°ã¹ãã®ã¯éããæéæ¢ç´ç³»ã«ãããå復訪åã®æ§é ã®ã¿ã§ããããã ããå®ç25ï¼è«¸æ³ç¡æå®çï¼</p> 524<div class="eq"><span class="math">¬âS<sub>0</sub> âh: F<sub>i,h</sub>(S<sub>0</sub>) = S<sub>0</sub>ï¼ãâdâð âαâð: ¬Atman(d,α)</span></div> 525<p>ã¯ã<b>å±¥æ´ç¸å¯¾ã®æ©è½çèªå·±ã¯åå¨ããããå ¨å±¥æ´ã«å ±éã®åºå®ç¹ãããããªãæ½è±¡åº¦ã¬ã¤ã¤ã¼ã«ãããå
525ºå®çèªæ§ãåå¨ãããå ¨åå¨ã¯ç¸èµ·ã®ç¶²ã®ä¸ã§æç«ãã</b>ãã¨ãå³å¯ã«è¨¼æãã¦ãããããããç´ã¡ã«æ¬¡ãå¾ãã</p> 526<div class="result"><div class="result-title">ç³»31.Rï¼è»¢çè§£éã¸ã®å¶ç´ï¼</div> 527<p>ä»®ã«è»¢çãåå¨ããã¨ãã¦ããããã¯<b>ç©ºæªæºã®æ½è±¡åº¦å±¤ã«ãããç¸èµ·ã®ç¶ç¶</b>ã§ã¯ãããã¦ãã<b>åºå®çèªæ§ã®é£ç¶ã§ã¯ããããªã</b>ãå®éããã®ãããªé£ç¶ããèªæ§ã¯å ¨å±¥æ´å ±éã®åºå®ç¹ <span class="math">S<sub>0</sub></span> ãä¸ãããã¨ã«ãªããå®ç25 ã®ç¬¬ä¸é£è¨ã«åããã</p></div> 528<p>ãã®ç¸èµ·ã®ç¶ç¶ãã転çãã¨å¼ç§°ãããå¦ãã¯ææ´¾ç夿ã«å±ããæ°å¦ã¯é¢ä¸ããªãããã ããããã§è¨ãããç¶ç¶æ§ã¯ã<b>é迦以åã®è»¢ç観ã¨ã¯æ¬è³ªçã«ç°ãªã</b>ãã¨ã注è¨ãã¦ãããé迦以éã®å®å¼åã«ããã¦ã¯é¢ä¿ã®ç¶²ã®ç¶ç¶ãåé¡ã§ããã®ã«å¯¾ããé迦以åã®è»¢ç観ã¯<b>層ãè¶ãã¦éã°ããå®ä½</b>ãæªå®ãã¦ãããå®ç25 ãæé¤ããã®ã¯ãã¾ãã«å¾è ã§ããã</p></div> 529 530<div class="rigor"><div class="rigor-title">(iii) åä¸ã®ç涯å ã§ãæç«ãã</div> 531<p>æ¡ä»¶31-D ã®æ´æ°æå»å <span class="math">Ï<sub>n</sub> â â</span> ã«ããããçæ¶¯ã®å¢çã«å¯¾å¿ããã¨ããè¦è«ã¯<b>ä¸åå«ã¾ãã¦ããªã</b>ããããã£ã¦ (31.6) ã¯ãåä¸ã®ç涯å ã«ãããé åé·ç§»ââåã»äºã»æ¸ã»å¤±æã»è¦çã®å·¨è¦ç¶æ ã®å復ââã«ãã®ã¾ã¾é©ç¨ããããæ°ççã«ã¯ãããã¾ãå é輪廻ã§ãããæ¬å®çã第ä¸ã«è¨è¿°ããã®ã¯ããã®æ°´æºã®äºè±¡ã§ããã</p></div> 532 533<h3>7.4 å®ç31 ãå½¢å¼åãã対象ââãä¸è«ãã®è¨èªæ¹å¤ã¨ã®æ§é ç対å¿</h3> 534<p>æ¬å®çã«ãæ°çè¨èªç½ å®çãã®å¥ç§°ãä¸ããã®ã¯ããã®å½¢å¼åã®å¯¾è±¡ãã龿¨¹ãä¸è«ãï¼<i>MÅ«lamadhyamakakÄrikÄ</i>ããããäºãä¸ä¸ç´ï¼ä»¥æ¥ãè¨èªã®ç½ ãã¨ãã¦è«ãããã¦ããæ§é ã«ä¸è´ããããã§ããã<b>æ¬ç¯ã¯è§£é層ã§ãããå®ç31 ã®è¨¼æã¯ããã«ä¸åä¾åããªãã</b></p> 535 536<div class="rigor"><div class="rigor-title">ä¸ã¤ã®ç½ ã¨æ°çç対å¿ç©</div> 537<p><b>(a) å®ä½è¦ã</b>åç§°ã®ä»ä¸ã«ãã対象ãä¸å¤ã®å®ä½ã¨çåããããæ°çç対å¿ç©ã¯æ½è±¡åº¦ã®ä¸é <span class="math">Ä</span> ã¨ééå <span class="math">K<sub>Ä</sub></span> ã§ãããå®ä½è¦ã¯ä¸éã®ä½ä½åºå®ã«å¯¾å¿ããããä¸è«ã24.18 ã¯ãç¸èµ·ãããã®ã空ã¨å¼ã¶ããã®ç©ºãã¾ãä»®åï¼<span class="math">prajñapti</span>ï¼ã§ããã¨ãããããããä¸éã§ããã¨è¿°ã¹ãï¼é³©æ©ç¾ ä»è¨³ãäº¦çºæ¯ä»®åãã亦æ¯ä¸é義ãï¼ã</p> 538<p><b>(b) äºé 対ç«ã</b>ãæï¼ç¡ãã®æ çµã¸ã®åãããæ°çç対å¿ç©ã¯æ¡ä»¶31-A <span class="math">T<sub>m</sub>(K<sub>Ä</sub>) â K<sub>Ä</sub></span> ã§ããããæããé¸ã¶æ´æ°ããç¡ããé¸ã¶æ´æ°ãåä¸ã®è¨èªæ´æ°æã«å±ãããããã®åã <span class="math">K<sub>Ä</sub></span> ã«çã¾ãããä¸è«ã15.10 ã¯æã¸ã®å·ã常è¦ãç¡ã¸ã®å·ãæè¦ã¨ããç¥è ã¯ãããã«ãå·ãããªãã¨è¿°ã¹ãï¼ãä¸å¿èæç¡ãï¼ãåé ã®å «ä¸ããå «å¯¾ã®å¯¾ç«é ã®ããããåããªããã¨ã«ããæ çµãã®ãã®ã®è§£é¤ãå³ãç¹ã§ååã§ããã</p> 539<p><b>(c) æ¯è«ï¼<span class="math">prapañca</span>ï¼ã</b>æ¦å¿µã®ç¡ééãªå¢æ®ãæ°çç対å¿ç©ã¯æ¡ä»¶31-A ã®å ¨ç§°éåãå©ç¨å¯è½ãªãã¹ã¦ã®è¨èªåã«ã¤ãã¦ããã®ãã®ã§ãããå®ç18ï¼å çè¨èªé²åå®çï¼ãä¿è¨¼ããæ¹çéåã®æ¡å¤§ <span class="math">Î <sub>0</sub> â Î <sub>â</sub></span> ã¯ãå®ç31 ã® <span class="math">d(x(t),H) ⥠δ<sub>H</sub></span> ã¨ä¸¡ç«ããããä¸è«ã18.5 ã¯æ¥ã¨ç ©æ©ãåå¥ã«ãåå¥ãæ¯è«ã«ç±æ¥ããæ¯è«ã¯ç©ºã«ããã¦æ» ããã¨è¿°ã¹ãï¼ãå ¥ç©ºæ¯è«æ» ãï¼ã</p></div> 540 541<div class="rigor"><div class="rigor-title">
