{"id":62131,"date":"2024-07-16T01:31:47","date_gmt":"2024-07-16T01:31:47","guid":{"rendered":"https:\/\/science-hub.click\/Thom-Boardman%E3%82%AF%E3%83%A9%E3%82%B9-%E5%AE%9A%E7%BE%A9\/"},"modified":"2024-07-16T01:31:47","modified_gmt":"2024-07-16T01:31:47","slug":"Thom-Boardman%E3%82%AF%E3%83%A9%E3%82%B9-%E5%AE%9A%E7%BE%A9","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=62131","title":{"rendered":"Thom-Boardman \u30af\u30e9\u30b9 – \u5b9a\u7fa9"},"content":{"rendered":"

\u5c0e\u5165<\/h2>

1956 \u5e74\u306b Ren\u00e9 Thom \u306b\u3088\u3063\u3066\u5c0e\u5165\u3055\u308c\u30011967 \u5e74\u306b JM Boardman \u306b\u3088\u3063\u3066\u4e00\u822c\u5316\u3055\u308c\u305fThom-Boardman<\/b> \u03a3<\/span>\u30af\u30e9\u30b9\u306f\u3001\u5fae\u5206\u53ef\u80fd\u306a\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3\u306e\u7279\u7570\u70b9\u3092\u7814\u7a76\u3059\u308b\u305f\u3081\u306e\u30c4\u30fc\u30eb\u3067\u3059\u3002

$$ {f: M \\rightarrow N} $$<\/div> \u3002\u3053\u308c\u3089\u306b\u3088\u308a\u3001 f<\/i><\/span> (\u306e\u63a5\u7dda\u30de\u30c3\u30d7) \u306e\u30ab\u30fc\u30cd\u30eb\u306e\u6b21\u5143\u3068\u305d\u306e\u5236\u9650\u306e\u4e00\u90e8\u306b\u5f93\u3063\u3066\u7279\u7570\u70b9\u3092\u533a\u5225\u3059\u308b\u3053\u3068\u304c\u53ef\u80fd\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u3089\u306f\u3001\u5e7e\u4f55\u5149\u5b66<\/a><\/span>\u306b\u304a\u3051\u308b\u30b3\u30fc\u30b9\u30c6\u30a3\u30af\u30b9\u306e\u8a08\u7b97\u306b\u91cd\u8981\u306a\u7528\u9014\u3092\u6301\u3063\u3066\u3044\u307e\u3059\u3002<\/p>
\n\"<\/figure>

\u30af\u30e9\u30b9 \u03a3I(f) (Thom\u30011956)<\/h2>

(\u7121\u9650\u306b) \u5fae\u5206\u53ef\u80fd\u306a\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3\u304c\u4e0e\u3048\u3089\u308c\u305f\u5834\u5408<\/span>

$$ {f: M^m\\rightarrow N^n} $$<\/div> \u3001\u3059\u3079\u3066\u306e\u70b9\u3067\u5b9a\u7fa9\u3057\u307e\u3059
$$ {x\\in M} $$<\/div> f<\/i><\/span>\u306e\u30e9\u30f3\u30afr<\/i><\/span>\u306f\u30bf\u30f3\u30b8\u30a7\u30f3\u30c8 \u30de\u30c3\u30d7T<\/i> f<\/i> x<\/i><\/sub><\/span>\u306e\u30e9\u30f3\u30af\u3067\u3059\u3002\u307e\u305f\u3001\u30bd\u30fc\u30b9\u306e\u30b3\u30e9\u30f3\u3092m<\/i> \u2212 r<\/i><\/span>\u3068\u3057\u3066\u5b9a\u7fa9\u3057\u3001\u30b4\u30fc\u30eb\u306e\u30b3\u30e9\u30f3\u3092n<\/i> \u2212 r<\/i><\/span>\u3068\u3057\u3066\u5b9a\u7fa9\u3057\u307e\u3059\u3002\u30bd\u30fc\u30b9\u306e\u30b3\u30e9\u30f3\u306f\u3001 T<\/i> f<\/i> x<\/i><\/sub><\/span> : i<\/i> = m<\/i> \u2212 r<\/i><\/span>\u306e\u30ab\u30fc\u30cd\u30eb\u306e\u6b21\u5143i<\/i><\/span>\u3067\u3082\u3042\u308a\u307e\u3059\u3002<\/p>

