鏁板鐗涗汉鐪嬫暟瀛︽暀鑲?
鏁板鏄墿鐞嗙殑涓€閮ㄥ垎銆?span class="t_tag" onclick="tagshow(event)" href="tools/ajax.aspx?t=topicswithsametag&tagid=123">鐗╃悊瀛?/span>鏄竴闂ㄥ疄楠岀瀛︼紝瀹冩槸鑷劧绉戝鐨勪竴閮ㄥ垎銆傝€屾暟瀛?
鏄?
鐗╃悊瀛︿腑鍙渶瑕佽姳璐硅緝灏戠殑浠ｄ环杩涜瀹為獙鐨勯偅涓€閮ㄥ垎銆備緥濡?Jacobi 鎭掔瓑寮?
锛堜繚璇佷笁瑙掑舰涓夋潯楂樹氦浜庝竴鐐癸級灏辨槸涓€涓疄楠屼簨瀹烇紝姝ｅ鍚屽湴鐞冩槸鍦嗙殑锛堝嵆鍚岃儦浜庣悆浣擄級
杩欐牱鐨勪簨瀹炰竴鏍枫€備絾鏄彂鐜板墠鑰呭嵈瑕佹瘮鍙戠幇鍚庤€呴渶瑕佽緝灏戠殑浠ｄ环銆?涓€涓兢鍙堟槸浠€涔堜笢涓滃憿锛熶唬鏁板瀹朵滑浼氳繖鏍锋潵鏁欏锛氳繖鏄竴涓亣璁剧殑闆嗗悎锛屽叿鏈変袱绉嶈繍
绠楋紝瀹冧滑婊¤冻涓€缁勫鏄撹浜哄繕璁扮殑鍏悊銆傝繖涓畾涔夊緢瀹规槗婵€璧蜂竴涓嚜鐒剁殑鎶楄锛氫换浣曚竴涓?
鏁忔劅鐨勪汉涓轰綍浼氶渶瑕佽繖涓€瀵硅繍绠楋紵“鍝︼紝杩欑鏁板鍘绘鍚?rdquo;锛嶏紞杩欏氨鏄鐢熺殑鍙嶅簲锛堜粬寰?
鍙兘灏嗘潵灏辨垚涓轰簡绉戝寮轰汉锛夈€?
濡傛灉鎴戜滑鐨勫嚭鍙戠偣涓嶆槸缇よ€屾槸鍙樻崲鐨勬蹇碉紙涓€涓泦鍚堝埌鑷韩鐨?锛?鏄犲皠锛夛紝鍒欐垜浠粷瀵?
灏嗗緱鍒颁笉鍚岀殑灞€闈紝杩欎篃鎵嶆洿鍍忓巻鍙茬殑鍙戝睍銆傛墍鏈夊彉鎹㈢殑闆嗗悎琚О涓轰竴涓兢锛屽叾涓换浣曚袱
涓彉鎹㈢殑澶嶅悎浠嶅湪姝ら泦鍚堝唴骞朵笖姣忎釜鍙樻崲鐨勯€嗗彉鎹篃濡傛銆?
杩欏氨鏄畾涔夌殑鍏抽敭鎵€鍦ㄣ€傞偅鎵€璋撶殑“鍏悊”浜嬪疄涓婁笉杩囨槸鍙樻崲缇ゆ墍鍏锋湁鐨勬樉鐒剁殑鎬ц川銆?
鍏悊鍖栫殑鍊″鑰呮墍绉扮殑“鎶借薄缇?rdquo;涓嶈繃鏄湪鍏佽鐩稿樊鍚屾瀯锛堜繚鎸佽繍绠楃殑1锛?鏄犲皠锛夋剰涔変笅
鐨勪笉鍚岄泦鍚堢殑鍙樻崲缇ゃ€傛濡?Cayley璇佹槑鐨勶紝鍦ㄨ繖涓笘鐣屼笂鏍规湰灏辨病鏈?ldquo;鏇存娊璞＄殑”缇ゃ€?
閭ｄ箞涓轰粈涔堥偅浜涗唬鏁板瀹朵粛瑕佺敤鎶借薄鐨勫畾涔夋潵鎶樼（杩欎簺楗卞彈鐥涜嫤鐨勫鐢熶滑鍛紵
椤轰究鎻愪竴鍙ワ紝鍦ㄤ笂涓栫邯60骞翠唬鎴戞浘缁欒帿鏂鐨勪腑灏忓鐢熶滑璁叉巿缇よ銆傛垜鍥為伩浜嗕换浣曠殑鍏?
鐞嗭紝灏藉彲鑳界殑璁╁唴瀹硅创杩戠墿鐞嗭紝鍦ㄥ崐骞村唴鎴戝氨鏁欑粰浜嗕粬浠叧浜庝竴鑸殑浜旀鏂圭▼涓嶅彲瑙ｆ€х殑
Abel 瀹氱悊锛堜互鍚屾牱鐨勬柟寮忥紝鎴戣繕鏁欑粰浜嗗皬瀛︾敓浠鏁帮紝榛庢浖鏇查潰锛屽熀鏈兢浠ュ強浠ｆ暟鍑芥暟
鐨刴onodromy 缇わ級銆傝繖闂ㄨ绋嬬殑鍐呭鍚庢潵鐢辨垜鐨勪竴涓惉浼?V. Alekseev 缁勭粐鍑虹増浜嗭紝鍚?