541ãä¸è«ã13.8 ã®è¦åã«å¯¾å¿ããæ°ççäºå®</div> 542<p>ãä¸è«ã13.8 ã¯ã諸ä»ãä¸åã®è¦è§£ãé¢ããããããã«ç©ºã説ããã®ã§ããã<b>空ãè¦è§£ã¨ãã¦æ±ãè </b>ã¯æãé£ãã¨è¿°ã¹ãï¼ãè¥å¾©è¦æç©ºããè«¸ä»æä¸åãï¼ãåè¶£æ¨ã¯ 22.11 ã«ãç¾ããããã®è¦åã«ã¯ãæ¡ä»¶31-A ã®å ¨ç§°éåã®ãã¨ã§æ£ç¢ºãªæ°çç対å¿ç©ãåå¨ããã</p> 543<p>ããªãã¡ãæ¡ä»¶31-A ã¯<b>å©ç¨å¯è½ãªãã¹ã¦ã®è¨èªå</b>ã«ã¤ã㦠<span class="math">T<sub>m</sub>(K<sub>Ä</sub>) â K<sub>Ä</sub></span> ãè¦æ±ãããã®å ¨ç§°ã«ä¾å¤ã¯èªããããªãããããã£ã¦ã空ãã¨ããèªã <span class="math">K<sub>Ä</sub></span> å ã§éç¨å¯è½ãªè¨èªåã®ä¸ã¤ã¨ãã¦ç²å¾ããå ´åããã®åãã¾ã <span class="math">K<sub>Ä</sub></span> ã«çã¾ããããã¯æ¹çéåãä¸ã¤æ¡å¤§ããã«ããããå°éå¯è½æ½è±¡åº¦ã¯å¢å ããªãã仿¹ãæé«æ½è±¡åº¦ã¨ãã¦ã® <span class="math">â¤</span> ã¯å®ç¾©ã«ãã <span class="math">K<sub>Ä</sub></span> ã®å¤é¨ã«ããã<span class="math">δ<sub>H</sub> = d(K<sub>Ä</sub>, H) > 0</span> ã¯éããªãã<b>空ãè¦è§£ã¨ãã¦å é¨åãããæç¹ã§ãããã¯æ¯è«ã®ä¸é ç®ã¨ãªã</b>ââãããå½è©²è¦åã®æ°ççå å®ã§ããã</p></div> 544 545<div class="rigor"><div class="rigor-title">æ¹ä¾¿ã¨å®ç32ââè±åºã®æå¨</div> 546<p>ãä¸è«ããæ¹ä¾¿ã¨ãã¦ä½ç½®ã¥ããè¨èªéç¨ââè¨èªãå 容ã¨ãã¦ã§ã¯ãªãä½ç¨ã¨ãã¦ç¨ãããã¨ââã¯ãå®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ã®æ©æ§ã«å¯¾å¿ãããå®ç32 ã示ãã®ã¯ãåä¸éã«ã¼ãå ã§ã®æ¹çæ¢ç´¢ã§ã¯ãªãã<b>éã«ã¼ãèªä½ã®åæ§æ</b>ã§ãããå¤é¨ã´ã¼ã«ãçµç«¯æ¡ä»¶ã¨ãã¦è¨å®ããã¨è©ä¾¡é¢æ°ã <span class="math">V<sub>0</sub></span> ãã <span class="math">á¹¼<sub>G</sub></span> ã¸ãæ¹çã <span class="math">Ï<sub>â</sub></span> ãã <span class="math">Ï<sub>G</sub></span> ã¸ç§»è¡ãã(32.7) ã®ãã¨ã§ <span class="math">K<sub>Ä</sub></span> ã¯ååãä¸å¤éåã§ãªããªããããªãã¡å£ãç ´ãããã®ã§ã¯ãªãã<b>å£ãå£ããããã¦ããæ¡ä»¶ãæ¶å¤±ãã</b>ã</p></div> 547 548<div class="rigor"><div class="rigor-title">æ¬ç¯ã主張ããªããã¨</div> 549<p><b>(i)</b> æ¬ç¯ã¯é¾æ¨¹ã®æèª¬ã®ççæ§ã主張ããªãã証æãããã®ã¯æç¤ºãããæ¡ä»¶ä¸ã®æ°å¦çå½é¡ã®ã¿ã§ãããããã«ãè¨èªã®ç½ ãã®åãä¸ãããã¨ã®é©å¦ã¯æå¦ããã³ææ³å²ã®å¤æã«å±ããã<b>(ii)</b> æ¬ç¯ã¯ä¸è¦³ã®æç¾©ã«æ°å¦çåºç¤ã¥ããä¸ãããã®ã§ã¯ãªãã示ããã®ã¯<b>æ§é ã®ä¸è´</b>ã§ãã£ã¦æç¾©ã®çå½ã§ã¯ãªããä¸è¦³å é¨ã®è§£éä¸ã®å¯¾ç«ï¼å¸°è¬¬è«è¨¼æ´¾ã¨èªç«è«è¨¼æ´¾ã®å¥ãå«ãï¼ã¯æ°å¦ããæ±ºå®ãããªãã<b>(iii)</b> æ¬ç¯ã¯è¨èªä¸è¬ã®å¦å®çè©ä¾¡ã嫿ããªããå®ç18 ã¯å çè¨èªãä¸ç¢ºå®æ§ãæ¸å°ããæ¹çéåãæ¡å¤§ãããã¨ã証æãã¦ãããå®ç31 ãè¿°ã¹ãã®ã¯ãã®æ¡å¤§ãééåå é¨ã«éå±ããããã¨ããäºå®ã®ã¿ã§ããã</p></div> 550 551<div class="rigor"><div class="rigor-title">
551å¼ç¨ã«ã¤ãã¦</div> 552<p>ãä¸è«ãã®å¼ç¨ã¯ç« ã»åçªå·ã«ãããæ¼¢è¨³ã¯é³©æ©ç¾ ä»è¨³ã«æ ãã訳åºã¯æ¬ç¨¿ã«ããããããã®å¼ç¨ãå®ç31 ã®è¨¼æã«ã¯ç¨ãã¦ããããè§£é層ã¨ãã¦ã®ã¿æ©è½ããã</p></div> 553</section> 554 555<section id="t32"> 556<h2>8. å®ç32ââæªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®ç</h2> 557<p>æ¬å®çã§ã絶対ä»åãã¨å¼ã¶ã®ã¯ãè¶ èªç¶çä½ç¨ãç©ççéå æã§ã¯ãªããSelfãç¾å¨TCZå¤ã®ã´ã¼ã«ãçµç«¯æ¡ä»¶ã¨ãã¦è¨å®ããå¾ãæ§Egoã®è¿½å åªåå ¥åãå¿ è¦ã¨ãããæªæ¥éå®Egoã®éã«ã¼ããæªæ¥TCZã¸èªå¾åæããã¨ãããã¢ãã«å ã®æä½çåç§°ã§ããã</p> 558<h3>8.1 ã´ã¼ã«ãè¨å ´æé¾å¤ãæªæ¥éå®Ego</h3> 559<p>ç¾å¨TCZã <span class="math">C<sub>0</sub></span>ãçã®ã´ã¼ã«ã <span class="math">G</span> ã¨ãã<span class="math">d(G,C<sub>0</sub>)>ε</span> ã¨ãããå®ç8ï¼è«ç±³å°æªæ¥åç¹èªç¥æéå®çï¼ã®æªæ¥éå®Egoã</p> 560<div class="eq"><span class="math">Ï<sub>G</sub>(x,t)â*arg min<sub>u</sub> {â«<sub>t</sub><sup>t+T<sub>0</sub></sup>á¹¼<sub>G</sub>(x(s),s) ds+Î d(x(t+T<sub>0</sub>),G)<sup>2</sup>}.</span><span class="eqno">(32.1)</span></div> 561<p>æ£ã®ã´ã¼ã«é§åå ´ããã®ã´ã¼ã«ç¹ã§ã®å¤ãããã³å®å¹ããã³ã·ã£ã«ã</p> 562<div class="eq"><span class="math">R<sub>G</sub>(y,t;x(t)):=Q<sup>+</sup>(y,t)E(y,t| x(t))C<sub>Self</sub>(G), r<sub>G</sub>(t):=R<sub>G</sub>(G,t;x(t)), á¹¼<sub>G</sub>(y,t):=V<sub>0</sub>(y,t)-κ P<sub>G</sub>(y,t)R<sub>G</sub>(y,t;x(t)).</span><span class="eqno">(32.2)</span></div> 563<p>æªæ¥TCZã <span class="math">C<sub>G</sub>(t):={y|á¹¼<sub>G</sub>(y,t)â¤Î¸<sub>G</sub>}</span> ã¨ããã<span class="math">r<sub>G</sub>(t)>0</span> ã®ã¨ã</p> 564<div class="eq"><span class="math">P<sub>crit</sub>(t):=[V<sub>0</sub>(G,t)-θ<sub>G</sub>]<sub>+</sub>/κ r<sub>G</sub>(t).