\u30ef\u30f3\u30dd\u30a4\u30f3\u30c8

$$ {x\\in M} $$<\/div>\u30e9\u30f3\u30af\u304c\u53ef\u80fd\u306a\u6700\u5927\u5024r<\/i> = min( m<\/i> , n<\/i> )<\/span>\u3092\u53d6\u308b\u5834\u5408\u3001\u306f\u6b63\u898f\u3067\u3042\u308b\u3068<\/i>\u8a00\u308f\u308c\u307e\u3059\u3002\u305d\u308c\u4ee5\u5916\u306e\u5834\u5408\u306f\u3001\u7279\u7570\u7684<\/i>\u307e\u305f\u306f\u81e8\u754c\u7684<\/i>\u3067\u3042\u308b\u3068\u8a00\u308f\u308c\u307e\u3059\u3002\u7279\u7570\u70b9\u306e\u30bb\u30c3\u30c8\u306f<\/a><\/span>\u03a3( f<\/i> )<\/span>\u3067\u8868\u3055\u308c\u307e\u3059\u3002 Thom-Boardman \u30af\u30e9\u30b9\u306f<\/a><\/span>\u03a3( f<\/i> )<\/span>\u306e\u69cb\u9020\u3092\u8a18\u8ff0\u3057\u307e\u3059\u3002 <\/p>

$$ {x\\in M} $$<\/div> T<\/i> f<\/i> x<\/i><\/sub><\/span>\u306e\u30ab\u30fc\u30cd\u30eb\u304c\u6b21\u5143i<\/i><\/span>\u3067\u3042\u308b\u5834\u5408\u3001 \u306f\u30af\u30e9\u30b9\u03a3 i<\/i><\/sup><\/span>\u3067\u3042\u308b\u3068\u8a00\u308f\u308c\u307e\u3059\u3002\u30af\u30e9\u30b9\u03a3 i<\/i><\/sup><\/span>\u306e\u70b9\u306e\u96c6\u5408\u3092\u03a3 i<\/i><\/sup> ( f<\/i> )<\/span>\u3067\u8868\u3057\u307e\u3059\u3002<\/p>

\u4e00\u822c\u7684\u306a\u30b1\u30fc\u30b9\u306f\u3001\u4efb\u610f\u306e\u6574\u6570\u65cf\u306b\u5bfe\u3057\u3066\u8a2d\u5b9a\u306b\u3088\u3063\u3066\u5e30\u7d0d\u7684\u306b\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002

$$ {I=(i_1,i_2,\\cdots,i_n)} $$<\/div> : <\/p>
$$ { \\Sigma^{i_1,…,i_{n-1},i_n}(f)=\\Sigma^{i_n}(f \\vert_{\\Sigma^{i_1,\\cdots,i_{n-1}}(f) })} $$<\/div> \u3002<\/center><\/center>

\u30a4\u30f3\u30af\u30eb\u30fc\u30b8\u30e7\u30f3\u304c\u3042\u308a\u307e\u3059

$$ {M \\supset \\Sigma^{i_1} \\supset \\Sigma^{i_1,i_2} \\supset \\Sigma^{i_1,i_2,i_3 } \\supset \\cdots} $$<\/div> \u3002<\/p>