涓篢he Abel theorem in problems.
鏇查潰鐨勫垎绫诲畾鐞嗗簲璇ヨ鍖呭惈鍦ㄩ珮涓暟瀛︾殑璇剧▼閲岋紙鍙互涓嶇敤璇佹槑锛夛紝浣嗕笉鐭ヤ负浠€涔堝氨杩?
澶у鏁板鐨勮绋嬮噷涔熸壘涓嶅埌锛堥『渚夸竴涓嬶紝鍦ㄦ硶鍥借繎鍑犲崄骞存潵璇存湁鐨勫嚑浣曡绋嬮兘琚姝級銆?
鍦ㄥ悇涓眰娆′笂锛屾暟瀛︽暀鑲茬敱瀛﹂櫌鐨勭壒寰佽浆鍥炲埌琛ㄨ堪鑷劧绉戝鐨勯噸瑕佹€х殑鐗瑰緛锛屽娉曞浗鑰?
瑷€鏄竴涓強鍏剁儹鐐圭殑闂銆備娇鎴戞劅鍒板緢闇囨儕鐨勬槸閭ｄ簺鏈€濂界殑涔熸槸鏈€閲嶈鐨勬潯鐞嗘竻鏅扮殑鏁板
涔︼紝鍦ㄨ繖鍎垮嚑涔庨兘涓嶄负瀛︾敓浠墍鐭ワ紙鑰屼緷鎴戠湅瀹冧滑杩樻病鏈夎璇戞垚娉曡锛夈€傝繖浜涗功涓湁Rad
emacher 鍜?Tö鍐欑殑 銆奛umbers and figures銆嬶紱Hilbert 鍜?Cohn-Vossen鍐欑殑銆妏
litz, Geometry and the imagination銆嬶紱Courant 鍜孯obbins 鍐欑殑銆奧hat is mathema
tics?銆嬶紱Polya 鍐欑殑銆奌ow to solve it銆?鍜?銆奙ath
ematics and plausible reasoning 銆嬶紱 F. Klein 鍐欑殑
銆奃evelopment of mathematics in the 19th century 銆嬨€?
鎴戞竻鏅板湴璁板緱鍦ㄥ鏍℃椂锛孒ermite 鍐欑殑寰Н鍒?span class="t_tag" onclick="tagshow(event)" href="tools/ajax.aspx?t=topicswithsametag&tagid=89">鏁欑▼锛堟湁淇勮璇戞湰锛夌粰鎴戠暀涓嬩簡澶氫箞寮虹儓
鐨勫嵃璞°€傛垜璁板緱鍦ㄥ叾鏈€寮€濮嬬殑涓€绡囪涔変腑灏卞嚭鐜颁簡榛庢浖鏇查潰锛堝綋鐒舵墍鏈夊垎鏋愮殑鍐呭閮芥槸閽?
瀵瑰鍙橀噺鐨勶紝涔熸湰璇ュ姝わ級銆傝€岀Н鍒嗘笎杩涚殑鍐呭鏄€氳繃榛庢浖鏇查潰涓婇亾璺舰鍙樼殑鏂规硶鏉ョ爺绌?
锛堝浠婏紝鎴戜滑绉版鏂规硶涓篜icard-Lefschetz 鐞嗚;椤轰究鎻愪竴涓嬶紝Picard鏄疕ermite鐨勫コ濠?
锛嶏紞鏁板鑳藉姏寰€寰€鏄敱濂冲┛鏉ヤ紶鎵匡細Hadamard - P. Levy - L. Schwarz - U. Frisch 杩?
涓帇鏈濆氨鏄反榛庣瀛﹂櫌涓彟涓€涓繖鏍风殑鑼冧緥锛夈€?
鐢盚ermite 涓€鐧惧骞村墠鎵€鍐欑殑鎵€璋撶殑“杩囨椂鐨?rdquo;鏁欑▼锛堜篃璁告棭灏辫娉曞浗澶у鐨勫鐢熷浘涔?
棣嗗綋鍨冨溇鎵旀帀浜嗭級瀹為檯涓婅姣旈偅浜涘浠婃姌纾ㄥ鐢熶滑鐨勬渶浠や汉鍘岀儲鐨勫井绉垎璇炬湰鐜颁唬鍖栫殑澶?
銆?
濡傛灉鏁板瀹朵滑鍐嶄笉鐫￠啋锛岄偅涔堥偅浜涘鐜颁唬鐨勶紙鏈€姝ｉ潰鎰忎箟涓婄殑锛夋暟瀛︾悊璁轰粛鏈夐渶瑕侊紝鍚?
