# Лекции о неподвижных точках - Лаборатория математической

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```&Iacute;&aring;&atilde;&icirc;&ntilde;&oacute;&auml;&agrave;&eth;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&aring; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&aring; &oacute;&divide;&eth;&aring;&aelig;&auml;&aring;&iacute;&egrave;&aring;
&ETH;&icirc;&ntilde;&ntilde;&egrave;&eacute;&ntilde;&ecirc;&agrave;&yuml; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml; &oslash;&ecirc;&icirc;&euml;&agrave;
&Acirc;.&Egrave;. &Auml;&agrave;&iacute;&egrave;&euml;&icirc;&acirc;
&Euml;&aring;&ecirc;&ouml;&egrave;&egrave; &icirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&ecirc;&agrave;&otilde;
&Igrave;&icirc;&ntilde;&ecirc;&acirc;&agrave;
2006
&Auml;&agrave;&iacute;&egrave;&euml;&icirc;&acirc; &Acirc;.&Egrave;., &Euml;&aring;&ecirc;&ouml;&egrave;&egrave; &icirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&ecirc;&agrave;&otilde;.
&oslash;&ecirc;&icirc;&euml;&agrave;, &Igrave;&icirc;&ntilde;&ecirc;&acirc;&agrave;, 2006 &atilde;. 32 &ntilde;.
&ETH;&icirc;&ntilde;&ntilde;&egrave;&eacute;&ntilde;&ecirc;&agrave;&yuml; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml;
&Yacute;&ograve;&egrave; &divide;&aring;&ograve;&ucirc;&eth;&aring; &euml;&aring;&ecirc;&ouml;&egrave;&egrave;, &iuml;&icirc;&ntilde;&acirc;&yuml;&ugrave;&aring;&iacute;&iacute;&ucirc;&aring; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&igrave; &ograve;&icirc;&divide;&ecirc;&agrave;&igrave;, &acirc;&otilde;&icirc;&auml;&egrave;&euml;&egrave; &acirc; &ecirc;&oacute;&eth;&ntilde; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;
&ograve;&egrave;&ecirc;&egrave; &auml;&euml;&yuml; &ntilde;&ograve;&oacute;&auml;&aring;&iacute;&ograve;&icirc;&acirc; &ETH;&icirc;&ntilde;&ntilde;&egrave;&eacute;&ntilde;&ecirc;&icirc;&eacute; &Yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &Oslash;&ecirc;&icirc;&euml;&ucirc;. &Acirc; &iuml;&aring;&eth;&acirc;&icirc;&eacute; &icirc;&aacute;&ntilde;&oacute;&aelig;&auml;&agrave;&aring;&ograve;&ntilde;&yuml; &iuml;&eth;&egrave;&iacute;
&ouml;&egrave;&iuml; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&egrave;&otilde; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute; &egrave; &aring;&atilde;&icirc; &iuml;&eth;&egrave;&igrave;&aring;&iacute;&aring;&iacute;&egrave;&yuml;. &Acirc;&ograve;&icirc;&eth;&agrave;&yuml; &iuml;&icirc;&ntilde;&acirc;&yuml;&ugrave;&aring;&iacute;&agrave; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&egrave;&eth;&icirc;&acirc;&ecirc;&aring;
&egrave; &eth;&agrave;&ccedil;&euml;&egrave;&divide;&iacute;&ucirc;&igrave; &igrave;&icirc;&auml;&egrave;&ocirc;&egrave;&ecirc;&agrave;&ouml;&egrave;&yuml;&igrave; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;. &Acirc; &ograve;&eth;&aring;&ograve;&uuml;&aring;&eacute; &iuml;&eth;&egrave;&acirc;&icirc;&auml;&yuml;&ograve;&ntilde;&yuml; &iuml;&eth;&egrave;&igrave;&aring;&iacute;&aring;&iacute;&egrave;&yuml;
&ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &ecirc; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&yuml;&igrave; &acirc; &egrave;&atilde;&eth;&agrave;&otilde; &egrave; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&ecirc;&agrave;&otilde;, &ecirc; &yuml;&auml;&eth;&agrave;&igrave; &ecirc;&icirc;&icirc;&iuml;&aring;&eth;&agrave;&ograve;&egrave;&acirc;&iacute;&ucirc;&otilde; &egrave;&atilde;&eth;.
&Acirc; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&eacute; &icirc;&aacute;&ntilde;&oacute;&aelig;&auml;&agrave;&thorn;&ograve;&ntilde;&yuml; &eth;&agrave;&ccedil;&euml;&egrave;&divide;&iacute;&ucirc;&aring; &iuml;&icirc;&auml;&otilde;&icirc;&auml;&ucirc; &ecirc; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&oacute; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;.
c &ETH;&icirc;&ntilde;&ntilde;&egrave;&eacute;&ntilde;&ecirc;&agrave;&yuml; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml; &oslash;&ecirc;&icirc;&euml;&agrave;, 2006
c &Auml;&agrave;&iacute;&egrave;&euml;&icirc;&acirc; &Acirc;.&Egrave;., 2006
&Icirc;&atilde;&euml;&agrave;&acirc;&euml;&aring;&iacute;&egrave;&aring;
1
&Iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&aring; &ograve;&icirc;&divide;&ecirc;&egrave; &egrave; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&egrave;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml;
2
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;: &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&egrave;&eth;&icirc;&acirc;&ecirc;&agrave; &egrave; &icirc;&aacute;&ntilde;&oacute;&aelig;&auml;&aring;&iacute;&egrave;&aring;
11
3
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;: &iuml;&eth;&egrave;&igrave;&aring;&iacute;&aring;&iacute;&egrave;&yuml;
17
4
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;: &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave; &egrave; &agrave;&euml;&atilde;&icirc;&eth;&egrave;&ograve;&igrave;&ucirc;
25
2
3
&Euml;&aring;&ecirc;&ouml;&egrave;&yuml; 1
&Iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&aring; &ograve;&icirc;&divide;&ecirc;&egrave; &egrave; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&egrave;&aring;
&icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml;
&Iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&aring; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave;
&Iacute;&aring;&ntilde;&icirc;&igrave;&iacute;&aring;&iacute;&iacute;&icirc;, &yacute;&ograve;&icirc; &icirc;&auml;&iacute;&icirc; &egrave;&ccedil; &ntilde;&agrave;&igrave;&ucirc;&otilde; &iuml;&eth;&icirc;&ntilde;&ograve;&ucirc;&otilde; &egrave; &ocirc;&oacute;&iacute;&auml;&agrave;&igrave;&aring;&iacute;&ograve;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&eacute;: &icirc;&iacute;&icirc; &ograve;&eth;&aring;&aacute;&oacute;&aring;&ograve; &euml;&egrave;&oslash;&uuml;
&iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&euml;&aring;&iacute;&egrave;&yuml; &icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&aring; &egrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&egrave;. &Iuml;&oacute;&ntilde;&ograve;&uuml; &auml;&agrave;&iacute;&icirc; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f : X → X &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave;
X &acirc; &ntilde;&aring;&aacute;&yuml;. &Iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&icirc;&eacute; f &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &euml;&thorn;&aacute;&icirc;&eacute; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve; x ∈ X , &auml;&euml;&yuml; &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&atilde;&icirc; f (x) = x.
&Egrave;&iacute;&agrave;&divide;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&yuml;, &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave; &icirc;&ntilde;&ograve;&agrave;&aring;&ograve;&ntilde;&yuml; &iacute;&agrave; &igrave;&aring;&ntilde;&ograve;&aring; &iuml;&eth;&egrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&egrave; f . &times;&oacute;&ograve;&uuml; &auml;&icirc;&iuml;&oacute;&ntilde;&ecirc;&agrave;&yuml;
&acirc;&icirc;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&uuml;, &igrave;&icirc;&aelig;&iacute;&icirc; &ntilde;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&aring; &ograve;&icirc;&divide;&ecirc;&egrave; f &yacute;&ograve;&icirc; &ograve;&icirc;&divide;&ecirc;&egrave; &iuml;&aring;&eth;&aring;&ntilde;&aring;&divide;&aring;&iacute;&egrave;&yuml; &atilde;&eth;&agrave;&ocirc;&egrave;&ecirc;&agrave; f
&ntilde; &auml;&egrave;&agrave;&atilde;&icirc;&iacute;&agrave;&euml;&uuml;&thorn; &acirc; X &times; X .
&Iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&aring; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave; &acirc;&ntilde;&ograve;&eth;&aring;&divide;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc;&icirc; &igrave;&iacute;&icirc;&atilde;&egrave;&otilde;, &divide;&oacute;&ograve;&uuml; &euml;&egrave; &iacute;&aring; &acirc;&icirc; &acirc;&ntilde;&aring;&otilde; &ccedil;&agrave;&auml;&agrave;&divide;&agrave;&otilde;. &Iacute;&agrave;
&iuml;&eth;&egrave;&igrave;&aring;&eth;, &euml;&thorn;&aacute;&icirc;&aring; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&aring; F (x) = 0 &igrave;&icirc;&aelig;&iacute;&icirc; &ntilde;&acirc;&aring;&ntilde;&ograve;&egrave; &ecirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&aring;, &iuml;&aring;&eth;&aring;&iuml;&egrave;&ntilde;&agrave;&acirc; &aring;&atilde;&icirc; &acirc;
&acirc;&egrave;&auml;&aring;
F (x) + x = x.
&Acirc;&icirc;&ograve; &igrave;&aring;&iacute;&aring;&aring; &ograve;&agrave;&acirc;&ograve;&icirc;&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&eacute; &iuml;&eth;&egrave;&igrave;&aring;&eth;. &Iuml;&oacute;&ntilde;&ograve;&uuml; &egrave;&igrave;&aring;&aring;&ograve;&ntilde;&yuml; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&icirc;&aring; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&aring;
dy/dt = ϕ(y, t),
&atilde;&auml;&aring; ϕ &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&agrave;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml;, &egrave; y(&middot;) &aring;&atilde;&icirc; &eth;&aring;&oslash;&aring;&iacute;&egrave;&aring;, &iuml;&eth;&icirc;&otilde;&icirc;&auml;&yuml;&ugrave;&aring;&aring; &divide;&aring;&eth;&aring;&ccedil; &ograve;&icirc;&divide;&ecirc;&oacute; (t0 , y0 ) (&yacute;&ograve;&icirc;
&ccedil;&iacute;&agrave;&divide;&egrave;&ograve; &iuml;&eth;&icirc;&ntilde;&ograve;&icirc;, &divide;&ograve;&icirc; y(t0 ) = y0 ). &Ograve;&icirc;&atilde;&auml;&agrave; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; t &egrave;&ccedil; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&oacute;&thorn;&ugrave;&aring;&atilde;&icirc; &egrave;&iacute;&ograve;&aring;&eth;&acirc;&agrave;&euml;&agrave; &acirc;&ucirc;&iuml;&icirc;&euml;
&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &ntilde;&icirc;&icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring;
Z
t
y(t) = y0 +
ϕ(y(s), s)ds.
t0
&Egrave;&iacute;&agrave;&divide;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&yuml;, &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml;
y &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&icirc;&eacute; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml; A, &ccedil;&agrave;&auml;&agrave;&iacute;&iacute;&icirc;&atilde;&icirc; &ocirc;&icirc;&eth;
Rt
&igrave;&oacute;&euml;&icirc;&eacute; (Ay)(t) = y0 + t0 ϕ(y(s), s)ds.
&szlig; &ograve;&oacute;&ograve; &ccedil;&agrave;&igrave;&yuml;&euml; &icirc;&auml;&iacute;&oacute; &acirc;&aring;&ugrave;&uuml; &ograve;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &egrave;&euml;&egrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;, &iacute;&agrave; &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&igrave; &ccedil;&agrave;&auml;&agrave;&iacute; &icirc;&iuml;&aring;&eth;&agrave;&ograve;&icirc;&eth; A.
&Yacute;&ograve;&icirc; &iacute;&aring; &ntilde;&euml;&oacute;&divide;&agrave;&eacute;&iacute;&icirc;. &Icirc;&aacute;&ucirc;&divide;&iacute;&icirc; &egrave;&euml;&egrave; &yuml;&ntilde;&iacute;&icirc;, &divide;&ograve;&icirc; &yacute;&ograve;&icirc; &ccedil;&agrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;, &egrave;&euml;&egrave;, &iacute;&agrave;&iuml;&eth;&icirc;&ograve;&egrave;&acirc;, &aring;&ntilde;&ograve;&uuml; &igrave;&iacute;&icirc;&atilde;&icirc; &eth;&agrave;&ccedil;&iacute;&ucirc;&otilde;
&acirc;&icirc;&ccedil;&igrave;&icirc;&aelig;&iacute;&icirc;&ntilde;&ograve;&aring;&eacute;, &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&igrave;&egrave; &igrave;&icirc;&aelig;&iacute;&icirc; &oacute;&auml;&agrave;&divide;&iacute;&icirc; &eth;&agrave;&ntilde;&iuml;&icirc;&eth;&yuml;&auml;&egrave;&ograve;&uuml;&ntilde;&yuml;. &Acirc; &iacute;&agrave;&oslash;&aring;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &aring;&ntilde;&ograve;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&aring;&aring; &acirc;&ntilde;&aring;&atilde;&icirc;
&acirc;&ccedil;&yuml;&ograve;&uuml; &acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute; (&iacute;&agrave; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&oacute;&thorn;&ugrave;&aring;&igrave; &egrave;&iacute;
&ograve;&aring;&eth;&acirc;&agrave;&euml;&aring;); &ograve;&icirc;&atilde;&auml;&agrave; &iacute;&agrave;&auml;&icirc; &aring;&ugrave;&aring; &oacute;&aacute;&aring;&auml;&egrave;&ograve;&uuml;&ntilde;&yuml;, &divide;&ograve;&icirc; &icirc;&iuml;&aring;&eth;&agrave;&ograve;&icirc;&eth; A &iuml;&eth;&aring;&icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&ograve; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&aring; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; &acirc;
&iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&aring;.
&times;&agrave;&ntilde;&ograve;&icirc; &acirc; &iuml;&eth;&egrave;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&yuml;&otilde; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&aring; &ograve;&icirc;&divide;&ecirc;&egrave; &acirc;&icirc;&ccedil;&iacute;&egrave;&ecirc;&agrave;&thorn;&ograve; &ograve;&agrave;&igrave;, &atilde;&auml;&aring; &egrave;&igrave;&aring;&aring;&ograve;&ntilde;&yuml; &auml;&egrave;&iacute;&agrave;&igrave;&egrave;&ecirc;&agrave; &acirc;&egrave;&auml;&agrave;
xt+1 = A(xt ). &Iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;, &ntilde;&divide;&egrave;&ograve;&agrave;&aring;&ograve;&ntilde;&yuml;, &divide;&ograve;&icirc; &ouml;&aring;&iacute;&ucirc; &iuml;&icirc;&auml;&iacute;&egrave;&igrave;&agrave;&thorn;&ograve;&ntilde;&yuml;, &aring;&ntilde;&euml;&egrave; &ntilde;&iuml;&eth;&icirc;&ntilde; &iuml;&eth;&aring;&acirc;&ucirc;&oslash;&agrave;&aring;&ograve; &iuml;&eth;&aring;&auml;
&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring;, &egrave; &iuml;&icirc;&auml;&iacute;&egrave;&igrave;&agrave;&thorn;&ograve;&ntilde;&yuml; &ograve;&aring;&igrave; &ntilde;&egrave;&euml;&uuml;&iacute;&aring;&aring;, &divide;&aring;&igrave; &aacute;&icirc;&euml;&uuml;&oslash;&aring; &yacute;&ograve;&icirc;&ograve; &yacute;&ecirc;&ntilde;&ouml;&aring;&ntilde;&ntilde; &ntilde;&iuml;&eth;&icirc;&ntilde;&agrave;. &Iacute;&agrave;&egrave;&aacute;&icirc;&euml;&aring;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&icirc;&eacute;
&ntilde;&iuml;&icirc;&ntilde;&icirc;&aacute; &ocirc;&icirc;&eth;&igrave;&agrave;&euml;&egrave;&ccedil;&icirc;&acirc;&agrave;&ograve;&uuml; &yacute;&ograve;&icirc; &ccedil;&agrave;&igrave;&aring;&divide;&agrave;&iacute;&egrave;&aring; (&iacute;&icirc; &iacute;&aring; &iacute;&agrave;&egrave;&aacute;&icirc;&euml;&aring;&aring; &iuml;&eth;&agrave;&acirc;&egrave;&euml;&uuml;&iacute;&ucirc;&eacute;, &egrave;&aacute;&icirc; &iacute;&egrave;&ecirc;&ograve;&icirc; &iacute;&aring; &ccedil;&iacute;&agrave;&aring;&ograve;, &ecirc;&agrave;&ecirc;
3
4
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 1.