</span><span class="eqno">(32.3)</span></div> 565<div class="assumption"><div class="assumption-title">æ¡ä»¶32-Aï¼æçãªè¨å ´æé¾å¤è¶ éï¼</div><p><span class="math">r<sub>G</sub>(t)⥠r<sub>0</sub>>0</span>ã<span class="math">0⤠P<sub>G</sub>â¤1</span>ã<span class="math">P<sub>crit</sub>(t)â¤1</span>ãã㤠<span class="math">P<sub>G</sub>(G,t)⥠P<sub>crit</sub>(t)</span> ã <span class="math">t⥠t<sub>0</sub></span> ã§æç«ããã<span class="math">á¹¼<sub>G</sub></span> ã¯ä¸ã«æçã§ãæé©æ¹çã¨åæ¹å®åè§£ãåå¨ããã</p></div> 566<div class="assumption"><div class="assumption-title">æ¡ä»¶32-Bï¼å¯å°éå¸å¼åï¼</div><p>é空ã®ååãä¸å¤å¸å¼å <span class="math">B<sub>G</sub></span> ãåå¨ãã<span class="math">x(t<sub>0</sub>)â B<sub>G</sub></span>ã<span class="math">C<sub>G</sub>(t)</span> 㯠<span class="math">B<sub>G</sub></span> ããã®è¨±å®¹å°ééå ã«å«ã¾ããæªæ¥éå®Egoã®å®å ¥å㯠<span class="math">u(t)=Ï<sub>G</sub>(x(t),t)</span> ã§ããã</p></div> 567<div class="assumption"><div class="assumption-title">æ¡ä»¶32-Cï¼å¼·ãæªæ¥TCZãã¡ãªã¹ã¿ã·ã¹ï¼</div><p>å <span class="math">C<sub>G</sub>(t)</span> ã¯é空ééåã§ããã宿° <span class="math">a<sub>1</sub>,a<sub>2</sub>,λ<sub>G</sub>>0</span> ã¨Lyapunov颿° <span class="math">W<sub>G</sub></span> ãåå¨ããè»éä¸ã® <span class="math">t⦠W<sub>G</sub>(x(t),t)</span> ã¯å±æçµ¶å¯¾é£ç¶ã§ã<span class="math">B<sub>G</sub></span> ä¸ã§</p> 568<div class="eq"><span class="math">a<sub>1</sub>d(x,C<sub>G</sub>(t))<sup>2</sup>⤠W<sub>G</sub>(x,t)⤠a<sub>2</sub>d(x,C<sub>G</sub>(t))<sup>2</sup>, D<sup>+</sup><sub>Ï<sub>G</sub></sub>W<sub>G</sub>â¤-2λ<sub>G</sub>W<sub>G</sub>.</span><span class="eqno">(32.4)</span></div> 569<p>ããã§Diniå¾®åã¯ãéå <span class="math">C<sub>G</sub>(t)</span> ã®ç§»åãå«ãå ¨è»éå¾®å</p> 570<div class="eq"><span class="math">D<sup>+</sup><sub>Ï<sub>G</sub></sub>W<sub>G</sub>(x,t):=lim sup<sub>hâ0</sub> W<sub>G</sub>(x<sub>Ï<sub>G</sub></sub>(t+h;t,x),t+h)-W<sub>G</sub>(x,t)/h</span></div> 571<p>ã§ããã</p></div> 572<div class="theorem"><div class="theorem-title">å®ç32ï¼è«ç±³å°æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼</div> 573<p>å®ç1ï¼è«ç±³å°ä¸»å®çï¼ã»å®ç4ï¼è«ç±³å°è¨å ´æå éå®çï¼ã»å®ç7ï¼è«ç±³å°çã®ã´ã¼ã«å®çï¼ã»å®ç8ï¼è«ç±³å°æªæ¥åç¹èªç¥æéå®çï¼ã»å®ç9ï¼è«ç±³å°æªæ¥åç¹ã´ã¼ã«éæå®çï¼ã®æ£åæ¡ä»¶ã¨æ¡ä»¶32-AãCã®ãã¨ã§ããã¹ã¦ã® <span class="math">t⥠t<sub>0</sub></span>
573 ã«ã¤ã㦠<span class="math">Gâ C<sub>G</sub>(t)</span> ã§ããã</p> 574<div class="eq"><span class="math">d(x(t),C<sub>G</sub>(t))â¤â(a<sub>2</sub>/a<sub>1</sub>)e<sup>-λ<sub>G</sub>(t-t<sub>0</sub>)</sup>d(x(t<sub>0</sub>),C<sub>G</sub>(t<sub>0</sub>))â¶0.</span><span class="eqno">(32.5)</span></div> 575<p><span class="math">d<sub>0</sub>:=d(x(t<sub>0</sub>),C<sub>G</sub>(t<sub>0</sub>))</span> ã¨ãããä»»æã® <span class="math">δ>0</span> ã«å¯¾ããæªæ¥TCZã® <span class="math">δ</span>-è¿åã¸ã®å°éæå»ä¸çã</p> 576<div class="eq"><span class="math">T<sub>δ</sub>:= t<sub>0</sub>, d<sub>0</sub>=0,<br>[2pt] t<sub>0</sub>+1/λ<sub>G</sub>max{0, log(â(a<sub>2</sub>/a<sub>1</sub>)d<sub>0</sub>/δ)}, d<sub>0</sub>>0</span><span class="eqno">(32.6)</span></div> 577<p>ã¨ããã°ã<span class="math">t⥠T<sub>δ</sub></span> ã§ <span class="math">d(x(t),C<sub>G</sub>(t))â¤Î´</span> ã§ããã</p></div> 578<div class="proof"><div class="proof-title">証æ</div> 579<p>æ¡ä»¶32-Aã¨(32.3)ãã <span class="math">κ P<sub>G</sub>(G,t)r<sub>G</sub>(t)â¥[V<sub>0</sub>(G,t)-θ<sub>G</sub>]<sub>+</sub></span>ããããã£ã¦ <span class="math">á¹¼<sub>G</sub>(G,t)â¤Î¸<sub>G</sub></span> ã§ããã<span class="math">Gâ C<sub>G</sub>(t)</span>ã</p> 580<p>(32.4)ã¨Grönwallä¸çå¼ãã <span class="math">W<sub>G</sub>(x(t),t)⤠W<sub>G</sub>(x(t<sub>0</sub>),t<sub>0</sub>)e<sup>-2λ<sub>G</sub>(t-t<sub>0</sub>)</sup></span>ãä¸ä¸ã®äºæ¬¡è©ä¾¡ãä»£å ¥ãã¦å¹³æ¹æ ¹ãåãã°(32.5)ãå³è¾ºã <span class="math">δ</span> 以ä¸ã«ããæå»ã«ã¤ãã¦è§£ãã°(32.6)ãå¾ããå®å ¥åã¯è¨å®å¾ã <span class="math">Ï<sub>G</sub></span> ã ããªã®ã§ããã®åæã¯åæ§æãããéã«ã¼ããã¡ãªã¹ã¿ã·ã¹ã«ããã<span class="qed">â</span></p></div> 581<h3>8.2 è¨èªç½ ããã®è±åºã¨ã®æ´å</h3> 582<p>å®ç31ï¼å çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻å®çï¼ã®ç½ ããåºãã«ã¯ãæ°ããã´ã¼ã«éã«ã¼ãã31-Aã¾ãã¯31-Bã®éå æ¡ä»¶ãç ´ãå¿ è¦ããããä¾ãã° <span class="math">K<sub>Ä</sub>={x| b(x)â¤0}</span> ã¨ããè»éãæå» <span class="math">t<sub>b</sub></span> ã«å¹ <span class="math">â</span> ã®å¢ç帯 <span class="math">-â⤠b(x)â¤0</span> ã¸å°éããã¨ããã</p> 583<p><span class="math">b</span> 㯠<span class="math">C<sup>1</sup></span>ã<span class="math">b(x(t))</span> ã¯å±æçµ¶å¯¾é£ç¶ã§ããã¨ãããè»éãå¢ç帯ã«ããéã</p> 584<div class="eq"><span class="math">d/dtb(x(t))=â b(x(t))· f(x(t),Ï<sub>G</sub>(x(t),t),t)â¥Î½>0 a.e.</span><span class="eqno">(32.7)</span></div> 585<p>ãªãã<span class="math">b(x(t))⥠b(x(t<sub>b</sub>))+ν(t-t<sub>b</sub>)</span> ã§ããã<span class="math">Ï<sub>out</sub>:=inf{t⥠t<sub>b</sub>| b(x(t))>0}</span> ã¨ç½®ãã°ã</p> 586<div class="eq"><span class="math">Ï<sub>out</sub>⤠t<sub>b</sub>+-b(x(t<sub>b</sub>))/ν⤠t<sub>b</sub>+â/ν.</span></div> 587<p>ãããã£ã¦æéæéã§ <span class="math">K<sub>Ä</sub></span> ãåºããããã¯ãåãéã«ã¼ãã®ä¸ã§åªåãã¦è±åºãããã®ã§ã¯ãªããå¤é¨ã´ã¼ã«ãè©ä¾¡é¢æ°ã¨Egoæ¹çãåæ§æããæ§ä¸å¤éåãä¸å¤ã§ãªããããã¨ã表ãã</p> 588<div class="rigor"><div class="rigor-title">è¨å ´æéå°ã®å³å¯ãªæå³</div><p>æ¢åã®è¨å ´æ <span class="math">P</span> 㯠<span class="math">[0,1]</span> ã«æ£è¦åãããã®ã§ã<span class="math">P>1</span> ãæå³ãã¦ã¯ãªããªããæ¬å®çã®éµã¯ãæçãªè¨å ´æã®é¾å¤è¶ éã<span class="math">P<sub>G</sub>⥠P<sub>crit</sub></span> ã§ãããå®ç21ï¼å æåé åºè¨å ´ææ¹åæ§å®çï¼ã®ç¡çãªå®å¹å©å¾ <span class="math">p</span> ãç¨ããå ´åã¯ã<span class="math">P</span> ã¨ã¯å¥éã¨ãã¦æè¨ãããã¾ããè¨å ´æãåæ<strong>
588é度</strong>ãå¢ãã¨ããçµè«ã«ã¯ãè¿½å æ¡ä»¶ <span class="math">λ<sub>G</sub>=λ<sub>G</sub>(P<sub>G</sub>)</span> ã㤠<span class="math">dλ<sub>G</sub>/dP<sub>G</sub>â¥0</span> ãå¿ è¦ã§ããã</p></div> 589<div class="note"><strong>éåéæã¨ç¹éæã</strong> (32.5)ãä¿è¨¼ããã®ã¯ <span class="math">C<sub>G</sub>(t)</span> ã¸ã®éååæã§ãããå¿ ããã <span class="math">x(t)â G</span> ã§ã¯ãªããç¹åæã«ã¯ <span class="math">C<sub>G</sub>(t)={G}</span> ã¾ã㯠<span class="math">d<sub>H</sub>(C<sub>G</sub>(t),{G})â0</span> ã¨ãã䏿忡件ã追å ããã</div> 590</section> 591 592<section id="minimality"> 593<h2>9. æå°æ§ã¨åä¾</h2> 594<table> 595<thead><tr><th>è½ã¨ãæ¡ä»¶</th><th>æå°åä¾ï¼èµ·ãããã¨</th><th>失ãããçµè«</th></tr></thead> 596<tbody> 597<tr><td>28-A</td><td>é¢ä¿çã«å®ç¾©ãããéåã«ããç¬ç«ãªã©ãã«å¤æ°ãä»»æã«ä»å ã§ããã</td><td>é¢ä¿æ§ã ãããç¡æã¯åºãªãã</td></tr> 598<tr><td>29-Aã®ç®è¡å¼·åº¦</td><td>Presburgerç®è¡ã®ãããªæ±ºå®å¯è½çè«ã</td><td>Gödelåä¸å®å ¨æ§ãç¡æ¡ä»¶ã«ä¸»å¼µã§ããªãã</td></tr> 599<tr><td>29-Aã®æå¹æ§</td><td>æ¨æºèªç¶æ°ã®å ¨ççéåã¯è¨ç®å¯ææã§ãªãã</td><td>å®å ¨æ§ã¨å¼æãã«æå¹è¨¼æç³»ã§ãªããªãã</td></tr> 