鏃跺張瀵归偅浜涙鏃犵敤澶勭殑鍏悊鍖栫壒寰佸叿鏈夊厤鐤姏锛堣繖鏄换浣曟晱閿愮殑浜烘墍鍏锋湁鐨勭壒寰侊級鐨勬秷璐?
鑰呬滑浼氭涓嶇姽璞殑灏嗚繖浜涘鏍￠噷鐨勫彈鏁欒偛涓嶈冻鐨勫绌朵滑鎵湴鍑洪棬銆?
涓€涓暟瀛︽暀甯堬紝濡傛灉鑷充粖杩樻病鏈夋帉鎻¤嚦灏戝嚑鍗稬andau 鍜?Lifshitz 钁楃殑鐗╃悊瀛︽暀绋嬶紝
浠栵紙濂癸級蹇呭皢鎴愪负涓€涓暟瀛︾晫鐨勫笇缃曠殑娈嬪瓨鑰咃紝灏卞ソ浼煎浠婁竴涓粛涓嶇煡閬撳紑闆嗕笌闂泦宸埆
鐨勪汉銆?br />
鍦?0涓栫邯涓彾锛屼汉浠瘯鍥句弗鏍煎湴鍖哄垎鐗╃悊涓庢暟瀛︺€傚叾閫犳垚鍦板悗鏋滄槸鐏鹃毦鎬х殑銆傛暣鏁翠竴
浠ｇ殑鏁板瀹跺湪瀵逛粬浠墍浠庝簨鐨勭瀛︾殑鍙︿竴鍗婂強鍏舵棤鐭ョ殑鎯呭喌涓嬫垚闀匡紝褰撶劧锛屽鍏朵粬鐨勭瀛?
灏辨洿鏃犵煡浜嗐€傝繖浜涗汉鍙堝紑濮嬫妸浠栦滑鐨勪笐闄嬬殑瀛﹂櫌寮忕殑浼暟瀛︽暀缁欎粬浠殑瀛︾敓锛屾帴鐫€杩欎簺涓?
闄嬬殑浼暟瀛﹀張琚氦缁欎腑灏忓鏍￠噷鐨勫瀛愪滑锛堜粬浠畬鍏ㄥ繕璁颁簡Hardy鐨勮鍛婏細涓戦檵鐨勬暟瀛﹀湪
闃冲厜涓嬩笉鍙兘鎬绘湁钘忚韩涔嬪锛夈€?
鏃㈢劧閭ｄ簺浠庣墿鐞嗗涓汉涓烘寲鍑烘潵鐨勫闄㈠紡鐨勬暟瀛︽棦鏃犵泭浜庢暀瀛︼紝鍙堝鍏朵粬鐨勭瀛︽鏃犵敤
澶勶紝缁撴灉鍙互鎯宠锛屽叏涓栫晫鐨勪汉閮借鍘屾暟瀛﹀锛堢敋鑷冲寘鎷偅浜涜浠栦滑鏁欏嚭鏉ョ殑鍙€滅殑瀛︽牎
閲岀殑瀛╁瓙浠互鍙婇偅浜涜繍鐢ㄨ繖浜涗笐闄嬫暟瀛︾殑浜猴級銆傝繖浜涘厛澶╀笉瓒崇殑鏁板瀹惰浠栦滑鎵€鎮ｇ殑浣庤兘
鐥囧€欑兢鎶樿吘鐨勭瓔鐤插姏灏斤紝浠栦滑鏃犺兘瀵圭墿鐞嗗鏈変釜璧风爜鐨勪簡瑙ｃ€備护浜轰滑璁板繂鐘规柊鐨勭敱浠栦滑寤?
閫犵殑涓€涓笐闄嬪缓绛戠墿灏辨槸“濂囨暟鐨勪弗鏍煎叕鐞嗗寲鐞嗚”銆?
寰堟樉鐒讹紝瀹屽叏鍙兘鍒涢€犺繖鏍蜂竴绉嶇悊璁猴紝浣垮緱骞肩鐨勫皬瀛︾敓浠暚鐣忓畠鐨勫畬缇庡強鍏跺唴閮ㄦ瀯閫?
鐨勫拰璋愶紙渚嬪锛岃繖绉嶇悊璁哄畾涔変簡濂囨暟涓」鐨勫拰浠ュ強浠绘剰涓洜瀛愮殑涔樼Н锛夈€備粠杩欑鍋忔墽鐙殬
鐨勮鐐规潵鐪嬶紝鍋舵暟鎴栬€呰璁や负鏄竴绫?ldquo;寮傜”锛屾垨鑰呴殢鐫€鏃堕棿娴侀€濓紝琚敤鏉ヤ綔涓鸿鐞嗚涓?
鍑犱釜“鐞嗘兂”瀵硅薄鐨勮ˉ鍏咃紙涓轰簡閬典粠鐗╃悊涓庣湡瀹炰笘鐣岀殑闇€瑕侊級銆傚緢涓嶅垢鐨勬槸锛岃繖绉嶇悊璁哄彧
鏄暟瀛︿腑涓€涓笐闄嬭€屽彉鎬佺殑鏋勯€狅紝浣嗗嵈缁熸不浜嗘垜浠殑鏁板鏁欒偛鏁板崄骞淬€傚畠棣栧厛婧愯嚜浜庢硶鍥?
