Первоначальные понятия симплектической геометрии

```&Iuml;&aring;&eth;&acirc;&icirc;&iacute;&agrave;&divide;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&yuml; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &atilde;&aring;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&egrave;
&Oacute;&divide;&aring;&aacute;&iacute;&icirc;&aring; &iuml;&icirc;&ntilde;&icirc;&aacute;&egrave;&aring; &auml;&euml;&yuml;
&ntilde;&ograve;&oacute;&auml;&aring;&iacute;&ograve;&icirc;&acirc;-&igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&ecirc;&icirc;&acirc;
&Atilde;&icirc;&otilde;&igrave;&agrave;&iacute; &Agrave;.&Acirc;.
&Ntilde;&agrave;&eth;&agrave;&ograve;&icirc;&acirc; 2009
2
&Ntilde;&icirc;&auml;&aring;&eth;&aelig;&agrave;&iacute;&egrave;&aring;
&Acirc;&acirc;&aring;&auml;&aring;&iacute;&egrave;&aring;
1. &Iuml;&icirc;&euml;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&aring; &ocirc;&icirc;&eth;&igrave;&ucirc; &acirc; &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&igrave;&aring;&eth;&iacute;&icirc;&igrave; &acirc;&aring;&ugrave;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&igrave; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&igrave;
&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring;
2. &Agrave;&euml;&atilde;&aring;&aacute;&eth;&agrave; &acirc;&iacute;&aring;&oslash;&iacute;&egrave;&otilde; &ocirc;&icirc;&eth;&igrave;.
3. &Ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc;.
3.1. &Iuml;&eth;&egrave;&igrave;&aring;&eth;&ucirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&otilde; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&otilde; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;.
&Ntilde;&iuml;&egrave;&ntilde;&icirc;&ecirc; &euml;&egrave;&ograve;&aring;&eth;&agrave;&ograve;&oacute;&eth;&ucirc;
3
3
6
7
10
12
3
&Acirc;&acirc;&aring;&auml;&aring;&iacute;&egrave;&aring;
&Ecirc;&agrave;&ecirc; &yacute;&ograve;&icirc; &ntilde;&euml;&aring;&auml;&oacute;&aring;&ograve; &egrave;&ccedil; &iacute;&agrave;&ccedil;&acirc;&agrave;&iacute;&egrave;&yuml;, &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc;
&iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&aring;&ograve; &ntilde;&icirc;&aacute;&icirc;&eacute; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &ntilde; &auml;&icirc;&iuml;&icirc;&euml;&iacute;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&eacute; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&icirc;&eacute;, &iuml;&icirc;&auml;&icirc;&aacute;&iacute;&icirc;
&ograve;&icirc;&igrave;&oacute;, &ecirc;&agrave;&ecirc; &aring;&acirc;&ecirc;&euml;&egrave;&auml;&icirc;&acirc;&icirc; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc;&igrave; &ntilde;
&ccedil;&agrave;&auml;&agrave;&iacute;&iacute;&ucirc;&igrave; &acirc; &iacute;&aring;&igrave; &ntilde;&ecirc;&agrave;&euml;&yuml;&eth;&iacute;&ucirc;&igrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&aring;&igrave; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&icirc;&acirc; - &ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&eacute; (&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;
&icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&iacute;&icirc;&eacute;) &iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&icirc;&eacute; &aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&icirc;&eacute;. &Acirc; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&atilde;&icirc;
&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; &yacute;&ograve;&agrave; &auml;&icirc;&iuml;&icirc;&euml;&iacute;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&agrave;&yuml; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&agrave; &ccedil;&agrave;&auml;&agrave;&aring;&ograve;&ntilde;&yuml; &ecirc;&icirc;&ntilde;&icirc;&ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&eacute;
&iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&icirc;&eacute; &aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&icirc;&eacute;. &Acirc; &ntilde;&acirc;&yuml;&ccedil;&egrave; &ntilde; &yacute;&ograve;&egrave;&igrave; &iacute;&agrave;&iuml;&icirc;&igrave;&iacute;&egrave;&igrave; &divide;&egrave;&ograve;&agrave;&ograve;&aring;&euml;&yuml;&igrave;
(&ntilde;&ograve;&oacute;&auml;&aring;&iacute;&ograve;&agrave;&igrave;&igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&ecirc;&agrave;&igrave;) &iacute;&aring;&icirc;&aacute;&otilde;&icirc;&auml;&egrave;&igrave;&ucirc;&aring; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&ucirc; &acirc;&iacute;&aring;&oslash;&iacute;&aring;&eacute; &agrave;&euml;&atilde;&aring;&aacute;&eth;&ucirc; (&ccedil;&iacute;&agrave;&ecirc;&icirc;&igrave;&ntilde;&ograve;&acirc;&icirc; &ntilde;
&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&eacute; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc; &icirc;&aacute;&ugrave;&aring;&igrave; &ecirc;&oacute;&eth;&ntilde;&aring; &Atilde;&aring;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&yuml; &egrave; &agrave;&euml;&atilde;&aring;&aacute;&eth;&agrave;) &iuml;&eth;&egrave;&igrave;&aring;&iacute;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc; &ecirc;
&acirc;&iacute;&aring;&oslash;&iacute;&egrave;&igrave; &ocirc;&icirc;&eth;&igrave;&agrave;&igrave; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave;.
&Iacute;&agrave;&ntilde;&ograve;&icirc;&yuml;&ugrave;&aring;&aring; &oacute;&divide;&aring;&aacute;&iacute;&icirc;&aring; &iuml;&icirc;&ntilde;&icirc;&aacute;&egrave;&aring; &iuml;&eth;&aring;&auml;&iacute;&agrave;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&ograve;&ntilde;&yuml; &auml;&euml;&yuml; &ntilde;&ograve;&oacute;&auml;&aring;&iacute;&ograve;&icirc;&acirc;&igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&ecirc;&icirc;&acirc;,
&iuml;&egrave;&oslash;&oacute;&ugrave;&egrave;&otilde; &ecirc;&oacute;&eth;&ntilde;&icirc;&acirc;&ucirc;&aring; &egrave; &auml;&egrave;&iuml;&euml;&icirc;&igrave;&iacute;&ucirc;&aring; &eth;&agrave;&aacute;&icirc;&ograve;&ucirc;, &ntilde;&acirc;&yuml;&ccedil;&agrave;&iacute;&iacute;&ucirc;&aring; &ntilde; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &atilde;&aring;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&aring;&eacute;
&egrave; &aring;&aring; &iuml;&eth;&egrave;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&yuml;&igrave;&egrave;.
1. &Iuml;&icirc;&euml;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&aring; &ocirc;&icirc;&eth;&igrave;&ucirc; &acirc; &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&igrave;&aring;&eth;&iacute;&icirc;&igrave; &acirc;&aring;&ugrave;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&igrave; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&igrave;
&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring;
&Iuml;&oacute;&ntilde;&ograve;&uuml; Vn = V - &acirc;&aring;&ugrave;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&aring; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave; n. &Iuml;&oacute;&ntilde;&ograve;&uuml; &auml;&agrave;&euml;&aring;&aring;
p
V &times; . . . &times; V = X V - &auml;&aring;&ecirc;&agrave;&eth;&ograve;&icirc;&acirc;&agrave; &ntilde;&ograve;&aring;&iuml;&aring;&iacute;&uuml; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; V &egrave; R &iuml;&icirc;&euml;&aring; &acirc;&aring;&ugrave;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&ucirc;&otilde;
p
&divide;&egrave;&ntilde;&aring;&euml;. &Icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; X V → R, &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&icirc; &ecirc;&agrave;&aelig;&auml;&icirc;&igrave;&oacute; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&oacute; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml;
&iuml;&icirc;&euml;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&icirc;&eacute; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; p &iacute;&agrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; V . &Igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &acirc;&ntilde;&aring;&otilde; &iuml;&icirc;&euml;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&otilde;
&ocirc;&icirc;&eth;&igrave; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; p &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&ograve; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; Lp &icirc;&ograve;&iacute;&icirc;&ntilde;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc; &aring;&ntilde;&ograve;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&ucirc;&otilde;
(&ecirc;&agrave;&ecirc;&egrave;&otilde; &egrave;&igrave;&aring;&iacute;&iacute;&icirc;?) &icirc;&iuml;&aring;&eth;&agrave;&ouml;&egrave;&eacute; &ntilde;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&yuml; &egrave; &oacute;&igrave;&iacute;&icirc;&aelig;&aring;&iacute;&egrave;&yuml; &iacute;&agrave; &acirc;&aring;&ugrave;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&ucirc;&aring; &divide;&egrave;&ntilde;&euml;&agrave;. &Acirc;
&divide;&agrave;&ntilde;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; L1 &aring;&ntilde;&ograve;&uuml; &iacute;&egrave;&divide;&ograve;&icirc; &egrave;&iacute;&icirc;&aring;, &ecirc;&agrave;&ecirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; V ∗ &ntilde;&icirc;&iuml;&eth;&yuml;&aelig;&aring;&iacute;&iacute;&icirc;&aring; &ecirc;
&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&igrave;&oacute; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&oacute; V .
&Acirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&aring; L &acirc;&ntilde;&aring;&otilde; &iuml;&icirc;&euml;&egrave;&euml;&aring;&eacute;&iacute;&ucirc;&otilde; &ocirc;&icirc;&eth;&igrave; &acirc;&acirc;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &icirc;&iuml;&aring;&eth;&agrave;&ouml;&egrave;&yuml; &ograve;&aring;&iacute;&ccedil;&icirc;&eth;&iacute;&icirc;&atilde;&icirc; &oacute;&igrave;&iacute;&icirc;&aelig;&aring;&iacute;&egrave;&yuml;,
&icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&yuml;&aring;&igrave;&agrave;&yuml; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;. &Aring;&ntilde;&euml;&egrave; α ∈ Lp , β ∈ Lq , &ograve;&icirc; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve; &ograve;&aring;&iacute;&ccedil;&icirc;&eth;&iacute;&icirc;&atilde;&icirc;
&oacute;&igrave;&iacute;&icirc;&aelig;&aring;&iacute;&egrave;&yuml; &iacute;&agrave;&auml; &iacute;&egrave;&igrave;&egrave;, &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&igrave;&ucirc;&eacute; α ⊗ β &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &iuml;&icirc;&euml;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&icirc;&eacute; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;
p + q:
α ⊗ β(v1 , . . . , vp , vp+1 , . . . , vp+q ) = α(v1 , . . . , vp ) &middot; β(vp+1 , . . . , vp+q ),
(1.1)
&atilde;&auml;&aring; vi &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; V . &Iacute;&aring;&iuml;&icirc;&ntilde;&eth;&aring;&auml;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc; &iuml;&eth;&icirc;&acirc;&aring;&eth;&yuml;&aring;&ograve;&ntilde;&yuml; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&icirc;&ntilde;&ograve;&uuml;
&ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&otilde; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc; &yacute;&ograve;&icirc;&eacute; &icirc;&iuml;&aring;&eth;&agrave;&ouml;&egrave;&egrave;. &Iuml;&oacute;&ntilde;&ograve;&uuml; α ∈ Lp , β ∈ Lq , γ ∈ Lr . &Ograve;&icirc;&atilde;&auml;&agrave;
(α ⊗ β) ⊗ γ = α ⊗ (β ⊗ γ), k(α ⊗ β) = (kα) ⊗ β = α ⊗ (kβ),
(α1 + α2 ) ⊗ γ = α1 ⊗ γ + α2 ⊗ γ, γ ⊗ (α1 + α2 ) = γ ⊗ α1 + γ ⊗ α2 .