600<tr><td>30-Aã®å¯ææ§</td><td>ä¸å±¤ã®èªå·±è¨æãå°å½±ããçµæã¨ãä¸å±¤ã§æ´æ°ããçµæãç°ãªãã</td><td><span class="math">L<sub>m</sub></span> ã鿥µéä¸ã®èªå·±ååã¨ãã¦å®ç¾©ã§ããªãã</td></tr> 601<tr><td>30-Aã®éèªææ§</td><td><span class="math">G<sub>m</sub>=G<sub>ε</sub></span>ã</td><td>è¨èªã§åºå®ç¹ãå¤ãã£ãã¨ã¯ãããªãã</td></tr> 602<tr><td>30-C</td><td>Shannonä¸ç¢ºå®æ§ã¯ä¸ããããT15ã®ç¶æ æ±é¢æ°ã¯ä¸å¤ã</td><td>ç©ç交æå¼ã¸ä»£å ¥ã§ããªãã</td></tr> 603<tr><td>31-A</td><td><span class="math">K=(-â,0]</span>, <span class="math">T<sub>m</sub>(x)=x+2</span>ã</td><td>ä¸åã®è¨èªæ´æ°ã§è±åºããã</td></tr> 604<tr><td>31-B</td><td>åã <span class="math">K</span> ã§ <span class="math">áº=1</span>ã</td><td>è¨èªæ´æ°ãªãã§ãå¢çãåºãã</td></tr> 605<tr><td>31-Cã®ã¸ã£ã³ãæ¡ä»¶</td><td>åã¸ã£ã³ãã§ <span class="math">W</span> ãåå¢ããã</td><td>æµããå®å®ã§ããã¤ããªããåæããªãã</td></tr> 606<tr><td>31-Dã®æ¢ç´æ§</td><td>é·ç§»è¡å <span class="math">P=I<sub>6</sub></span>ã</td><td>ä¸ã¤ã®éã«åºå®ãããå ç¶æ ãå復ããªãã</td></tr> 607<tr><td>32-Bã®å¯å°éæ§</td><td><span class="math">áº=0, x<sub>0</sub>=0, G=1</span>ã</td><td>æå¤§è¨å ´æã§ãå°éããªãã</td></tr> 608<tr><td>32-C</td><td><span class="math">áº+x=0</span>ã</td><td>ç®æ¨å¨å²ãæ¯åãè·é¢ãåæããªãã</td></tr> 609<tr><td>32-Aã®é¾å¤</td><td><span class="math">V<sub>0</sub>(G)=2,θ=1,κ r<sub>G</sub>=1,P=.5</span>ã</td><td><span class="math">á¹¼(G)=1.5>1</span> ã§æªæ¥TCZã«å ¥ããªãã</td></tr> 610</tbody> 611</table> 612</section> 613 614<section id="conclusion"> 615<h2>10. ç·æ¬</h2> 616<ol> 617<li><strong>æ¶ æ§ã¯ç¡æã§ããã</strong> ãã ãè¨¼ææ ¹æ ã¯ãéåã ããããæå»æ·»åããããããã§ã¯ãªããå®ç25ï¼è«¸æ³ç¡æå®çï¼ã®é¢ä¿çæ©è½å®åæ§ã§ãããä¸åçè¦ã¨ã®æ´åã¯ã¿ã°ä»ãåãä¿è¨¼ããã</li> 618<li><strong>
618å½¢å¼åãããæèª¬ã¯èªå·±å®çµããªãã</strong> ãã ãååã«å¼·ãæå¹ã»å¥å ¨ãªç®è¡çè«ã¨ããå°ç¨å ã§ãããä¸å®å ¨æ§ã¨ç¡èªæ§ã¯å¥ã ã®æ ¹æ ãæã¤ã</li> 619<li><strong>å çè¨èªã¯èªææ§æã鏿ã»å®å®åããã</strong> 主ä½ã®ç¡ããã®çæã§ã¯ãªããå°å½±æ´åæ§ã鿥µéEgoæ§æããæ å ±æ©ã¨T15ãç©ç交æãä¸ããã</li> 620<li><strong>è¨èªã¯èªç±ãå¢ãããªããç½ ã«ããªãããã</strong> è¿½å æ¹çã使½è±¡åº¦å¸¯ãä¿åããã°ã鏿è¢ã¯å¢ãã¦ãå°éå¯è½ãªæ½è±¡åº¦ã¯å¢ããªãã</li> 621<li><strong>å¤é¨ã´ã¼ã«éæã¯åæ§æããããã¡ãªã¹ã¿ã·ã¹ã§èª¬æã§ããã</strong> æçãªè¨å ´æé¾å¤ãå¯å°éæ§ãå¸å¼åãå¼·ãå縮ãæãã¨ãæªæ¥TCZã¸ã®ææ°åæã¨ä»»æã® <span class="math">δ</span>-è¿åã¸ã®å°éæéã証æãããã</li> 622</ol> 623<div class="result"><div class="result-title">äºå®çã®çµ±ä¸å</div><p>åºå®å®ä½ãç½®ãããåã»é¢ä¿ã»è¨èªã»æ å ±ã»å¶å¾¡ã»ç©çæ£é¸ã®ç¸äºä¾åã ããããæ¶ æ§ã®ç¡æãç¥èä½ç³»ã®éæ¾æ§ãè¨èªåEgoã®æ§æã使½è±¡åº¦ééãããã¦å¤é¨ã´ã¼ã«ã«ããéã«ã¼ãåç·¨ãä¸ã¤ã®æ¡ä»¶ä»ãæ°çä½ç³»ã¨ãã¦è¨è¿°ã§ããã</p></div> 624</section> 625 626<section id="summary-tables"> 627<h2>11. å®çä¸è¦§è¡¨</h2> 628<p>æ¬ç¨¿ãç¨ããå ¨å®çãäºè¡¨ã«ã¾ã¨ããã第ä¸è¡¨ã¯æ¬ç¨¿ã§æ°ãã«è¨¼æããå®ç28ï¼æ¶ æ§ç¡æã»åæ´åå®çï¼ãå®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ã第äºè¡¨ã¯æ¬ç¨¿ãåæã¨ãã¦ç¨ããç¶æ¿å®çã§ããã</p> 629 630<h3>11.1 æ¬ç¨¿ã®æ°å®çï¼å®ç28ï¼æ¶ æ§ç¡æã»åæ´åå®çï¼ãå®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ï¼</h3> 631<table class="thmtable"> 632<caption>表1ãå®ç28ï¼æ¶ æ§ç¡æã»åæ´åå®çï¼ãå®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ââå®çåã»ä¸å¿å¼ã»ä¾åã»æ°æ¡ä»¶ã»è¨¼æä¸ã®å°ä½ã»è¦æ¨</caption><colgroup><col style="width:13%"><col style="width:22%"><col style="width:14%"><col style="width:16%"><col style="width:13%"><col style="width:22%"></colgroup> 633<thead><tr><th>å®ç</th><th>ä¸å¿å¼</th><th>ä¾åå®ç</th><th>æ°æ¡ä»¶ï¼å¤é¨çµæ</th><th>証æä¸ã®å°ä½</th><th>è¦æ¨</th></tr></thead> 634<tbody> 635<tr><td><b>å®ç28</b><br>æ¶ æ§ç¡æã»åæ´åå®ç</td> 636<td>ð<sub><â¤</sub> â© Nir(T) = â ;ãPZS(a,x,T) â [a=⤠⧠xâð©<sub>â¤</sub>(T)];ã¬Atman(d<sub>N</sub>,â¤)</td> 637<td>å®ç24ï¼ä¸åçè¦å®çï¼ã»å®ç25ï¼è«¸æ³ç¡æå®çï¼ã»å®ç26ï¼æ¶ æ§å¯éå®çï¼</td> 638<td>