&Iacute;&Aring;&Iuml;&Icirc;&Auml;&Acirc;&Egrave;&AElig;&Iacute;&Ucirc;&Aring; &Ograve;&Icirc;&times;&Ecirc;&Egrave; &Egrave; &Ntilde;&AElig;&Egrave;&Igrave;&Agrave;&THORN;&Ugrave;&Egrave;&Aring; &Icirc;&Ograve;&Icirc;&Aacute;&ETH;&Agrave;&AElig;&Aring;&Iacute;&Egrave;&szlig;
&yacute;&ograve;&icirc; &ntilde;&auml;&aring;&euml;&agrave;&ograve;&uuml;) &ntilde;&divide;&egrave;&ograve;&agrave;&ograve;&uuml; &ccedil;&agrave;&acirc;&egrave;&ntilde;&egrave;&igrave;&icirc;&ntilde;&ograve;&uuml; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute;. &Ograve;&agrave;&ecirc; &igrave;&ucirc; &iuml;&eth;&egrave;&otilde;&icirc;&auml;&egrave;&igrave; &ecirc; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&thorn; &acirc;&egrave;&auml;&agrave;
pt+1 = pt + DE(pt ),
&atilde;&auml;&aring; E(pt ) &yacute;&ograve;&icirc; &yacute;&ecirc;&ntilde;&ouml;&aring;&ntilde;&ntilde;, &egrave;&euml;&egrave; &egrave;&ccedil;&aacute;&ucirc;&ograve;&icirc;&ecirc; &ntilde;&iuml;&eth;&icirc;&ntilde;&agrave; (&ograve;. &aring;. &eth;&agrave;&ccedil;&iacute;&icirc;&ntilde;&ograve;&uuml; &igrave;&aring;&aelig;&auml;&oacute; &ntilde;&iuml;&eth;&icirc;&ntilde;&icirc;&igrave; &egrave; &iuml;&eth;&aring;&auml;&euml;&icirc;&aelig;&aring;&iacute;&egrave;
&aring;&igrave;), &agrave; D &auml;&egrave;&agrave;&atilde;&icirc;&iacute;&agrave;&euml;&uuml;&iacute;&agrave;&yuml; &igrave;&agrave;&ograve;&eth;&egrave;&ouml;&agrave; &ntilde; &iacute;&aring;&icirc;&ograve;&eth;&egrave;&ouml;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&igrave;&egrave; &ecirc;&icirc;&yacute;&ocirc;&ocirc;&egrave;&ouml;&egrave;&aring;&iacute;&ograve;&agrave;&igrave;&egrave;. &Acirc; &yacute;&ograve;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring;
&iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave; &auml;&agrave;&aring;&ograve; &ntilde;&ograve;&agrave;&ouml;&egrave;&icirc;&iacute;&agrave;&eth;&iacute;&ucirc;&aring;, &iacute;&aring;&egrave;&ccedil;&igrave;&aring;&iacute;&iacute;&ucirc;&aring; &iuml;&icirc; &acirc;&eth;&aring;&igrave;&aring;&iacute;&egrave; &ouml;&aring;&iacute;&ucirc;. &Iacute;&agrave; &ntilde;&agrave;&igrave;&icirc;&igrave; &auml;&aring;&euml;&aring;, &iacute;&egrave;
&ecirc;&agrave;&ecirc;&icirc;&eacute; &ograve;&icirc;&divide;&iacute;&icirc;&eacute; &auml;&egrave;&iacute;&agrave;&igrave;&egrave;&ecirc;&egrave; &ouml;&aring;&iacute; &iacute;&aring;&egrave;&ccedil;&acirc;&aring;&ntilde;&ograve;&iacute;&icirc;, &iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &ograve;&eth;&oacute;&auml;&iacute;&icirc; &ntilde;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &iuml;&eth;&icirc;&egrave;&ntilde;&otilde;&icirc;&auml;&egrave;&ograve; &ntilde; &ouml;&aring;&iacute;&agrave;&igrave;&egrave;
&acirc; &icirc;&aacute;&ugrave;&aring;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring;, &egrave; &egrave;&igrave;&aring;&aring;&ograve; &ntilde;&igrave;&ucirc;&ntilde;&euml; &atilde;&icirc;&acirc;&icirc;&eth;&egrave;&ograve;&uuml; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &icirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&ecirc;&agrave;&otilde; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml; (&egrave;&euml;&egrave;
&icirc; &iacute;&oacute;&euml;&yuml;&otilde; E ). &Ograve;&agrave;&ecirc;&egrave;&aring; &ouml;&aring;&iacute;&ucirc; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&iacute;&ucirc;&igrave;&egrave;.
&Acirc;&icirc;&iuml;&eth;&icirc;&ntilde;&ucirc; &iuml;&eth;&icirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&aring; &ograve;&icirc;&divide;&ecirc;&egrave;
&Acirc; &ntilde;&acirc;&yuml;&ccedil;&egrave; &ntilde; &acirc;&acirc;&aring;&auml;&aring;&iacute;&iacute;&ucirc;&igrave; &acirc;&ucirc;&oslash;&aring; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&aring;&igrave; &icirc;&aacute;&ucirc;&divide;&iacute;&icirc; &icirc;&aacute;&ntilde;&oacute;&aelig;&auml;&agrave;&thorn;&ograve;&ntilde;&yuml; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&aring; &acirc;&icirc;&iuml;&eth;&icirc;&ntilde;&ucirc;:
1. &Ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&thorn;&ograve; &euml;&egrave; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&aring; &ograve;&icirc;&divide;&ecirc;&egrave;?
2. &Ntilde;&ecirc;&icirc;&euml;&uuml;&ecirc;&icirc; &egrave;&otilde;? &Icirc;&auml;&iacute;&agrave;, &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&aring; &divide;&egrave;&ntilde;&euml;&icirc;?
3. &Oacute;&ntilde;&ograve;&icirc;&eacute;&divide;&egrave;&acirc;&icirc;&ntilde;&ograve;&uuml; &acirc; &ecirc;&agrave;&ecirc;&icirc;&igrave;-&iacute;&egrave;&aacute;&oacute;&auml;&uuml; &ntilde;&igrave;&ucirc;&ntilde;&euml;&aring;. &Acirc;&aring;&eth;&iacute;&icirc; &euml;&egrave;, &iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;, &divide;&ograve;&icirc; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave; x
&iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&uuml; xn = f n (x) &ntilde;&otilde;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &ecirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&aring; x∗ ? &Acirc; &yacute;&ograve;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &atilde;&icirc;
&acirc;&icirc;&eth;&yuml;&ograve; &icirc; &atilde;&euml;&icirc;&aacute;&agrave;&euml;&uuml;&iacute;&icirc;&eacute; &ntilde;&otilde;&icirc;&auml;&egrave;&igrave;&icirc;&ntilde;&ograve;&egrave;. &Acirc;&aring;&eth;&iacute;&icirc; &euml;&egrave; &yacute;&ograve;&icirc; &auml;&euml;&yuml; &ograve;&icirc;&divide;&aring;&ecirc; x, &auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc; &aacute;&euml;&egrave;&ccedil;&ecirc;&egrave;&otilde; &ecirc; x?
&Aacute;&icirc;&euml;&aring;&aring; &iuml;&icirc;&auml;&eth;&icirc;&aacute;&iacute;&icirc; &ograve;&agrave;&ecirc;&egrave;&aring; &acirc;&icirc;&iuml;&eth;&icirc;&ntilde;&ucirc; &icirc;&aacute;&ntilde;&oacute;&aelig;&auml;&agrave;&thorn;&ograve;&ntilde;&yuml; &acirc; &ecirc;&oacute;&eth;&ntilde;&aring; &icirc; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;
&yuml;&otilde;.
4. &Ecirc;&agrave;&ecirc; &igrave;&icirc;&aelig;&iacute;&icirc; &iacute;&agrave;&eacute;&ograve;&egrave; (&acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&egrave;&ograve;&uuml;) &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&aring; &ograve;&icirc;&divide;&ecirc;&egrave;, &ograve;&icirc;&divide;&iacute;&icirc; &egrave;&euml;&egrave; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&iacute;&icirc;?
&Aacute;&euml;&egrave;&ccedil;&ecirc;&egrave;&aring; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&yuml;
1. &Acirc; &auml;&agrave;&iacute;&iacute;&icirc;&igrave; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&egrave; &icirc;&divide;&aring;&iacute;&uuml; &acirc;&agrave;&aelig;&iacute;&icirc;, &divide;&ograve;&icirc; f &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&agrave;&aring;&ograve; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; X &acirc; &ntilde;&aring;&aacute;&yuml;. &Icirc;&auml;&iacute;&agrave;&ecirc;&icirc;
&aring;&ntilde;&euml;&egrave; &egrave;&igrave;&aring;&thorn;&ograve;&ntilde;&yuml; &auml;&acirc;&agrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml; f &egrave; g &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; X &acirc; &auml;&eth;&oacute;&atilde;&icirc;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; Y , &igrave;&icirc;&aelig;&iacute;&icirc;
&atilde;&icirc;&acirc;&icirc;&eth;&egrave;&ograve;&uuml; &icirc; &ograve;&icirc;&divide;&ecirc;&agrave;&otilde; &ntilde;&icirc;&acirc;&iuml;&agrave;&auml;&aring;&iacute;&egrave;&yuml; f &egrave; g , &ograve;. &aring;. &ograve;&agrave;&ecirc;&egrave;&otilde; &ograve;&icirc;&divide;&ecirc;&agrave;&otilde; x∗ ∈ X , &divide;&ograve;&icirc; f (x∗ ) = g(x∗ ).
2. &Egrave;&iacute;&icirc;&atilde;&auml;&agrave; &iacute;&agrave;&eth;&yuml;&auml;&oacute; &ntilde; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&igrave;&egrave; &ograve;&icirc;&divide;&ecirc;&agrave;&igrave;&egrave; f , &agrave; &icirc;&ntilde;&icirc;&aacute;&aring;&iacute;&iacute;&icirc; &ecirc;&icirc;&atilde;&auml;&agrave; &egrave;&otilde; &iacute;&aring;&ograve;, &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&icirc; &eth;&agrave;&ntilde;
&ntilde;&igrave;&icirc;&ograve;&eth;&aring;&ograve;&uuml; &ouml;&egrave;&ecirc;&euml;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&aring; &ograve;&icirc;&divide;&ecirc;&egrave;, &ograve;. &aring;. &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&aring; &ograve;&icirc;&divide;&ecirc;&egrave; &egrave;&ograve;&aring;&eth;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&icirc;&atilde;&icirc; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml;
f n , &atilde;&auml;&aring; n &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &iacute;&agrave;&ograve;&oacute;&eth;&agrave;&euml;&uuml;&iacute;&icirc;&aring; &divide;&egrave;&ntilde;&euml;&icirc;. &Yacute;&ograve;&icirc; &ouml;&egrave;&ecirc;&euml;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&aring; &ograve;&icirc;&divide;&ecirc;&egrave; n-&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;. &times;&agrave;
&ntilde;&ograve;&icirc; &egrave; &ograve;&agrave;&ecirc;&egrave;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; &iacute;&aring;&ograve;, &egrave; &ograve;&icirc;&atilde;&auml;&agrave; &iuml;&eth;&egrave;&otilde;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &iuml;&icirc;&euml;&uuml;&ccedil;&icirc;&acirc;&agrave;&ograve;&uuml;&ntilde;&yuml; &divide;&aring;&igrave;-&ograve;&icirc; &acirc;&eth;&icirc;&auml;&aring; &frac34;&iuml;&eth;&aring;&auml;&aring;&euml;&uuml;&iacute;&ucirc;&otilde;&iquest;
&ouml;&egrave;&ecirc;&euml;&icirc;&acirc;. &Aacute;&icirc;&euml;&aring;&aring; &ograve;&icirc;&divide;&iacute;&icirc;, &igrave;&icirc;&aelig;&iacute;&icirc; &atilde;&icirc;&acirc;&icirc;&eth;&egrave;&ograve;&uuml; &icirc;&aacute; &egrave;&iacute;&acirc;&agrave;&eth;&egrave;&agrave;&iacute;&ograve;&iacute;&ucirc;&otilde; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave;&otilde;, &ograve;. &aring;. &icirc; &ograve;&agrave;&ecirc;&egrave;&otilde;
&iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave;&otilde; Y ⊂ X , &auml;&euml;&yuml; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; f (Y ) = Y . &Iuml;&eth;&egrave; &yacute;&ograve;&icirc;&igrave; &egrave;&iacute;&ograve;&aring;&eth;&aring;&ntilde;&iacute;&ucirc; &egrave;&iacute;&acirc;&agrave;&eth;&egrave;&agrave;&iacute;&ograve;&iacute;&ucirc;&aring;
&iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave;, &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &acirc; &ograve;&icirc;&igrave; &egrave;&euml;&egrave; &egrave;&iacute;&icirc;&igrave; &ntilde;&igrave;&ucirc;&ntilde;&euml;&aring;.
3. &times;&agrave;&ntilde;&ograve;&icirc; &iuml;&eth;&egrave;&otilde;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &egrave;&igrave;&aring;&ograve;&uuml; &auml;&aring;&euml;&icirc; &ntilde; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&igrave;&egrave; &ograve;&icirc;&divide;&ecirc;&agrave;&igrave;&egrave; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&eacute;, &egrave;&euml;&egrave; &igrave;&iacute;&icirc;&atilde;&icirc;&ccedil;&iacute;&agrave;&divide;
&iacute;&ucirc;&otilde; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute;. &times;&ograve;&icirc; &ograve;&agrave;&ecirc;&icirc;&aring; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring;? &Iacute;&agrave;&iuml;&icirc;&igrave;&iacute;&egrave;&igrave;, &divide;&ograve;&icirc; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &igrave;&aring;&aelig;&auml;&oacute; &igrave;&iacute;&icirc;
&aelig;&aring;&ntilde;&ograve;&acirc;&agrave;&igrave;&egrave; X &egrave; Y &yacute;&ograve;&icirc; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; F = &atilde;&eth;&agrave;&ocirc;&egrave;&ecirc;(f ) &acirc; X &times; Y , &icirc;&aacute;&euml;&agrave;&auml;&agrave;&thorn;&ugrave;&aring;&aring; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc;&igrave;
&ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&icirc;&iacute;&agrave;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&egrave;: &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; x ∈ X &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &egrave; &aring;&auml;&egrave;&iacute;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc; y ∈ Y , &ograve;&agrave;&ecirc;&icirc;&aring; &divide;&ograve;&icirc;
(x, y) ∈ F . &Aring;&ntilde;&euml;&egrave; &icirc;&ograve;&aacute;&eth;&icirc;&ntilde;&egrave;&ograve;&uuml; &yacute;&ograve;&icirc; &ograve;&eth;&aring;&aacute;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&icirc;&iacute;&agrave;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&egrave;, &igrave;&ucirc; &iuml;&icirc;&euml;&oacute;&divide;&egrave;&igrave; &icirc;&aacute;&ugrave;&aring;&aring;
&iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&aring; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&yuml;. &Egrave;&ograve;&agrave;&ecirc;: &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; &igrave;&aring;&aelig;&auml;&oacute; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave;&igrave;&egrave; X &egrave; Y &yacute;&ograve;&icirc; &iuml;&eth;&icirc;&egrave;&ccedil;
&acirc;&icirc;&euml;&uuml;&iacute;&icirc;&aring; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; F &acirc; &auml;&aring;&ecirc;&agrave;&eth;&ograve;&icirc;&acirc;&icirc;&igrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&egrave; X &times; Y .
&Icirc;&aacute;&ucirc;&divide;&iacute;&icirc; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; &iuml;&icirc;&iacute;&egrave;&igrave;&agrave;&thorn;&ograve; &ecirc;&agrave;&ecirc; &frac34;&igrave;&iacute;&icirc;&atilde;&icirc;&ccedil;&iacute;&agrave;&divide;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring;&iquest; &egrave;&ccedil; X &acirc; Y . &Icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;
&ograve;&icirc;&divide;&ecirc;&egrave; x ∈ X &iuml;&eth;&egrave; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&egrave; F &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;
F (x) = {y ∈ Y, (x, y) ∈ F }.
5
&Icirc;&iacute;&icirc; &igrave;&icirc;&aelig;&aring;&ograve; &ntilde;&icirc;&ntilde;&ograve;&icirc;&yuml;&ograve;&uuml; &egrave;&ccedil; &iacute;&aring;&ntilde;&ecirc;&icirc;&euml;&uuml;&ecirc;&egrave;&otilde; &ograve;&icirc;&divide;&aring;&ecirc;, &agrave; &igrave;&icirc;&aelig;&aring;&ograve; &aacute;&ucirc;&ograve;&uuml; &iuml;&oacute;&ntilde;&ograve;&ucirc;&igrave;. &Iuml;&eth;&egrave;&auml;&aring;&eth;&aelig;&egrave;&acirc;&agrave;&yuml;&ntilde;&uuml;
&ograve;&agrave;&ecirc;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave; &ccedil;&eth;&aring;&iacute;&egrave;&yuml;, &igrave;&ucirc; &aacute;&oacute;&auml;&aring;&igrave; &egrave;&ccedil;&icirc;&aacute;&eth;&agrave;&aelig;&agrave;&ograve;&uuml; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; &ecirc;&agrave;&ecirc; F : X ⇒ Y (&auml;&acirc;&icirc;&eacute;&iacute;&agrave;&yuml;
&ntilde;&ograve;&eth;&aring;&euml;&ecirc;&agrave; &oacute;&ecirc;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve; &iacute;&agrave; &frac34;&igrave;&iacute;&icirc;&atilde;&icirc;&ccedil;&iacute;&agrave;&divide;&iacute;&icirc;&ntilde;&ograve;&uuml;&iquest;). &Iacute;&agrave; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&yuml; &iuml;&aring;&eth;&aring;&iacute;&icirc;&ntilde;&yuml;&ograve;&ntilde;&yuml; &igrave;&iacute;&icirc;&atilde;&egrave;&aring; &iuml;&icirc;&iacute;&yuml;
&ograve;&egrave;&yuml;, &egrave;&ccedil;&acirc;&aring;&ntilde;&ograve;&iacute;&ucirc;&aring; &auml;&euml;&yuml; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute;; &igrave;&ucirc; &ecirc; &yacute;&ograve;&icirc;&igrave;&oacute; &aring;&ugrave;&aring; &acirc;&aring;&eth;&iacute;&aring;&igrave;&ntilde;&yuml; &acirc; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&eacute; &euml;&aring;&ecirc;&ouml;&egrave;&egrave;. &Acirc;
&divide;&agrave;&ntilde;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave;, &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&icirc;&eacute; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&yuml; F : X ⇒ X &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &euml;&thorn;&aacute;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave;
x∗ ∈ X , &ograve;&agrave;&ecirc;&agrave;&yuml; &divide;&ograve;&icirc; x∗ ∈ F (x∗ ).