\u5e7e\u4f55\u5149\u5b66\u3078\u306e\u5fdc\u7528<\/h2>

\u5e7e\u4f55\u5149\u5b66\u306e\u30b3\u30fc\u30b9\u30c6\u30a3\u30af\u30b9\u306f\u3001\u7279\u7570\u70b9\u3068\u3057\u3066\u3001\u3088\u308a\u6b63\u78ba\u306b\u306f\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u7279\u7570\u70b9\u3068\u3057\u3066\u6570\u5b66\u7684\u306b\u30e2\u30c7\u30eb\u5316\u3055\u308c\u307e\u3059\u3002\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u7279\u7570\u70b9\u306e\u7406\u8ad6\u306f<\/a><\/span>\u3001\u79c1\u305f\u3061\u306e\u7269\u7406<\/a><\/span>\u7a7a\u9593\u306b\u306f 5 \u3064\u306e\u4e00\u822c\u7684\u306a\u30bf\u30a4\u30d7\u306e\u82db\u6027\u70b9\u304c\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u307e\u3059: \u6298\u308a\u76eeA<\/i> 2<\/sub><\/span> \u3001\u30ae\u30e3\u30b6\u30fcA<\/i> 3<\/sub><\/span> \u3001\u30c0\u30d6\u30c6\u30fc\u30ebA<\/i> 4<\/sub><\/span> \u3001\u53cc\u66f2\u3078\u305d

$$ {D_4^+} $$<\/div>\u305d\u3057\u3066\u6955\u5186\u5f62\u306e\u304a\u3078\u305d
$$ {D_4^-} $$<\/div> \u3002<\/p>

\u5fae\u5206\u53ef\u80fd\u306a\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3\u306e\u7279\u7570\u70b9\u3068\u306e\u95a2\u9023\u6027\u306f\u3001\u6b21\u306e\u8a18\u8ff0\u304b\u3089\u5f97\u3089\u308c\u307e\u3059\u3002\u5149\u7dda\u306e\u96c6\u5408 (\u307e\u305f\u306f\u5408\u540c) \u3092\u8003\u3048\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u5404\u5149\u7dda\u306f\u3001\uff12\u3064\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\uff58<\/i> \uff11<\/sub><\/span> \u3001 \uff58<\/i> \uff12<\/sub><\/span>\u306b\u3088\u3063\u3066\u5b9a\u7fa9\u3055\u308c\u3001\u3053\u308c\u3089\u306f\u3001\u4f8b\u3048\u3070\u3001\u5149\u7dda\u304c\u767a\u3059\u308b\u6ce2\u9762<\/a><\/span>\uff37<\/i><\/span>\u306e\u70b9\uff31<\/i><\/span>\u306e\u5ea7\u6a19\u3067\u3042\u308b\u3002\u5149\u7dda\u7cfb\u306e\u3059\u3079\u3066\u306e\u70b9P \u3092<\/i><\/span>\u8a18\u8ff0\u3059\u308b\u305f\u3081\u306b\u3001\u5ea7\u6a19x<\/i> 1<\/sub><\/span>\u3068x<\/i> 2<\/sub><\/span>\u306b\u5149\u7dda\u306b\u6cbf\u3063\u305f 3 \u756a\u76ee\u306e\u5ea7\u6a19x<\/i> 3<\/sub><\/span>\u3092\u8ffd\u52a0\u3057\u307e\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u5149\u7dda( x<\/i> 1<\/sub> , x<\/i> 2<\/sub> )<\/span>\u306b\u6cbf\u3063\u3066\u6e2c\u5b9a\u3055\u308c\u305f\u8ddd\u96e2Q<\/i> P<\/i><\/span>\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u534a\u5f84\u306b\u6cbf\u3063\u305f\u5ea7\u6a19 (\u8ddd\u96e2) x<\/i> 3<\/sub><\/span>\u306b\u4f4d\u7f6e\u3059\u308b\u7269\u7406\u7a7a\u9593\u306e\u70b9( y<\/i> 1<\/sub> , y<\/i> 2<\/sub> , y<\/i> 3<\/sub> )<\/span>\u306e\u4e09\u91cd\u9805( x<\/i> 1<\/sub> , x<\/i> 2<\/sub> , x<\/i> 3<\/sub> )<\/span>\u306b\u5bfe\u5fdc\u3059\u308b\u30de\u30c3\u30d7f<\/i><\/span>\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002 ( x<\/i> 1<\/sub> \u3001 x<\/i> 2<\/sub> )<\/span> \u3002\u5149\u7dda\u304c\u300c\u4e92\u3044\u306b\u4ea4\u5dee\u3059\u308b\u300d\u3068\u3044\u3046\u3053\u3068\u306f\u3001 f<\/i><\/span>\u304c\u5358\u5c04\u7684\u3067\u306f\u306a\u3044\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002 f<\/i><\/span>\u306e\u975e\u5168\u5c04\u6027\u306f\u30b0\u30ec\u30fc\u9818\u57df\u306e\u5b58\u5728\u3092\u8868\u3057\u307e\u3059\u3002\u5149\u7dda\u30b7\u30b9\u30c6\u30e0\u306e\u30b3\u30fc\u30b9\u30c6\u30a3\u30af\u30b9<\/a><\/span>K<\/i><\/span>\u306ff<\/i><\/span>\u306e\u7279\u7570\u96c6\u5408\u03a3<\/span> \u3001\u3088\u308a\u6b63\u78ba\u306b\u306f\u7269\u7406\u7a7a\u9593\u306b\u304a\u3051\u308b\u305d\u306e\u30a4\u30e1\u30fc\u30b8\u3067\u3059: K<\/i> = f<\/i> (\u03a3)<\/span> \u3002<\/p>