&Iuml;&eth;&aring;&auml;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring; 1.1 &Aring;&ntilde;&euml;&egrave; dim V = n, &ograve;&icirc; dim Lp = np .
(1.2)
4
&Auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;, &iuml;&oacute;&ntilde;&ograve;&uuml; (εi ) &ecirc;&agrave;&ecirc;&icirc;&eacute;-&iacute;&egrave;&aacute;&oacute;&auml;&uuml; &aacute;&agrave;&ccedil;&egrave;&ntilde; &acirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; V ∗ . &Ograve;&icirc;&atilde;&auml;&agrave;
&oacute;&iuml;&icirc;&eth;&yuml;&auml;&icirc;&divide;&aring;&iacute;&iacute;&agrave;&yuml; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;&agrave; &ocirc;&icirc;&eth;&igrave; (εi1 ⊗ . . . ⊗ εip ) &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; p, &atilde;&auml;&aring; 1 ≤ ik ≤ n &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&ograve;
&aacute;&agrave;&ccedil;&egrave;&ntilde; &acirc; Lp , &ntilde;&icirc;&ntilde;&ograve;&icirc;&yuml;&ugrave;&egrave;&eacute; &egrave;&ccedil; np &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;.
&Iuml;&oacute;&ntilde;&ograve;&uuml; Sp &atilde;&eth;&oacute;&iuml;&iuml;&agrave; &iuml;&icirc;&auml;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&icirc;&ecirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; p. &Icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&egrave;&igrave; &aring;&aring; &auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&aring; &iacute;&agrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring;
p
L &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;. &Iuml;&oacute;&ntilde;&ograve;&uuml; s ∈ Sp &egrave; α ∈ Lp . &Ograve;&icirc;&atilde;&auml;&agrave;
(sα)(v1 , . . . , vp ) = α(vs−1 (1) , . . . , vs−1 (1) ).
(1.3)
&Iacute;&aring;&iuml;&icirc;&ntilde;&eth;&aring;&auml;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc; &iuml;&eth;&icirc;&acirc;&aring;&eth;&yuml;&aring;&ograve;&ntilde;&yuml;, &divide;&ograve;&icirc;
(s2 ◦ s1 )α = s2 (s1 α),
(1.4)
s(l1 α1 + l2 α2 ) = l1 (sα1 ) + l2 (sα2 ).
(1.5)
&egrave;
&Acirc; &ecirc;&agrave;&aelig;&auml;&icirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; Lp &acirc;&ucirc;&auml;&aring;&euml;&thorn;&thorn;&ograve;&ntilde;&yuml; 2 &acirc;&agrave;&aelig;&iacute;&ucirc;&otilde; &ecirc;&euml;&agrave;&ntilde;&ntilde;&agrave; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;.
&Icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; 1.2. &Iuml;&icirc;&euml;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&agrave;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave; α ∈ Lp &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&eacute;, &aring;&ntilde;&euml;&egrave;
&auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&eacute; &iuml;&icirc;&auml;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&ecirc;&egrave; s ∈ Sp
sα = α
(1.6)
&egrave; &ecirc;&icirc;&ntilde;&icirc;&ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&eacute;, &aring;&ntilde;&euml;&egrave;
sα = (signs)α.
(1.7)
&Igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &acirc;&ntilde;&aring;&otilde; &ecirc;&icirc;&ntilde;&icirc;&ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&ucirc;&otilde; &ocirc;&icirc;&eth;&igrave; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; p &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&ograve; &iuml;&icirc;&auml;&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &acirc; Lp ,
&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&ograve;&ntilde;&yuml; Ap .
&Iuml;&eth;&egrave;&igrave;&aring;&eth;&icirc;&igrave; &ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&eacute; &aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&ucirc; &ntilde;&euml;&oacute;&aelig;&egrave;&ograve; &ntilde;&ecirc;&agrave;&euml;&yuml;&eth;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&aring; &acirc;
&ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&iacute;&icirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; Rn , &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &acirc; &ecirc;&agrave;&iacute;&icirc;&iacute;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;&aring; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve; &ccedil;&agrave;&auml;&agrave;&aring;&ograve;&ntilde;&yuml;
&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&icirc;&eacute;
n
X
α(x̄, ȳ) =
xi y i , &atilde;&auml;&aring; x̄ = (x1 , . . . , xn ).
i=1
&Iuml;&eth;&egrave;&igrave;&aring;&eth;&icirc;&igrave; &ecirc;&icirc;&ntilde;&icirc;&ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&ucirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; n &acirc; &yacute;&ograve;&icirc;&igrave; &aelig;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml;
&icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&egrave;&ograve;&aring;&euml;&uuml;
&macr;
&macr;
&macr; x1 . . . x n &macr;
&macr; 1
1&macr;
&macr;
&macr;
δ(x̄1 , . . . , x̄n ) = &macr;. . . . . . . . . . .&macr; .
&macr;
&macr; 1
&macr;xn . . . xnn &macr;
&Ccedil;&agrave;&igrave;&aring;&divide;&agrave;&iacute;&egrave;&aring;. &Ecirc;&icirc;&ntilde;&icirc;&ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&ucirc;&aring; &ocirc;&icirc;&eth;&igrave;&ucirc; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve; &aring;&ugrave;&aring; &agrave;&iacute;&ograve;&egrave;&ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&ucirc;&igrave;&egrave;,
&ccedil;&iacute;&agrave;&ecirc;&icirc;&iuml;&aring;&eth;&aring;&igrave;&aring;&iacute;&iacute;&ucirc;&igrave;&egrave; &egrave;&euml;&egrave; &acirc;&iacute;&aring;&oslash;&iacute;&egrave;&igrave;&egrave;.
&Auml;&euml;&yuml; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc;&aacute;&ucirc; &iuml;&icirc;&euml;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&agrave;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave; &aacute;&ucirc;&euml;&agrave; &ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&eacute;, &iacute;&aring;&icirc;&aacute;&otilde;&icirc;&auml;&egrave;&igrave;&icirc; &egrave;
&auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc;, &divide;&ograve;&icirc;&aacute;&ucirc; &icirc;&iacute;&agrave; &iacute;&aring; &egrave;&ccedil;&igrave;&aring;&iacute;&yuml;&euml;&agrave;&ntilde;&uuml; &iuml;&eth;&egrave; &iuml;&aring;&eth;&aring;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&ecirc;&aring; &euml;&thorn;&aacute;&icirc;&eacute; &iuml;&agrave;&eth;&ucirc; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;,
&egrave; &ecirc;&icirc;&ntilde;&icirc;&ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&eacute;, &iacute;&aring;&icirc;&aacute;&otilde;&icirc;&auml;&egrave;&igrave;&icirc; &egrave; &auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc;, &divide;&ograve;&icirc;&aacute;&ucirc; &yacute;&ograve;&agrave; &ocirc;&icirc;&eth;&igrave;&agrave; &igrave;&aring;&iacute;&yuml;&euml;&agrave; &ccedil;&iacute;&agrave;&ecirc; &iuml;&eth;&egrave;
&iuml;&aring;&eth;&aring;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&ecirc;&aring; &euml;&thorn;&aacute;&icirc;&eacute; &iuml;&agrave;&eth;&ucirc; &aring;&aring; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;. &Aacute;&icirc;&euml;&aring;&aring; &ograve;&icirc;&atilde;&icirc;, &ocirc;&icirc;&eth;&igrave;&agrave; α &ecirc;&icirc;&ntilde;&icirc;&ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&agrave;, &aring;&ntilde;&euml;&egrave;
&icirc;&iacute;&agrave; &icirc;&aacute;&eth;&agrave;&ugrave;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc; &iacute;&oacute;&euml;&uuml; &iuml;&eth;&egrave; &euml;&thorn;&aacute;&ucirc;&otilde; &auml;&acirc;&oacute;&otilde; &icirc;&auml;&egrave;&iacute;&agrave;&ecirc;&icirc;&acirc;&ucirc;&otilde; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&agrave;&otilde;. &Ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;, &yacute;&ograve;&icirc; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&aring;
&yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &egrave; &iacute;&aring;&icirc;&aacute;&otilde;&icirc;&auml;&egrave;&igrave;&ucirc;&igrave;.
5
&Acirc; &auml;&agrave;&euml;&uuml;&iacute;&aring;&eacute;&oslash;&aring;&igrave; &icirc;&ecirc;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&ucirc;&igrave; &ograve;&agrave;&ecirc; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&igrave;&ucirc;&eacute; &icirc;&iuml;&aring;&eth;&agrave;&ograve;&icirc;&eth; &agrave;&euml;&uuml;&ograve;&aring;&eth;&iacute;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml;
(&agrave;&euml;&uuml;&ograve;&aring;&eth;&iacute;&agrave;&ograve;&icirc;&eth;) Alt, &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&eacute; &iuml;&aring;&eth;&aring;&acirc;&icirc;&auml;&egrave;&ograve; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&oacute;&thorn; &iuml;&icirc;&euml;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&oacute;&thorn; &ocirc;&icirc;&eth;&igrave;&oacute; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;
p &acirc;&icirc; &acirc;&iacute;&aring;&oslash;&iacute;&thorn;&thorn; &ocirc;&icirc;&eth;&igrave;&oacute; &ograve;&icirc;&atilde;&icirc; &aelig;&aring; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;. &Agrave;&euml;&uuml;&ograve;&aring;&eth;&iacute;&agrave;&ograve;&icirc;&eth; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&eacute;
&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&icirc;&eacute;
X
Alt(α) =
(signs)sα.
(1.8)
s∈ Sp
&Icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&icirc;, &iacute;&agrave;&auml;&icirc; &iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve;&icirc;&igrave; &auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&yuml; &agrave;&euml;&uuml;&ograve;&aring;&eth;&iacute;&agrave;&ograve;&icirc;&eth;&agrave; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &acirc;&iacute;&aring;&oslash;&iacute;&yuml;&yuml;
&ocirc;&icirc;&eth;&igrave;&agrave;. &szlig;&ntilde;&iacute;&icirc;, &divide;&ograve;&icirc; Alt(α) &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; p. &Acirc;&icirc;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&oacute;&aring;&igrave;&ntilde;&yuml; &auml;&agrave;&euml;&aring;&aring;
&icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring;&igrave; &acirc;&iacute;&aring;&oslash;&iacute;&aring;&eacute; &ocirc;&icirc;&eth;&igrave;&ucirc; (1.7), &agrave; &egrave;&igrave;&aring;&iacute;&iacute;&icirc;, &iuml;&icirc;&ecirc;&agrave;&aelig;&aring;&igrave;, &divide;&ograve;&icirc; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&eacute; &iuml;&icirc;&auml;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&ecirc;&egrave;
t ∈ Sp &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&aring;
t(Alt(α)) = signt(Alt(α)).
(1.9)
&Egrave;&ograve;&agrave;&ecirc;, &eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &acirc;&ucirc;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; t(Alt(α)) &egrave; &ccedil;&agrave;&eacute;&igrave;&aring;&igrave;&ntilde;&yuml; &aring;&atilde;&icirc; &iuml;&eth;&aring;&icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&acirc;&agrave;&iacute;&egrave;&aring;&igrave;.