锛岃繖鑲℃椋庡緢蹇紶鎾埌瀵规暟瀛﹀熀纭€鐨勬暀瀛﹂噷锛屽厛鏄瘨瀹冲ぇ瀛︾敓锛屾帴鐫€涓皬瀛︾敓涔熼毦鍏嶆鐏?
锛堣€岀伨鍖烘渶鍏堟槸娉曞浗锛屾帴鐫€鏄叾浠栧浗瀹讹紝鍖呮嫭淇勭綏鏂級銆?to be continued)
濡傛灉浣犻棶涓€涓硶鍥界殑灏忓鐢燂細“2锛?绛変簬鍑狅紵”锛屼粬锛堝ス锛変細杩欐牱鍥炵瓟锛?
“绛変簬3锛?锛屽洜涓哄姞娉曡繍绠楁槸鍙氦鎹㈢殑”銆備粬锛堝ス锛夋牴鏈笉鐭ラ亾杩欎釜鍜岀瓑浜庡嚑锛岀敋鑷虫牴鏈?
涓嶈兘鐞嗚В浣犲湪闂粬锛堝ス锛変粈涔堬紒
杩樻湁鐨勬硶鍥藉皬瀛︾敓浼氳繖鏍峰畾涔夋暟瀛︼紙鑷冲皯鎴戣涓哄緢鏈夊彲鑳斤級锛?ldquo;瀛樺湪涓€涓鏂瑰舰锛屼絾鍗?
杩樻病鏈夎璇佹槑”銆?
鎹垜鍦ㄦ硶鍥芥暀瀛︾殑缁忛獙锛屽ぇ瀛﹂噷鐨勫鐢熷鏁板鐨勮璇嗕笌杩欎簺灏忓鐢熶篃宸笉澶?
锛堢敋鑷冲寘鎷偅浜涘湪'楂樼瓑甯堣寖瀛︽牎'锛圗NS锛夐噷瀛︿範鏁板鐨勫鐢燂紞锛嶆垜涓鸿繖浜涙樉鐒跺緢鑱槑浣?
鍗磋姣掑棰囨繁鐨勫瀛愪滑鎰熷埌鏋佸害鐨勬儖鎯滐級銆?
渚嬪锛岃繖浜涘鐢熶粠鏈杩囦竴涓姏鐗╅潰锛岃€屼笖涓€涓繖鏍风殑闂锛氭弿杩扮敱鏂圭▼xy=z^2鎵€缁欏嚭
鐨勬洸闈㈢殑褰㈢姸锛屽氨鑳戒娇閭ｄ簺鍦‥NS涓爺绌剁殑鏁板瀹朵滑鍙戝憜鍗婂ぉ锛涜€屽涓嬮棶棰橈細鐢诲嚭骞抽潰涓?
鐢卞弬鏁版柟绋嬶紙渚嬪x = t^3 - 3t, y = t^4 - 2t^2锛夌粰鍑虹殑鏇茬嚎锛屽瀛︾敓鏉ヨ鏄笉鍙兘瀹?
鎴愮殑锛堢敋鑷冲澶у鏁版硶鍥界殑鏁板鏁欐巿涔熶竴鏍凤級銆備粠寰Н鍒嗙殑鍏ラ棬鏁欑涔︾洿鍒癎oursat鍐欑殑
璇炬湰锛岃В杩欎簺闂鐨勮兘鍔涢兘琚涓烘槸姣忎釜鏁板瀹跺簲鍏峰鐨勫熀鏈妧鑳姐€?
閭ｄ簺鍠滄鎸戞垬澶ц剳鐨勬墍璋?ldquo;鎶借薄鏁板”鐨勭媯鐑€呬滑锛屾妸鎵€鏈夊湪鏁板涓兘涓庣墿鐞嗗拰鐜板疄缁?
甯稿彂鐢熻仈绯荤殑鍑犱綍缁熺粺鎺掗櫎鍦ㄦ暀瀛︿箣澶栥€傜敱Goursat, Hermite, Picard绛変汉鍐欑殑寰Н鍒嗘暀
绋嬭璁や负鏄湁瀹崇殑锛屾渶杩戝樊鐐硅宸撮粠绗?鍜岀7澶у鐨勫浘涔﹂褰撳瀮鍦句涪鎺夛紝鍙槸鍦ㄦ垜鐨勫共棰?
涓嬫墠寰椾互淇濆瓨銆?
ENS鐨勫惉瀹屾墍鏈夊井鍒嗗嚑浣曚笌浠ｆ暟鍑犱綍璇剧▼鐨勫鐢燂紙鍒嗗埆琚笉鍚岀殑鏁板瀹舵暀鐨勶級锛?
鍗存棦涓嶇啛鎮夌敱妞渾鏇茬嚎 y^2 = x^3 + ax + b 鍐冲畾鐨勯粠鏇兼洸闈紝涔熶笉鐭ラ亾鏇查潰鐨勬嫇鎵戝垎绫?