\u5fae\u5206\u53ef\u80fd\u306a\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3f<\/i><\/span>\u306b\u3088\u308b\u5149\u7dda\u306e\u3053\u306e\u30e2\u30c7\u30ea\u30f3\u30b0\u306f\u3001\u30b3\u30fc\u30b9\u30c6\u30a3\u30af\u30b9\u304c\u8936\u66f2\u9762A<\/i> 2<\/sub> = \u03a3 1<\/sup> ( f<\/i> )<\/span> \u3001\u30ae\u30e3\u30b6\u30fc\u30e9\u30a4\u30f3A<\/i> 3<\/sub> = \u03a3 1.1<\/sup> ( f<\/i> )<\/span> \u3001\u304a\u3088\u3073\u5c3e\u70b9 d’round \u3067\u69cb\u6210\u3055\u308c\u3066\u3044\u308b\u3053\u3068\u3092\u8aac\u660e\u3057\u307e\u3059\u3002 A<\/i> 4<\/sub> = \u03a3 1,1,1<\/sup> ( f<\/i> )<\/span> \u3002\u3057\u304b\u3057\u3001\u305d\u308c\u306f\u3078\u305d\u306e\u5b58\u5728\u3092\u8aac\u660e\u3059\u308b\u3082\u306e\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002

$$ {\\{D_4^+,D_4^-\\}=\\Sigma^{2}(f)} $$<\/div>\u4e00\u822c\u7406\u8ad6\u3067\u306f\u3001\u3053\u308c\u306f\u5171\u6b21\u5143 4 \u3067\u3059\u3002<\/p>

Thom-Boardman \u30af\u30e9\u30b9\u306b\u3088\u308b\u82db\u6027\u70b9\u306e\u7279\u6027\u8a55\u4fa1\u306b\u3088\u308a\u3001\u307b\u3068\u3093\u3069\u306e\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3\u3067\u52b9\u679c\u7684\u306a\u8a08\u7b97\u304c\u53ef\u80fd\u306b\u306a\u308a\u307e\u3059\u3002<\/p>