X
X
t(Alt(α)) = t(
(signs)sα) =
(signs)t(sα) =
= (signt)2
X
s∈ Sp
s∈ Sp
(signs)(t ◦ s)α = signt
s∈ Sp
= signt
X
X
(signt)(signs)(t ◦ s)α =
s∈ Sp
(sign(t ◦ s))(t ◦ s)α.
s∈ Sp
&Oacute;&divide;&egrave;&ograve;&ucirc;&acirc;&agrave;&yuml;, &divide;&ograve;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &acirc;&ntilde;&aring;&otilde; &iuml;&icirc;&auml;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&icirc;&ecirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; p &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&ograve; &atilde;&eth;&oacute;&iuml;&iuml;&oacute; &egrave;,
&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;, &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&eacute; t ◦ s &iuml;&eth;&egrave; &ocirc;&egrave;&ecirc;&ntilde;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&icirc;&igrave; t &egrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&icirc;&igrave;
s &iuml;&eth;&icirc;&aacute;&aring;&atilde;&agrave;&aring;&ograve; &acirc;&ntilde;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &iuml;&icirc;&auml;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&icirc;&ecirc; Sp &iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&igrave;
X
t(Alt(α)) = signt
(signr)rα = signtAlt(α).
r∈ Sp
&Ccedil;&agrave;&igrave;&aring;&ograve;&egrave;&igrave;, &divide;&ograve;&icirc; &aring;&ntilde;&euml;&egrave; α ∈ Ap &acirc;&iacute;&aring;&oslash;&iacute;&yuml;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; p, &ograve;&icirc;
Alt(α) = p!α.
(1.12)
&ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &aring;&ugrave;&aring; &icirc;&auml;&iacute;&icirc; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc; &agrave;&euml;&uuml;&ograve;&aring;&eth;&iacute;&agrave;&ograve;&icirc;&eth;&agrave;, &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &icirc;&ecirc;&agrave;&aelig;&aring;&ograve;&ntilde;&yuml; &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&ucirc;&igrave; &acirc;
&auml;&agrave;&euml;&uuml;&iacute;&aring;&eacute;&oslash;&aring;&igrave;.
&Iuml;&eth;&aring;&auml;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring; 1.3.&Iuml;&oacute;&ntilde;&ograve;&uuml; α ∈ Lp &egrave; β ∈ Lq . &Ograve;&icirc;&atilde;&auml;&agrave;
Alt((Alt(α)) ⊗ β) = p!Alt(α ⊗ β).
P
Alt(Alt(α) ⊗ β)
=
sign(s)s(Alt(α)
(1.13)
&Auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;,
⊗ β)
=
s∈ Sp+q
P
P
sign(s)s(
sign(t)(tα ⊗ β). &times;&ograve;&icirc;&aacute;&ucirc; &ntilde;&icirc;&acirc;&aring;&eth;&oslash;&egrave;&ograve;&uuml; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&aring; &iuml;&eth;&aring;&icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml;,
s∈ Sp+q
t∈ S p
&acirc;&acirc;&aring;&auml;&aring;&igrave; &acirc; &eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&aring;&iacute;&egrave;&aring; &iuml;&icirc;&auml;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&ecirc;&egrave; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; p + q &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&atilde;&icirc; &acirc;&egrave;&auml;&agrave;
&Atilde;
!
1
.
.
.
p
p
+
1
.
.
.
p
+
q
t0 =
.
t(1) . . . t(p) p + 1 . . . p + q
6
P P
&Ograve;&icirc;&atilde;&auml;&agrave; tα ⊗ β = t0 (α ⊗ β) &egrave; Alt((Alt(α)) ⊗ β) =
sign(s ◦ t0 )s(t0 (α ⊗ β)) =
0
s∈ Sp+q t
P P
P P
=
sign(s ◦ t0 )s ◦ t0 (α ⊗ β) =
sign(r)r(α ⊗ β) = p!Alt(α ⊗ β).
t0 s∈ Sp+q
t0 r∈ Sp+q
&Acirc; &auml;&agrave;&euml;&uuml;&iacute;&aring;&eacute;&oslash;&aring;&igrave; &iacute;&agrave;&ntilde; &aacute;&oacute;&auml;&oacute;&ograve; &egrave;&iacute;&ograve;&aring;&eth;&aring;&ntilde;&icirc;&acirc;&agrave;&ograve;&uuml; &acirc; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&icirc;&igrave; &aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&aring; &ocirc;&icirc;&eth;&igrave;&ucirc;. &Yacute;&ograve;&egrave; &ocirc;&icirc;&eth;&igrave;&ucirc;
&otilde;&agrave;&eth;&agrave;&ecirc;&ograve;&aring;&eth;&egrave;&ccedil;&oacute;&thorn;&ograve;&ntilde;&yuml; &ntilde;&acirc;&icirc;&egrave;&igrave; &eth;&agrave;&iacute;&atilde;&icirc;&igrave;. &Aring;&ntilde;&euml;&egrave; (ei ) &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&eacute; &aacute;&agrave;&ccedil;&egrave;&ntilde; &acirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; Vn , &agrave; εi
&aring;&igrave;&oacute; &acirc;&ccedil;&agrave;&egrave;&igrave;&iacute;&ucirc;&eacute; &aacute;&agrave;&ccedil;&egrave;&ntilde; &acirc; &ntilde;&icirc;&iuml;&eth;&yuml;&aelig;&aring;&iacute;&iacute;&icirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; V ∗ , &ograve;&icirc; &ecirc;&agrave;&ecirc; &aacute;&ucirc;&euml;&icirc; &iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&iacute;&icirc; &acirc;&ucirc;&oslash;&aring;
&auml;&euml;&yuml; α ∈ L2
α = αij εi ⊗ εj , αij = α(ei , ej ).
&Igrave;&agrave;&ograve;&eth;&egrave;&ouml;&agrave; &ecirc;&icirc;&yacute;&ocirc;&ocirc;&egrave;&ouml;&egrave;&aring;&iacute;&ograve;&icirc;&acirc; &eth;&agrave;&ccedil;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&yuml; (αij ) &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &igrave;&agrave;&ograve;&eth;&egrave;&ouml;&aring;&eacute; &ocirc;&icirc;&eth;&igrave;&ucirc;, &ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml;,
&ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;, &ccedil;&agrave;&acirc;&egrave;&ntilde;&egrave;&ograve; &icirc;&ograve; &acirc;&ucirc;&aacute;&icirc;&eth;&agrave; &aacute;&agrave;&ccedil;&egrave;&ntilde;&agrave; (ei ). &Icirc;&auml;&iacute;&agrave;&ecirc;&icirc;, &ecirc;&agrave;&ecirc; &otilde;&icirc;&eth;&icirc;&oslash;&icirc; &egrave;&ccedil;&acirc;&aring;&ntilde;&ograve;&iacute;&icirc;, &eth;&agrave;&iacute;&atilde; &yacute;&ograve;&icirc;&eacute;
&igrave;&agrave;&ograve;&eth;&egrave;&ouml;&ucirc; &icirc;&ntilde;&ograve;&agrave;&aring;&ograve;&ntilde;&yuml; &iuml;&icirc;&ntilde;&ograve;&icirc;&yuml;&iacute;&iacute;&ucirc;&igrave;. &Icirc;&iacute; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &eth;&agrave;&iacute;&atilde;&icirc;&igrave; &ocirc;&icirc;&eth;&igrave;&ucirc;. &Aring;&ntilde;&euml;&egrave; &eth;&agrave;&iacute;&atilde; &igrave;&agrave;&ograve;&eth;&egrave;&ouml;&ucirc;
&eth;&agrave;&acirc;&aring;&iacute; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; n, &ograve;&icirc; &auml;&agrave;&iacute;&iacute;&agrave;&yuml; &aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&agrave;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml;
&iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&icirc;&eacute;.
2. &Agrave;&euml;&atilde;&aring;&aacute;&eth;&agrave; &acirc;&iacute;&aring;&oslash;&iacute;&egrave;&otilde; &ocirc;&icirc;&eth;&igrave;.
&Iuml;&oacute;&ntilde;&ograve;&uuml; Ap &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &acirc;&ntilde;&aring;&otilde; &acirc;&iacute;&aring;&oslash;&iacute;&egrave;&otilde; &ocirc;&icirc;&eth;&igrave; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; p &egrave; &iuml;&oacute;&ntilde;&ograve;&uuml; A
&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &acirc;&ntilde;&aring;&otilde; &acirc;&iacute;&aring;&oslash;&iacute;&egrave;&otilde; &ocirc;&icirc;&eth;&igrave;. &Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &acirc; n-&igrave;&aring;&eth;&iacute;&icirc;&igrave; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; V
&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; Ap &iuml;&eth;&egrave; p &gt; n &egrave;&igrave;&aring;&thorn;&ograve; &iacute;&oacute;&euml;&aring;&acirc;&oacute;&thorn; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&uuml;, &ograve;&icirc; &igrave;&icirc;&aelig;&iacute;&icirc; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&egrave;&ograve;&uuml; &iuml;&eth;&yuml;&igrave;&oacute;&thorn;
&ntilde;&oacute;&igrave;&igrave;&oacute; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&otilde; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;, &icirc;&ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&euml;&yuml;&yuml; &icirc;&auml;&iacute;&icirc;&eth;&icirc;&auml;&iacute;&oacute;&thorn; &ocirc;&icirc;&eth;&igrave;&oacute; α ∈ Ap &ntilde; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&igrave;
&iuml;&eth;&yuml;&igrave;&icirc;&eacute; &ntilde;&oacute;&igrave;&igrave;&ucirc; &acirc;&egrave;&auml;&agrave; (0, . . . , 0, α , 0, . . .). &Acirc; &yacute;&ograve;&icirc;&eacute; &iuml;&eth;&yuml;&igrave;&icirc;&eacute; &ntilde;&oacute;&igrave;&igrave;&aring; &igrave;&icirc;&aelig;&iacute;&icirc; &acirc;&acirc;&aring;&ntilde;&ograve;&egrave; &iacute;&icirc;&acirc;&oacute;&thorn;
p
&icirc;&iuml;&aring;&eth;&agrave;&ouml;&egrave;&thorn; &ograve;&agrave;&ecirc; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&igrave;&icirc;&atilde;&icirc; &acirc;&iacute;&aring;&oslash;&iacute;&aring;&atilde;&icirc; &oacute;&igrave;&iacute;&icirc;&aelig;&aring;&iacute;&egrave;&yuml; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;. &Iuml;&eth;&egrave; &yacute;&ograve;&icirc;&igrave; &igrave;&ucirc;
&iuml;&icirc;&euml;&oacute;&divide;&egrave;&igrave; &agrave;&euml;&atilde;&aring;&aacute;&eth;&oacute; &acirc;&iacute;&aring;&oslash;&iacute;&egrave;&otilde; &ocirc;&icirc;&eth;&igrave;.
&Iuml;&oacute;&ntilde;&ograve;&uuml; α ∈ Ap &egrave; β ∈ Aq . &Egrave;&otilde; &acirc;&iacute;&aring;&oslash;&iacute;&egrave;&igrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&aring;&igrave; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave; α∧β ∈ Ap+q ,
&icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&yuml;&aring;&igrave;&agrave;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&icirc;&eacute;
1 1
α∧β =
Alt(α ⊗ β).