4. &Egrave;&iacute;&ocirc;&egrave;&iacute;&egrave;&ograve;&aring;&ccedil;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&igrave; &agrave;&iacute;&agrave;&euml;&icirc;&atilde;&icirc;&igrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml; &ntilde;&euml;&oacute;&aelig;&egrave;&ograve; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&iacute;&icirc;&aring; &iuml;&icirc;&euml;&aring;. &Iuml;&oacute;&ntilde;&ograve;&uuml; v &acirc;&aring;&ecirc;
&ograve;&icirc;&eth;&iacute;&icirc;&aring; &iuml;&icirc;&euml;&aring; &iacute;&agrave; X , &ograve;. &aring;. &auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave; x ∈ X &oacute;&ecirc;&agrave;&ccedil;&agrave;&iacute; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&eacute; &acirc;&aring;&ecirc;&ograve;&icirc;&eth; v(x) &egrave;&ccedil;
&ecirc;&agrave;&ntilde;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; TX (x). &Ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;, &yacute;&ograve;&icirc; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&aring;&ograve;, &divide;&ograve;&icirc; X &atilde;&euml;&agrave;&auml;&ecirc;&icirc;&aring; &igrave;&iacute;&icirc;&atilde;&icirc;
&icirc;&aacute;&eth;&agrave;&ccedil;&egrave;&aring;. &Acirc;&aring;&ecirc;&ograve;&icirc;&eth;&iacute;&icirc;&aring; &iuml;&icirc;&euml;&aring; &iacute;&oacute;&aelig;&iacute;&icirc; &iuml;&icirc;&iacute;&egrave;&igrave;&agrave;&ograve;&uuml; &ecirc;&agrave;&ecirc; &aacute;&aring;&ntilde;&ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc; &igrave;&agrave;&euml;&ucirc;&eacute; &ntilde;&auml;&acirc;&egrave;&atilde; &iacute;&agrave; &igrave;&iacute;&icirc;&atilde;&icirc;&icirc;&aacute;&eth;&agrave;
&ccedil;&egrave;&egrave; X . &Iuml;&eth;&egrave; &ograve;&agrave;&ecirc;&icirc;&igrave; &acirc;&ccedil;&atilde;&euml;&yuml;&auml;&aring; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&icirc;&eacute; &iuml;&icirc;&euml;&yuml; v &iacute;&oacute;&aelig;&iacute;&icirc; &ntilde;&divide;&egrave;&ograve;&agrave;&ograve;&uuml; &ograve;&icirc;&divide;&ecirc;&oacute; x∗ ∈ X ,
&auml;&euml;&yuml; &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&eacute; v(x∗ ) = 0. &Iuml;&eth;&egrave; &agrave;&iacute;&agrave;&euml;&egrave;&ccedil;&aring; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&iacute;&ucirc;&otilde; &iuml;&icirc;&euml;&aring;&eacute; &aacute;&icirc;&euml;&uuml;&oslash;&oacute;&thorn;
&eth;&icirc;&euml;&uuml; &egrave;&atilde;&eth;&agrave;&aring;&ograve; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&aring; &egrave;&iacute;&auml;&aring;&ecirc;&ntilde;&agrave; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&iacute;&icirc;&atilde;&icirc; &iuml;&icirc;&euml;&yuml;.
&Iuml;&eth;&egrave;&iacute;&ouml;&egrave;&iuml; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&egrave;&otilde; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute;
&Atilde;&euml;&agrave;&acirc;&iacute;&ucirc;&eacute; &acirc;&icirc;&iuml;&eth;&icirc;&ntilde;, &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&igrave; &igrave;&ucirc; &aacute;&oacute;&auml;&aring;&ograve; &ccedil;&agrave;&iacute;&egrave;&igrave;&agrave;&ograve;&uuml;&ntilde;&yuml; &acirc; &yacute;&ograve;&egrave;&otilde; &euml;&aring;&ecirc;&ouml;&egrave;&yuml;&otilde; &yacute;&ograve;&icirc; &acirc;&icirc;&iuml;&eth;&icirc;&ntilde; &icirc; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;
&iacute;&egrave;&egrave; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc;. &Ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; (&egrave; &auml;&eth;&oacute;&atilde;&egrave;&aring; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&agrave;) &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;
&aelig;&aring;&iacute;&egrave;&yuml; f : X → X &ccedil;&agrave;&acirc;&egrave;&ntilde;&yuml;&ograve; &ecirc;&agrave;&ecirc; &icirc;&ograve; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml; f , &ograve;&agrave;&ecirc; &egrave; &icirc;&ograve; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; X .
&Iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;, &iuml;&eth;&agrave;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave; &acirc;&ntilde;&aring;&atilde;&auml;&agrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&aring;&ograve;&ntilde;&yuml; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&igrave;. &Igrave;&ucirc; &eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;
&eth;&egrave;&igrave; &iacute;&aring;&ntilde;&ecirc;&icirc;&euml;&uuml;&ecirc;&icirc; &eth;&aring;&agrave;&euml;&egrave;&ccedil;&agrave;&ouml;&egrave;&eacute; &yacute;&ograve;&icirc;&atilde;&icirc; &icirc;&aacute;&ugrave;&aring;&atilde;&icirc; &ccedil;&agrave;&igrave;&aring;&divide;&agrave;&iacute;&egrave;&yuml;. &Iacute;&agrave;&divide;&iacute;&aring;&igrave; &ntilde; &iuml;&eth;&icirc;&ntilde;&ograve;&aring;&eacute;&oslash;&aring;&atilde;&icirc; &ntilde;&euml;&oacute;&divide;&agrave;&yuml;, &ecirc;&icirc;&atilde;&auml;&agrave;
&ograve;&eth;&aring;&aacute;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &ecirc; X &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;, &iacute;&icirc; &ccedil;&agrave;&ograve;&icirc; &igrave;&iacute;&icirc;&atilde;&icirc;&aring; &ograve;&eth;&aring;&aacute;&oacute;&aring;&ograve;&ntilde;&yuml; &icirc;&ograve; f .
&Iuml;&oacute;&ntilde;&ograve;&uuml; X &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &ntilde; &igrave;&aring;&ograve;&eth;&egrave;&ecirc;&icirc;&eacute; ρ. &Iacute;&agrave;&iuml;&icirc;&igrave;&iacute;&thorn;, &divide;&ograve;&icirc; &igrave;&aring;&ograve;&eth;&egrave;&ecirc;&icirc;&eacute; &iacute;&agrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&aring; X &iacute;&agrave;
&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; ρ : X &times; X → R+ (ρ(x, y) &egrave;&iacute;&ograve;&aring;&eth;&iuml;&eth;&aring;&ograve;&egrave;&eth;&oacute;&aring;&ograve;&ntilde;&yuml; &ecirc;&agrave;&ecirc; &frac34;&eth;&agrave;&ntilde;&ntilde;&ograve;&icirc;&yuml;&iacute;&egrave;&aring;&iquest; &igrave;&aring;&aelig;&auml;&oacute;
&ograve;&icirc;&divide;&ecirc;&agrave;&igrave;&egrave; x, y ∈ X ), &ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&eth;&yuml;&aring;&ograve; &ograve;&eth;&aring;&igrave; &agrave;&ecirc;&ntilde;&egrave;&icirc;&igrave;&agrave;&igrave;:
1) &ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&ntilde;&ograve;&egrave;: ρ(x, y) = ρ(y, x);
2) ρ(x, y) = 0 &ograve;&icirc;&atilde;&auml;&agrave; &egrave; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &ograve;&icirc;&atilde;&auml;&agrave;, &ecirc;&icirc;&atilde;&auml;&agrave; x = y ;
3) &iacute;&aring;&eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&oacute; &ograve;&eth;&aring;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave;: ρ(x, z) ≥ ρ(x, y) + ρ(y, z) &auml;&euml;&yuml; &euml;&thorn;&aacute;&ucirc;&otilde; x, y &egrave; z .
&Icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f &igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; &acirc; &ntilde;&aring;&aacute;&yuml; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&egrave;&igrave;, &aring;&ntilde;&euml;&egrave; &ntilde;&oacute;&ugrave;&aring;
&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ograve;&agrave;&ecirc;&agrave;&yuml; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&agrave; K &lt; 1, &divide;&ograve;&icirc; &auml;&euml;&yuml; &euml;&thorn;&aacute;&ucirc;&otilde; &auml;&acirc;&oacute;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; x &egrave; y &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &iacute;&aring;&eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&icirc;
ρ(f (x), f (y)) ≤ Kρ(x, y).
&Icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&icirc;, &divide;&ograve;&icirc; &iuml;&eth;&egrave; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&aring;&igrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&egrave; &eth;&agrave;&ntilde;&ntilde;&ograve;&icirc;&yuml;&iacute;&egrave;&yuml; &igrave;&aring;&aelig;&auml;&oacute; &ograve;&icirc;&divide;&ecirc;&agrave;&igrave;&egrave; &ntilde;&ograve;&eth;&icirc;&atilde;&icirc; &oacute;&igrave;&aring;&iacute;&uuml;&oslash;&agrave;
&thorn;&ograve;&ntilde;&yuml;, &iacute;&icirc; &yacute;&ograve;&icirc;&atilde;&icirc; &igrave;&agrave;&euml;&icirc;! &Icirc;&aacute;&eth;&agrave;&ograve;&egrave;&ograve;&aring; &acirc;&iacute;&egrave;&igrave;&agrave;&iacute;&egrave;&aring;, &divide;&ograve;&icirc; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&agrave; K &igrave;&aring;&iacute;&uuml;&oslash;&aring; &aring;&auml;&egrave;&iacute;&egrave;&ouml;&ucirc; &egrave; &icirc;&aacute;&ntilde;&euml;&oacute;&aelig;&egrave;&acirc;&agrave;&aring;&ograve;
&ntilde;&eth;&agrave;&ccedil;&oacute; &acirc;&ntilde;&aring; &iuml;&agrave;&eth;&ucirc; &ograve;&icirc;&divide;&aring;&ecirc;. &Aacute;&icirc;&euml;&aring;&aring; &iuml;&eth;&agrave;&acirc;&egrave;&euml;&uuml;&iacute;&icirc; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&aring;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&ograve;&uuml; &ntilde;&ograve;&eth;&icirc;&atilde;&icirc; &ntilde;&aelig;&egrave;&igrave;&agrave;
&thorn;&ugrave;&egrave;&igrave;. &Ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&egrave;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml; &icirc;&aacute;&euml;&agrave;&auml;&agrave;&thorn;&ograve; &eth;&yuml;&auml;&icirc;&igrave; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;, &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&ucirc;&otilde; &auml;&euml;&yuml; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde;
&ograve;&icirc;&divide;&aring;&ecirc;.
&Oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&aring;&iacute;&egrave;&aring;. &Iuml;&oacute;&ntilde;&ograve;&uuml; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&aring;&aring;. &Ograve;&icirc;&atilde;&auml;&agrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &iacute;&aring; &aacute;&icirc;&euml;&aring;&aring; &icirc;&auml;&iacute;&icirc;&eacute;
&iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave;.
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;. &Acirc; &ntilde;&agrave;&igrave;&icirc;&igrave; &auml;&aring;&euml;&aring;, &aring;&ntilde;&euml;&egrave; x &egrave; y &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&aring; &ograve;&icirc;&divide;&ecirc;&egrave; f , &ograve;&icirc;
0 ≤ ρ(x, y) = ρ(f (x), f (y)) ≤ Kρ(x, y),
&divide;&ograve;&icirc; &iuml;&eth;&egrave; K &lt; 1 &igrave;&icirc;&aelig;&aring;&ograve; &aacute;&ucirc;&ograve;&uuml; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &iuml;&eth;&egrave; ρ(x, y) = 0. &Iacute;&icirc; &ograve;&icirc;&atilde;&auml;&agrave; x = y .
6
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 1.
&Iacute;&Aring;&Iuml;&Icirc;&Auml;&Acirc;&Egrave;&AElig;&Iacute;&Ucirc;&Aring; &Ograve;&Icirc;&times;&Ecirc;&Egrave; &Egrave; &Ntilde;&AElig;&Egrave;&Igrave;&Agrave;&THORN;&Ugrave;&Egrave;&Aring; &Icirc;&Ograve;&Icirc;&Aacute;&ETH;&Agrave;&AElig;&Aring;&Iacute;&Egrave;&szlig;
&Auml;&euml;&yuml; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; &iacute;&oacute;&aelig;&iacute;&icirc; &iacute;&agrave;&euml;&icirc;&aelig;&egrave;&ograve;&uuml; &iacute;&agrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; X &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&aring;
&iuml;&icirc;&euml;&iacute;&icirc;&ograve;&ucirc;. &Iacute;&agrave;&iuml;&icirc;&igrave;&iacute;&egrave;&igrave;, &divide;&ograve;&icirc; &igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &iuml;&icirc;&euml;&iacute;&icirc;&aring;, &aring;&ntilde;&euml;&egrave; &euml;&thorn;&aacute;&agrave;&yuml; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&uuml;
&Ecirc;&icirc;&oslash;&egrave; &egrave;&igrave;&aring;&aring;&ograve; &iuml;&eth;&aring;&auml;&aring;&euml; &acirc; X . &Igrave;&icirc;&aelig;&iacute;&icirc; &ntilde;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; X &iacute;&aring; &egrave;&igrave;&aring;&aring;&ograve; &frac34;&igrave;&egrave;&ecirc;&eth;&icirc;&auml;&ucirc;&eth;&iquest; &egrave;&euml;&egrave; &iuml;&eth;&icirc;&ecirc;&icirc;&euml;&icirc;&acirc;.
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; 1. &Iuml;&oacute;&ntilde;&ograve;&uuml; f &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&aring;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &iuml;&icirc;&euml;&iacute;&icirc;&atilde;&icirc; &igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave;
X &acirc; &ntilde;&aring;&aacute;&yuml;. &Ograve;&icirc;&atilde;&auml;&agrave; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave; x ∈ X &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&uuml;
x, f (x), f 2 (x) = f (f (x)), f 3 (x), . . .
&ntilde;&otilde;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &ecirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&aring;. &Acirc; &divide;&agrave;&ntilde;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave;, f &egrave;&igrave;&aring;&aring;&ograve; (&aring;&auml;&egrave;&iacute;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&oacute;&thorn;) &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&oacute;&thorn;
&ograve;&icirc;&divide;&ecirc;&oacute;.
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;. &Iuml;&oacute;&ntilde;&ograve;&uuml; d = ρ(x, f (x)). &Ograve;&icirc;&atilde;&auml;&agrave;
ρ(f n x, f n+1 x) ≤ K n d,
&egrave; &acirc;&icirc;&icirc;&aacute;&ugrave;&aring;,
K nd
.
1−K
&Ccedil;&iacute;&agrave;&divide;&egrave;&ograve; &iacute;&agrave;&oslash;&agrave; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&uuml; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn; &Ecirc;&icirc;&oslash;&egrave; &egrave; &iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &egrave;&igrave;&aring;&aring;&ograve;
&iuml;&eth;&aring;&auml;&aring;&euml;, &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&eacute; &igrave;&ucirc; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&egrave;&igrave; x∗ . &Igrave;&ucirc; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&agrave;&aring;&igrave;, &divide;&ograve;&icirc; &ograve;&icirc;&divide;&ecirc;&agrave; x∗ &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&agrave;. &Auml;&euml;&yuml; &yacute;&ograve;&icirc;&atilde;&icirc;
&iuml;&icirc;&ecirc;&agrave;&aelig;&aring;&igrave;, &divide;&ograve;&icirc; f (x∗ ) &ograve;&agrave;&ecirc;&aelig;&aring; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &iuml;&eth;&aring;&auml;&aring;&euml;&icirc;&igrave; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&egrave; xn = f n (x). &Acirc; &ntilde;&agrave;&igrave;&icirc;&igrave; &auml;&aring;&euml;&aring;,
ρ(f k x, f k+l x) ≤ (K n + K n+1 + &middot; &middot; &middot; + K n+l−1 )d ≤
ρ(f (x∗ ), xn+1 ) = ρ(f (x∗ ), f (xn )) &lt; ρ(x∗ , xn ) → 0 &iuml;&eth;&egrave; n → ∞.
&Acirc; &ntilde;&egrave;&euml;&oacute; &aring;&auml;&egrave;&iacute;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&egrave; &iuml;&eth;&aring;&auml;&aring;&euml;&agrave; f (x∗ ) = x∗ .
&Ograve;&icirc;&divide;&ecirc;&egrave; x, f (x), f 2 (x), . . . &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&igrave;&egrave; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&egrave;&yuml;&igrave;&egrave; &ecirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;
&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&aring;. &Igrave;&ucirc; &acirc;&egrave;&auml;&egrave;&igrave;, &divide;&ograve;&icirc; &acirc; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&aring;&atilde;&icirc; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml; &igrave;&icirc;&aelig;&iacute;&icirc; &iacute;&agrave;&divide;&egrave;&iacute;&agrave;&ograve;&uuml; &ntilde; &euml;&thorn;&aacute;&icirc;&atilde;&icirc;
&yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave; x, &egrave; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&aring; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&egrave;&yuml; &ntilde;&otilde;&icirc;&auml;&yuml;&ograve;&ntilde;&yuml; &ecirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&aring;. &Euml;&aring;&atilde;&ecirc;&icirc; &icirc;&ouml;&aring;
&iacute;&egrave;&ograve;&uuml; &egrave; &ograve;&icirc;&divide;&iacute;&icirc;&ntilde;&ograve;&uuml; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&egrave;&yuml;:
Kkd
ρ(f k x, x∗ ) ≤
.