\u03a3I(f)\u306e\u8a08\u7b97\u4f8b<\/h2>

\u901a\u5e38\u30dd\u30a4\u30f3\u30c8<\/span><\/h3>

\u6a5f\u80fd\u3055\u305b\u307e\u3057\u3087\u3046

$$ {f: R \\rightarrow R} $$<\/div> \u3001 f<\/i> ( x<\/i> )= x<\/i><\/span> \u3002\u305d\u306e\u5c0e\u95a2\u6570\u306f<\/a><\/span>\u6c7a\u3057\u3066\u30bc\u30ed\u306b\u306f\u306a\u308a\u307e\u305b\u3093\u3002\u95a2\u6570f \u306b<\/i><\/span>\u306f\u6b63\u5247\u70b9\u306e\u307f\u304c\u3042\u308a\u307e\u3059: \u03a3 0<\/sup> ( f<\/i> ) = R\u3002<\/i><\/span><\/p>

\u6298\u308a\u76ee<\/span><\/h3>

\u6a5f\u80fd\u3055\u305b\u307e\u3057\u3087\u3046

$$ {f: R \\rightarrow R} $$<\/div> \u3001 f<\/i> (<\/sup> x<\/i> )= x2<\/i><\/span> \u3002\u305d\u306e\u5c0e\u95a2\u6570\u306f2 x<\/i><\/span>\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u79c1\u305f\u3061\u306f
$$ {\\Sigma^0(f)=R\\backslash \\{0\\}} $$<\/div>\u304a\u3088\u3073\u03a3 1<\/sup> ( f<\/i> ) = {0}<\/span> \u3002\u5358\u4e00\u306e\u81e8\u754c\u70b9\u306f\u30d5\u30a9\u30fc\u30eb\u30c9<\/i>\u3068\u547c\u3070\u308c\u307e\u3059\u3002<\/p>
\n\"<\/figure>

\u3057\u304b\u3081\u3063\u9762<\/span><\/h3>

\u3069\u3061\u3089\u304b\u306e\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3

$$ {f: R^2 \\rightarrow R^2} $$<\/div> \u3001
$$ {f(x_1,x_2)=(y_1,y_2)=(x_1^3+x_1 x_2,x_2)} $$<\/div> \u3002\u30e4\u30b3\u30d3\u30a2\u30f3\u884c\u5217\u5f0f\u3092\u8a08\u7b97\u3057\u307e\u3059
$$ {J(x_1,x_2)=\\det \\partial (y_1,y_2)\/\\partial (x_1,x_2)} $$<\/div> \u3002\u79c1\u305f\u3061\u306f\u898b\u3064\u3051\u307e\u3059
$$ {J(x_1,x_2)=3x_1^2 +x_2} $$<\/div> \u3002\u6b21\u306b\u3001 \u03a3 1<\/sup> ( f<\/i> ) \u306f<\/span>\u3001 J<\/i> ( x<\/i> 1<\/sub> , x<\/i> 2<\/sub> ) = 0 \u3068<\/span>\u66f8\u304f\u3053\u3068\u306b\u3088\u3063\u3066\u53d6\u5f97\u3055\u308c\u307e\u3059\u3002
$$ {x_2=-3x_1^2} $$<\/div> \u3002\u305d\u308c\u306f\u305f\u3068\u3048\u8a71\u3067\u3059\u3002 \u03a31.1<\/sup> ( f<\/i> )<\/span>\u3092\u6c42\u3081\u308b\u306b\u306f\u3001 f<\/i><\/span>\u306e\u5236\u7d04g \u3092<\/i><\/span>\u03a31<\/sup> ( f<\/i> )<\/span>\u306b\u66f8\u304d\u8fbc\u3080\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u7b49\u4fa1\u6027\u306e\u4f7f\u7528
$$ {x_2=-3x_1^2} $$<\/div> \u3001\u79c1\u305f\u3061\u306f\u898b\u3064\u3051\u307e\u3059
$$ {g: R \\rightarrow R^2} $$<\/div> \u3001
$$ {g(x_1)=(-2x_1^3,-3x_1^2)} $$<\/div> \u3002\u6d3e\u751f\u95a2\u6570
$$ {g'(x_1)=(-6x_1^2,-6x_1)} $$<\/div>\u539f\u70b9\u3067\u306e\u307f\u30ad\u30e3\u30f3\u30bb\u30eb\u3055\u308c\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001 \u03a3 1.1<\/sup> ( f<\/i> ) = (0.0)<\/span>\u3068\u306a\u308a\u307e\u3059\u3002\u5143\u306e\u7279\u7570\u70b9\u306f<\/a><\/span>france<\/i>\u3068\u547c\u3070\u308c\u307e\u3059\u3002<\/p>