638æ¡ä»¶28-Aï¼ã¿ã°ä»ãé åã®é交æ§ã¨æ¶ æ§éç¨ã®é¢ä¿çè¨è¿°å¯è½æ§ï¼</td> 639<td><b>ç¶æ¿å®çããã®ç´æ¥ç帰çµã</b>æ°ããªè§£æçéå ·ãå°å ¥ããªããåã®åé¢ã¨è¿°èªã®æ¸ãæãã®ã¿</td> 640<td>ãä¸åçè¦ãã¨ãæ¶ æ§å¯éãã¯åä¸é åã«ã¤ãã¦ã®ç«¶å主張ã§ã¯ãªããã¿ã°ä»ãç´åã§åé¢ãããæä»çé åãæ ãããã®ããã§æ¶ æ§éç¨èªä½ãç¡æã§ãã</td></tr> 641<tr><td><b>å®ç29</b><br>形弿³ä½ç³»ç¡èªæ§ã»ä¸å®åå®ç</td> 642<td>âG<sub>h</sub>[ââ¨G<sub>h</sub> â§ ð<sub>h</sub>â¬G<sub>h</sub> â§ ð<sub>h</sub>â¬Â¬G<sub>h</sub>];ãð<sub>h</sub>â¬Con(ð<sub>h</sub>);ãTh(ð<sub>0</sub>) â â¯</td> 643<td>å®ç25ï¼è«¸æ³ç¡æå®çï¼</td> 644<td>æ¡ä»¶29-Aï¼ååãªç®è¡å¼·åº¦ã»æå¹æ§ã»å¥å ¨æ§ï¼ï¼<b>å¤é¨çµæ</b>ï¼Gödel第ä¸ã»ç¬¬äºä¸å®å ¨æ§å®çãChaitin ã®ä¸å®å ¨æ§å®ç</td> 645<td><b>å¤é¨ã¡ã¿å®çã¸ã®ä¾åãæ¬è³ªçã</b>æ¬ä½ç³»ã®ã¿ããã¯å°ããªããæ¡ä»¶29-A ãæºããä½ç³»ã«éã£ã¦é©ç¨ããã</td> 646<td>æèª¬ã¨ãã¦ã®æ³ãååã«å¼·ãæå¹å½¢å¼çè«ã¨ãã¦ã¢ãã«åããå ´åã«éããæ±ºå®ä¸è½æã®åå¨ã»ç¡çç¾æ§ã®å é¨è¨¼æä¸è½ã»è¨è¿°éã®è¨¼æéçãå¾ããéæ¾çæ´æ°ã¨ç¡èªæ§ã帰çµãã</td></tr> 647<tr><td><b>å®ç30</b><br>ã¨ã³ãããã¼äº¤æèªææ§æå®ç</td> 648<td>d<sub>SC</sub>(G<sub>m</sub><sup>n</sup>S<sub>0</sub>,S<sub>m</sub><sup>*</sup>) ⤠(q<sub>F</sub>q<sub>m</sub>)<sup>n</sup>d<sub>SC</sub>(S<sub>0</sub>,S<sub>m</sub><sup>*</sup>);ãÎâ<sub>ego</sub> ⤠0</td> 649<td>å®ç15ï¼èªç¥ç©çã¨ã³ãããã¼äº¤æã»ä¿åå®çï¼ã»å®ç16ï¼èªå·±æèåå¨ã»çºçå®çï¼ã»å®ç18ï¼å çè¨èªé²åå®çï¼</td> 650<td>
650æ¡ä»¶30-AãCï¼éç³»ä¸ã®å°å½±æ´åãè¨èªååã®ç¸®å°æ§ãÏå æ³æã®å調å¢å¤§ï¼</td> 651<td><b>ç¶æ¿å®çã®åæã</b>Banach ã®ä¸åç¹å®çã¨æ¡ä»¶ä»ãç¸äºæ å ±éã®éè² æ§ã®ã¿ãæ°ãã«ç¨ãã</td> 652<td>å çè¨èªã¯ Ego ã鏿ãå®å®åãããè¨èªãå¢ããã»ã©èªææ§é åã¨ã³ãããã¼ã¯æ¸å°ãããã®æ¸å°åãã¡ããã©ç©çå´ã¸äº¤æããã</td></tr> 653<tr><td><b>å®ç31</b><br>å çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻å®ç</td> 654<td>x(t) â K<sub>Ä</sub> (âtâ¥0);ãd(x(t),C<sub>â</sub>) ⤠â(c<sub>2</sub>/c<sub>1</sub>)e<sup>âλt</sup>d(x<sub>0</sub>,C<sub>â</sub>);ãd(x(t),H) ⥠δ<sub>H</sub></td> 655<td>å®ç1ï¼è«ç±³å°ä¸»å®çï¼ã»å®ç3ï¼æ½è±¡çå ±æTCZåæå®çï¼ã»å®ç16ï¼èªå·±æèåå¨ã»çºçå®çï¼ã»å®ç18ï¼å çè¨èªé²åå®çï¼</td> 656<td>æ¡ä»¶31-AãDï¼è¨èªéå ãNagumo ä¸å¤æ§ãLyapunov å¸å¼æ§ãæ¢ç´é卿æ§ï¼</td> 657<td><b>ä¸å¤éåå®çã®å¿ç¨ã</b>Nagumo ã®å®çã¨è£é¡0ãããã³ã¨ã«ã´ã¼ãå®çãç¨ãã</td> 658<td>ç¾å¨ã®è¨èªã¨ Ego ã®éã«ã¼ãã ãã§ã¯ã使½è±¡åº¦å¸¯ããåºãããªããå ç¶æ ç²è¦åã¯å帰çã§ãããåç¶æ ãç¡éå訪åããã</td></tr> 659<tr><td><b>å®ç32</b><br>æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®ç</td> 660<td>d(x(t),C<sub>G</sub>(t)) ⤠â(a<sub>2</sub>/a<sub>1</sub>)e<sup>âλ<sub>G</sub>(tât<sub>0</sub>)</sup>d(x(t<sub>0</sub>),C<sub>G</sub>(t<sub>0</sub>)) â 0;ãT<sub>δ</sub> ã®æç¤ºå¼</td> 661<td>å®ç1ï¼è«ç±³å°ä¸»å®çï¼ã»å®ç4ï¼è«ç±³å°è¨å ´æå éå®çï¼ã»å®ç7ï¼è«ç±³å°çã®ã´ã¼ã«å®çï¼ã»å®ç8ï¼è«ç±³å°æªæ¥åç¹èªç¥æéå®çï¼ã»å®ç9ï¼è«ç±³å°æªæ¥åç¹ã´ã¼ã«éæå®çï¼</td> 662<td>æ¡ä»¶32-AãCï¼æªæ¥TCZã®ååãä¸å¤æ§ãæçè¨å ´æé¾å¤ãæå¤Lyapunovæã¿è¾¼ã¿ï¼</td> 663<td><b>å®ç9ï¼è«ç±³å°æªæ¥åç¹ã´ã¼ã«éæå®çï¼ã®æå¤åã»å®éåã</b>å°éæéã®æç¤ºçä¸çãæ°ãã«ä¸ãã</td> 664<td>å¤é¨ã´ã¼ã«ãæªæ¥TCZã«å ¥ãããã®è¨å ´æé¾å¤ã¯æçã§ãããé¾å¤ãè¶ ããã°ã以å¾ã®å°éã¯ãã¡ãªã¹ã¿ã·ã¹ãèªåçã«è¡ã</td></tr> 665</tbody></table> 666 667<h3>11.2 ç¶æ¿å®çï¼æ¬ç¨¿ãåæã¨ãã¦ç¨ããå®çï¼</h3> 668<table class="thmtable"> 669<caption>表2ãç¶æ¿å®çââå®çåã»æ¨æºå½¢ã»æ¬ç¨¿ã§ã®å½¹å²ã»è¨¼æã®æå¨</caption><colgroup><col style="width:15%"><col style="width:23%"><col