1−K
&Yacute;&ograve;&icirc; &iuml;&icirc;&ccedil;&acirc;&icirc;&euml;&yuml;&aring;&ograve; &icirc;&ouml;&aring;&iacute;&egrave;&ograve;&uuml; &divide;&egrave;&ntilde;&euml;&icirc; &oslash;&agrave;&atilde;&icirc;&acirc;, &iacute;&oacute;&aelig;&iacute;&icirc;&aring; &auml;&euml;&yuml; &iacute;&agrave;&otilde;&icirc;&aelig;&auml;&aring;&iacute;&egrave;&yuml; x∗ &ntilde; &ccedil;&agrave;&auml;&agrave;&iacute;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn;.
&Iuml;&eth;&egrave;&igrave;&aring;&iacute;&aring;&iacute;&egrave;&aring; &ecirc; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&igrave; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml;&igrave;
&Iuml;&eth;&egrave;&iacute;&ouml;&egrave;&iuml; &ntilde;&aelig;&agrave;&ograve;&ucirc;&otilde; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute; &egrave;&igrave;&aring;&aring;&ograve; &igrave;&iacute;&icirc;&atilde;&icirc;&divide;&egrave;&ntilde;&euml;&aring;&iacute;&iacute;&ucirc;&aring; &iuml;&eth;&egrave;&igrave;&aring;&iacute;&aring;&iacute;&egrave;&yuml; &ecirc; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave;&igrave; &ntilde;&oacute;&ugrave;&aring;
&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &eth;&aring;&oslash;&aring;&iacute;&egrave;&eacute; &icirc;&aacute;&ucirc;&ecirc;&iacute;&icirc;&acirc;&aring;&iacute;&iacute;&ucirc;&otilde; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&eacute; &egrave; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&eacute; &acirc; &divide;&agrave;&ntilde;&ograve;&iacute;&ucirc;&otilde;
&iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&auml;&iacute;&ucirc;&otilde;, &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &iacute;&aring;&yuml;&acirc;&iacute;&ucirc;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute;, &igrave;&aring;&ograve;&icirc;&auml;&agrave;&igrave; &eth;&aring;&oslash;&aring;&iacute;&egrave;&yuml; &ntilde;&egrave;&ntilde;&ograve;&aring;&igrave; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&otilde; &oacute;&eth;&agrave;&acirc;
&iacute;&aring;&iacute;&egrave;&eacute;. &Iuml;&eth;&icirc;&auml;&aring;&igrave;&icirc;&iacute;&ntilde;&ograve;&eth;&egrave;&eth;&oacute;&aring;&igrave; &aring;&atilde;&icirc; &iacute;&agrave; &iuml;&eth;&egrave;&igrave;&aring;&eth;&aring; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&eacute;. &Iuml;&oacute;&ntilde;&ograve;&uuml; &egrave;&igrave;&aring;&aring;&ograve;&ntilde;&yuml;
&auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&icirc;&aring; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&aring;
dx
= ϕ(x, t),
dt
x(t0 ) = x0 .
(1.1)
&Ccedil;&auml;&aring;&ntilde;&uuml; x(t) &divide;&egrave;&ntilde;&euml;&icirc;&acirc;&agrave;&yuml; (&egrave;&euml;&egrave; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&iacute;&agrave;&yuml;) &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; &icirc;&ograve; t, &atilde;&auml;&aring; t &igrave;&aring;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &acirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&igrave; &egrave;&iacute;&ograve;&aring;&eth;&acirc;&agrave;&euml;&aring;
[a, b] &acirc;&icirc;&ecirc;&eth;&oacute;&atilde; t0 . &Ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; ϕ &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&aring;&ograve;&ntilde;&yuml; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&eacute;.
&Auml;&euml;&yuml; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&egrave;&yuml; &eth;&aring;&oslash;&aring;&iacute;&egrave;&yuml; &eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; C &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&otilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute; &iacute;&agrave; &icirc;&ograve;&eth;&aring;&ccedil;&ecirc;&aring;
[a, b] &ntilde; &eth;&agrave;&acirc;&iacute;&icirc;&igrave;&aring;&eth;&iacute;&icirc;&eacute; &igrave;&aring;&ograve;&eth;&egrave;&ecirc;&icirc;&eacute;. &Iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&aring; &ccedil;&iacute;&agrave;&divide;&egrave;&ograve;, &divide;&ograve;&icirc; &eth;&agrave;&ntilde;&ntilde;&ograve;&icirc;&yuml;&iacute;&egrave;&aring; &igrave;&aring;&aelig;&auml;&oacute; &auml;&acirc;&oacute;&igrave;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml;&igrave;&egrave; x
&egrave; y (&icirc;&aacute;&aring; &egrave;&ccedil; [a, b] &acirc; R (&egrave;&euml;&egrave; Rn )) &eth;&agrave;&acirc;&iacute;&icirc; max |x(t)−y(t)|, &atilde;&auml;&aring; t ∈ [a, b]. &Egrave;&ccedil; &ecirc;&oacute;&eth;&ntilde;&agrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&icirc;&iacute;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc;
&agrave;&iacute;&agrave;&euml;&egrave;&ccedil;&agrave; &acirc;&ucirc; &ccedil;&iacute;&agrave;&aring;&ograve;&aring;, &divide;&ograve;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; C &ntilde; &yacute;&ograve;&icirc;&eacute; &igrave;&aring;&ograve;&eth;&egrave;&ecirc;&icirc;&eacute; &iuml;&icirc;&euml;&iacute;&icirc;&aring;. &Ntilde;&acirc;&yuml;&aelig;&aring;&igrave; &ntilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&aring;&eacute; ϕ
(&iacute;&aring;&euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&eacute;) &icirc;&iuml;&aring;&eth;&agrave;&ograve;&icirc;&eth; A : C → C , &ccedil;&agrave;&auml;&agrave;&iacute;&iacute;&ucirc;&eacute; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&icirc;&eacute;
Z t
(Ax)(t) = x0 +
ϕ(x(s), t)ds.
t0
7
&Icirc;&iacute; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &icirc;&iuml;&aring;&eth;&agrave;&ograve;&icirc;&eth;&icirc;&igrave; &Iuml;&egrave;&ecirc;&agrave;&eth;&agrave;.
&Igrave;&ucirc; &oacute;&aelig;&aring; &acirc;&egrave;&auml;&aring;&euml;&egrave; &eth;&agrave;&iacute;&uuml;&oslash;&aring;, &divide;&ograve;&icirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave; &icirc;&iuml;&aring;&eth;&agrave;&ograve;&icirc;&eth;&agrave; A &auml;&agrave;&aring;&ograve; &eth;&aring;&oslash;&aring;&iacute;&egrave;&aring; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;
&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml; (1.1). &Aring;&ntilde;&ograve;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc; &egrave;&ntilde;&ecirc;&agrave;&ograve;&uuml; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute; &ecirc;&agrave;&ecirc; &iuml;&eth;&aring;&auml;&aring;&euml; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;
&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&otilde; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&egrave;&eacute; z, Az, A2 z, . . . , &acirc;&ucirc;&aacute;&eth;&agrave;&acirc; &acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring; &iacute;&oacute;&euml;&aring;&acirc;&icirc;&atilde;&icirc; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&egrave;&yuml; &iuml;&icirc;&ntilde;&ograve;&icirc;&yuml;&iacute;
&iacute;&oacute;&thorn; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&thorn; z ≡ x0 (&igrave;&icirc;&aelig;&iacute;&icirc;, &acirc;&iuml;&eth;&icirc;&divide;&aring;&igrave;, &iacute;&agrave;&divide;&egrave;&iacute;&agrave;&ograve;&uuml; &egrave; &ntilde; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&atilde;&icirc; &iacute;&oacute;&euml;&yuml;).
&Iuml;&eth;&egrave;&igrave;&aring;&eth;. &ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&aring; dx/dt = x, x(0) = 1. &Iacute;&agrave;&divide;&egrave;&iacute;&agrave;&yuml; &ntilde; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; x ≡ 1, &aacute;&oacute;&auml;&aring;&igrave;
&ntilde;&ograve;&eth;&icirc;&egrave;&ograve;&uuml; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&aring; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&egrave;&yuml;:
R
Ax = 1 + Rdt = 1 + t,
A2 x = 1 + (1 + s)ds = 1 + t + t2 /2,
...
tn
t2
An x = 1 + t + + &middot; &middot; &middot; + .
2
n!
&Acirc; &iuml;&eth;&aring;&auml;&aring;&euml;&aring; &iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&igrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&thorn;
x∗ = 1 + t +
t2
tn
+ &middot;&middot;&middot; +
+ &middot; &middot; &middot; = et .
2
n!
&Icirc;&ntilde;&ograve;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc;&ucirc;&yuml;&ntilde;&iacute;&egrave;&ograve;&uuml;, &ecirc;&icirc;&atilde;&auml;&agrave; &icirc;&iuml;&aring;&eth;&agrave;&ograve;&icirc;&eth; &Iuml;&egrave;&ecirc;&agrave;&eth;&agrave; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&egrave;&eacute;. &Auml;&euml;&yuml; &yacute;&ograve;&icirc;&atilde;&icirc; &iacute;&agrave; &iuml;&eth;&agrave;&acirc;&oacute;&thorn; &divide;&agrave;&ntilde;&ograve;&uuml;
ϕ(&middot;, &middot;) &iacute;&oacute;&aelig;&iacute;&icirc; &iacute;&agrave;&euml;&icirc;&aelig;&egrave;&ograve;&uuml; &ograve;&agrave;&ecirc; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&igrave;&icirc;&aring; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&aring; &Euml;&egrave;&iuml;&oslash;&egrave;&ouml;&agrave;, &aacute;&icirc;&euml;&aring;&aring; &ntilde;&egrave;&euml;&uuml;&iacute;&icirc;&aring;, &divide;&aring;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&icirc; &iacute;&aring;&iuml;&eth;&aring;
&eth;&ucirc;&acirc;&iacute;&icirc;&ntilde;&ograve;&uuml;.
&Icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring;. &Icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f : X → Y &auml;&acirc;&oacute;&otilde; &igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&otilde; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc; &oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&eth;&yuml;&aring;&ograve;
&oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&thorn; &Euml;&egrave;&iuml;&oslash;&egrave;&ouml;&agrave; &ntilde; &iuml;&icirc;&ntilde;&ograve;&icirc;&yuml;&iacute;&iacute;&icirc;&eacute; L, &aring;&ntilde;&euml;&egrave;
ρX (f x, f z) ≤ LρY (x, z)
&auml;&euml;&yuml; &euml;&thorn;&aacute;&ucirc;&otilde; x, z ∈ X .
&Iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;, &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&aring;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &euml;&egrave;&iuml;&oslash;&egrave;&ouml;&aring;&acirc;&icirc; &ntilde; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&icirc;&eacute;, &igrave;&aring;&iacute;&uuml;&oslash;&aring;&eacute; 1.
&Ograve;&agrave;&ecirc; &acirc;&icirc;&ograve;, &iuml;&icirc;&ograve;&eth;&aring;&aacute;&oacute;&aring;&igrave;, &divide;&ograve;&icirc;&aacute;&ucirc; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; ϕ(&middot;, &middot;) &aacute;&ucirc;&euml;&agrave; &euml;&egrave;&iuml;&oslash;&egrave;&ouml;&aring;&acirc;&icirc;&eacute; &iuml;&icirc; &iuml;&aring;&eth;&acirc;&icirc;&eacute; &iuml;&aring;&eth;&aring;&igrave;&aring;&iacute;&iacute;&icirc;&eacute;, &ograve;. &aring;.
|ϕ(v, t) − ϕ(w, t)| ≤ L|v − w|
&auml;&euml;&yuml; &euml;&thorn;&aacute;&ucirc;&otilde; v, w ∈ R (&egrave;&euml;&egrave; Rn ) &egrave; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; t ∈ [a, b]. &Acirc; &yacute;&ograve;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring;, &aring;&ntilde;&euml;&egrave; &egrave;&iacute;&ograve;&aring;&eth;&acirc;&agrave;&euml; [a, b] &auml;&icirc;&ntilde;&ograve;&agrave;
&ograve;&icirc;&divide;&iacute;&icirc; &igrave;&agrave;&euml;, &icirc;&iuml;&aring;&eth;&agrave;&ograve;&icirc;&eth; &Iuml;&egrave;&ecirc;&agrave;&eth;&agrave; &aacute;&oacute;&auml;&aring;&ograve; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&egrave;&igrave;. &Acirc; &ntilde;&agrave;&igrave;&icirc;&igrave; &auml;&aring;&euml;&aring;, &acirc;&icirc;&ccedil;&uuml;&igrave;&aring;&igrave; &auml;&acirc;&aring; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; x &egrave; y
&egrave;&ccedil; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; C . &Ograve;&icirc;&atilde;&auml;&agrave;
ρ(Ax, Ay) = max |Ax(t) − Ay(t)| =
a≤t≤b
Z
Z
= max [ϕ(x(s), s)) − ϕ(y(s), s))]ds ≤ max
L|x(s) − y(s)|ds.
a≤t≤b
a≤t≤b
&Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; |x(s)−y(s)| &iuml;&icirc; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&thorn; &iacute;&aring; &aacute;&icirc;&euml;&uuml;&oslash;&aring;, &divide;&aring;&igrave; ρ(x, y), &iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&aring; &acirc;&ucirc;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &igrave;&icirc;&aelig;&iacute;&icirc;
&icirc;&ouml;&aring;&iacute;&egrave;&ograve;&uuml; &ntilde;&acirc;&aring;&eth;&otilde;&oacute; &divide;&egrave;&ntilde;&euml;&icirc;&igrave;
Z
L max
a≤t≤b
ρ(x, y)ds ≤ Lρ(x, y)|b − a|.
&Ograve;&agrave;&ecirc;&egrave;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave; &igrave;&ucirc; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&euml;&egrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&oacute;&thorn; &euml;&aring;&igrave;&igrave;&oacute;:
&Euml;&aring;&igrave;&igrave;&agrave;. &Aring;&ntilde;&euml;&egrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; ϕ(&middot;, &middot;) &euml;&egrave;&iuml;&oslash;&egrave;&ouml;&aring;&acirc;&agrave; &iuml;&icirc; &iuml;&aring;&eth;&acirc;&icirc;&igrave;&oacute; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&oacute; &ntilde; &iuml;&icirc;&ntilde;&ograve;&icirc;&yuml;&iacute;&iacute;&icirc;&eacute; L, &ograve;&icirc; &icirc;&ograve;&icirc;&aacute;
&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &Iuml;&egrave;&ecirc;&agrave;&eth;&agrave; A &ograve;&icirc;&aelig;&aring; &euml;&egrave;&iuml;&oslash;&egrave;&ouml;&aring;&acirc;&icirc; &ntilde; &iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&aring;&igrave; L|b−a|. &Acirc; &divide;&agrave;&ntilde;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave;, &aring;&ntilde;&euml;&egrave; L|b−a| &lt; 1,
&ograve;&icirc; &icirc;&iuml;&aring;&eth;&agrave;&ograve;&icirc;&eth; A &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&egrave;&eacute;.
8
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 1.
&Iacute;&Aring;&Iuml;&Icirc;&Auml;&Acirc;&Egrave;&AElig;&Iacute;&Ucirc;&Aring; &Ograve;&Icirc;&times;&Ecirc;&Egrave; &Egrave; &Ntilde;&AElig;&Egrave;&Igrave;&Agrave;&THORN;&Ugrave;&Egrave;&Aring; &Icirc;&Ograve;&Icirc;&Aacute;&ETH;&Agrave;&AElig;&Aring;&Iacute;&Egrave;&szlig;
&Ntilde;&euml;&aring;&auml;&ntilde;&ograve;&acirc;&egrave;&aring;. &Aring;&ntilde;&euml;&egrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; ϕ(&middot;, &middot;) &euml;&egrave;&iuml;&oslash;&egrave;&ouml;&aring;&acirc;&agrave; &iuml;&icirc; &iuml;&aring;&eth;&acirc;&icirc;&igrave;&oacute; &agrave;&eth;&atilde;&oacute;&igrave;&aring;&iacute;&ograve;&oacute;, &ograve;&icirc; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &egrave;
&aring;&auml;&egrave;&iacute;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc; &eth;&aring;&oslash;&aring;&iacute;&egrave;&aring; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml; (1.1).
&Ccedil;&agrave;&igrave;&aring;&divide;&agrave;&iacute;&egrave;&aring;. &Aring;&ntilde;&euml;&egrave; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&aring; &Euml;&egrave;&iuml;&oslash;&egrave;&ouml;&agrave; &iacute;&agrave;&eth;&oacute;&oslash;&agrave;&aring;&ograve;&ntilde;&yuml;, &ograve;&icirc; &eth;&aring;&oslash;&aring;&iacute;&egrave;&aring; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml;
√
&igrave;&icirc;&aelig;&aring;&ograve; &iacute;&aring; &aacute;&ucirc;&ograve;&uuml; &aring;&auml;&egrave;&iacute;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&ucirc;&igrave;. &Ograve;&agrave;&ecirc; &oacute; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml; dx/dt = 2 x &divide;&aring;&eth;&aring;&ccedil;
&ograve;&icirc;&divide;&ecirc;&oacute; (0, 0) &iuml;&eth;&icirc;&otilde;&icirc;&auml;&yuml;&ograve; &eth;&aring;&oslash;&aring;&iacute;&egrave;&yuml; x ≡ 0 &egrave; x = t2 .
&Agrave;&iacute;&agrave;&euml;&icirc;&atilde;&egrave;&divide;&iacute;&ucirc;&igrave; &ntilde;&iuml;&icirc;&ntilde;&icirc;&aacute;&icirc;&igrave; &igrave;&icirc;&aelig;&iacute;&icirc; &oacute;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&egrave;&ograve;&uuml; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&oacute;&thorn; &ccedil;&agrave;&acirc;&egrave;&ntilde;&egrave;&igrave;&icirc;&ntilde;&ograve;&uuml; &eth;&aring;&oslash;&aring;&iacute;&egrave;&yuml; &auml;&egrave;&ocirc;&ocirc;&aring;
&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml; &icirc;&ograve; &iacute;&agrave;&divide;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &auml;&agrave;&iacute;&iacute;&ucirc;&otilde; &egrave; &icirc;&ograve; &iuml;&eth;&agrave;&acirc;&icirc;&eacute; &divide;&agrave;&ntilde;&ograve;&egrave; (&ntilde;&igrave;. &oacute;&iuml;&eth;. 1.7).