(2.1)
p! q!
&Iacute;&agrave; &icirc;&ntilde;&iacute;&icirc;&acirc;&agrave;&iacute;&egrave;&egrave; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&yuml; &ograve;&aring;&iacute;&ccedil;&icirc;&eth;&iacute;&icirc;&atilde;&icirc; &oacute;&igrave;&iacute;&icirc;&aelig;&aring;&iacute;&egrave;&yuml; &egrave; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&agrave; &agrave;&euml;&uuml;&ograve;&aring;&eth;&iacute;&agrave;&ograve;&icirc;&eth;&agrave;
&ccedil;&agrave;&ecirc;&euml;&thorn;&divide;&agrave;&aring;&igrave;, &divide;&ograve;&icirc; α ∧ β &acirc;&iacute;&aring;&oslash;&iacute;&yuml;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; p + q . &Iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;, &aring;&ntilde;&euml;&egrave; α, β &ocirc;&icirc;&eth;&igrave;&ucirc;
&iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;, &ograve;&icirc;
&macr;
&macr;
&macr;α(v) α(w)&macr;
1 1 X
&macr;
&macr;
α ∧ β(v, w) =
sign(s)(α ⊗ β)(v, w) = &macr;
&macr;.
&macr;β(v) β(w)&macr;
1! 1! s∈ S
2
&Egrave;&ccedil; &eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&aring;&iacute;&iacute;&icirc;&eacute; &acirc; &icirc;&aacute;&ugrave;&aring;&igrave; &ecirc;&oacute;&eth;&ntilde;&aring; &acirc;&iacute;&aring;&oslash;&iacute;&aring;&eacute; &agrave;&euml;&atilde;&aring;&aacute;&eth;&ucirc; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ograve; &ocirc;&icirc;&eth;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&agrave;
&acirc;&iacute;&aring;&oslash;&iacute;&aring;&atilde;&icirc; &oacute;&igrave;&iacute;&icirc;&aelig;&aring;&iacute;&egrave;&yuml;. &Aring;&ntilde;&euml;&egrave; α ∈ Ap , β ∈ Aq , γ ∈ Aγ , &ograve;&icirc;
1)
2)
3)
4)
(α ∧ β) ∧ γ = α ∧ (β ∧ γ),
(α1 + α2 ) ∧ β = α1 ∧ β + α2 ∧ β
k(α ∧ β) = (kα) ∧ β = α ∧ (kβ),
α ∧ β = (−1)pq β ∧ α.
(2.2)
7
&Acirc;&iacute;&aring;&oslash;&iacute;&yuml;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave; α &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &eth;&agrave;&ccedil;&euml;&icirc;&aelig;&egrave;&igrave;&icirc;&eacute;, &aring;&ntilde;&euml;&egrave; &aring;&aring; &igrave;&icirc;&aelig;&iacute;&icirc; &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&egrave;&ograve;&uuml; &ecirc;&agrave;&ecirc; &acirc;&iacute;&aring;&oslash;&iacute;&aring;&aring;
&iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&aring; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&otilde; &ocirc;&icirc;&eth;&igrave;: α = α1 ∧ α2 ∧ . . . ∧ αp .
&Iuml;&eth;&aring;&auml;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring; 2.1. &Aring;&ntilde;&euml;&egrave; α &eth;&agrave;&ccedil;&euml;&icirc;&aelig;&egrave;&igrave;&agrave;&yuml; p &ocirc;&icirc;&eth;&igrave;&agrave;, &ograve;&icirc;
α(v1 , . . . , vp ) = α1 ∧ . . . ∧ αp (v1 , . . . , vp ) = det(αi (vj )).
(2.3)
&Ntilde;&iacute;&agrave;&divide;&agrave;&euml;&agrave; &auml;&icirc;&ecirc;&agrave;&aelig;&aring;&igrave; &euml;&aring;&igrave;&igrave;&oacute;
α1 ∧ α2 ∧ . . . ∧ αp = Alt(α1 ⊗ α2 ⊗ . . . ⊗ αp ).
(2.4)
&Auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;, &oacute;&divide;&egrave;&ograve;&ucirc;&acirc;&agrave;&yuml; &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&icirc; α1 ∧ α2 = Alt(α1 ⊗ α2 ) &egrave; &iuml;&eth;&aring;&auml;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring; (1.3),
&acirc;&icirc;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&oacute;&aring;&igrave;&ntilde;&yuml; &igrave;&aring;&ograve;&icirc;&auml;&icirc;&igrave; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &egrave;&iacute;&auml;&oacute;&ecirc;&ouml;&egrave;&egrave;: α1 ∧ α2 ∧ . . . ∧ αp−1 ∧ αp = (α1 ∧
1
1
Alt(Alt(α1 ⊗ . . . ⊗ αp−1 ) ⊗ αp ) =
. . . ∧ αp−1 ) ∧ αp = (Alt(α1 ⊗ . . . ⊗ αp−1 )) ∧ αp =
(p − 1)! 1!
Alt((α1 ⊗ . . . ⊗ αp−1 ) ⊗ αp ) = Alt(α1 ⊗ . . . ⊗ αp−1 ⊗ αp ).
&Iuml;&aring;&eth;&aring;&eacute;&auml;&aring;&igrave; &ecirc; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&oacute; &iuml;&eth;&aring;&auml;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&yuml;:
α1 ∧ . . . ∧ αp (v1 , . . . , vp )
=
P
Alt(α1 ⊗ α2 ⊗ . . . ⊗ αp )(v1 , . . . , vp ) =
sign(s)s(α1 ⊗ . . . ⊗ αp )(v1 , . . . , vp ) =
s∈ Sp
P
sign(s)α1 (vs(1) ) &middot; . . . &middot; αp (vs(p) ) = det(αi (vj )), i, j = 1 . . . p.
s∈ Sp
&Ograve;&aring;&iuml;&aring;&eth;&uuml; &igrave;&ucirc; &igrave;&icirc;&aelig;&aring;&igrave; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;
&Iuml;&eth;&aring;&auml;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring; 2.2. &Aring;&ntilde;&euml;&egrave; dim V = n &egrave; p ≤ n, &ograve;&icirc; dim Ap = Cnp .
&Auml;&euml;&yuml; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave; &iuml;&eth;&aring;&auml;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&yuml; &iuml;&icirc;&ecirc;&agrave;&aelig;&aring;&igrave;, &divide;&ograve;&icirc; &oacute;&iuml;&icirc;&eth;&yuml;&auml;&icirc;&divide;&aring;&iacute;&iacute;&agrave;&yuml; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;&agrave;
i1
(ε ∧ . . . ∧ εip ), i1 &lt; i2 &lt; . . . &lt; ip &acirc;&iacute;&aring;&oslash;&iacute;&egrave;&otilde; p-&ocirc;&icirc;&eth;&igrave; &ntilde;&icirc;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&aring;&ograve; &aacute;&agrave;&ccedil;&egrave;&ntilde; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave;
Ap , &atilde;&auml;&aring; (εi ) &aacute;&agrave;&ccedil;&egrave;&ntilde; &acirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; V ∗ .
&Auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;, &iuml;&oacute;&ntilde;&ograve;&uuml; li1 ...ip εi1 ∧ . . . ∧ εip = 0. &Icirc;&ograve;&ntilde;&thorn;&auml;&agrave; &ntilde;&euml;&aring;&auml;&oacute;&aring;&ograve;
(li1 ...ip εi1 ∧ . . . ∧ εip )(ej1 , . . . , ejp ) = 0,
&atilde;&auml;&aring; j1 &lt; . . . &lt; jp
&auml;&euml;&yuml; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&icirc;&acirc; &aacute;&agrave;&ccedil;&egrave;&ntilde;&agrave; (ek ) &acirc;&ccedil;&agrave;&egrave;&igrave;&iacute;&icirc;&atilde;&icirc; &ecirc; &aacute;&agrave;&ccedil;&egrave;&ntilde;&oacute; (εi ). &Iacute;&icirc; &ntilde;&icirc;&atilde;&euml;&agrave;&ntilde;&iacute;&icirc; &iuml;&eth;&aring;&auml;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&thorn; (2.2)
&macr;
&macr;
&macr;εj1 (e ) . . . εj1 (e )&macr;
j1
jp &macr;
&macr;
&macr;
&macr;
εi1 ∧ . . . ∧ εip (ej1 , . . . , ejp ) = &macr;. . . . . . . . . . . . . . . . . . . .&macr; .
&macr; j
&macr;
&macr;ε p (ej1 ) . . . εjp (ejp )&macr;
&Iuml;&eth;&egrave; &yacute;&ograve;&icirc;&igrave; &iuml;&icirc;&euml;&oacute;&divide;&aring;&iacute;&iacute;&ucirc;&eacute; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&egrave;&ograve;&aring;&euml;&uuml; &eth;&agrave;&acirc;&aring;&iacute; &iacute;&oacute;&euml;&thorn; &iuml;&eth;&egrave; (j1 . . . jp ) 6= (i1 . . . ip ) &egrave; &aring;&auml;&egrave;&iacute;&egrave;&ouml;&aring;
&acirc; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&iacute;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring;. &Icirc;&ograve;&ntilde;&thorn;&auml;&agrave; &ntilde;&euml;&aring;&auml;&oacute;&aring;&ograve;, &divide;&ograve;&icirc; li1 ...ip = 0.
&Ograve;&aring;&iuml;&aring;&eth;&uuml; &iuml;&icirc;&ecirc;&agrave;&aelig;&aring;&igrave;, &divide;&ograve;&icirc; &acirc;&ntilde;&yuml;&ecirc;&oacute;&thorn; p &ocirc;&icirc;&eth;&igrave;&oacute; &igrave;&icirc;&aelig;&iacute;&icirc; &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&egrave;&ograve;&uuml; &acirc; &acirc;&egrave;&auml;&aring; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute;
&ecirc;&icirc;&igrave;&aacute;&egrave;&iacute;&agrave;&ouml;&egrave;&egrave; &ocirc;&icirc;&eth;&igrave; εi1 ∧ . . . ∧ εip . &Auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;, &aring;&ntilde;&euml;&egrave; α ∈ Ap , &ograve;&icirc; α = α(ei1 , . . . , eip )εi1 ∧
. . . ∧ εip , i1 &lt; . . . &lt; ip .
&Auml;&agrave;&euml;&aring;&aring; &icirc;&ntilde;&ograve;&agrave;&aring;&ograve;&ntilde;&yuml; &ccedil;&agrave;&igrave;&aring;&ograve;&egrave;&ograve;&uuml;, &divide;&ograve;&icirc; &acirc; &eth;&agrave;&ntilde;&ntilde;&igrave;&agrave;&ograve;&eth;&egrave;&acirc;&agrave;&aring;&igrave;&icirc;&igrave; &aacute;&agrave;&ccedil;&egrave;&ntilde;&aring; Cnp &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;.
3. &Ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc;.
&Icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; 3.1. &Ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&igrave; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc;&igrave; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml;
&divide;&aring;&ograve;&iacute;&icirc;&igrave;&aring;&eth;&iacute;&icirc;&aring; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; Vn , &acirc; &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&igrave; &ccedil;&agrave;&auml;&agrave;&iacute;&agrave; &acirc;&iacute;&aring;&oslash;&iacute;&yuml;&yuml; &iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&agrave;&yuml;
&aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&agrave;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave; ω .