锛堟洿鍒彁绗竴绫绘き鍦嗙Н鍒嗗拰妞渾鏇茬嚎鐨勭兢鎬ц川浜嗭紝鍗?Euler-Abel 鍔犳硶瀹氱悊锛夈€備粬浠粎浠?
瀛﹀埌浜咹odge 鏋勯€犱互鍙?Jacobi 绨囷紒杩欐牱鐨勭幇璞＄珶鐒朵細鍦ㄦ硶鍥藉嚭鐜帮紒杩欎釜鍥藉鍙槸涓烘暣涓?
涓栫晫璐＄尞浜嗚濡?Lagrange 锛孡aplace, Cauchy 浠ュ強 Poincaré, Leray 杩樻湁 Thom 杩欎簺
椤剁骇鐨勪紵澶т汉鐗╁晩锛佸鎴戣€岃█锛屼竴涓悎鐞嗙殑瑙ｉ噴鏉ヨ嚜 I.G. Petrovskii, 浠栧湪1966骞存浘鏁?
瀵艰繃鎴? 鐪熸鐨勬暟瀛﹀鍐充笉浼氭媺甯粨娲撅紝鍙湁寮辫€呬负浜嗙敓瀛樻墠浼氬姞鍏ュ府娲俱€備粬浠彲浠ヨ仈缁?
寰堝鐨勬柟闈紙鍙兘浼氭槸瓒呯骇鐨勬娊璞★紝鍙嶇姽澶富涔夋垨鑰?ldquo;搴旂敤鐨勫拰宸ヤ笟涓婄殑”闂锛夛紝
浣嗗叾鏈川鎬绘槸涓轰簡瑙ｅ喅绀句細鐢熷瓨闂銆?to be continued)
鎴戝湪姝ゅ悜澶у椤轰究鎻愪竴涓?L. Pasteur 鐨勫繝鍛婏細浠庢潵娌℃湁涔熷喅涓嶄細鏈変换浣曟墍璋撶殑“搴?
鐢ㄧ瀛?rdquo;锛岃€屼粎浠呮湁鐨勬槸绉戝鐨勫簲鐢紙鍗佸垎鏈夌敤鐨勪笢涓滃晩锛侊級
闀夸箙浠ユ潵鎴戜竴鐩村 Petrovskii 鐨勮瘽蹇冨瓨鐤戣檻锛屼絾鏄幇浠婃垜瓒婃潵瓒婅偗瀹氫粬璇寸殑涓€鐐规病
閿欍€傞偅浜涜秴绾ф娊璞℃椿鍔ㄧ殑鐩稿綋澶х殑閮ㄥ垎姝ｅ湪鍫曡惤鍒颁互宸ヤ笟鍖栫殑妯″紡鏃犺€荤殑鎺犲ず閭ｄ簺鍙戠幇鑰?
鐨勬垚鏋滐紝鐒跺悗鍐嶅姞浠ョ郴缁熷湴缁勭粐璁捐浣胯嚜宸辨垚涓轰竾鑳界殑鎺ㄥ箍鑰呫€傚氨褰蜂經缇庝附鍧氭墍鍦ㄧ殑鏂板ぇ
闄嗕笉浠ュ摜浼﹀竷鍛藉悕涓€鏍凤紝鏁板缁撴灉涔熷嚑涔庝粠鏈互瀹冧滑鐪熸鐨勫彂鐜拌€呮潵鍛藉悕銆?
涓洪伩鍏嶈璁や负鎴戝湪鑳¤鍏亾锛屾垜涓嶅緱涓嶅湪姝ゅ０鏄庢垜鑷繁鐨勪竴浜涙垚鏋滅敱浜庤帿鍚嶅叾濡欑殑鍘熷洜
灏辫浠ヤ笂杩版柟寮忔棤鍋垮緛鐢紝鍏跺疄杩欐牱鐨勪簨鎯呯粡甯稿湪鎴戠殑鑰佸笀锛圞olmogorov, Petrovskii,
Pontryagin, Rokhlin锛夊拰瀛︾敓韬笂鍙戠敓銆?
M. Berry 鏁欐巿鏇剧粡鎻愬嚭杩囧涓嬩袱涓師鐞嗭細
Arnold 鍘熺悊锛氬鏋滄煇涓悊蹇典腑鍑虹幇浜嗘煇涓汉鍚嶏紝鍒欒繖涓汉鍚嶅繀闈炲彂鐜版鐞?
蹇佃€呯殑鍚嶅瓧銆?
Berry 鍘熺悊锛欰rnold 鍘熺悊閫傜敤浜庤嚜韬€?
涓嶈繃锛屾垜浠繕鏄鍥炴硶鍥界殑鏁板鏁欒偛涓婃潵銆傚綋鎴戣繕鏄帿鏂澶у鏁板姏绯荤殑涓€骞寸骇鏂扮敓
鏃讹紝闆嗗悎璁虹殑鎷撴墤瀛﹀ L.A. Tumarkin 鏁欐垜浠井绉垎锛屼粬鍦ㄨ鍫備笂寰堣皑鎱庡湴涓€閬嶅張涓€閬?