&Oacute;&iuml;&eth;&agrave;&aelig;&iacute;&aring;&iacute;&egrave;&yuml;
1.1. &Iuml;&oacute;&ntilde;&ograve;&uuml; X &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &ntilde; n &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&igrave;&egrave;. &Ecirc;&agrave;&ecirc;&icirc;&acirc;&agrave; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; &acirc;&ccedil;&yuml;&ograve;&icirc;&aring;
&iacute;&agrave;&oacute;&atilde;&agrave;&auml; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; X &acirc; &ntilde;&aring;&aacute;&yuml; &egrave;&igrave;&aring;&aring;&ograve; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute;. &Ecirc; &divide;&aring;&igrave;&oacute; &ntilde;&ograve;&eth;&aring;&igrave;&egrave;&ograve;&ntilde;&yuml; &yacute;&ograve;&agrave; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml;
&iuml;&eth;&egrave; &aacute;&icirc;&euml;&uuml;&oslash;&icirc;&igrave; n?
1.2. &Iuml;&oacute;&ntilde;&ograve;&uuml; f : R → R &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&eth;&oacute;&aring;&igrave;&agrave;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml;. &Iuml;&eth;&egrave; &ecirc;&agrave;&ecirc;&icirc;&igrave; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&egrave; &iacute;&agrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&auml;&iacute;&oacute;&thorn;
&icirc;&iacute;&icirc; &aacute;&oacute;&auml;&aring;&ograve; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&egrave;&igrave;?
1.3. &Aacute;&oacute;&auml;&oacute;&ograve; &euml;&egrave; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&egrave;&igrave;&egrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; sin x, arctg x, x2 ,
p
&iuml;&eth;&yuml;&igrave;&icirc;&eacute; 1 ≤ x?
|x| &iacute;&agrave; &iuml;&eth;&yuml;&igrave;&icirc;&eacute; R, x + 1/x &iacute;&agrave; &iuml;&icirc;&euml;&oacute;
1.4. &Iuml;&oacute;&ntilde;&ograve;&uuml; A &euml;&egrave;&iacute;&aring;&eacute;&iacute;&ucirc;&eacute; &icirc;&iuml;&aring;&eth;&agrave;&ograve;&icirc;&eth; &egrave;&ccedil; Rn &acirc; &ntilde;&aring;&aacute;&yuml; &ntilde; &igrave;&agrave;&ograve;&eth;&egrave;&ouml;&aring;&eacute; aij .
&agrave;) &Iuml;&oacute;&ntilde;&ograve;&uuml; &igrave;&aring;&ograve;&eth;&egrave;&ecirc;&agrave; &acirc; Rn &ccedil;&agrave;&auml;&agrave;&aring;&ograve;&ntilde;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&icirc;&eacute;
ρ(x, y) = max |xi − yi |
i
(&ograve;. &aring;. &yacute;&ograve;&icirc; l1 -&igrave;&aring;&ograve;&eth;&egrave;&ecirc;&agrave;). &Iuml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc; &aring;&ntilde;&euml;&egrave;
max |aij | &lt; 1,
i,j
&ograve;&icirc; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; A &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&aring;&aring;.
&aacute;) &Iuml;&oacute;&ntilde;&ograve;&uuml; &igrave;&aring;&ograve;&eth;&egrave;&ecirc;&agrave; &acirc; Rn &ccedil;&agrave;&auml;&agrave;&aring;&ograve;&ntilde;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&icirc;&eacute;
ρ(x, y) =
X
(xi − yi )2
1/2
i
(&ograve;. &aring;. &yacute;&ograve;&icirc; l2 -&igrave;&aring;&ograve;&eth;&egrave;&ecirc;&agrave;). &Iuml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc; &aring;&ntilde;&euml;&egrave;
X
a2ij ≤ 1,
i,j
&ograve;&icirc; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; A &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&aring;&aring;.
&acirc;)* &Iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave;, &divide;&ograve;&icirc; &acirc;&ntilde;&aring; &ntilde;&icirc;&aacute;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&ucirc;&aring; &divide;&egrave;&ntilde;&euml;&agrave; &icirc;&iuml;&aring;&eth;&agrave;&ograve;&icirc;&eth;&agrave; A &iuml;&icirc; &igrave;&icirc;&auml;&oacute;&euml;&thorn; &ntilde;&ograve;&eth;&icirc;&atilde;&icirc; &igrave;&aring;&iacute;&uuml;&oslash;&aring;
1. &Iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &acirc; Rn &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ograve;&agrave;&ecirc;&agrave;&yuml; &igrave;&aring;&ograve;&eth;&egrave;&ecirc;&agrave; (&egrave; &auml;&agrave;&aelig;&aring; &aring;&acirc;&ecirc;&euml;&egrave;&auml;&icirc;&acirc;&agrave;), &divide;&ograve;&icirc; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; A
&ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&aring;&aring;.
1.5. &Oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&eth;&yuml;&thorn;&ograve;
p &euml;&egrave; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&thorn; &Euml;&egrave;&iuml;&oslash;&egrave;&ouml;&agrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&aring; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; &iacute;&agrave; R (&igrave;&aring;&ograve;&eth;&egrave;&ecirc;&agrave; &acirc;&ntilde;&thorn;&auml;&oacute; &aring;&acirc;&ecirc;&euml;&egrave;
&auml;&icirc;&acirc;&agrave;): 1) x2 ; 2)
|x|; 3) sin x?
1.6. &Auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&eth;&oacute;&aring;&igrave;&agrave;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; f &euml;&egrave;&iuml;&oslash;&egrave;&ouml;&aring;&acirc;&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; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&auml;&iacute;&agrave;&yuml;
&icirc;&atilde;&eth;&agrave;&iacute;&egrave;&divide;&aring;&iacute;&agrave;. &times;&ograve;&icirc; &acirc;&ccedil;&yuml;&ograve;&uuml; &acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring; &iuml;&icirc;&ntilde;&ograve;&icirc;&yuml;&iacute;&iacute;&icirc;&eacute; L?
9
1.7. &Ntilde;&ecirc;&agrave;&aelig;&aring;&igrave;, &divide;&ograve;&icirc; &auml;&acirc;&agrave; &icirc;&iuml;&aring;&eth;&agrave;&ograve;&icirc;&eth;&agrave; f &egrave; g &iacute;&agrave; &igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; (X, ρ) &yuml;&acirc;&euml;&yuml;&thorn;&ograve;&ntilde;&yuml; ε-&aacute;&euml;&egrave;&ccedil;-
&ecirc;&egrave;&igrave;&egrave;, &aring;&ntilde;&euml;&egrave; ρ(f (x), g(x)) ≤ ε &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; x ∈ X . &Iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &aring;&ntilde;&euml;&egrave; f &egrave; g &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&egrave;&aring;
&icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml; (&ntilde; &iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&aring;&igrave; K ), &ograve;&icirc; &egrave;&otilde; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&aring; &ograve;&icirc;&divide;&ecirc;&egrave; &iacute;&agrave;&otilde;&icirc;&auml;&yuml;&ograve;&ntilde;&yuml; &auml;&eth;&oacute;&atilde; &icirc;&ograve; &auml;&eth;&oacute;&atilde;&agrave; &iacute;&agrave;
&eth;&agrave;&ntilde;&ntilde;&ograve;&icirc;&yuml;&iacute;&egrave;&egrave; &iacute;&aring; &aacute;&icirc;&euml;&uuml;&oslash;&aring;&igrave; ε/(1 − K).
1.8. &Iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &aring;&ntilde;&euml;&egrave; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &ntilde;&ograve;&aring;&iuml;&aring;&iacute;&uuml; f n &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml; f &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&agrave;&yuml;, &ograve;&icirc; f &egrave;&igrave;&aring;&aring;&ograve; &eth;&icirc;&acirc;&iacute;&icirc;
&icirc;&auml;&iacute;&oacute; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute;.
1.9.* &Icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f &igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; X &acirc; &ntilde;&aring;&aacute;&yuml; &ntilde;&euml;&agrave;&aacute;&icirc; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&aring;&aring;, &aring;&ntilde;&euml;&egrave; &auml;&euml;&yuml; &euml;&thorn;
&aacute;&ucirc;&otilde; &eth;&agrave;&ccedil;&euml;&egrave;&divide;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; x, y ∈ X &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&aring;&iacute;&icirc; &iacute;&aring;&eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&icirc; ρ(f (x), f (y)) &lt; ρ(x, y). &Iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc;
&euml;&thorn;&aacute;&icirc;&aring; &ntilde;&euml;&agrave;&aacute;&icirc; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&aring;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&agrave; &acirc; &ntilde;&aring;&aacute;&yuml; &egrave;&igrave;&aring;&aring;&ograve; &aring;&auml;&egrave;&iacute;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&oacute;&thorn; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&oacute;&thorn;
&ograve;&icirc;&divide;&ecirc;&oacute;.
1.10. &Iuml;&eth;&icirc;&acirc;&aring;&eth;&egrave;&ograve;&uuml;, &divide;&ograve;&icirc; &auml;&euml;&yuml; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&eacute; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; x &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; Ax (&atilde;&auml;&aring; A &icirc;&iuml;&aring;&eth;&agrave;&ograve;&icirc;&eth; &Iuml;&egrave;&ecirc;&agrave;&eth;&agrave;)
&ograve;&icirc;&aelig;&aring; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&agrave;.
&ETH;&aring;&ecirc;&icirc;&igrave;&aring;&iacute;&auml;&oacute;&aring;&igrave;&agrave;&yuml; &euml;&egrave;&ograve;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave;: [1, 3, 9].
10
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 1.
&Iacute;&Aring;&Iuml;&Icirc;&Auml;&Acirc;&Egrave;&AElig;&Iacute;&Ucirc;&Aring; &Ograve;&Icirc;&times;&Ecirc;&Egrave; &Egrave; &Ntilde;&AElig;&Egrave;&Igrave;&Agrave;&THORN;&Ugrave;&Egrave;&Aring; &Icirc;&Ograve;&Icirc;&Aacute;&ETH;&Agrave;&AElig;&Aring;&Iacute;&Egrave;&szlig;
&Euml;&aring;&ecirc;&ouml;&egrave;&yuml; 2
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;: &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&egrave;&eth;&icirc;&acirc;&ecirc;&agrave; &egrave;
&icirc;&aacute;&ntilde;&oacute;&aelig;&auml;&aring;&iacute;&egrave;&aring;
&Iuml;&eth;&icirc;&auml;&icirc;&euml;&aelig;&egrave;&igrave; &egrave;&ntilde;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &ccedil;&agrave;&auml;&agrave;&divide;&egrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml; f : X →
X . &Iuml;&icirc; &ntilde;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&thorn; &ntilde; &iuml;&eth;&aring;&auml;&ucirc;&auml;&oacute;&ugrave;&aring;&eacute; &euml;&aring;&ecirc;&ouml;&egrave;&aring;&eacute; &igrave;&ucirc; &icirc;&ntilde;&euml;&agrave;&aacute;&egrave;&igrave; &ograve;&eth;&aring;&aacute;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &ecirc; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&thorn; f &ccedil;&agrave; &ntilde;&divide;&aring;&ograve;
&oacute;&ntilde;&egrave;&euml;&aring;&iacute;&egrave;&yuml; &ograve;&eth;&aring;&aacute;&icirc;&acirc;&agrave;&iacute;&egrave;&eacute; &iacute;&agrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; X . &Aring;&ntilde;&euml;&egrave; &eth;&agrave;&iacute;&uuml;&oslash;&aring; X &aacute;&ucirc;&euml;&icirc; &iuml;&icirc;&divide;&ograve;&egrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&icirc;&eacute; &iuml;&eth;&egrave;&eth;&icirc;
&auml;&ucirc;, &ograve;&icirc; &ograve;&aring;&iuml;&aring;&eth;&uuml; &igrave;&ucirc; &aacute;&oacute;&auml;&aring;&igrave; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; X &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&eacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;.
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;
&Iuml;&oacute;&ntilde;&ograve;&uuml; V &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; (&iacute;&agrave;&auml; &iuml;&icirc;&euml;&aring;&igrave; &acirc;&aring;&ugrave;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&ucirc;&otilde; &divide;&egrave;&ntilde;&aring;&euml; R). &Iacute;&agrave;&iuml;&icirc;&igrave;&iacute;&egrave;&igrave;, &divide;&ograve;&icirc;
&iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; X ⊂ V &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&igrave;, &aring;&ntilde;&euml;&egrave; &auml;&euml;&yuml; &euml;&thorn;&aacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; x, y ∈ X &egrave; &divide;&egrave;&ntilde;&euml;&agrave; α,
0 ≤ α ≤ 1, &ograve;&icirc;&divide;&ecirc;&agrave; αx + (1 − α)y &ograve;&agrave;&ecirc;&aelig;&aring; &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; X . &Egrave;&iacute;&agrave;&divide;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&yuml;, &ntilde; &euml;&thorn;&aacute;&ucirc;&igrave;&egrave; &auml;&acirc;&oacute;&igrave;&yuml;
&ograve;&icirc;&divide;&ecirc;&agrave;&igrave;&egrave; X &ntilde;&icirc;&auml;&aring;&eth;&aelig;&egrave;&ograve; &egrave; &ntilde;&icirc;&aring;&auml;&egrave;&iacute;&yuml;&thorn;&ugrave;&egrave;&eacute; &egrave;&otilde; &icirc;&ograve;&eth;&aring;&ccedil;&icirc;&ecirc;. &Ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&ntilde;&ograve;&egrave; &acirc;&euml;&aring;&divide;&aring;&ograve;, &divide;&ograve;&icirc; X &iacute;&aring;
&egrave;&igrave;&aring;&aring;&ograve; &frac34;&auml;&ucirc;&eth;&icirc;&ecirc;&iquest;, &agrave; &yacute;&ograve;&icirc; &auml;&agrave;&aring;&ograve; &iacute;&agrave;&auml;&aring;&aelig;&auml;&oacute; &iacute;&agrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; &auml;&euml;&yuml; &oslash;&egrave;&eth;&icirc;&ecirc;&icirc;&atilde;&icirc;
&ecirc;&euml;&agrave;&ntilde;&ntilde;&agrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute;.
&Aacute;&icirc;&euml;&aring;&aring; &ograve;&icirc;&divide;&iacute;&icirc;, &igrave;&ucirc; &aacute;&oacute;&auml;&aring;&igrave; &eth;&agrave;&ntilde;&ntilde;&igrave;&agrave;&ograve;&eth;&egrave;&acirc;&agrave;&ograve;&uuml; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml;. &Yacute;&ograve;&icirc; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&aring;&ograve;,
&divide;&ograve;&icirc; X &ntilde;&iacute;&agrave;&aacute;&aelig;&aring;&iacute;&icirc; &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&aring;&eacute;. &times;&ograve;&icirc;&aacute;&ucirc; &iacute;&aring; &oacute;&ntilde;&euml;&icirc;&aelig;&iacute;&yuml;&ograve;&uuml; &egrave;&ccedil;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring;, &igrave;&ucirc; &aacute;&oacute;&auml;&aring;&igrave; &ntilde;&divide;&egrave;&ograve;&agrave;&ograve;&uuml; &acirc;&ntilde;&thorn;&auml;&oacute;, &divide;&ograve;&icirc;
&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; V &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&igrave;&aring;&eth;&iacute;&icirc;&aring; (&egrave;&ccedil;&icirc;&igrave;&icirc;&eth;&ocirc;&iacute;&icirc; Rn ) &egrave; &ntilde;&iacute;&agrave;&aacute;&aelig;&aring;&iacute;&icirc; &icirc;&aacute;&ucirc;&divide;&iacute;&icirc;&eacute; &aring;&acirc;&ecirc;&euml;&egrave;&auml;&icirc;&acirc;&icirc;&eacute; &igrave;&aring;&ograve;&eth;&egrave;&ecirc;&icirc;&eacute;,
&egrave; &divide;&ograve;&icirc; X &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&iacute;&icirc;&aring; (&ograve;. &aring;. &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;&aring; &egrave; &icirc;&atilde;&eth;&agrave;&iacute;&egrave;&divide;&aring;&iacute;&iacute;&icirc;&aring;) &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; V . &Auml;&euml;&yuml; &ecirc;&eth;&agrave;&ograve;&ecirc;&icirc;&ntilde;&ograve;&egrave;
&igrave;&ucirc; &aacute;&oacute;&auml;&aring;&igrave; &atilde;&icirc;&acirc;&icirc;&eth;&egrave;&ograve;&uuml;, &divide;&ograve;&icirc; X &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&eacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;.
&Ccedil;&iacute;&agrave;&igrave;&aring;&iacute;&egrave;&ograve;&agrave;&yuml; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&agrave;&aring;&ograve;, &divide;&ograve;&icirc; &euml;&thorn;&aacute;&icirc;&aring; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &acirc;&ucirc;&iuml;&oacute;&ecirc;
&euml;&icirc;&atilde;&icirc; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&agrave; &acirc; &ntilde;&aring;&aacute;&yuml; &egrave;&igrave;&aring;&aring;&ograve; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute;.
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; (&Aacute;&eth;&agrave;&oacute;&yacute;&eth;, 1910). &Iuml;&oacute;&ntilde;&ograve;&uuml; X &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&aring; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&iacute;&icirc;&aring; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&igrave;&aring;&eth;
&iacute;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave;, &agrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f : X → X &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;. &Ograve;&icirc;&atilde;&auml;&agrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;
&iacute;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave; f .