8
&Acirc;&acirc;&egrave;&auml;&oacute; &aacute;&icirc;&euml;&uuml;&oslash;&icirc;&atilde;&icirc; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&yuml;, &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &acirc; &auml;&agrave;&iacute;&iacute;&icirc;&igrave; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&egrave; &egrave;&igrave;&aring;&aring;&ograve; &iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml;
&ocirc;&icirc;&eth;&igrave;&ucirc; ω , &iacute;&agrave;&iuml;&icirc;&igrave;&iacute;&egrave;&igrave; &icirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; &egrave;&ccedil;&acirc;&aring;&ntilde;&ograve;&iacute;&ucirc;&otilde; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml;&otilde; &iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&egrave; &aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute;
&acirc;&iacute;&aring;&oslash;&iacute;&aring;&eacute; &ocirc;&icirc;&eth;&igrave;&ucirc;.
1. &Aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&agrave;&yuml; &acirc;&iacute;&aring;&oslash;&iacute;&yuml;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave; &iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&agrave; &ograve;&icirc;&atilde;&auml;&agrave; &egrave; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &ograve;&icirc;&atilde;&auml;&agrave;, &ecirc;&icirc;&atilde;&auml;&agrave; &auml;&euml;&yuml;
&euml;&thorn;&aacute;&icirc;&atilde;&icirc; &iacute;&aring;&iacute;&oacute;&euml;&aring;&acirc;&icirc;&atilde;&icirc; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave; v &iacute;&agrave;&eacute;&auml;&aring;&ograve;&ntilde;&yuml; &acirc;&aring;&ecirc;&ograve;&icirc;&eth; w &ograve;&agrave;&ecirc;&icirc;&eacute;, &divide;&ograve;&icirc; ω(v, w) 6= 0.
2. &Aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&agrave;&yuml; &acirc;&iacute;&aring;&oslash;&iacute;&yuml;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave; α &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&icirc;&eacute; &ograve;&icirc;&atilde;&auml;&agrave; &egrave; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &ograve;&icirc;&atilde;&auml;&agrave;,
&ecirc;&icirc;&atilde;&auml;&agrave; &atilde;&icirc;&igrave;&icirc;&igrave;&icirc;&eth;&ocirc;&egrave;&ccedil;&igrave; ϕα : V → V ∗ , &ccedil;&agrave;&auml;&agrave;&acirc;&agrave;&aring;&igrave;&ucirc;&eacute; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&icirc;&eacute; (ϕα (v))(x) = α(x, v)
&yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &egrave;&ccedil;&icirc;&igrave;&icirc;&eth;&ocirc;&egrave;&ccedil;&igrave;&icirc;&igrave;.
&Iuml;&icirc; &agrave;&iacute;&agrave;&euml;&icirc;&atilde;&egrave;&egrave; &ntilde; &aring;&acirc;&ecirc;&euml;&egrave;&auml;&icirc;&acirc;&ucirc;&igrave; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc;&igrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave;
ω &ograve;&eth;&agrave;&ecirc;&ograve;&oacute;&aring;&ograve;&ntilde;&yuml; &ecirc;&agrave;&ecirc; &aacute;&egrave;&iacute;&agrave;&eth;&iacute;&agrave;&yuml; &icirc;&iuml;&aring;&eth;&agrave;&ouml;&egrave;&yuml; &iacute;&agrave;&auml; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave;&igrave;&egrave;, &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&igrave;&agrave;&yuml; &ecirc;&icirc;&ntilde;&ucirc;&igrave; &egrave;&euml;&egrave;
&ecirc;&icirc;&ntilde;&ecirc;&agrave;&euml;&yuml;&eth;&iacute;&ucirc;&igrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&aring;&igrave;:
hv, wi = ω(v, w).
(3.1)
&Icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&icirc;, &egrave;&ccedil; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc; (1.3) &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ograve; &ocirc;&icirc;&eth;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&agrave; &yacute;&ograve;&icirc;&eacute; &icirc;&iuml;&aring;&eth;&agrave;&ouml;&egrave;&egrave;:
1)
2)
3)
4)
hv, wi = −hw, vi;
hv1 + v2 , wi = hv1 , wi + hv2 , wi;
hkv, wi = khv, wi, k ∈ R;
(∀v 6= 0)(∃w) hv, wi 6= 0.
(3.2)
&Aring;&ntilde;&euml;&egrave; hv, wi
=
0 &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&ucirc; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &ecirc;&icirc;&ntilde;&icirc;&icirc;&eth;&ograve;&icirc;&atilde;&icirc;&iacute;&agrave;&euml;&uuml;&iacute;&ucirc;&igrave;&egrave;. &Iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;,
&auml;&acirc;&agrave; &ecirc;&icirc;&euml;&euml;&egrave;&iacute;&aring;&agrave;&eth;&iacute;&ucirc;&otilde; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave; &ecirc;&icirc;&ntilde;&icirc;&icirc;&eth;&ograve;&icirc;&atilde;&icirc;&iacute;&agrave;&euml;&uuml;&iacute;&ucirc; &acirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&igrave; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&igrave;
&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring;.
&Acirc;&agrave;&aelig;&iacute;&ucirc;&igrave; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc;&igrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&atilde;&icirc; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &iacute;&agrave;&euml;&egrave;&divide;&egrave;&aring;
&egrave;&ccedil;&icirc;&igrave;&icirc;&eth;&ocirc;&egrave;&ccedil;&igrave;&agrave; ϕω : V → V ∗ , &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&eacute; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve; &ecirc;&agrave;&iacute;&icirc;&iacute;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&igrave; &egrave; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&thorn;&ograve; &ntilde;&egrave;&igrave;&acirc;&icirc;&euml;&icirc;&igrave;
Iω . &Acirc; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&agrave;&otilde; &ecirc;&agrave;&iacute;&icirc;&iacute;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&eacute; &egrave;&ccedil;&icirc;&igrave;&icirc;&eth;&ocirc;&egrave;&ccedil;&igrave; &ccedil;&agrave;&auml;&agrave;&aring;&ograve;&ntilde;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&icirc;&eacute;
(Iω (v))k = v i ωki = ωki v i
(3.3)
&icirc;&ograve;&iacute;&icirc;&ntilde;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &aacute;&agrave;&ccedil;&egrave;&ntilde;&agrave; (ek ) &egrave; &aring;&atilde;&icirc; &ntilde;&icirc;&iuml;&eth;&yuml;&aelig;&aring;&iacute;&iacute;&icirc;&atilde;&icirc;.
&Auml;&eth;&oacute;&atilde;&egrave;&igrave; &acirc;&agrave;&aelig;&iacute;&ucirc;&igrave; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc;&igrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&atilde;&icirc; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave;,
&agrave;&iacute;&agrave;&euml;&icirc;&atilde;&egrave;&divide;&iacute;&ucirc;&igrave; &aring;&acirc;&ecirc;&euml;&egrave;&auml;&icirc;&acirc;&oacute; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&oacute; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &acirc; &iacute;&aring;&igrave; &ecirc;&euml;&agrave;&ntilde;&ntilde;&agrave;
&ntilde;&iuml;&aring;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &aacute;&agrave;&ccedil;&egrave;&ntilde;&icirc;&acirc;. &Yacute;&ograve;&icirc;&ograve; &ocirc;&agrave;&ecirc;&ograve; &icirc;&iuml;&egrave;&eth;&agrave;&aring;&ograve;&ntilde;&yuml; &iacute;&agrave; &egrave;&ccedil;&acirc;&aring;&ntilde;&ograve;&iacute;&oacute;&thorn;, &ograve;&agrave;&ecirc; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&igrave;&oacute;&thorn;
&euml;&egrave;&iacute;&aring;&eacute;&iacute;&oacute;&thorn; &ograve;&aring;&icirc;&eth;&aring;&igrave;&oacute; &Auml;&agrave;&eth;&aacute;&oacute;.
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; 3.2 (&Euml;&egrave;&iacute;&aring;&eacute;&iacute;&agrave;&yuml; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Auml;&agrave;&eth;&aacute;&oacute;) &Auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&eacute; &acirc;&iacute;&aring;&oslash;&iacute;&aring;&eacute; &aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&ucirc;
α &acirc; n-&igrave;&aring;&eth;&iacute;&icirc;&igrave; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; V &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ouml;&aring;&euml;&icirc;&aring; &iacute;&aring;&icirc;&ograve;&eth;&egrave;&ouml;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&aring; &divide;&egrave;&ntilde;&euml;&icirc;
s, 2s ≤ n &egrave; &ograve;&agrave;&ecirc;&icirc;&eacute; &aacute;&agrave;&ccedil;&egrave;&ntilde; (e1 , e2 , . . . , en ) &acirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; V , &divide;&ograve;&icirc;
α = ε1 ∧ ε2 + ε3 ∧ ε4 + . . . + ε2s−1 ∧ ε2s ,
&atilde;&auml;&aring; εi (ek ) = δki .
(3.4)
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;. &Oacute;&divide;&egrave;&ograve;&ucirc;&acirc;&agrave;&yuml;, &divide;&ograve;&icirc; αij = α(ei , ej ) &iacute;&agrave;&igrave; &iacute;&oacute;&aelig;&iacute;&icirc; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&aring;
&ograve;&agrave;&ecirc;&icirc;&atilde;&icirc; &aacute;&agrave;&ccedil;&egrave;&ntilde;&agrave;, &auml;&euml;&yuml; &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&atilde;&icirc; he1 , e2 i = he3 , e4 i = . . . = he2s−1 , e2s i = 1, he2 , e1 i =
he4 , e3 i = . . . = he2s , e2s−1 i = −1 &egrave; &auml;&euml;&yuml; &icirc;&ntilde;&ograve;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &iuml;&agrave;&eth; &aacute;&agrave;&ccedil;&egrave;&ntilde;&iacute;&ucirc;&otilde; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&icirc;&acirc; &egrave;&otilde; &ecirc;&icirc;&ntilde;&icirc;&aring;
9
&iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&aring; &eth;&agrave;&acirc;&iacute;&icirc; &iacute;&oacute;&euml;&thorn;. &Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc; &iuml;&eth;&icirc;&acirc;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &igrave;&aring;&ograve;&icirc;&auml;&icirc;&igrave; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute;
&egrave;&iacute;&auml;&oacute;&ecirc;&ouml;&egrave;&egrave; &iuml;&icirc; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; V .
&Auml;&euml;&yuml; n = 0, 1 &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&agrave;, &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &acirc; &yacute;&ograve;&egrave;&otilde; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave;&otilde; &egrave;&igrave;&aring;&thorn;&ograve;&ntilde;&yuml;
&ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &iacute;&oacute;&euml;&aring;&acirc;&ucirc;&aring; &aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&aring; &acirc;&iacute;&aring;&oslash;&iacute;&egrave;&aring; &ocirc;&icirc;&eth;&igrave;&ucirc;. &Iuml;&oacute;&ntilde;&ograve;&uuml;, &ograve;&aring;&iuml;&aring;&eth;&uuml; n ≥ 2. &Iuml;&eth;&egrave; &yacute;&ograve;&icirc;&igrave; &aacute;&oacute;&auml;&aring;&igrave;
&iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &eth;&agrave;&ntilde;&ntilde;&igrave;&agrave;&ograve;&eth;&egrave;&acirc;&agrave;&aring;&igrave;&ucirc;&aring; &ocirc;&icirc;&eth;&igrave;&ucirc; &icirc;&ograve;&euml;&egrave;&divide;&iacute;&ucirc; &icirc;&ograve; &iacute;&oacute;&euml;&yuml;, &egrave;&aacute;&icirc; &acirc; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&iacute;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring;
&ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&icirc; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&agrave;.