\u30c0\u30d6\u30c6\u30fc\u30eb<\/span><\/h3>

\u3069\u3061\u3089\u304b\u306e\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3

$$ {f: R^3 \\rightarrow R^3} $$<\/div> \u3001
$$ {f(x_1,x_2,x_3)=(y_1,y_2,y_3)=(x_1^4+x_1^2 x_2 +x_1 x_3,x_2,x_3)} $$<\/div> \u3002\u30e4\u30b3\u30d3\u30a2\u30f3\u306e\u6c7a\u5b9a\u5b50
$$ {J(x_1,x_2,x_3)=\\det \\partial (y_1,y_2,y_3)\/\\partial (x_1,x_2,x_3)} $$<\/div>\u66f8\u304b\u308c\u3066\u3044\u307e\u3059
$$ {J(x_1,x_2,x_3)=4 x_1^3 + 2 x_1 x_2 +x_3} $$<\/div> \u3002\u6b21\u306b\u3001 \u03a3 1<\/sup> ( f<\/i> ) \u306f<\/span>\u3001 J<\/i> ( x<\/i> 1<\/sub> , x<\/i> 2<\/sub> , x<\/i> 3<\/sub> ) = 0 \u3068<\/span>\u66f8\u304f\u3053\u3068\u306b\u3088\u3063\u3066\u53d6\u5f97\u3055\u308c\u307e\u3059\u3002
$$ {x_3=-4x_1^3-2x_1 x_2} $$<\/div> \u3002 R<\/i> 3<\/sup><\/span>\u306e\u6b63\u66f2\u9762\u3067\u3059\u3002 \u03a31.1<\/sup> ( f<\/i> )<\/span>\u3092\u6c42\u3081\u308b\u306b\u306f\u3001 f<\/i><\/span>\u306e\u5236\u7d04g \u3092<\/i><\/span>\u03a31<\/sup> ( f<\/i> )<\/span>\u306b\u66f8\u304d\u8fbc\u3080\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u7b49\u4fa1\u6027\u306e\u4f7f\u7528
$$ {x_3=-4x_1^3-2x_1 x_2} $$<\/div> \u3001\u79c1\u305f\u3061\u306f\u898b\u3064\u3051\u307e\u3059
$$ {g: R^2 \\rightarrow R^3} $$<\/div> \u3001
$$ {g(x_1,x_2)=(-3x_1^4-x_1^2 x_2,x_2,-4x_1^3-2x_1 x_2)} $$<\/div> \u3002\u5c0e\u51fa\u3055\u308c\u305f\u884c\u5217\u3092\u8a08\u7b97\u3057\u307e\u3059
$$ {g'(x_1,x_2)= \\partial (y_1,y_2,y_3)\/\\partial (x_1,x_2)} $$<\/div> \u3002 \u03a3 1.1<\/sup> ( f<\/i> ) \u3092<\/span>\u7279\u5fb4\u4ed8\u3051\u308b\u6761\u4ef6\u306f\u3001\u6b21\u6570 2 \u306e 3 \u3064\u306e\u30de\u30a4\u30ca\u30fc\u3092\u30ad\u30e3\u30f3\u30bb\u30eb\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u5f97\u3089\u308c\u307e\u3059\u3002
$$ {x_2=-6x_1^2} $$<\/div> \u3002\u3057\u305f\u304c\u3063\u3066\u3001 \u03a3 1.1<\/sup> ( f<\/i> )<\/span>\u306fR<\/i> 3<\/sup><\/span>\u306e\u6b63\u898f\u66f2\u7dda<\/a><\/span>\u3067\u3042\u308a\u3001\u6b21\u306e\u65b9\u7a0b\u5f0f<\/a><\/span>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002
$$ {x_2=-6x_1^2} $$<\/div> \u3001
$$ {x_3=8x_1^3} $$<\/div> \u3002\u3053\u306e\u65b9\u7a0b\u5f0f\u3092\u4f7f\u7528\u3059\u308b\u3068\u3001 f<\/i><\/span>\u306e\u5236\u9650h \u3092<\/i><\/span>\u03a3 1.1<\/sup> ( f<\/i> )<\/span>\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002
$$ {h(x_1)=(3x_1^4,-6x_1^2,8x_1^3)} $$<\/div> \u3002\u6b21\u306b\u3001 h<\/i> ‘( x<\/i> 1<\/sub> )<\/span>\u304c\u30bc\u30ed\u3001\u3064\u307e\u308ax<\/i> 1<\/sub> = 0<\/span>\u3067\u3042\u308b\u3068\u66f8\u304f\u3053\u3068\u3067\u3001 \u03a3 1,1,1<\/sup> ( f<\/i> )<\/span>\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u539f\u70b9\u306e\u7279\u7570\u70b9\u306f\u30c0\u30d6\u30c6\u30fc\u30eb<\/i>\u3068\u547c\u3070\u308c\u307e\u3059\u3002<\/p>