&Yacute;&ograve;&agrave; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &aacute;&oacute;&auml;&aring;&ograve; &ocirc;&icirc;&ecirc;&oacute;&ntilde;&icirc;&igrave; &acirc;&ntilde;&aring;&atilde;&icirc; &auml;&agrave;&euml;&uuml;&iacute;&aring;&eacute;&oslash;&aring;&atilde;&icirc;. &Acirc; &yacute;&ograve;&icirc;&eacute; &euml;&aring;&ecirc;&ouml;&egrave;&egrave; &igrave;&ucirc; &icirc;&aacute;&ntilde;&oacute;&auml;&egrave;&igrave; &aring;&aring; &divide;&agrave;&ntilde;&ograve;&iacute;&ucirc;&aring;
&ntilde;&euml;&oacute;&divide;&agrave;&egrave; &egrave; &eth;&agrave;&ccedil;&euml;&egrave;&divide;&iacute;&ucirc;&aring; &iuml;&aring;&eth;&aring;&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&egrave;&eth;&icirc;&acirc;&ecirc;&egrave;, &acirc;&agrave;&eth;&egrave;&agrave;&iacute;&ograve;&ucirc; &egrave; &icirc;&aacute;&icirc;&aacute;&ugrave;&aring;&iacute;&egrave;&yuml;. &Acirc; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&eacute; &iuml;&icirc;&atilde;&icirc;&acirc;&icirc;&eth;&egrave;&igrave;
&icirc; &iuml;&eth;&egrave;&igrave;&aring;&iacute;&aring;&iacute;&egrave;&yuml;&otilde; &yacute;&ograve;&egrave;&otilde; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve;&icirc;&acirc; &ecirc; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&igrave; &ccedil;&agrave;&auml;&agrave;&divide;&agrave;&igrave;. &Egrave; &iacute;&agrave;&ecirc;&icirc;&iacute;&aring;&ouml; &acirc; &euml;&aring;&ecirc;&ouml;&egrave;&egrave; 4 &eth;&agrave;&ntilde;&ntilde;&ecirc;&agrave;&aelig;&aring;&igrave;
&icirc; &aring;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&aring;(&agrave;&otilde;) &egrave; &agrave;&euml;&atilde;&icirc;&eth;&egrave;&ograve;&igrave;&aring; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&iacute;&icirc;&atilde;&icirc; &iacute;&agrave;&otilde;&icirc;&aelig;&auml;&aring;&iacute;&egrave;&yuml; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc;.
&Iuml;&icirc;&iuml;&eth;&icirc;&aacute;&oacute;&aring;&igrave; &euml;&oacute;&divide;&oslash;&aring; &iuml;&icirc;&iacute;&yuml;&ograve;&uuml; &ntilde;&igrave;&ucirc;&ntilde;&euml; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc;. &Iuml;&oacute;&ntilde;&ograve;&uuml; X &icirc;&auml;&iacute;&icirc;&igrave;&aring;&eth;&iacute;&icirc; (&ograve;. &aring;. &iuml;&icirc;&iuml;&eth;&icirc;&ntilde;&ograve;&oacute; &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;
&ograve;&ucirc;&eacute; &icirc;&ograve;&eth;&aring;&ccedil;&icirc;&ecirc; [a, b]), &egrave; &iuml;&oacute;&ntilde;&ograve;&uuml; f &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &yacute;&ograve;&icirc;&atilde;&icirc; &icirc;&ograve;&eth;&aring;&ccedil;&ecirc;&agrave; &acirc; &ntilde;&aring;&aacute;&yuml;. &Aring;&ntilde;&euml;&egrave; &igrave;&ucirc;
&eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&thorn; f (x) − x, &ograve;&icirc; &icirc;&iacute;&agrave; &igrave;&aring;&iacute;&yuml;&aring;&ograve; &ccedil;&iacute;&agrave;&ecirc; &iacute;&agrave; [a, b]. &Agrave; &egrave;&ccedil; &agrave;&iacute;&agrave;&euml;&egrave;&ccedil;&agrave; &otilde;&icirc;&eth;&icirc;&oslash;&icirc; &egrave;&ccedil;&acirc;&aring;&ntilde;&ograve;&iacute;&icirc;,
&divide;&ograve;&icirc; &atilde;&auml;&aring;-&ograve;&icirc; &acirc;&iacute;&oacute;&ograve;&eth;&egrave; &icirc;&iacute;&agrave; &icirc;&aacute;&eth;&agrave;&ugrave;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc; &iacute;&oacute;&euml;&uuml;, &divide;&ograve;&icirc; &yacute;&ecirc;&acirc;&egrave;&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&icirc; &ograve;&icirc;&igrave;&oacute;, &divide;&ograve;&icirc; f (x) = x.
11
12
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 2.
&Ograve;&Aring;&Icirc;&ETH;&Aring;&Igrave;&Agrave; &Aacute;&ETH;&Agrave;&Oacute;&Yacute;&ETH;&Agrave;: &Ocirc;&Icirc;&ETH;&Igrave;&Oacute;&Euml;&Egrave;&ETH;&Icirc;&Acirc;&Ecirc;&Agrave; &Egrave; &Icirc;&Aacute;&Ntilde;&Oacute;&AElig;&Auml;&Aring;&Iacute;&Egrave;&Aring;
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&aring;&ograve; &igrave;&iacute;&icirc;&atilde;&icirc;&igrave;&aring;&eth;&iacute;&icirc;&aring; &icirc;&aacute;&icirc;&aacute;&ugrave;&aring;&iacute;&egrave;&aring; &yacute;&ograve;&icirc;&atilde;&icirc; &iuml;&icirc;&divide;&ograve;&egrave; &icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&icirc;&atilde;&icirc; &oacute;&ograve;&acirc;&aring;&eth;
&aelig;&auml;&aring;&iacute;&egrave;&yuml;. &Icirc;&auml;&iacute;&agrave;&ecirc;&icirc; &oacute;&aelig;&aring; &acirc; &auml;&acirc;&oacute;&igrave;&aring;&eth;&iacute;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&aring;&iacute;&egrave;&aring; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &acirc;&ucirc;&atilde;&euml;&yuml;&auml;&egrave;&ograve; &ntilde;&icirc;&acirc;&ntilde;&aring;&igrave; &iacute;&aring;&icirc;&divide;&aring;&acirc;&egrave;&auml;
&iacute;&ucirc;&igrave; &egrave; &auml;&agrave;&aelig;&aring; &igrave;&agrave;&euml;&icirc;&iuml;&eth;&agrave;&acirc;&auml;&icirc;&iuml;&icirc;&auml;&icirc;&aacute;&iacute;&ucirc;&igrave;. &Iuml;&icirc;&divide;&aring;&igrave;&oacute; &yacute;&ograve;&icirc; &euml;&thorn;&aacute;&icirc;&aring; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring;, &ntilde;&ecirc;&agrave;&aelig;&aring;&igrave;,
&ecirc;&eth;&oacute;&atilde;&agrave; &acirc; &ntilde;&aring;&aacute;&yuml; &auml;&icirc;&euml;&aelig;&iacute;&icirc; &egrave;&igrave;&aring;&ograve;&uuml; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute;? &Igrave;&icirc;&aelig;&aring;&ograve; &aacute;&ucirc;&ograve;&uuml;, &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&aring;&iacute;&egrave;&aring; &ntilde;&ograve;&agrave;&iacute;&aring;&ograve; &egrave;&iacute;&ograve;&oacute;&egrave;
&ograve;&egrave;&acirc;&iacute;&icirc; &aacute;&icirc;&euml;&aring;&aring; &iuml;&eth;&agrave;&acirc;&auml;&icirc;&iuml;&icirc;&auml;&icirc;&aacute;&iacute;&ucirc;&igrave; &iuml;&icirc;&ntilde;&euml;&aring; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&eacute; &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&icirc;&eacute; &atilde;&aring;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &ecirc;&icirc;&iacute;&ntilde;&ograve;&eth;&oacute;&ecirc;&ouml;&egrave;&egrave;. &Auml;&icirc;
&iuml;&oacute;&ntilde;&ograve;&egrave;&igrave; (&acirc;&icirc;&iuml;&eth;&aring;&ecirc;&egrave; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&aring;&iacute;&egrave;&thorn;), &divide;&ograve;&icirc; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f : D → D &iacute;&aring; &egrave;&igrave;&aring;&aring;&ograve; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc;
(&ccedil;&auml;&aring;&ntilde;&uuml; D &auml;&egrave;&ntilde;&ecirc; (&oslash;&agrave;&eth;) &euml;&thorn;&aacute;&icirc;&eacute; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave;). &Ograve;&icirc;&atilde;&auml;&agrave; &acirc;&ucirc;&iuml;&oacute;&ntilde;&ograve;&egrave;&igrave; &egrave;&ccedil; &ograve;&icirc;&divide;&ecirc;&egrave; f (x) &euml;&oacute;&divide; &acirc; &ograve;&icirc;&divide;&ecirc;&oacute; x,
&egrave; &iuml;&eth;&icirc;&auml;&icirc;&euml;&aelig;&egrave;&igrave; &aring;&atilde;&icirc; &auml;&icirc; &iuml;&aring;&eth;&aring;&ntilde;&aring;&divide;&aring;&iacute;&egrave;&yuml; &ntilde; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&aring;&eacute; D, &ograve;. &aring;. &ntilde;&icirc; &ntilde;&ocirc;&aring;&eth;&icirc;&eacute; ∂D = S . &Iuml;&icirc;&euml;&oacute;&divide;&aring;&iacute;&iacute;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute;
&iuml;&aring;&eth;&aring;&ntilde;&aring;&divide;&aring;&iacute;&egrave;&yuml; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&egrave;&igrave; g(x).
&Ograve;&agrave;&ecirc; &igrave;&ucirc; &iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&igrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; g : D → ∂D &auml;&egrave;&ntilde;&ecirc;&agrave; D &iacute;&agrave; &aring;&atilde;&icirc; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&oacute;. &Ecirc;&agrave;&ecirc; &euml;&aring;&atilde;&ecirc;&icirc; &iuml;&icirc;&iacute;&yuml;&ograve;&uuml;,
&icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; g &aacute;&oacute;&auml;&aring;&ograve; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&igrave; &egrave; &icirc;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&ograve;&uuml; &ograve;&icirc;&divide;&ecirc;&egrave; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&ucirc; &iacute;&agrave; &igrave;&aring;&ntilde;&ograve;&aring;. &Ograve;&agrave;&ecirc;&egrave;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml;
&iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&yuml;&igrave;&egrave;. &Aacute;&icirc;&euml;&aring;&aring; &ograve;&icirc;&divide;&iacute;&icirc;, &iuml;&oacute;&ntilde;&ograve;&uuml; X &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;, &egrave; Y ⊂ X &aring;&atilde;&icirc; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;
&ntilde;&ograve;&acirc;&icirc;; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; g : X → Y &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&aring;&eacute; &iacute;&agrave; Y , &aring;&ntilde;&euml;&egrave; f (y) = y &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave;
y ∈Y.
&Egrave;&iacute;&ograve;&oacute;&egrave;&ograve;&egrave;&acirc;&iacute;&icirc; &auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc; &yuml;&ntilde;&iacute;&icirc;, &divide;&ograve;&icirc; &iacute;&aring;&euml;&uuml;&ccedil;&yuml; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc; &iuml;&aring;&eth;&aring;&ograve;&yuml;&iacute;&oacute;&ograve;&uuml; &auml;&egrave;&ntilde;&ecirc; &iacute;&agrave; &aring;&atilde;&icirc; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&oacute;.
&times;&ograve;&icirc;&aacute;&ucirc; &ntilde;&auml;&aring;&euml;&agrave;&ograve;&uuml; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&aring;&iacute;&egrave;&aring; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &aring;&ugrave;&aring; &aacute;&icirc;&euml;&aring;&aring; &iuml;&eth;&agrave;&acirc;&auml;&icirc;&iuml;&icirc;&auml;&icirc;&aacute;&iacute;&ucirc;&igrave;, &igrave;&ucirc; &icirc;&aacute;&ntilde;&oacute;&auml;&egrave;&igrave; &aring;&atilde;&icirc; &aring;&ugrave;&aring; &acirc;
&icirc;&auml;&iacute;&icirc;&igrave; &divide;&agrave;&ntilde;&ograve;&iacute;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring;, &ecirc;&icirc;&atilde;&auml;&agrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f &iacute;&aring; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;, &iacute;&icirc; &egrave; &agrave;&ocirc;&ocirc;&egrave;&iacute;&iacute;&icirc;. &Icirc;&ograve;&icirc;&aacute;
&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f &icirc;&auml;&iacute;&icirc;&atilde;&icirc; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&atilde;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; &acirc; &auml;&eth;&oacute;&atilde;&icirc;&aring; &agrave;&ocirc;&ocirc;&egrave;&iacute;&iacute;&icirc;, &aring;&ntilde;&euml;&egrave; &auml;&euml;&yuml; &euml;&thorn;&aacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; x, y &egrave;
&divide;&egrave;&ntilde;&euml;&agrave; α, 0 ≤ α ≤ 1,
f (αx + (1 − α)y) = αf (x) + (1 − α)f (y).
&Egrave;&auml;&aring;&yuml; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave; &acirc; &agrave;&ocirc;&ocirc;&egrave;&iacute;&iacute;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&agrave;. &Acirc;&icirc;&ccedil;&uuml;&igrave;&aring;&igrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute; x0 &egrave;
&icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&igrave; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&aring; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&egrave;&yuml; xn = f n x0 . &Icirc;&auml;&iacute;&agrave;&ecirc;&icirc; &acirc; &icirc;&ograve;&euml;&egrave;&divide;&egrave;&egrave; &icirc;&ograve; &ntilde;&aelig;&egrave;&igrave;&agrave;&thorn;&ugrave;&aring;&atilde;&icirc;
&icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml;, &ntilde;&aring;&eacute;&divide;&agrave;&ntilde; &iacute;&aring;&ograve; &iacute;&egrave;&ecirc;&agrave;&ecirc;&egrave;&otilde; &icirc;&ntilde;&iacute;&icirc;&acirc;&agrave;&iacute;&egrave;&eacute; &ntilde;&divide;&egrave;&ograve;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&uuml; xn &ntilde;&otilde;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml;.
&Iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;, f &igrave;&icirc;&aelig;&aring;&ograve; &aacute;&ucirc;&ograve;&uuml; &iuml;&icirc;&acirc;&icirc;&eth;&icirc;&ograve;&icirc;&igrave; &auml;&acirc;&oacute;&igrave;&aring;&eth;&iacute;&icirc;&atilde;&icirc; &auml;&egrave;&ntilde;&ecirc;&agrave;, &egrave; &aring;&ntilde;&euml;&egrave; x0 &aacute;&ucirc;&euml;&agrave; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&aring;, &icirc;&iacute;&agrave; &ograve;&agrave;&ecirc;
&egrave; &aacute;&oacute;&auml;&aring;&ograve; &ecirc;&eth;&oacute;&ograve;&egrave;&ograve;&uuml;&ntilde;&yuml; &iuml;&icirc; &atilde;&eth;&agrave;&iacute;&egrave;&divide;&iacute;&icirc;&eacute; &icirc;&ecirc;&eth;&oacute;&aelig;&iacute;&icirc;&ntilde;&ograve;&egrave;. &Icirc;&auml;&iacute;&agrave;&ecirc;&icirc; &aring;&ntilde;&euml;&egrave; &aacute;&eth;&agrave;&ograve;&uuml; &ouml;&aring;&iacute;&ograve;&eth;&ucirc; &ograve;&yuml;&aelig;&aring;&ntilde;&ograve;&egrave; &ograve;&icirc;&divide;&aring;&ecirc;
x0 , x1 , . . . , xn , &ograve;. &aring;. &ograve;&icirc;&divide;&ecirc;&egrave;
zn = (x0 + x1 + &middot; &middot; &middot; + xn−1 )/n,
&ograve;&icirc; &egrave;&igrave;&aring;&thorn;&ograve;&ntilde;&yuml; &acirc;&ntilde;&aring; &icirc;&ntilde;&iacute;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &eth;&agrave;&ntilde;&ntilde;&divide;&egrave;&ograve;&ucirc;&acirc;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &icirc;&iacute;&egrave; &aacute;&oacute;&auml;&oacute;&ograve; &acirc;&ntilde;&aring; &aacute;&icirc;&euml;&aring;&aring; &egrave; &aacute;&icirc;&euml;&aring;&aring; &frac34;&iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&igrave;&egrave;&iquest;.
&Acirc; &ntilde;&agrave;&igrave;&icirc;&igrave; &auml;&aring;&euml;&aring;,
f (zn ) − zn = (f (x0 ) + f (x1 ) + &middot; &middot; &middot; + f (xn−1 ) − x0 − x1 − &middot; &middot; &middot; − xn−1 )/n =
= (x1 + x2 + &middot; &middot; &middot; + xn − x0 − x1 − &middot; &middot; &middot; − xn−1 )/n = (xn − x0 )/n,
&agrave; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&aring; &ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&egrave;&ograve;&ntilde;&yuml; &acirc;&ntilde;&aring; &igrave;&aring;&iacute;&uuml;&oslash;&aring; &egrave; &igrave;&aring;&iacute;&uuml;&oslash;&aring; &ntilde; &eth;&icirc;&ntilde;&ograve;&icirc;&igrave; n.