鍦拌杩板彜鑰佽€岀粡鍏哥殑Goursat 鐗堢殑娉曡寰Н鍒嗘暀绋嬨€備粬鍛婅瘔鎴戜滑鏈夌悊鍑芥暟娌跨潃涓€鏉′唬鏁版洸
绾跨殑绉垎鍙互姹傚嚭鏉ュ鏋滆浠ｆ暟鏇茬嚎瀵瑰簲鐨勯粠鏇奸潰鏃朵竴涓悆闈€傝€屼竴鑸潵璇达紝濡傛灉璇ユ洸闈?
鐨勪簭鏍兼洿楂樿繖鏍风殑绉垎灏嗕笉鍙眰锛屼笉杩囧鐞冮潰鑰岃█锛屽彧瑕佸湪涓€涓粰瀹氬害鏁扮殑鏇茬嚎涓婃湁鍏呭垎
澶氱殑double points 灏辫冻澶熶簡锛堝嵆瑕佹眰璇ユ洸绾挎槸unicursal 锛氬嵆鍙互灏嗗叾瀹炵偣鍦ㄥ皠褰卞钩闈?
涓婁竴绗旂敾鍑烘潵锛夈€?
杩欎簺浜嬪疄缁欐垜浠€犳垚澶氫箞娣卞埢鐨勫嵃璞″晩锛堝嵆浣挎病鏈夌粰鍑鸿瘉鏄庯級锛屽畠浠粰浜嗘垜浠潪甯镐紭
缇庤€屾纭殑鐜颁唬鏁板鐨勬€濇兂锛屾瘮閭ｄ簺闀跨瘒绱墠鐨凚ourbaki瀛︽淳鐨勮钁椾笉鐭ラ亾濂藉埌鍝噷鍘讳簡
銆傝鐪熺殑锛屾垜浠湪杩欓噷鐪嬪埌浜嗛偅浜涜〃闈笂瀹屽叏涓嶅悓鐨勪簨鐗╀箣闂村瓨鍦ㄧ潃浠や汉鎯婂鐨勮仈绯伙細涓€
鏂归潰锛屽浜庣浉搴旂殑榛庢浖闈笂鐨勭Н鍒嗕笌鎷撴墤瀛樺湪鐫€鏄惧紡鐨勮〃杈惧紡锛岃€屽彟涓€鏂归潰锛屽湪 double
points 鐨勪釜鏁颁笌鐩稿簲鐨勯粠鏇奸潰鐨勪簭鏍间箣闂翠篃鏈夐噸瑕佺殑鑱旂郴銆?
杩欐牱鐨勪緥瀛愬苟涓嶉矞瑙侊紝浣滀负鏁板涓渶杩蜂汉鐨勬€ц川涔嬩竴锛孞acobi鏇炬寚鍑猴細鐢ㄥ悓涓€涓嚱鏁?
灏辨棦鍙互鐞嗚В鑳借〃绀轰负4涓暟骞虫柟鍜岀殑鏁存暟鐨勬€ц川锛屽張鍙互鎻忚堪涓€涓崟鎽嗙殑杩愬姩銆?
杩欎簺涓嶅悓绉嶇被鐨勬暟瀛﹀璞′箣闂磋仈绯荤殑鍙戠幇锛屽氨濂芥瘮鍦ㄧ墿鐞嗗涓數涓庣涔嬮棿鑱旂郴鐨勫彂鐜帮紝
涔熺被鍚屼簬鍦拌川瀛︿笂瀵圭編娲插ぇ闄嗙殑涓滄捣宀镐笌闈炴床澶ч檰鐨勮タ娴峰哺涔嬮棿鐩镐技鎬х殑鍙戠幇銆?
杩欎簺鍙戠幇瀵逛簬鏁欏鎵€鍏锋湁鐨勪护浜烘縺鍔ㄧ殑闈炲嚒鎰忎箟鏄棤娉曚及閲忕殑銆傛鏄畠浠寚寮曠潃鎴戜滑鍘?
鐮旂┒鍜屽彂鐜?span class="t_tag" onclick="tagshow(event)" href="tools/ajax.aspx?t=topicswithsametag&tagid=575">瀹囧畽涓拰璋愯€岀簿褰╃殑鐜拌薄銆?
鐒惰€岋紝鏁板鏁欒偛鐨勯潪鍑犱綍鍖栦互鍙婁笌鐗╃悊瀛︾殑鍒嗙鍗村壊鏂簡杩欑鑱旂郴銆備緥濡傦紝涓嶄粎浠呭涔?
鏁板鐨勫鐢熻€屼笖缁濆ぇ閮ㄥ垎鐨勪唬鏁板嚑浣曞瀹堕兘瀵逛互涓嬫彁鍙婄殑Jacobi 浜嬪疄涓€鏃犳墍鐭ワ細涓€涓
涓€绫诲瀷鐨勬き鍦嗙Н鍒嗚〃绀轰簡鐩稿簲鐨勫搱瀵嗛】绯荤粺涓部鏌愪釜妞渾鐩告洸绾跨殑杩愬姩鎵€璧扮殑鏃堕棿銆?