&Egrave;&ograve;&agrave;&ecirc;, &iuml;&oacute;&ntilde;&ograve;&uuml; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&agrave; &auml;&euml;&yuml; &acirc;&ntilde;&aring;&otilde; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&aring;&eacute; &igrave;&aring;&iacute;&uuml;&oslash;&egrave;&otilde; n, &auml;&icirc;&ecirc;&agrave;&aelig;&aring;&igrave;
&aring;&aring; &ntilde;&iuml;&eth;&agrave;&acirc;&aring;&auml;&euml;&egrave;&acirc;&icirc;&ntilde;&ograve;&uuml; &auml;&euml;&yuml; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave; n. &Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &ocirc;&icirc;&eth;&igrave;&agrave; α 6= 0, &ograve;&icirc; &iacute;&agrave;&eacute;&auml;&oacute;&ograve;&ntilde;&yuml; &auml;&acirc;&agrave;
&iacute;&aring;&ecirc;&icirc;&euml;&euml;&egrave;&iacute;&aring;&agrave;&eth;&iacute;&ucirc;&otilde; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave; (&iuml;&eth;&egrave;&divide;&aring;&igrave;, &iuml;&aring;&eth;&acirc;&ucirc;&eacute; &egrave;&ccedil; &iacute;&egrave;&otilde; &igrave;&icirc;&aelig;&aring;&ograve; &aacute;&ucirc;&ograve;&uuml; &euml;&thorn;&aacute;&ucirc;&igrave; &icirc;&ograve;&euml;&egrave;&divide;&iacute;&ucirc;&igrave; &icirc;&ograve;
&iacute;&oacute;&euml;&yuml;) x0 , y0 &ograve;&agrave;&ecirc;&egrave;&otilde;, &divide;&ograve;&icirc; α(x0 , y0 ) 6= 0. &Acirc;&icirc;&ccedil;&uuml;&igrave;&aring;&igrave; &ograve;&icirc;&atilde;&auml;&agrave; &acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring; &iuml;&aring;&eth;&acirc;&ucirc;&otilde; &auml;&acirc;&oacute;&otilde; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&ucirc;
y0
e1 = x0 ,
e2 =
(3.5)
α(x0 , y0 )
&egrave; &ccedil;&agrave;&igrave;&aring;&ograve;&egrave;&igrave;, &divide;&ograve;&icirc; he1 , e2 i = 1.
&ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &auml;&agrave;&euml;&aring;&aring; &yuml;&auml;&eth;&agrave; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&otilde; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute; ϕα (e1 ), ϕα (e2 ). &Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc;
dim ϕα (e1 ) = dim ϕα (e2 ) = n − 1 &egrave; &yacute;&ograve;&egrave; &yuml;&auml;&eth;&agrave; &iacute;&aring; &ntilde;&icirc;&acirc;&iuml;&agrave;&auml;&agrave;&thorn;&ograve;, &ograve;&icirc; dim P =
T
dim(ϕα (e1 ) ϕα (e2 )) = n − 2. &Ecirc;&eth;&icirc;&igrave;&aring; &ograve;&icirc;&atilde;&icirc;, &iuml;&aring;&eth;&aring;&ntilde;&aring;&divide;&aring;&iacute;&egrave;&aring; &iuml;&icirc;&auml;&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; P &egrave;
&iuml;&icirc;&auml;&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; B , &iacute;&agrave;&ograve;&yuml;&iacute;&oacute;&ograve;&icirc;&atilde;&icirc; &iacute;&agrave; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave; e1 , e2 &iuml;&oacute;&ntilde;&ograve;&icirc;. &Ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;,
V = P ⊕ B.
(3.6)
&Iuml;&icirc; &iuml;&eth;&aring;&auml;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&thorn; &egrave;&iacute;&auml;&oacute;&ecirc;&ouml;&egrave;&egrave; &acirc; &iuml;&icirc;&auml;&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; P &auml;&euml;&yuml; &ocirc;&icirc;&eth;&igrave;&ucirc; α|p &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &egrave;&ntilde;&ecirc;&icirc;&igrave;&ucirc;&eacute;
&aacute;&agrave;&ccedil;&egrave;&ntilde; (e3 , e4 , . . . , e2s ). &Oacute;&divide;&egrave;&ograve;&ucirc;&acirc;&agrave;&yuml;, &divide;&ograve;&icirc; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave; e3 , e4 , . . . , e2s &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&agrave;&ograve; &yuml;&auml;&eth;&agrave;&igrave;
ϕα (e1 ), ϕα (e2 ), &iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&igrave;, &divide;&ograve;&icirc; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;&agrave; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&icirc;&acirc; (e1 , e2 , e3 , e4 , . . . , e2s ) &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&ograve; &egrave;&ntilde;&ecirc;&icirc;&igrave;&ucirc;&eacute;
&aacute;&agrave;&ccedil;&egrave;&ntilde;, &ograve;&icirc; &aring;&ntilde;&ograve;&uuml; &egrave;&igrave;&aring;&aring;&ograve; &igrave;&aring;&ntilde;&ograve;&icirc; &eth;&agrave;&ccedil;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring;
α = ε1 ∧ ε2 + ε3 ∧ ε4 + . . . + ε2s−1 ∧ ε2s .
&curren;
&Iuml;&icirc;&euml;&oacute;&divide;&aring;&iacute;&iacute;&ucirc;&eacute; &aacute;&agrave;&ccedil;&egrave;&ntilde; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve; &ecirc;&agrave;&iacute;&icirc;&iacute;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&igrave;.
&Ntilde;&euml;&aring;&auml;&ntilde;&ograve;&acirc;&egrave;&yuml; &egrave;&ccedil; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc;.
1. &Igrave;&agrave;&ograve;&eth;&egrave;&ouml;&agrave; &aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&ucirc; &acirc; &ecirc;&agrave;&iacute;&icirc;&iacute;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&igrave; &aacute;&agrave;&ccedil;&egrave;&ntilde;&aring; &egrave;&igrave;&aring;&aring;&ograve; &acirc;&egrave;&auml;


0 1 0 0 ... 0 0 ... 0 0
−1 0 0 0 . . . 0 0 . . . 0 0




 0 0 0 1 . . . 0 0 . . . 0 0


 0 0 −1 0 . . . 0 0 . . . 0 0


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


,

 0 0 0 0 . . . 0 1 . . . 0 0


 0 0 0 0 . . . −1 0 . . . 0 0


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .




 0 0 0 0 . . . 0 0 . . . 0 0
0 0 0 0 ... 0 0 ... 0 0
(3.7)
10
&atilde;&auml;&aring; &iacute;&aring;&iacute;&oacute;&euml;&aring;&acirc;&ucirc;&igrave;&egrave; &yuml;&acirc;&euml;&yuml;&thorn;&ograve;&ntilde;&yuml; &iuml;&aring;&eth;&acirc;&ucirc;&aring; 2s &ntilde;&ograve;&eth;&icirc;&divide;&aring;&ecirc; &egrave; &ntilde;&ograve;&icirc;&euml;&aacute;&ouml;&icirc;&acirc;.
2. &Acirc; &iacute;&aring;&divide;&aring;&ograve;&iacute;&icirc;&igrave;&aring;&eth;&iacute;&icirc;&igrave; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; &aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&agrave;&yuml; &acirc;&iacute;&aring;&oslash;&iacute;&yuml;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave; &iacute;&aring; &igrave;&icirc;&aelig;&aring;&ograve;
&aacute;&ucirc;&ograve;&uuml; &iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&icirc;&eacute;.
3. &Aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&agrave;&yuml; &acirc;&iacute;&aring;&oslash;&iacute;&yuml;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave; &acirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; V2m &iacute;&aring;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&agrave; &ograve;&icirc;&atilde;&auml;&agrave; &egrave; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc;
&ograve;&icirc;&atilde;&auml;&agrave;, &ecirc;&icirc;&atilde;&auml;&agrave; &aring;&aring; &acirc;&iacute;&aring;&oslash;&iacute;&yuml;&yuml; m-&agrave;&yuml; &ntilde;&ograve;&aring;&iuml;&aring;&iacute;&uuml; &iacute;&aring; &eth;&agrave;&acirc;&iacute;&agrave; &iacute;&oacute;&euml;&thorn;.
&Auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;, &aring;&ntilde;&euml;&egrave; &ocirc;&icirc;&eth;&igrave;&agrave; α &iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&agrave;, &ograve;&icirc; α = ε1 ∧ ε2 + . . . + ε2m−1 ∧ ε2m &egrave;
αm = kε1 ∧ ε2 ∧ . . . ∧ ε2m , &atilde;&auml;&aring; k &icirc;&ograve;&euml;&egrave;&divide;&iacute;&icirc; &icirc;&ograve; &iacute;&oacute;&euml;&yuml; &egrave;&ccedil;-&ccedil;&agrave; &divide;&aring;&ograve;&iacute;&icirc;&atilde;&icirc; &divide;&egrave;&ntilde;&euml;&agrave; &iuml;&aring;&eth;&aring;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&icirc;&ecirc; &iuml;&eth;&egrave;
&iuml;&eth;&aring;&icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&acirc;&agrave;&iacute;&egrave;&egrave; (ε1 ∧ ε2 + . . . + ε2m−1 ∧ ε2m )m &ecirc; &acirc;&egrave;&auml;&oacute; kε1 ∧ ε2 ∧ . . . ∧ ε2m .
&Iuml;&oacute;&ntilde;&ograve;&uuml; &ograve;&aring;&iuml;&aring;&eth;&uuml; αm 6= 0. &Iuml;&icirc;&ecirc;&agrave;&aelig;&aring;&igrave;, &divide;&ograve;&icirc; &ocirc;&icirc;&eth;&igrave;&agrave; α &iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&agrave;. &Auml;&icirc;&iuml;&oacute;&ntilde;&ograve;&egrave;&igrave; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&iacute;&icirc;&aring;,
&ograve;&icirc; &aring;&ntilde;&ograve;&uuml; ε1 ∧ ε2 + . . . + ε2s−1 ∧ ε2s , &atilde;&auml;&aring; s &lt; m. &Ograve;&icirc;&atilde;&auml;&agrave; &iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&igrave;, &divide;&ograve;&icirc; &acirc; &ecirc;&agrave;&aelig;&auml;&icirc;&igrave;
&ntilde;&euml;&agrave;&atilde;&agrave;&aring;&igrave;&icirc;&igrave; &ocirc;&icirc;&eth;&igrave;&ucirc; αm &aacute;&oacute;&auml;&oacute;&ograve; &ntilde;&icirc;&auml;&aring;&eth;&aelig;&agrave;&ograve;&uuml;&ntilde;&yuml; &icirc;&auml;&egrave;&iacute;&agrave;&ecirc;&icirc;&acirc;&ucirc;&aring; &ntilde;&icirc;&igrave;&iacute;&icirc;&aelig;&egrave;&ograve;&aring;&euml;&egrave;. &Ograve;&icirc;&atilde;&auml;&agrave; αm = 0,
&divide;&ograve;&icirc; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&ograve; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&thorn;.