\u304a\u3078\u305d<\/span><\/h3>

\u3069\u3061\u3089\u304b\u306e\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3

$$ {f: R^4 \\rightarrow R^4} $$<\/div> \u3001
$$ {f(x_1,x_2,x_3,x_4)=(y_1,y_2,y_3,y_4)= (x_1,x_2,x_3^2\\pm x_4^2 +x_1 x_3 + x_2 x_4, x_3 x_4)} $$<\/div> \u3002\u4ee5\u524d\u306e\u3088\u3046\u306b\u03a3 1<\/sup> ( f<\/i> )<\/span>\u3068\u03a3 1.1<\/sup> ( f<\/i> )<\/span>\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002 \u03a3 2<\/sup> ( f<\/i> ) \u3092<\/span>\u898b\u3064\u3051\u308b\u305f\u3081\u306b\u3001\u30e4\u30b3\u30d3\u884c\u5217\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002
$$ {\\partial (y_1,y_2,y_3,y_4)\/\\partial (x_1,x_2,x_3,x_4)} $$<\/div> \u3002\u6700\u521d\u306e 2 \u3064\u306e\u5217\u30d9\u30af\u30c8\u30eb\u304c
$$ {\\partial y_i\/\\partial x_1} $$<\/div>\u305d\u3057\u3066
$$ {\\partial y_i\/\\partial x_2} $$<\/div>\u30d9\u30af\u30c8\u30eb\u5e73\u9762\u3092\u5f62\u6210\u3057\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u4ed6\u306e 2 \u3064\u306e\u5217\u30d9\u30af\u30c8\u30eb\u304c\u3053\u306e\u5e73\u9762\u306b\u542b\u307e\u308c\u3001\u6b21\u6570 3 \u306e\u30de\u30a4\u30ca\u30fc\u884c\u5217\u5f0f\u304c 8 \u56de\u30ad\u30e3\u30f3\u30bb\u30eb\u3055\u308c\u308b\u3068\u66f8\u304d\u307e\u3059\u3002\u6700\u7d42\u7684\u306b\u03a3 2<\/sup> ( f<\/i> ) = (0,0,0,0)<\/span>\u304c\u898b\u3064\u304b\u308a\u307e\u3059\u3002\u539f\u70b9\u306e\u7279\u7570\u70b9\u306f\u81cd\u3068\u547c\u3070\u308c\u307e\u3059\u3002 +<\/span>\u8a18\u53f7\u306e\u5834\u5408\u306f\u53cc\u66f2\u72b6\u81cd<\/i>\u3001 –<\/span>\u8a18\u53f7\u306e\u5834\u5408\u306f\u6955\u5186\u72b6\u81cd\u3067\u3059<\/i>\u3002\u53cc\u66f2\u72b6\u304a\u3088\u3073\u6955\u5186\u72b6\u306e\u3078\u305d\u306f\u3001\u540c\u3058 Thom-Boardman \u30af\u30e9\u30b9\u03a32<\/sup> ( f<\/i> )<\/span>\u306b\u5c5e\u3057\u307e\u3059\u3002<\/p>
\n\"<\/figure>