鎴戜滑鐭ラ亾涓€涓?hypocycloid 灏卞鍚屽椤瑰紡鐜腑鐨勭悊鎯充竴鏍锋槸鏃犵┓鏃犲敖鐨勩€備絾鏄鏋滆
鎶婄悊鎯宠繖涓蹇垫暀缁欎竴涓粠鏈杩囦换浣?hypocycloid 鐨勫鐢燂紝灏卞ソ姣旀妸鍒嗘暟鐨勫姞娉曟暀缁?
涓€涓粠鏉ユ病鏈夋妸铔嬬硶鎴栬嫻鏋滅瓑鍒嗗垏鍓茶繃锛堣嚦灏戝湪鑴戝瓙閲屽垏杩囷級鐨勫鐢熴€傛鏃犵枒闂瀛愪滑灏?
浼氬€惧悜浜庡悓鏃跺垎瀛愬姞鍒嗗瓙鍒嗘瘝鍔犲垎姣嶃€?
浠庢垜鐨勬硶鍥芥湅鍙嬮偅閲屾垜鍚杩欑瓒呯骇鎶借薄鐨勪竴鑸寲姝ｆ槸浠栦滑鍥藉鐨勪紶缁熺壒鑹层€?
濡傛灉璇磋繖鍙兘鏄竴涓笘琚殑缂洪櫡锛屾垜鍊掍笉浼氫笉璧炴垚锛屼笉杩囨垜杩樻槸鎰挎剰寮鸿皟閭ｄ釜浠?
Poincaré 閭ｅ効鍊熸潵鐨?ldquo;铔嬬硶涓庤嫻鏋?rdquo;鐨勪簨瀹炪€?
鏋勯€犳暟瀛︾悊璁虹殑鏂瑰紡涓庡叾瀹冪殑鑷劧绉戝骞舵病鏈変粈涔堜笉鍚屻€傞鍏堬紝鎴戜滑瑕佽€冭檻涓€浜涘璞″苟
瀵逛竴浜涚壒娈婄殑浜嬩緥杩涜瑙傚療銆傛帴鐫€鎴戜滑璇曞浘瑕佹壘鍒颁竴浜涙垜浠墍瑙傚療鍒扮殑缁撴灉鍦ㄥ簲鐢ㄤ笂鐨勯檺
鍒讹紝鍗冲鎵鹃偅浜涢槻姝㈡垜浠笉姝ｇ‘鍦版妸鎴戜滑鎵€瑙傚療鐨勭粨鏋滄墿灞曞埌鏇村箍娉涢鍩熺殑鍙嶄緥銆備綔涓轰竴
涓粨鏋滄垜浠敖鍙兘鍦版槑纭彁鍑洪偅鐢辩粡楠屽緱鏉ョ殑鍙戠幇
锛堝璐归┈鐚滄兂鍜屽簽鍔犺幈鐚滄兂锛夈€傝繖涔嬪悗灏嗘槸妫€楠屾垜浠殑缁撹鍒板簳鏈夊鍙潬鐨勫洶闅剧殑闃舵銆?
灏辫繖涓€鐐规潵璇达紝鏁板鐣屽凡缁忓彂灞曞嚭浜嗕竴濂楃壒鍒殑鎶€鏈€傝繖绉嶆妧鏈紝褰撹杩愮敤浜庣幇瀹炰笘鐣?
鏃讹紝鏈夋椂鍊欏緢鏈夌敤锛屼絾鏈夋椂鍊欎篃浼氬鑷磋嚜娆烘浜恒€傝繖鏍风殑鎶€鏈绉颁负“寤烘ā”銆傚綋鏋勯€犱竴
涓ā鍨嬫椂锛岃杩涜濡備笅鐨勭悊鎯冲寲锛氭煇浜涘彧鑳戒互涓€瀹氭鐜囨垨涓€瀹氱殑绮剧‘鎬т簡瑙ｇ殑浜嬪疄锛屽線寰€
琚涓烘槸“缁濆”姝ｇ‘鐨勫苟琚綋浣?ldquo;鍏悊”鏉ユ帴鍙椼€傝繖绉?ldquo;缁濆鎬?rdquo;鐨勬剰涔夋伆鎭版槸锛屽湪鎶?
鎵€鏈夋垜浠彲浠ュ€熷姪杩欎簺浜嬪疄寰楀嚭鐨勭粨璁虹О涓哄畾鐞嗙殑杩囩▼涓紝鎴戜滑鍏佽鑷繁渚濇嵁褰㈠紡閫昏緫鐨?