&Egrave;&ograve;&agrave;&ecirc;, &acirc;&icirc; &acirc;&ntilde;&yuml;&ecirc;&icirc;&igrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&igrave; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; V2m &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve;
&ecirc;&agrave;&iacute;&icirc;&iacute;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&eacute; &aacute;&agrave;&ccedil;&egrave;&ntilde;. &Iuml;&eth;&egrave; &yacute;&ograve;&icirc;&igrave; &iuml;&aring;&eth;&acirc;&ucirc;&eacute; &acirc;&aring;&ecirc;&ograve;&icirc;&eth; &aacute;&agrave;&ccedil;&egrave;&ntilde;&agrave; &igrave;&icirc;&aelig;&aring;&ograve; &acirc;&ucirc;&aacute;&egrave;&eth;&agrave;&ograve;&uuml;&ntilde;&yuml; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&icirc;.
&Iuml;&eth;&egrave; &egrave;&ccedil;&oacute;&divide;&aring;&iacute;&egrave;&egrave; &egrave; &egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&icirc;&acirc;&agrave;&iacute;&egrave;&egrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&atilde;&icirc; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; Vn =
V2m &ecirc;&agrave;&iacute;&icirc;&iacute;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&eacute; &aacute;&agrave;&ccedil;&egrave;&ntilde; (e1 , . . . , en ), &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&iacute;&ucirc;&eacute; &acirc;&ucirc;&oslash;&aring;, &divide;&agrave;&ntilde;&ograve;&icirc; &ccedil;&agrave;&igrave;&aring;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &aacute;&agrave;&ccedil;&egrave;&ntilde;&icirc;&igrave;
(e0 1 , . . . , e0 n ) &iuml;&oacute;&ograve;&aring;&igrave; &iuml;&aring;&eth;&aring;&iacute;&oacute;&igrave;&aring;&eth;&icirc;&acirc;&ecirc;&egrave; &ograve;&agrave;&ecirc;, &divide;&ograve;&icirc;
m+1
α = 0 ε1 ∧ 0 ε
2
m+2
+ 0ε ∧ 0ε
m
+ . . . + 0ε ∧ 0ε
2m
.
(3.8)
&Yacute;&ograve;&icirc;&ograve; &aacute;&agrave;&ccedil;&egrave;&ntilde; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&igrave;. &Ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&oacute;&thorn;&ugrave;&agrave;&yuml; &aring;&igrave;&oacute; &igrave;&agrave;&ograve;&eth;&egrave;&ouml;&agrave; &ocirc;&icirc;&eth;&igrave;&ucirc;
&egrave;&igrave;&aring;&aring;&ograve; &acirc;&egrave;&auml;
&Atilde;
!
0 E
.
(3.9)
−E 0
&Acirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&igrave; &aacute;&agrave;&ccedil;&egrave;&ntilde;&aring; &ecirc;&icirc;&ntilde;&icirc;&aring; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&aring; hx, yi &acirc;&ucirc;&eth;&agrave;&aelig;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&agrave;&otilde;
&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&icirc;&eacute;
hx, yi = (x1 y 2 − x2 y 1 ) + (x3 y 4 − x4 y 3 ) + . . . + (x2m−1 y 2m − x2m y 2m−1 ).
(3.10
&Icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; 3.3 &Igrave;&agrave;&ograve;&eth;&egrave;&ouml;&agrave; &iuml;&aring;&eth;&aring;&otilde;&icirc;&auml;&agrave; &icirc;&ograve; &icirc;&auml;&iacute;&icirc;&atilde;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&atilde;&icirc; &aacute;&agrave;&ccedil;&egrave;&ntilde;&agrave; &ecirc;
&auml;&eth;&oacute;&atilde;&icirc;&igrave;&oacute; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute;. &Acirc;&ntilde;&aring; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&aring; &igrave;&agrave;&ograve;&eth;&egrave;&ouml;&ucirc; &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&thorn;&ograve;
&ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&oacute;&thorn; &atilde;&eth;&oacute;&iuml;&iuml;&oacute;.
3.1. &Iuml;&eth;&egrave;&igrave;&aring;&eth;&ucirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&otilde; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&otilde; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;. 1. &Iuml;&oacute;&ntilde;&ograve;&uuml; R2m
&acirc;&aring;&ugrave;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&aring; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &ntilde; &ecirc;&agrave;&iacute;&icirc;&iacute;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&icirc;&eacute; &egrave;
&ntilde;&ograve;&agrave;&iacute;&auml;&agrave;&eth;&ograve;&iacute;&ucirc;&igrave; &aacute;&agrave;&ccedil;&egrave;&ntilde;&icirc;&igrave; e1 = (1, 0, . . . , 0), e2 = (0, 1, . . . , 0), . . . , e2m = (0, 0, . . . , 1).
&Ntilde;&icirc;&iuml;&eth;&yuml;&aelig;&aring;&iacute;&iacute;&ucirc;&eacute; &ecirc; &iacute;&aring;&igrave;&oacute; &aacute;&agrave;&ccedil;&egrave;&ntilde; &acirc; (R2m )∗ &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&acirc;&agrave;&iacute; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&iacute;&ucirc;&igrave;&egrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml;&igrave;&egrave; ε1 =
x1 , ε2 = x2 , . . . , ε2m = x2m . &Ograve;&icirc;&atilde;&auml;&agrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&agrave; (&ocirc;&icirc;&eth;&igrave;&agrave;) &ccedil;&agrave;&auml;&agrave;&aring;&ograve;&ntilde;&yuml;
&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&icirc;&eacute;
ω = x1 ∧ x2 + x3 ∧ x4 + . . . + x2m−1 ∧ x2m ,
(3.11)
&ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml;, &icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&icirc; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &ecirc;&icirc;&ntilde;&icirc;&ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&eacute; &iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&icirc;&eacute; &aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&icirc;&eacute;
(&iuml;&eth;&egrave;&divide;&aring;&igrave; &ccedil;&agrave;&auml;&agrave;&iacute;&iacute;&icirc;&eacute; &acirc; &ecirc;&agrave;&iacute;&icirc;&iacute;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&igrave; &aacute;&agrave;&ccedil;&egrave;&ntilde;&aring;).
11
&Yacute;&ograve;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve; &ntilde;&ograve;&agrave;&iacute;&auml;&agrave;&eth;&ograve;&iacute;&ucirc;&igrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc;&igrave;
&eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave; n = 2m.
2. &Iuml;&oacute;&ntilde;&ograve;&uuml; E2 &icirc;&eth;&egrave;&aring;&iacute;&ograve;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&icirc;&aring; &auml;&acirc;&oacute;&igrave;&aring;&eth;&iacute;&icirc;&aring; &aring;&acirc;&ecirc;&euml;&egrave;&auml;&icirc;&acirc;&icirc; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &ntilde;&icirc;
&ntilde;&ecirc;&agrave;&euml;&yuml;&eth;&iacute;&ucirc;&igrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&aring;&igrave; (v, w). &Ograve;&icirc;&atilde;&auml;&agrave; &acirc; &iacute;&aring;&igrave; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml; &ocirc;&icirc;&eth;&igrave;&agrave;
ω(v, w) = (v , , w) = |v| |w| sin vw,
c
(3.12)
&atilde;&auml;&aring; v , &ograve;&agrave;&ecirc; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&igrave;&ucirc;&eacute; &iuml;&icirc;&acirc;&aring;&eth;&iacute;&oacute;&ograve;&ucirc;&eacute; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;. &Igrave;&ucirc; &acirc;&egrave;&auml;&egrave;&igrave;, &divide;&ograve;&icirc; &ecirc;&icirc;&ntilde;&ecirc;&agrave;&euml;&yuml;&eth;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&aring;
&acirc;&aring;&ecirc;&ograve;&icirc;&eth;&icirc;&acirc; hv, wi &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&yuml;&aring;&ograve; &icirc;&eth;&egrave;&aring;&iacute;&ograve;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&oacute;&thorn; &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&uuml; &iuml;&agrave;&eth;&agrave;&euml;&euml;&aring;&euml;&icirc;&atilde;&eth;&agrave;&igrave;&igrave;&agrave;,
&iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&iacute;&icirc;&atilde;&icirc; &iacute;&agrave; &yacute;&ograve;&egrave;&otilde; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave;&otilde;. &Ograve;&agrave;&ecirc;&egrave;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;, &auml;&acirc;&oacute;&igrave;&aring;&eth;&iacute;&icirc;&aring; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring;
&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; - &yacute;&ograve;&icirc; &auml;&acirc;&oacute;&igrave;&aring;&eth;&iacute;&icirc;&aring; &yacute;&ecirc;&acirc;&egrave;&ouml;&aring;&iacute;&ograve;&eth;&icirc;&agrave;&ocirc;&ocirc;&egrave;&iacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &egrave;
&iacute;&agrave;&icirc;&aacute;&icirc;&eth;&icirc;&ograve;. &Ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;, &acirc; &ntilde;&egrave;&euml;&oacute; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; ω m 6= 0, &acirc;&ntilde;&yuml;&ecirc;&icirc;&aring; 2m-&igrave;&aring;&eth;&iacute;&icirc;&aring; &ntilde;&egrave;&igrave;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring;
&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &yacute;&ecirc;&acirc;&egrave;&ouml;&aring;&iacute;&ograve;&eth;&icirc;&agrave;&ocirc;&ocirc;&egrave;&iacute;&iacute;&ucirc;&igrave;, &iacute;&icirc; &acirc; &auml;&agrave;&iacute;&iacute;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring; (m &gt; 1) &ntilde;
&auml;&icirc;&iuml;&icirc;&euml;&iacute;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&igrave;&egrave; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&agrave;&igrave;&egrave;.
&Iuml;&oacute;&ntilde;&ograve;&uuml; V &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&icirc;&aring; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &egrave; V ∗ &aring;&atilde;&icirc; &ntilde;&icirc;&iuml;&eth;&yuml;&aelig;&aring;&iacute;&iacute;&icirc;&aring;. &ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave;
&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; V ∗ ⊕ V &egrave; &ccedil;&agrave;&auml;&agrave;&auml;&egrave;&igrave; &acirc; &iacute;&aring;&igrave; &ntilde;&egrave;&igrave;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&oacute;&thorn; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&oacute; &ocirc;&icirc;&eth;&igrave;&icirc;&eacute;
Ω((α, v), (β, w)) = α(w) − β(v).