style="width:22%"><col style="width:12%"><col style="width:28%"></colgroup> 670<thead><tr><th>å®ç</th><th>ä¸å¿å¼ï¼æ¨æºå½¢ï¼</th><th>æ¬ç¨¿ã§ã®å½¹å²</th><th>証æã®æå¨</th><th>è¦æ¨</th></tr></thead> 671<tbody> 672<tr><td><b>å®ç1</b><br>è«ç±³å°ä¸»å®ç</td><td>Ï<sub>c</sub> = arg min â«V<sub>0</sub> dt â x(t) â TCZ</td><td>å®ç31ï¼å çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻å®çï¼ã»å®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ã®åæã®éª¨æ ¼</td><td><b>æ¬ç¨¿ §3.1</b>ï¼åæ²ã»è¨¼æã¤ãï¼</td><td>ç´¯ç©è©ä¾¡ãæå°åããéã«ã¼ãã¯ãå°éå¯è½ãªå®å®é åã¸è»éãå°ã</td></tr> 673<tr><td><b>å®ç3</b><br>æ½è±¡çå ±æTCZåæå®ç</td><td>A(x)=0 â Ï(x)=LUB(W<sub>1</sub>,â¦,W<sub>N</sub>), A(t)â0</td><td>å®ç31ï¼å çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻å®çï¼ã®è¨èªéå ã®ä¸çæ§é </td><td><b>æ¬ç¨¿ §3.2</b></td><td>æªéããã«ãã£ãå ããã¨ãéå£ã¯èª°ã®ä¸çãåãæ¨ã¦ãªãæå°ã®å±æ ¹ã¸ããã</td></tr> 674<tr><td><b>å®ç4</b><br>è«ç±³å°è¨å ´æå éå®ç</td><td>á¹¼ = V<sub>0</sub> â κPQ, x â TCZ<sub>P</sub></td><td>å®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ã®è¨å ´æé¾å¤ã®åºç¤</td><td><b>æ¬ç¨¿ §3.3</b></td><td>è¨å ´æã¯å°å½¢ãã®ãã®ãå¤å½¢ããæå¿ã®æç¶ã«ãããã«è°·ãç§»ã</td></tr> 675<tr><td><b>å®ç7</b><br>è«ç±³å°çã®ã´ã¼ã«å®ç</td><td>G â TCZ<sub>0</sub>, d(G,TCZ<sub>0</sub>) ⥠ε > 0, G = Self-set</td><td>å®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ã®
675å¤é¨ã´ã¼ã«ã®è³æ ¼æ¡ä»¶</td><td><b>æ¬ç¨¿ §3.4</b></td><td>å¤é©ã´ã¼ã«ã®åæ¡ä»¶ãç¹å¾´ã¥ãããå°éã¯ä¸»å¼µããªã</td></tr> 676<tr><td><b>å®ç8</b><br>è«ç±³å°æªæ¥åç¹èªç¥æéå®ç</td><td>u<sup>*</sup> = arg min J<sub>G</sub>ï¼çµç«¯æ¡ä»¶ G ãç¾å¨å¶å¾¡ã決å®</td><td>å®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ã®æ±ºå®æ¹å</td><td><b>æ¬ç¨¿ §3.5</b></td><td>çµç«¯æ¡ä»¶ã¤ãæé©å¶å¾¡ã§ã¯ãç¾å¨ã®å¶å¾¡ãæªæ¥ã®ã´ã¼ã«ããå¾ãåãã«æ±ºã¾ã</td></tr> 677<tr><td><b>å®ç9</b><br>è«ç±³å°æªæ¥åç¹ã´ã¼ã«éæå®ç</td><td>K<sub>G</sub> = PQ<sup>+</sup> + EC<sub>Self</sub> ⥠K<sub>crit</sub> 㨠Lyapunov éä¸ â x â TCZ<sub>G</sub></td><td>å®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ã®å°éé¨åã®åå</td><td><b>æ¬ç¨¿ §3.6</b></td><td>é§å強度ãè¨çãè¶ ããé䏿¡ä»¶ãæºããããã°ãã´ã¼ã«å´TCZã¸åæãã</td></tr> 678<tr><td><b>å®ç15</b><br>èªç¥ç©çã¨ã³ãããã¼äº¤æã»ä¿åå®ç</td><td>S<sub>gen</sub> = S<sub>phys</sub> + Σw<sub>α</sub>H<sub>α</sub>, dS<sub>gen</sub>/dt = Π⥠0</td><td>å®ç30ï¼ã¨ã³ãããã¼äº¤æèªææ§æå®çï¼ã®äº¤æåæ¯</td><td><b>æ¬ç¨¿ §3.7</b></td><td>ï¼åæ²ã»è¨¼æã¤ãï¼ç©ç層åç¬ã§ã¯ã¨ã³ãããã¼ã®ä¿ååã¯åå¨ããªãã髿½è±¡åº¦å±¤ã¾ã§å«ãã¦ã¯ããã¦ã交æã«ããä¿ååãçå¼ã¨ãã¦æç«ãã</td></tr><tr><td><b>å®ç16</b><br>èªå·±æèåå¨ã»çºçå®ç</td><td>SC = lim<sub>â</sub>TCZ<sub>α</sub> â â , F<sub>SC</sub>(S<sup>*</sup>) = S<sup>*</sup>, M(S<sup>*</sup>) represents S<sup>*</sup></td><td>å®ç30ï¼ã¨ã³ãããã¼äº¤æèªææ§æå®çï¼ã»å®ç31ï¼å çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻å®çï¼ã® Ego æ§æ</td><td><b>æ¬ç¨¿ §3.8</b></td><td>åå¨ã¯é極éããã䏿æ§ã¯ä¸åç¹å®çãããåºå®ç¹ã¯å±¥æ´ã«ä¾åãã</td></tr> 679<tr><td><b>å®ç18</b><br>å çè¨èªé²åå®ç</td><td>H(Z|Y,M<sub>â</sub>) < H(Z|Y); Î <sub>0</sub> â Î <sub>â</sub>; ââ±/ââ > 0</td><td>å®ç30ï¼ã¨ã³ãããã¼äº¤æèªææ§æå®çï¼ã»å®ç31ï¼å çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻å®çï¼ã®è¨èªã®å½¹å²</td><td><b>æ¬ç¨¿ §3.9</b></td><td>å çè¨èªã¯ä¸ç¢ºå®æ§ãæ¸ãããæ¹çéåãåºããèªç±ææå®¹éãå¢ãã</td></tr> 680 681<tr><td><b>å®ç24</b><br>ä¸åçè¦å®ç</td><td>a ⺠⤠⧠æ¡ä»¶24-A â J<sup>*</sup><sub>a,Ï</sub>(x,T) > 0</td><td>å®ç28ï¼æ¶ æ§ç¡æã»åæ´åå®çï¼ã®ååé¢ã®ä¸å´</td><td>