\u30db\u30a4\u30c3\u30c8\u30cb\u30fc\u30fb\u30b1\u30a4\u30ea\u30fc\u306e\u5098<\/span><\/h3>

\u3069\u3061\u3089\u304b\u306e\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3

$$ {f: R^2 \\rightarrow R^3} $$<\/div> \u3001
$$ {f(x_1,x_2)=(y_1,y_2,y_3)=(x_1,x_1x_2,x_2^2)} $$<\/div> \u3002\u30e4\u30b3\u30d3\u884c\u5217\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002 <\/p>

$$ {\\dfrac{\\partial (y_1,y_2,y_3)}{\\partial (x_1,x_2)} = \\begin{pmatrix} 1 & 0 \\\\ x_2 & x_1 \\\\ 0 & 2 x_2 \\end{pmatrix} } $$<\/div> \u3002<\/p>

\u03a3 1<\/sup> ( f<\/i> ) \u306f<\/span>\u3001\u6b21\u6570 2 \u306e 3 \u3064\u306e\u30de\u30a4\u30ca\u30fc\u3092\u30ad\u30e3\u30f3\u30bb\u30eb\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u53d6\u5f97\u3055\u308c\u307e\u3059\u3002\u3064\u307e\u308a\u3001 x<\/i> 1<\/sub> = 0<\/span> \u3001 2 x<\/i> 2<\/sub> = 0<\/span> \u3001

$$ {2x_2^2=0} $$<\/div> \u3002\u3057\u305f\u304c\u3063\u3066\u3001 \u03a3 1<\/sup> ( f<\/i> ) = (0,0)<\/span>\u3068\u306a\u308a\u307e\u3059\u3002\u5143\u306e\u7279\u7570\u70b9\u306f\u30db\u30a4\u30c3\u30c8\u30cb\u30fc\u30fb\u30b1\u30a4\u30ea\u30fc\u5098<\/i>\u3068\u547c\u3070\u308c\u307e\u3059\u3002<\/p><\/div>
\n\"<\/figure>

\u53c2\u8003\u8cc7\u6599<\/h2>
  1. Classe \u2013 catalan<\/a><\/li>
  2. Classe \u2013 anglais<\/a><\/li>
  3. Classe \u2013 italien<\/a><\/li>
  4. Classe \u2013 portugais<\/a><\/li>
  5. \u062a\u0639\u0631\u064a\u0641 \u2013 arabe<\/a><\/li>
  6. T\u0259rif (m\u0259ntiq) \u2013 azerba\u00efdjanais<\/a><\/li><\/ol><\/div>\n
    \n

    \n Thom-Boardman \u30af\u30e9\u30b9 – \u5b9a\u7fa9\u30fb\u95a2\u9023\u52d5\u753b\n <\/h2>\n
    \n \n
    \n
    \n