(3.13)
&Ecirc;&icirc;&ntilde;&icirc;&ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&ntilde;&ograve;&uuml; &ocirc;&icirc;&eth;&igrave;&ucirc; &icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&agrave;, &iuml;&eth;&icirc;&acirc;&aring;&eth;&egrave;&igrave; &iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml;. &Iuml;&oacute;&ntilde;&ograve;&uuml; (α, v) ∈
V ∗ ⊕V &icirc;&ograve;&euml;&egrave;&divide;&aring;&iacute; &icirc;&ograve; &iacute;&oacute;&euml;&yuml;. &Yacute;&ograve;&icirc; &ccedil;&iacute;&agrave;&divide;&egrave;&ograve;, &divide;&ograve;&icirc; &euml;&egrave;&aacute;&icirc; α 6= 0, &euml;&egrave;&aacute;&icirc; v 6= 0. &Iacute;&oacute;&aelig;&iacute;&icirc; &iacute;&agrave;&eacute;&ograve;&egrave; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;
(β, w) &ograve;&agrave;&ecirc;&icirc;&eacute;, &divide;&ograve;&icirc;&aacute;&ucirc; α(w) − β(v) 6= 0. &Aring;&ntilde;&euml;&egrave; α 6= 0, &ograve;&icirc; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &acirc;&aring;&ecirc;&ograve;&icirc;&eth; w &ograve;&agrave;&ecirc;&icirc;&eacute;, &divide;&ograve;&icirc;
α(w) 6= 0. &Ntilde; &auml;&eth;&oacute;&atilde;&icirc;&eacute; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute;&ucirc; &acirc;&icirc;&ccedil;&uuml;&igrave;&aring;&igrave; &ocirc;&icirc;&eth;&igrave;&oacute; β = 0, &ograve;&icirc;&atilde;&auml;&agrave; α(w) − β(v) 6= 0. &Iuml;&oacute;&ntilde;&ograve;&uuml;
&ograve;&aring;&iuml;&aring;&eth;&uuml; v 6= 0, &ograve;&icirc;&atilde;&auml;&agrave; &iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave; w = 0, &agrave; &acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring; &ocirc;&icirc;&eth;&igrave;&ucirc; β &eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &iuml;&icirc;&auml;&otilde;&icirc;&auml;&yuml;&ugrave;&oacute;&thorn;
&aacute;&agrave;&ccedil;&egrave;&ntilde;&iacute;&oacute;&thorn; &ocirc;&icirc;&eth;&igrave;&oacute; &egrave;&ccedil; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; V ∗ .
4. &ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; m-&igrave;&aring;&eth;&iacute;&icirc;&aring; &ecirc;&icirc;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&iacute;&icirc;&aring; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; W m &ntilde; &iuml;&icirc;&euml;&icirc;&aelig;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;
&icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&iacute;&icirc;&eacute; &yacute;&eth;&igrave;&egrave;&ograve;&icirc;&acirc;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&icirc;&eacute; α &egrave; &acirc; &iuml;&icirc;&auml;&euml;&aring;&aelig;&agrave;&ugrave;&aring;&igrave; &acirc;&aring;&ugrave;&aring;&ntilde;&ograve;&acirc;&iacute;&iacute;&icirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; V2m
&eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &igrave;&iacute;&egrave;&igrave;&oacute;&thorn; &divide;&agrave;&ntilde;&ograve;&uuml; &ocirc;&icirc;&eth;&igrave;&ucirc; α : (Imα)(x, y) = Im(α(x, y)). &Aacute;&egrave;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&ntilde;&ograve;&uuml;
&egrave; &ecirc;&icirc;&ntilde;&icirc;&ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&ntilde;&ograve;&uuml; &ocirc;&icirc;&eth;&igrave;&ucirc; &iuml;&eth;&icirc;&acirc;&aring;&eth;&yuml;&thorn;&ograve;&ntilde;&yuml; &iacute;&aring;&iuml;&icirc;&ntilde;&eth;&aring;&auml;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;. &times;&ograve;&icirc;&aacute;&ucirc; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;
&iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml; &auml;&euml;&yuml; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave; x 6= 0, &eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; Imα(x, ix) = Im(α(x, ix)) =
−Im(iα(x, x)) = −α(x, x) 6= 0. &Iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;, &acirc; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&iacute;&icirc;&igrave; &ecirc;&icirc;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&iacute;&icirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring;
C2 &ecirc;&agrave;&iacute;&icirc;&iacute;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml; &yacute;&eth;&igrave;&egrave;&ograve;&icirc;&acirc;&agrave; &ocirc;&icirc;&eth;&igrave;&agrave; &ccedil;&agrave;&auml;&agrave;&aring;&ograve;&ntilde;&yuml; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;
α(z, z 0 ) = z z̄ 0 .
&Ograve;&icirc;&atilde;&auml;&agrave;
(3.14)
&macr;
&macr;
&macr;x y &macr;
&macr;
&macr;
Im z z̄ 0 = Im(x + iy)(x0 − iy 0 ) = Im(xx0 + yy 0 + i(yx0 − xy 0 )) = − &macr; 0 0 &macr; =
&macr;x y &macr;
= −p ∧ q(z, z 0 ), &atilde;&auml;&aring; p(z) = p(x + iy) = x, q(z) = q(x + iy) = y.
5. &Iuml;&oacute;&ntilde;&ograve;&uuml; V &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &ntilde; &ocirc;&icirc;&eth;&igrave;&icirc;&eacute; ω . &Ograve;&icirc;&atilde;&auml;&agrave; &acirc; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&igrave;
&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; V ⊕ V &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&oacute;&thorn; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&oacute; &igrave;&icirc;&aelig;&iacute;&icirc; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&egrave;&ograve;&uuml; &ocirc;&icirc;&eth;&igrave;&icirc;&eacute;
Ω((x1 , x2 ), (y1 , y2 )) = ω(x1 , y1 ) − ω(x2 , y2 ).
(3.15)
12
&Iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml;: &iuml;&oacute;&ntilde;&ograve;&uuml; (x1 , x2 ) 6= 0, &iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;, x1 6= 0. &Ograve;&icirc;&atilde;&auml;&agrave; &acirc;&ucirc;&aacute;&aring;&eth;&aring;&igrave; &iuml;&agrave;&eth;&oacute; (y1 , y2 )
&ograve;&agrave;&ecirc;, &divide;&ograve;&icirc;&aacute;&ucirc; ω(x1 , y1 ) 6= 0, &agrave; y2 = 0.
&Euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &aacute;&egrave;&aring;&ecirc;&ograve;&egrave;&acirc;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; h &icirc;&auml;&iacute;&icirc;&atilde;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave;
(V1 , ω1 ) &acirc; &auml;&eth;&oacute;&atilde;&icirc;&aring; (V2 , ω2 ) &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&igrave; &egrave;&ccedil;&icirc;&igrave;&icirc;&eth;&ocirc;&egrave;&ccedil;&igrave;&icirc;&igrave;, &aring;&ntilde;&euml;&egrave;
ω2 (hx, hy) = ω1 (x, y).
(3.16)
&Iuml;&eth;&aring;&auml;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring; 3.4. &Euml;&thorn;&aacute;&ucirc;&aring; &auml;&acirc;&agrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&otilde; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; &icirc;&auml;&egrave;&iacute;&agrave;&ecirc;&icirc;&acirc;&icirc;&eacute;
&eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave; &egrave;&ccedil;&icirc;&igrave;&icirc;&eth;&ocirc;&iacute;&ucirc;.
&Auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;, &auml;&euml;&yuml; &oacute;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&euml;&aring;&iacute;&egrave;&yuml; &ograve;&agrave;&ecirc;&icirc;&atilde;&icirc; &egrave;&ccedil;&icirc;&igrave;&icirc;&eth;&ocirc;&egrave;&ccedil;&igrave;&agrave; &auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc; &acirc; &ecirc;&agrave;&aelig;&auml;&icirc;&igrave; &egrave;&ccedil;
&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc; &acirc;&ucirc;&aacute;&eth;&agrave;&ograve;&uuml; &iuml;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&igrave;&oacute; &aacute;&agrave;&ccedil;&egrave;&ntilde;&oacute;. &Acirc; &divide;&agrave;&ntilde;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave;, &acirc;&ntilde;&aring; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&aring;
&euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave; 2m &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave; &egrave;&ccedil;&icirc;&igrave;&icirc;&eth;&ocirc;&iacute;&ucirc; &ntilde;&ograve;&agrave;&iacute;&auml;&agrave;&eth;&ograve;&iacute;&icirc;&igrave;&oacute;
&ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&igrave;&oacute; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&oacute; R2m .
&Ntilde;&iuml;&egrave;&ntilde;&icirc;&ecirc; &euml;&egrave;&ograve;&aring;&eth;&agrave;&ograve;&oacute;&eth;&ucirc;
[1] &Agrave;&eth;&iacute;&icirc;&euml;&uuml;&auml; &Acirc;.&Egrave;. &Igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&aring; &igrave;&aring;&ograve;&icirc;&auml;&ucirc; &ecirc;&euml;&agrave;&ntilde;&ntilde;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &igrave;&aring;&otilde;&agrave;&iacute;&egrave;&ecirc;&egrave;. &Igrave;. &quot;&Iacute;&agrave;&oacute;&ecirc;&agrave;&quot;1989&atilde;.
[2] &Agrave;&eth;&iacute;&icirc;&euml;&uuml;&auml; &Acirc;. &Egrave;., &Atilde;&egrave;&acirc;&aring;&iacute;&ograve;&agrave;&euml;&uuml; &Agrave;.&Aacute;. &Ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml; &atilde;&aring;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&yuml;, &Ntilde;&icirc;&acirc;&eth;&aring;&igrave;&aring;&iacute;&iacute;&ucirc;&aring; &iuml;&eth;&icirc;&aacute;&euml;&aring;&igrave;&ucirc;
&igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&ecirc;&egrave;. &Ocirc;&oacute;&iacute;&auml;&agrave;&igrave;&aring;&iacute;&ograve;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &iacute;&agrave;&iuml;&eth;&agrave;&acirc;&euml;&aring;&iacute;&egrave;&yuml;. &Auml;&egrave;&iacute;&agrave;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&aring; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave;&ucirc; - 4,&Igrave;., 1985&atilde;.
[3] &Atilde;&icirc;&auml;&aacute;&egrave;&eacute;&icirc;&iacute; &Ecirc;. &Auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&agrave;&yuml; &atilde;&aring;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&yuml; &egrave; &agrave;&iacute;&agrave;&euml;&egrave;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml; &igrave;&aring;&otilde;&agrave;&iacute;&egrave;&ecirc;&agrave;. &Igrave;., &quot;&Igrave;&egrave;&eth;&quot;1973&atilde;.
[4] &Iacute;&icirc;&acirc;&egrave;&ecirc;&icirc;&acirc; &Ntilde;.&Iuml;.,&Ograve;&agrave;&eacute;&igrave;&agrave;&iacute;&icirc;&acirc; &Egrave;.&Agrave;. &Ntilde;&icirc;&acirc;&eth;&aring;&igrave;&aring;&iacute;&iacute;&ucirc;&aring; &atilde;&aring;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&aring; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&ucirc; &egrave; &iuml;&icirc;&euml;&yuml;. &Igrave;., 2005.
[5] &Iuml;&icirc;&ntilde;&ograve;&iacute;&egrave;&ecirc;&icirc;&acirc; &Igrave;.&Igrave;. &Euml;&egrave;&iacute;&aring;&eacute;&iacute;&agrave;&yuml; &agrave;&euml;&atilde;&aring;&aacute;&eth;&agrave;. &Igrave;.&quot;&Iacute;&agrave;&oacute;&ecirc;&agrave;&quot;, 1986&atilde;.
[6] &Ocirc;&icirc;&igrave;&aring;&iacute;&ecirc;&icirc; &Agrave;.&Ograve; &Ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml; &atilde;&aring;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&yuml;. &Igrave;&aring;&ograve;&icirc;&auml;&ucirc; &egrave; &iuml;&eth;&egrave;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&yuml;. &Igrave;., &Egrave;&ccedil;&auml; &Igrave;&Atilde;&Oacute;, 1988&atilde;.
```