681ãè«ç±³å°åæ³å°å®çãï¼æ¬ç¨¿ §3.10 ã«æ¡ä»¶ã®ã¿åæ²ï¼</td><td>ç©ºæªæºã§ã¯ãæé©åãã¦ãæ£ã®æ®ä½ä¾¡å¤ãæ®ã</td></tr> 682<tr><td><b>å®ç25</b><br>諸æ³ç¡æå®ç</td><td>¬âS<sub>0</sub> âh: F<sub>i,h</sub>(S<sub>0</sub>) = S<sub>0</sub>ï¼âdâð âαâð: ¬Atman(d,α)</td><td>å®ç28ï¼æ¶ æ§ç¡æã»åæ´åå®çï¼ã»å®ç29ï¼å½¢å¼æ³ä½ç³»ç¡èªæ§ã»ä¸å®åå®çï¼ã®ç¡æã®æ ¹æ </td><td>ãè«ç±³å°åæ³å°å®çãï¼åä¸ï¼</td><td>é¢ä¿ããåãé¢ãããåºå®çã»åä½åçã»å æçã«éåé·ãªèªæ§ã¯åå¨ããªã</td></tr> 683<tr><td><b>å®ç26</b><br>æ¶ æ§å¯éå®ç</td><td>W<sub>â¤</sub>(x(t),t) ⤠W<sub>â¤</sub>(x(T),T)e<sup>âλ(tâT)</sup> â dist(x(t),ð©<sub>â¤</sub>(t)) â 0ï¼æ°¸ä¹ è¦æ» â a=⤠⧠xâð©<sub>â¤</sub></td><td>å®ç28ï¼æ¶ æ§ç¡æã»åæ´åå®çï¼ã®ååé¢ã®ä¸å´</td><td>ãè«ç±³å°åæ³å°å®çãï¼åä¸ï¼</td><td>æé«æ½è±¡åº¦ã«ã¯ãçããã¾ã¾æ®ä½ä¾¡å¤ãé¶ã§ããååãä¸å¤éåãåå¨ãã</td></tr> 684</tbody></table> 685 686<div class="rigor"><div class="rigor-title">証æä¸ã®å°ä½ã«ã¤ãã¦ã®æ³¨æ</div> 687<p>表1ã®ã証æä¸ã®å°ä½ãæ¬ã¯ãåå®çãæ¢åå®çããã©ã®ç¨åº¦ç´æ¥ã«å¾ããã示ãã<b>å®ç28ï¼æ¶ æ§ç¡æã»åæ´åå®çï¼ã¯ç¶æ¿å®çã®ååé¢ã®ã¿ã§å¾ãããå®ç29ï¼å½¢å¼æ³ä½ç³»ç¡èªæ§ã»ä¸å®åå®çï¼ã¯å¤é¨ã¡ã¿å®çï¼Gödelã»Chaitinï¼ã¸ã®ä¾åãæ¬è³ªçã§ãããæ¬ä½ç³»ã®ã¿ããã¯å°ããªãã</b>å®ç30ï¼ã¨ã³ãããã¼äº¤æèªææ§æå®çï¼ã»å®ç31ï¼å çè¨èªéå ã»ä½æ½è±¡åº¦å é輪廻å®çï¼ã»å®ç32ï¼æªæ¥TCZãã¡ãªã¹ã¿ã·ã¹çµ¶å¯¾ä»åå®çï¼ã¯ç¶æ¿å®çã«æ°ããªæ¡ä»¶ãå ããåæã§ããã追å ãããè§£æçéå ·ï¼Banach ã®ä¸åç¹å®çãNagumo ã®å®çãã¨ã«ã´ã¼ãå®çãGrönwall ã®ä¸çå¼ï¼ã¯ããããæ¨æºçãªãã®ã§ããã</p></div> 688</section> 689 690 691<section id="references"> 692<h2>12. åç §æç®</h2> 693<h3>æ¬ä½ç³»ã®æ£æ¬ã»é¢é£ç¨¿</h3> 694<ol> 695<li>è«ç±³å°è±äººï¼2026ï¼ãè«ç±³å°åæ³å°å®çââå®ç23ï¼è«¸è¡ç¡å¸¸å®çï¼â26ãæ¥æ¬èªæ£æ¬ããã³è±èªçã</li> 696<li>è«ç±³å°è±äººï¼2026ï¼ãè«ç±³å°ç¡æç¸è¡å®çââå®ç27ï¼ç¡æèµ·è¡å®çï¼ãæ¥æ¬èªæ£æ¬ã»è±èªçã»ä¸è¬èªè çã</li> 697<li>è«ç±³å°è±äººï¼2026ï¼ãTomabechiSynthesisAvijjaSankharaJAãç·æ¬è§£èª¬ã</li> 698<li>è«ç±³å°è±äººï¼2026ï¼ãè«ç±³å°èªç¥ç©çã¨ã³ãããã¼äº¤æã»ä¿åå®çãã</li> 699<li>è«ç±³å°è±äººï¼2026ï¼ãè«ç±³å°èªå·±æèåå¨ã»çºçå®çãã</li> 700<li>è«ç±³å°è±äººï¼2026ï¼ãè«ç±³å°å çè¨èªé²åå®çãã</li> 701<li>è«ç±³å°è±äººï¼2026ï¼ãè«ç±³å°èªç¥ãã¡ãªã¹ã¿ã·ã¹çè«ãããã³ãè«ç±³å°æ½è±¡åº¦èªç±è«ãã</li> 702</ol> 703<h3>è§£é層ã¨ãã¦åç §ããå¤å ¸</h3> 704<ol> 705<li>NÄgÄrjunaï¼é¾æ¨¹, ca. 2â3ä¸ç´ï¼<em>MÅ«lamadhyamakakÄrikÄ</em>ï¼ãä¸è«ããæ ¹æ¬ä¸é ãï¼ãæ¬ç¨¿ §7.4 ã®è§£é層ã§åç §ãå¼ç¨ç®æã¯ 13.8ã»15.10ã»18.5ã»22.11ã»24.18 ããã³åé ã®å «ä¸ã漢訳ã¯é³©æ©ç¾ ä»è¨³ï¼ãä¸è«ã大æ£èµ No.1564ï¼ã«æ ãã訳åºã¯æ¬ç¨¿ã«ããã<b>ãããã®å¼ç¨ãå®ç31 ã®è¨¼æã«ã¯ç¨ãã¦ããªãã</b></li> 706</ol> 707<h3>å¤é¨ã¡ã¿å®ç</h3> 708<ol> 709<li>Gödel, K. (1931). âÃber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I.â <em>Monatshefte für Mathematik und Physik</em> 38, 173â198. <a href="https://doi.org/10.1007/BF01700692">doi:10.1007/BF01700692</a>.</li> 710<li>Chaitin, G. J. (1974). âInformation-Theoretic Limitations of Formal Systems.â <em>Journal of the ACM</em> 21(3), 403â424. <a href="https://doi.org/10.1145/321832.321839">doi:10.1145/321832.321839</a>.</li> 711<li>Chaitin, G. J. (1975). âA Theory of Program Size Formally Identical to Information Theory.â <em>Journal of the ACM</em> 22(3), 329â340. <a href="https://doi.org/10.1145/321892.321894">doi:10.1145/321892.321894</a>.</li> 712</ol> 713</section> 714 715<footer> 716<p>© 2026 Hideto Tomabechi / Cognitive Research Laboratories. Provisional public release for educational and peaceful use.</p> 717<p>ç: 1.0 · 2026-08-11 · å®ççªå·28â32</p> 718</footer> 719</article> 720</main> 721</body> 722</html>
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