瑙勫垯鏉ヨ繍鐢ㄨ繖浜?ldquo;浜嬪疄”銆?鏄剧劧鍦ㄤ换浣曠幇瀹炵殑鏃ュ父鐢熸椿涓紝鎴戜滑鐨勬椿鍔ㄨ瀹屽叏渚濊禆浜庤繖
鏍风殑鍖栧噺鏄笉鍙兘鐨勩€傚師鍥犺嚦灏戝湪浜庢墍鐮旂┒鐨勭幇璞＄殑鍙傛暟鍐充笉鍙兘琚粷瀵瑰噯纭殑鐭ユ檽锛屽苟
涓斿弬鏁扮殑寰皬鍙樺寲锛堜緥濡備竴涓繃绋嬪垵濮嬫潯浠剁殑寰皬鏀瑰彉锛夊氨浼氬畬鍏ㄥ湴鏀瑰彉缁撴灉銆傜敱浜庤繖涓?
鍘熷洜鎴戜滑鍙互璇翠换浣曢暱鏈熺殑澶╂皵棰勬姤閮芥槸涓嶅彲鑳界殑锛屾棤璁烘垜浠妸璁＄畻鏈洪€犵殑鏈夊楂樼骇鎴栨槸
璁板綍鍒濆鏉′欢鍦颁华鍣ㄦ湁澶氱伒鏁忥紝杩欐案杩滀篃鍔炰笉鍒般€?
涓庢瀹屽叏涓€鏍风殑鏄紝鍏悊锛堥偅浜涙垜浠笉鑳藉畬鍏ㄧ‘瀹氱殑锛夌殑涓€涓皬灏忕殑鏀瑰彉铏芥槸瀹硅鐨?
锛屼竴鑸潵璇达紝鐢遍偅浜涜鎺ュ彈鐨勫叕鐞嗘帹鍑虹殑瀹氱悊鍗村皢瀵煎嚭瀹屽叏涓嶅悓鐨勭粨璁恒€傛帹瀵肩殑閾撅紙鍗虫墍
璋撶殑“璇佹槑”锛夎秺闀胯秺澶嶆潅锛屾渶鍚庡緱鍒扮殑缁撹鍙潬鎬ц秺浣庛€傚鏉傜殑妯″瀷鍑犱箮姣棤鐢ㄥ锛堥櫎
浜嗗閭ｄ簺鏃犺亰鐨勪笓鍐?span class="t_tag" onclick="tagshow(event)" href="tools/ajax.aspx?t=topicswithsametag&tagid=240">璁烘枃鐨勪汉锛夈€?
鏁板寤烘ā鐨勬妧鏈杩欑楹荤儲涓€鏃犳墍鐭ワ紝骞朵笖杩樹笉鏂湴鍚瑰槝浠栦滑寰楀埌鐨勬ā鍨嬶紝浼间箮瀹冧滑鐪?
鐨勫氨涓庣幇瀹炰笘鐣屽惢鍚堛€備簨瀹炰笂锛屼粠鑷劧绉戝鐨勮鐐圭湅, 杩欑閫斿緞鏄樉鐒朵笉姝ｇ‘鐨勶紝浣嗗嵈缁?
甯稿鑷村緢澶氱墿鐞嗕笂鏈夌敤鐨勮绉颁负“鏈変笉鍙€濊鐨勬湁鏁堟€х殑鏁板”缁撴灉锛堟垨鍙仛“Wigner
鍘熺悊”锛夈€?
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浣撶幇鍦ㄧ敱鐜板疄涓栫晫鎶界鍑虹殑琚粷瀵瑰寲浜嗙殑妯″瀷锛屽苟涓斿畠涓庣幇瀹炲凡涓嶅啀鐩哥銆傝繖鍎挎槸涓€涓畝
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璁板彿鏈夊崐鏁伴兘浼氬嚭闂銆?to be continued)
涓庤繖鏍风殑璋鐩告姉鐨勬妧鏈篃鍚屾牱瀛樺湪浜庝换浣曞疄楠岀瀛﹂噷锛岃€屼笖搴旇鏁欑粰姣忎竴涓ぇ瀛︿綆骞?
绾х殑瀛︾敓銆?
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杩欎簺鏈夌姜鐨?ldquo;浠ｆ暟锛嶅叕鐞嗗瀹?rdquo;銆備緥濡傦紝浠栦滑鎬绘槸鎯崇敤闀夸箻瑙勫垯鐨勬墜娈垫潵瀹氫箟鑷劧鏁扮殑涔?
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鏄彁鍗囪繖闂ㄥ绉戜互鍙婃暀鎺堝畠鐨勪汉鐨勭ぞ浼氬湴浣嶏級銆傛樉鐒讹紝杩欑瀹氫箟鍜岃繖鏍风殑璇佹槑瀵规暀瀛﹀拰瀹?
闄呭伐浣滄湁鐧惧鑰屾棤涓€鐩娿€?
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涓轰竴椤规湁鐢ㄧ殑浜虹被娲诲姩鐨勭編濂藉嵃璞°€?
鎴戝皢鍐嶆彮绀哄嚑涓繖鏍风殑绉樺瘑锛堝彲鎬滅殑瀛︾敓浠姝ゅ緢鏈夊叴瓒ｏ級銆?
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鐞嗚繖浜涢涓滆タ銆?
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