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Ecirc;&agrave;&ecirc;&oacute;&ograve;&agrave;&iacute;&egrave;
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &egrave;&igrave;&aring;&aring;&ograve; &igrave;&agrave;&ntilde;&ntilde;&oacute; &yacute;&ecirc;&acirc;&egrave;&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&ucirc;&otilde; &iuml;&aring;&eth;&aring;&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&egrave;&eth;&icirc;&acirc;&icirc;&ecirc;, &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&ucirc;&otilde; &acirc; &ograve;&aring;&otilde; &egrave;&euml;&egrave; &egrave;&iacute;&ucirc;&otilde;
&ntilde;&egrave;&ograve;&oacute;&agrave;&ouml;&egrave;&yuml;&otilde;. &Iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &iuml;&aring;&eth;&aring;&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&egrave;&eth;&icirc;&acirc;&ecirc;&egrave; &iuml;&eth;&egrave;&acirc;&aring;&auml;&aring;&iacute;&ucirc; &acirc; &oacute;&iuml;&eth;&agrave;&aelig;&iacute;&aring;&iacute;&egrave;&yuml;&otilde;. &Ccedil;&auml;&aring;&ntilde;&uuml; &aelig;&aring; &igrave;&ucirc; &aacute;&icirc;&euml;&aring;&aring;
&iuml;&icirc;&auml;&eth;&icirc;&aacute;&iacute;&icirc; &icirc;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&egrave;&igrave;&ntilde;&yuml; &iacute;&agrave; &acirc;&agrave;&aelig;&iacute;&icirc;&igrave; &auml;&euml;&yuml; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&otilde; &iuml;&eth;&egrave;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&eacute; &icirc;&aacute;&icirc;&aacute;&ugrave;&aring;&iacute;&egrave;&egrave; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Aacute;&eth;&agrave;&oacute;
&yacute;&eth;&agrave;, &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&agrave;&ugrave;&aring;&igrave; &Ecirc;&agrave;&ecirc;&oacute;&ograve;&agrave;&iacute;&egrave;. &Icirc;&aacute;&icirc;&aacute;&ugrave;&aring;&iacute;&egrave;&aring; &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &acirc; &ograve;&icirc;&igrave;, &divide;&ograve;&icirc; &acirc;&igrave;&aring;&ntilde;&ograve;&icirc; &icirc;&auml;&iacute;&icirc;&ccedil;&iacute;&agrave;&divide;&iacute;&ucirc;&otilde; &iacute;&aring;&iuml;&eth;&aring;
&eth;&ucirc;&acirc;&iacute;&ucirc;&otilde; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute; &auml;&icirc;&iuml;&oacute;&ntilde;&ecirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &egrave; &igrave;&iacute;&icirc;&atilde;&icirc;&ccedil;&iacute;&agrave;&divide;&iacute;&ucirc;&aring;. &Auml;&aring;&euml;&icirc; &acirc; &ograve;&icirc;&igrave;, &divide;&ograve;&icirc; &igrave;&iacute;&icirc;&atilde;&egrave;&aring; &aring;&ntilde;&ograve;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&ucirc;&aring;
&icirc;&aacute;&uacute;&aring;&ecirc;&ograve;&ucirc; &acirc; &igrave;&agrave;&ograve;&yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&ecirc;&aring; &iuml;&icirc;&yuml;&acirc;&euml;&yuml;&thorn;&ograve;&ntilde;&yuml; &ecirc;&agrave;&ecirc; &eth;&aring;&oslash;&aring;&iacute;&egrave;&yuml; &ccedil;&agrave;&auml;&agrave;&divide;&egrave; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&egrave;&ccedil;&agrave;&ouml;&egrave;&egrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute; (&egrave;&euml;&egrave; &iuml;&eth;&aring;&auml;
&iuml;&icirc;&divide;&ograve;&aring;&iacute;&egrave;&eacute;), &agrave; &eth;&aring;&oslash;&aring;&iacute;&egrave;&aring; &ograve;&agrave;&ecirc;&egrave;&otilde; &ccedil;&agrave;&auml;&agrave;&divide; &acirc; &icirc;&aacute;&ugrave;&aring;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &igrave;&iacute;&icirc;&atilde;&icirc;&ccedil;&iacute;&agrave;&divide;&iacute;&icirc;&aring;.
&Igrave;&ucirc; &oacute;&aelig;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&egrave;&euml;&egrave; &acirc; &euml;&aring;&ecirc;&ouml;&egrave;&egrave; 1, &divide;&ograve;&icirc; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring;&igrave; (&egrave;&euml;&egrave; &igrave;&iacute;&icirc;&atilde;&icirc;&ccedil;&iacute;&agrave;&divide;&iacute;&ucirc;&igrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring;&igrave;)
&egrave;&ccedil; X &acirc; Y &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; F &acirc; &auml;&aring;&ecirc;&agrave;&eth;&ograve;&icirc;&acirc;&icirc;&igrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&egrave; X &times; Y . &Egrave;&iacute;&icirc;&atilde;&auml;&agrave; F &iacute;&agrave;&ccedil;&ucirc;
&acirc;&agrave;&thorn;&ograve; &ograve;&agrave;&ecirc;&aelig;&aring; &atilde;&eth;&agrave;&ocirc;&egrave;&ecirc;&icirc;&igrave; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&yuml;. &Icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave; &ograve;&icirc;&divide;&ecirc;&egrave; x ∈ X &iuml;&eth;&egrave; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&egrave; F &ntilde;&divide;&egrave;&ograve;&agrave;&aring;&ograve;&ntilde;&yuml;
&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; F (x) = {y ∈ Y, (x, y) ∈ F }. &Iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&icirc;&eacute; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&yuml; F : X ⇒ X
&iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ograve;&agrave;&ecirc;&icirc;&eacute; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve; x∗ ∈ X , &divide;&ograve;&icirc; x∗ ∈ F (x∗ ).
13
&Iacute;&agrave; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&yuml; &acirc; &ograve;&aring;&icirc;&eth;&aring;&igrave;&aring; &Ecirc;&agrave;&ecirc;&oacute;&ograve;&agrave;&iacute;&egrave; &iacute;&agrave;&ecirc;&euml;&agrave;&auml;&ucirc;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &auml;&acirc;&agrave; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml;. &Iuml;&aring;&eth;&acirc;&icirc;&aring;: &acirc;&ntilde;&aring; &aring;&atilde;&icirc; &icirc;&aacute;
&eth;&agrave;&ccedil;&ucirc; F (x) &auml;&icirc;&euml;&aelig;&iacute;&ucirc; &aacute;&ucirc;&ograve;&uuml; &iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&ucirc;&igrave;&egrave; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&igrave;&egrave; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave;&igrave;&egrave; X . &Acirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&igrave; &ntilde;&igrave;&ucirc;&ntilde;&euml;&aring;
&acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; &iuml;&icirc;&otilde;&icirc;&aelig;&egrave; &iacute;&agrave; &ograve;&icirc;&divide;&ecirc;&oacute;: &acirc; &iacute;&egrave;&otilde; &ograve;&icirc;&aelig;&aring; &iacute;&aring;&ograve; &frac34;&auml;&ucirc;&eth;&iquest;. &Acirc;&ograve;&icirc;&eth;&icirc;&aring; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&aring; &egrave;&igrave;&aring;&aring;&ograve; &ograve;&icirc;&iuml;&icirc;
&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&eacute; &otilde;&agrave;&eth;&agrave;&ecirc;&ograve;&aring;&eth; &egrave; &iuml;&eth;&egrave;&ccedil;&acirc;&agrave;&iacute;&icirc; &ccedil;&agrave;&igrave;&aring;&iacute;&egrave;&ograve;&uuml; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&ntilde;&ograve;&uuml;. &Ntilde;&ecirc;&agrave;&aelig;&aring;&igrave; &icirc; &iacute;&aring;&igrave; &iuml;&icirc;&auml;&eth;&icirc;&aacute;&iacute;&aring;&aring;.
&Egrave;&igrave;&aring;&aring;&ograve;&ntilde;&yuml; &igrave;&iacute;&icirc;&atilde;&icirc; &ntilde;&iuml;&icirc;&ntilde;&icirc;&aacute;&icirc;&acirc; &iuml;&aring;&eth;&aring;&iacute;&aring;&ntilde;&ograve;&egrave; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&aring; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&ntilde;&ograve;&egrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml; &iacute;&agrave; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;
&ntilde;&ograve;&acirc;&egrave;&yuml;. &Yacute;&ograve;&icirc; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&aring; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&ntilde;&ograve;&egrave;, &iuml;&icirc;&euml;&oacute;&iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&ntilde;&ograve;&egrave; &ntilde;&iacute;&egrave;&ccedil;&oacute; &egrave; &ntilde;&acirc;&aring;&eth;&otilde;&oacute;, &agrave; &ograve;&agrave;&ecirc;&aelig;&aring; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&aring;
&ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;&atilde;&icirc; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&yuml;. &Iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&aring; &auml;&euml;&yuml; &iacute;&agrave;&ntilde; &icirc;&ntilde;&icirc;&aacute;&aring;&iacute;&iacute;&icirc; &acirc;&agrave;&aelig;&iacute;&icirc;.
&Icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring;. &Iuml;&oacute;&ntilde;&ograve;&uuml; X &egrave; Y &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave;. &Ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; F &egrave;&ccedil; X &acirc; Y
&iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&ucirc;&igrave;, &aring;&ntilde;&euml;&egrave; F &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc; &ecirc;&agrave;&ecirc; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &acirc; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&egrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;
X &times;Y.
&Egrave;&iacute;&agrave;&divide;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&yuml;, &aring;&ntilde;&euml;&egrave; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&uuml; &ograve;&icirc;&divide;&aring;&ecirc; (xn , yn ) &egrave;&ccedil; F &ntilde;&otilde;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &ecirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&aring;
(x, y) ∈ X &times; Y , &ograve;&icirc; &iuml;&eth;&aring;&auml;&aring;&euml;&uuml;&iacute;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave; (x, y) &ograve;&agrave;&ecirc;&aelig;&aring; &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; F .
&Iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&aring; &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;&atilde;&icirc; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&yuml; &icirc;&divide;&aring;&iacute;&uuml; &aacute;&euml;&egrave;&ccedil;&ecirc;&icirc; &ecirc; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&thorn; &iuml;&icirc;&euml;&oacute;&iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&atilde;&icirc; &ntilde;&acirc;&aring;&eth;&otilde;&oacute;
&ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&yuml;, &otilde;&icirc;&ograve;&yuml; &egrave; &iacute;&aring; &ntilde;&icirc;&acirc;&iuml;&agrave;&auml;&agrave;&aring;&ograve; &ntilde; &iacute;&egrave;&igrave;. &Iuml;&eth;&egrave;&acirc;&aring;&auml;&aring;&igrave; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&atilde;&icirc; &acirc; &ntilde;&euml;&oacute;&divide;&agrave;&aring;, &ecirc;&icirc;&atilde;&auml;&agrave;
&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave; X &egrave; Y &igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&aring;. &Iacute;&agrave;&iuml;&icirc;&igrave;&iacute;&egrave;&igrave;, &divide;&ograve;&icirc; ε-&icirc;&ecirc;&eth;&aring;&ntilde;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; A ⊂ Y
&iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;
Uε (A) = {y ∈ Y, ρ(y, A) &lt; ε}
&ograve;&icirc;&divide;&aring;&ecirc;, &oacute;&auml;&agrave;&euml;&aring;&iacute;&iacute;&ucirc;&otilde; &icirc;&ograve; A &igrave;&aring;&iacute;&aring;&aring; &divide;&aring;&igrave; &iacute;&agrave; ε. &Ograve;&agrave;&ecirc; &acirc;&icirc;&ograve;, &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; F &igrave;&aring;&aelig;&auml;&oacute; X &egrave; Y &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml;
&iuml;&icirc;&euml;&oacute;&iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&igrave; &ntilde;&acirc;&aring;&eth;&otilde;&oacute; &acirc; &ograve;&icirc;&divide;&ecirc;&aring; x ∈ X , &aring;&ntilde;&euml;&egrave; ∀ ε &gt; 0 ∃ δ &gt; 0, &ograve;&agrave;&ecirc;&egrave;&aring; &divide;&ograve;&icirc; F (x0 ) ⊂ Uε (F (x)),
&ecirc;&agrave;&ecirc; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; ρ(x, x0 ) &lt; ρ.
&Egrave;&iacute;&ograve;&oacute;&egrave;&ograve;&egrave;&acirc;&iacute;&icirc; &yacute;&ograve;&icirc; &ograve;&agrave;&ecirc;&egrave;&aring; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&yuml;, &divide;&ograve;&icirc; &icirc;&aacute;&eth;&agrave;&ccedil;&ucirc; &igrave;&icirc;&atilde;&oacute;&ograve; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &oacute;&acirc;&aring;&euml;&egrave;&divide;&egrave;&acirc;&agrave;&ograve;&uuml;&ntilde;&yuml;.
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; (&Ecirc;&agrave;&ecirc;&oacute;&ograve;&agrave;&iacute;&egrave;, 1941). &Iuml;&oacute;&ntilde;&ograve;&uuml; X &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&eacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;, &agrave; F &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;&aring; &ntilde;&icirc;&icirc;&ograve;
&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; X &acirc; &ntilde;&aring;&aacute;&yuml; &ntilde; &iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&ucirc;&igrave;&egrave; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&igrave;&egrave; &icirc;&aacute;&eth;&agrave;&ccedil;&agrave;&igrave;&egrave;. &Ograve;&icirc;&atilde;&auml;&agrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ograve;&agrave;&ecirc;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave;
x∗ ∈ X , &divide;&ograve;&icirc; x∗ ∈ F (x∗ ).
&Ecirc;&ntilde;&ograve;&agrave;&ograve;&egrave;, &egrave;&ccedil; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Ecirc;&agrave;&ecirc;&oacute;&ograve;&agrave;&iacute;&egrave; &euml;&aring;&atilde;&ecirc;&icirc; &iuml;&icirc;&euml;&oacute;&divide;&egrave;&ograve;&uuml; &egrave; &ograve;&agrave;&ecirc;&icirc;&aring; &icirc;&aacute;&icirc;&aacute;&ugrave;&aring;&iacute;&egrave;&aring; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &iacute;&agrave;
&ntilde;&euml;&oacute;&divide;&agrave;&eacute; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&ucirc;&otilde; (&eth;&agrave;&ccedil;&eth;&ucirc;&acirc;&iacute;&ucirc;&otilde;) &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute; f &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&atilde;&icirc; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&agrave; &acirc; &ntilde;&aring;&aacute;&yuml;. &Agrave; &egrave;&igrave;&aring;&iacute;&iacute;&icirc;,
&ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ograve;&agrave;&ecirc;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave; x∗ , &divide;&ograve;&icirc; f (x∗ ) &icirc;&ograve;&ntilde;&ograve;&icirc;&egrave;&ograve; &icirc;&ograve; x∗ &iacute;&agrave; &eth;&agrave;&ntilde;&ntilde;&ograve;&icirc;&yuml;&iacute;&egrave;&aring;, &iacute;&aring; &iuml;&eth;&aring;&acirc;&ucirc;&oslash;&agrave;&thorn;&ugrave;&aring;&aring; &frac34;&acirc;&aring;&euml;&egrave;
&divide;&egrave;&iacute;&oacute; &eth;&agrave;&ccedil;&eth;&ucirc;&acirc;&iacute;&icirc;&ntilde;&ograve;&egrave;&iquest; f &acirc; &ograve;&icirc;&divide;&ecirc;&aring; x∗ .
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;. &Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Ecirc;&agrave;&ecirc;&oacute;&ograve;&agrave;&iacute;&egrave; &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &acirc; &ntilde;&acirc;&aring;&auml;&aring;&iacute;&egrave;&egrave; &aring;&aring; &ecirc; &ograve;&aring;&icirc;&eth;&aring;&igrave;&aring; &Aacute;&eth;&agrave;&oacute;
&yacute;&eth;&agrave;. &Egrave;&auml;&aring;&yuml; &ntilde;&acirc;&aring;&auml;&aring;&iacute;&egrave;&yuml; &ccedil;&agrave;&ecirc;&euml;&thorn;&divide;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc; &ograve;&icirc;&igrave;, &divide;&ograve;&icirc;&aacute;&ucirc; &agrave;&iuml;&iuml;&eth;&icirc;&ecirc;&ntilde;&egrave;&igrave;&egrave;&eth;&icirc;&acirc;&agrave;&ograve;&uuml; (&iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&agrave;&ograve;&uuml;) &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring;
F &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&igrave;&egrave; &icirc;&auml;&iacute;&icirc;&ccedil;&iacute;&agrave;&divide;&iacute;&ucirc;&igrave;&egrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml;&igrave;&egrave;. &Icirc;&auml;&iacute;&agrave;&ecirc;&icirc; &eth;&aring;&agrave;&euml;&egrave;&ccedil;&agrave;&ouml;&egrave;&yuml; &yacute;&ograve;&icirc;&eacute; &egrave;&auml;&aring;&egrave; &ograve;&eth;&aring;&aacute;&oacute;&aring;&ograve;
&iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&eacute; &acirc;&icirc;&ccedil;&iacute;&egrave;.
&Auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&atilde;&icirc; &divide;&egrave;&ntilde;&euml;&agrave; ε &gt; 0 &igrave;&ucirc; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&egrave;&igrave; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &acirc;&ntilde;&iuml;&icirc;&igrave;&icirc;&atilde;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&aring; &icirc;&auml;&iacute;&icirc;&ccedil;&iacute;&agrave;&divide;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;
&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f ε : X → X . &times;&ograve;&icirc;&aacute;&ucirc; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&egrave;&ograve;&uuml; f ε , &igrave;&ucirc; &acirc;&eth;&aring;&igrave;&aring;&iacute;&iacute;&icirc; &ccedil;&agrave;&ocirc;&egrave;&ecirc;&ntilde;&egrave;&eth;&oacute;&aring;&igrave; ε, &egrave; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&egrave;&igrave;
&divide;&aring;&eth;&aring;&ccedil; U (x) ε-&icirc;&ecirc;&eth;&aring;&ntilde;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave; x ∈ X . &Igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; U (x), x ∈ X , &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&thorn;&ograve;
&icirc;&ograve;&ecirc;&eth;&ucirc;&ograve;&icirc;&aring; &iuml;&icirc;&ecirc;&eth;&ucirc;&ograve;&egrave;&aring; X , &egrave; &acirc; &ntilde;&egrave;&euml;&oacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&atilde;&icirc; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&aring; &divide;&egrave;&ntilde;&euml;&icirc; &ograve;&icirc;
&divide;&aring;&ecirc; x1 , . . . , xm ∈ X , &ograve;&agrave;&ecirc;&egrave;&otilde; &divide;&ograve;&icirc; &oslash;&agrave;&eth;&ucirc; Ui = U (xi ) &iuml;&icirc;&ecirc;&eth;&ucirc;&acirc;&agrave;&thorn;&ograve; X . &Auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&atilde;&icirc; i &icirc;&ograve; 1 &auml;&icirc; m
&eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&thorn; ui : X → R, &ccedil;&agrave;&auml;&agrave;&iacute;&iacute;&oacute;&thorn; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&icirc;&eacute;
ui (x) = max(0, ε − r(x, xi )).
&Icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&icirc;, &divide;&ograve;&icirc; ui &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&agrave;, &eth;&agrave;&acirc;&iacute;&agrave; &iacute;&oacute;&euml;&thorn; &acirc;&iacute;&aring; &oslash;&agrave;&eth;&agrave; Ui &egrave; &icirc;&ograve;&euml;&egrave;&divide;&iacute;&agrave; &icirc;&ograve; &iacute;&oacute;&euml;&yuml; &acirc;&iacute;&oacute;&ograve;&eth;&egrave; Ui .
&Acirc;&ucirc;&aacute;&aring;&eth;&aring;&igrave; &iuml;&icirc; &ograve;&icirc;&divide;&ecirc;&aring; yi ∈ F (xi ), i = 1, . . . , m. &Iacute;&agrave;&ecirc;&icirc;&iacute;&aring;&ouml;, &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&igrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f ε : X → X &iuml;&icirc;
&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&aring;
P
ui (x)yi
ε
f (x) = Pi
.
i ui (x)
14
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 2.
&Ograve;&Aring;&Icirc;&ETH;&Aring;&Igrave;&Agrave; &Aacute;&ETH;&Agrave;&Oacute;&Yacute;&ETH;&Agrave;: &Ocirc;&Icirc;&ETH;&Igrave;&Oacute;&Euml;&Egrave;&ETH;&Icirc;&Acirc;&Ecirc;&Agrave; &Egrave; &Icirc;&Aacute;&Ntilde;&Oacute;&AElig;&Auml;&Aring;&Iacute;&Egrave;&Aring;
P
&Ccedil;&auml;&aring;&ntilde;&uuml; &acirc;&agrave;&aelig;&iacute;&icirc; &icirc;&ograve;&igrave;&aring;&ograve;&egrave;&ograve;&uuml;, &divide;&ograve;&icirc; &ccedil;&iacute;&agrave;&igrave;&aring;&iacute;&agrave;&ograve;&aring;&euml;&uuml; i ui (x) &gt; 0 &iuml;&eth;&egrave; &euml;&thorn;&aacute;&icirc;&igrave; x, &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &ecirc;&agrave;&aelig;&auml;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave;
x &iuml;&icirc;&iuml;&agrave;&auml;&agrave;&aring;&ograve; &acirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&eacute; &oslash;&agrave;&eth; Ui .
ε
&Ntilde;&igrave;&ucirc;&ntilde;&euml; &yacute;&ograve;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc;
&ecirc;&icirc;&igrave;&aacute;&egrave;&iacute;&agrave;&ouml;&egrave;&yuml; &ograve;&icirc;&divide;&aring;&ecirc; yi &egrave;&ccedil; F (xi )
P&acirc; &ograve;&icirc;&igrave;, &divide;&ograve;&icirc; &ograve;&icirc;&divide;&ecirc;&agrave; f (x) &aring;&ntilde;&ograve;&uuml; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&agrave;&yuml;
&acirc;&aring;&ntilde;&agrave;&igrave;&egrave; αi (x) = ui (x)/( j uj (x)). &Iacute;&agrave; &ntilde;&agrave;&igrave;&icirc;&igrave; &auml;&aring;&euml;&aring;, &ograve;&icirc;&divide;&ecirc;&agrave; f ε (x) &aring;&ntilde;&ograve;&uuml; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&agrave;&yuml; &ecirc;&icirc;&igrave;&aacute;&egrave;&iacute;&agrave;&ouml;&egrave;&yuml;
&ograve;&agrave;&ecirc;&egrave;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; yi , &auml;&euml;&yuml; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; ρ(x, xi ) &lt; ε, &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &auml;&euml;&yuml; &auml;&eth;&oacute;&atilde;&egrave;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; &acirc;&aring;&ntilde;&icirc;&acirc;&ucirc;&aring; &ecirc;&icirc;&yacute;&ocirc;&ocirc;&egrave;&ouml;&egrave;&aring;&iacute;&ograve;&ucirc;
αi (x) &icirc;&aacute;&eth;&agrave;&ugrave;&agrave;&thorn;&ograve;&ntilde;&yuml; &acirc; &iacute;&oacute;&euml;&uuml;.
&Ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; f ε &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&agrave; &egrave; &iuml;&icirc; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &egrave;&igrave;&aring;&aring;&ograve; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute; xε = f ε (xε ). &Egrave;
&yacute;&ograve;&icirc; &iuml;&eth;&egrave; &ecirc;&agrave;&aelig;&auml;&icirc;&igrave; ε. &Acirc; &ntilde;&egrave;&euml;&oacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave; X &igrave;&icirc;&aelig;&iacute;&icirc; &iacute;&agrave;&eacute;&ograve;&egrave; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&uuml; ε → 0, &ograve;&agrave;&ecirc;&oacute;&thorn;
&divide;&ograve;&icirc; &ograve;&icirc;&divide;&ecirc;&egrave; xε &ntilde;&otilde;&icirc;&auml;&yuml;&ograve;&ntilde;&yuml; &ecirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&aring; x∗ ∈ X . &Igrave;&ucirc; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&agrave;&aring;&igrave;, &divide;&ograve;&icirc; &ograve;&icirc;&divide;&ecirc;&agrave; x∗ &aacute;&oacute;&auml;&aring;&ograve;
&iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &auml;&euml;&yuml; F , &ograve;. &aring;. x∗ ∈ F (x∗ ).
&Iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&iacute;&icirc;&aring;, &ograve;. &aring;. &divide;&ograve;&icirc; x∗ &iacute;&aring; &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; F (x∗ ). &Acirc; &ntilde;&egrave;&euml;&oacute; &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;&ntilde;&ograve;&egrave; F (x∗ )
&yacute;&ograve;&icirc; &icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&ograve;, &divide;&ograve;&icirc; &eth;&agrave;&ntilde;&ntilde;&ograve;&icirc;&yuml;&iacute;&egrave;&aring; &icirc;&ograve; x∗ &auml;&icirc; F (x∗ ) &aacute;&icirc;&euml;&uuml;&oslash;&aring; &iacute;&oacute;&euml;&yuml;. &Iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave; δ = ρ(x∗ , F (x∗ ))/2 &egrave;
&eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; δ -&eth;&agrave;&ntilde;&oslash;&egrave;&eth;&aring;&iacute;&egrave;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; F (x∗ ), &ograve;. &aring;. &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; F (x∗ )δ = {y ∈ X, ρ(y, F (x∗ )) &lt;
δ}. &Acirc; &ntilde;&egrave;&euml;&oacute; &iuml;&icirc;&euml;&oacute;&iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&ntilde;&ograve;&egrave; F &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ograve;&agrave;&ecirc;&icirc;&aring; &divide;&egrave;&ntilde;&euml;&icirc; ε &gt; 0 (&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &igrave;&icirc;&aelig;&iacute;&icirc; &ntilde;&divide;&egrave;&ograve;&agrave;&ograve;&uuml;
&iacute;&aring; &iuml;&eth;&aring;&acirc;&ucirc;&oslash;&agrave;&thorn;&ugrave;&egrave;&igrave; δ ), &divide;&ograve;&icirc; F (x0 ) ⊂ F (x∗ )δ &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave; x0 , &oacute;&auml;&agrave;&euml;&aring;&iacute;&iacute;&icirc;&eacute; &icirc;&ograve; x∗ &igrave;&aring;&iacute;&aring;&aring; &divide;&aring;&igrave;
&iacute;&agrave; ε. &Icirc;&aacute;&eth;&agrave;&ograve;&egrave;&igrave;&ntilde;&yuml; &ograve;&aring;&iuml;&aring;&eth;&uuml; &ecirc; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&iacute;&icirc;&eacute; &eth;&agrave;&iacute;&aring;&aring; &ograve;&icirc;&divide;&ecirc;&aring; xε/2 , &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml;
f ε/2 , &ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &oacute;&auml;&agrave;&euml;&aring;&iacute;&agrave; &icirc;&ograve; x∗ &igrave;&aring;&iacute;&aring;&aring; &divide;&aring;&igrave; &iacute;&agrave; ε/2. &Igrave;&ucirc; &oacute;&aelig;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&egrave;&euml;&egrave;, &divide;&ograve;&icirc; xε/2 &aring;&ntilde;&ograve;&uuml; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&agrave;&yuml;
&ecirc;&icirc;&igrave;&aacute;&egrave;&iacute;&agrave;&ouml;&egrave;&yuml; &ograve;&icirc;&divide;&aring;&ecirc; yi ∈ F (xi ), &iuml;&eth;&egrave;&divide;&aring;&igrave; &ograve;&icirc;&divide;&ecirc;&egrave; xi &oacute;&auml;&agrave;&euml;&aring;&iacute;&ucirc; &icirc;&ograve; xε/2 &igrave;&aring;&iacute;&aring;&aring; &divide;&aring;&igrave; &iacute;&agrave; ε/2. &Acirc; &ntilde;&egrave;&euml;&oacute;
&iacute;&aring;&eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&agrave; &ograve;&eth;&aring;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave; &ograve;&icirc;&divide;&ecirc;&egrave; xi &oacute;&auml;&agrave;&euml;&aring;&iacute;&ucirc; &icirc;&ograve; x∗ &igrave;&aring;&iacute;&aring;&aring; &divide;&aring;&igrave; &iacute;&agrave; ε, &iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&oacute;&thorn;&ugrave;&egrave;&aring;
&ograve;&icirc;&divide;&ecirc;&egrave; yi &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&agrave;&ograve; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&oacute; F (x∗ )δ . &Ecirc;&agrave;&ecirc; &euml;&aring;&atilde;&ecirc;&icirc; &iuml;&icirc;&iacute;&yuml;&ograve;&uuml;, &euml;&thorn;&aacute;&agrave;&yuml; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&agrave;&yuml; &ecirc;&icirc;&igrave;&aacute;&egrave;&iacute;&agrave;&ouml;&egrave;&yuml;
&ograve;&icirc;&divide;&aring;&ecirc; yi &ograve;&agrave;&ecirc;&aelig;&aring; &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&oacute; F (x∗ )δ (&egrave;&aacute;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; F (x∗ )δ &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;, &ecirc;&agrave;&ecirc; &egrave;
F (x∗ )); &iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; xε/2 ∈ F (x∗ )δ . &Ccedil;&iacute;&agrave;&divide;&egrave;&ograve;
ρ(x∗ , F (x∗ )) ≤ ρ(x∗ , xε/2 ) + (xε/2 , F (x∗ )) &lt; ε/2 + δ ≤ 2δ = ρ(x∗ , F (x∗ )),
&divide;&ograve;&icirc; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&aring;&igrave;.
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Ograve;&agrave;&eth;&ntilde;&ecirc;&icirc;&atilde;&icirc;
&Iuml;&eth;&egrave;&acirc;&aring;&auml;&aring;&igrave; &aring;&ugrave;&aring; &icirc;&auml;&egrave;&iacute; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve; &icirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&ecirc;&agrave;&otilde;, &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&eacute; &iacute;&agrave; &iuml;&aring;&eth;&acirc;&ucirc;&eacute; &acirc;&ccedil;&atilde;&euml;&yuml;&auml; &icirc;&ograve;&iacute;&icirc;&ntilde;&egrave;&ograve;
&ntilde;&yuml; &ecirc; &ntilde;&icirc;&acirc;&aring;&eth;&oslash;&aring;&iacute;&iacute;&icirc; &auml;&eth;&oacute;&atilde;&icirc;&eacute; &ntilde;&egrave;&ograve;&oacute;&agrave;&ouml;&egrave;&egrave;, &divide;&aring;&igrave; &igrave;&ucirc; &eth;&agrave;&ntilde;&ntilde;&igrave;&agrave;&ograve;&eth;&egrave;&acirc;&agrave;&euml;&egrave; &acirc;&ucirc;&oslash;&aring;. &Acirc; &iacute;&aring;&igrave; &eth;&aring;&divide;&uuml; &iuml;&icirc;&eacute;&auml;&aring;&ograve; &iacute;&aring; &icirc;
&iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&otilde; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml;&otilde; &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&otilde; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;, &agrave; &icirc; &igrave;&icirc;&iacute;&icirc;&ograve;&icirc;&iacute;&iacute;&ucirc;&otilde; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml;&otilde;
&oacute;&iuml;&icirc;&eth;&yuml;&auml;&icirc;&divide;&aring;&iacute;&iacute;&ucirc;&otilde; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;. &Ntilde;&iacute;&agrave;&divide;&agrave;&euml;&agrave; &iacute;&aring;&ntilde;&ecirc;&icirc;&euml;&uuml;&ecirc;&icirc; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&eacute;.
&Icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring;. &Oacute;&iuml;&icirc;&eth;&yuml;&auml;&icirc;&divide;&aring;&iacute;&iacute;&icirc;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &yacute;&ograve;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; X , &ntilde;&iacute;&agrave;&aacute;&aelig;&aring;&iacute;&iacute;&icirc;&aring; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring;&igrave;
(&divide;&agrave;&ntilde;&ograve;&egrave;&divide;&iacute;&icirc;&atilde;&icirc;) &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; ≥.
&Egrave;&iacute;&agrave;&divide;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&yuml;, &auml;&euml;&yuml; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; (&iacute;&aring; &icirc;&aacute;&yuml;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc; &acirc;&ntilde;&aring;&otilde;) &iuml;&agrave;&eth; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&acirc; (x, y) &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml;
&ntilde;&icirc;&icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; x ≥ y . &Iuml;&eth;&egrave; &yacute;&ograve;&icirc;&igrave; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&aring;&ograve;&ntilde;&yuml;, &divide;&ograve;&icirc; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; ≥ &eth;&aring;&ocirc;&euml;&aring;&ecirc;&ntilde;&egrave;&acirc;&iacute;&icirc; (&ograve;. &aring;. x ≥ x
∀x), &ograve;&eth;&agrave;&iacute;&ccedil;&egrave;&ograve;&egrave;&acirc;&iacute;&icirc; (&ograve;. &aring;. &egrave;&ccedil; x ≥ y &egrave; y ≥ z &ntilde;&euml;&aring;&auml;&oacute;&aring;&ograve;, &divide;&ograve;&icirc; x ≥ z ), &egrave; &agrave;&iacute;&ograve;&egrave;&ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc; (&ograve;. &aring;. &egrave;&ccedil;
x ≥ y &egrave; y ≥ x &ntilde;&euml;&aring;&auml;&oacute;&aring;&ograve; x = y ).
&Iuml;&eth;&egrave;&igrave;&aring;&eth;&ucirc; &oacute;&iuml;&icirc;&eth;&yuml;&auml;&icirc;&divide;&aring;&iacute;&iacute;&ucirc;&otilde; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;: &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; Z &ouml;&aring;&euml;&ucirc;&otilde; &divide;&egrave;&ntilde;&aring;&euml;, &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; R &acirc;&aring;&ugrave;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;
&iacute;&ucirc;&otilde; &divide;&egrave;&ntilde;&aring;&euml; &ntilde; &icirc;&aacute;&ucirc;&divide;&iacute;&ucirc;&igrave;&egrave; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;&igrave;&egrave;; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&atilde;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; &ntilde; &icirc;&ograve;&iacute;&icirc;
&oslash;&aring;&iacute;&egrave;&aring;&igrave; &acirc;&ecirc;&euml;&thorn;&divide;&aring;&iacute;&egrave;&yuml; ⊂.
&Icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f : X → Y &oacute;&iuml;&icirc;&eth;&yuml;&auml;&icirc;&divide;&aring;&iacute;&iacute;&icirc;&atilde;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; X &acirc; &oacute;&iuml;&icirc;&eth;&yuml;&auml;&icirc;&divide;&aring;&iacute;&iacute;&icirc;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; Y
&igrave;&icirc;&iacute;&icirc;&ograve;&icirc;&iacute;&iacute;&icirc; (&acirc;&icirc;&ccedil;&eth;&agrave;&ntilde;&ograve;&agrave;&thorn;&ugrave;&aring;&aring;), &aring;&ntilde;&euml;&egrave; &egrave;&ccedil; x ≥ x0 &ntilde;&euml;&aring;&auml;&oacute;&aring;&ograve;, &divide;&ograve;&icirc; f (x) ≥ f (x0 ).
&Iuml;&oacute;&ntilde;&ograve;&uuml; A &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &oacute;&iuml;&icirc;&eth;&yuml;&auml;&icirc;&divide;&aring;&iacute;&iacute;&icirc;&atilde;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; X . &Ograve;&icirc;&divide;&ecirc;&agrave; x ∈ X &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ograve;&icirc;&divide;
&iacute;&icirc;&eacute; &acirc;&aring;&eth;&otilde;&iacute;&aring;&eacute; &atilde;&eth;&agrave;&iacute;&uuml;&thorn; A &egrave; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&ograve;&ntilde;&yuml; sup(A), &aring;&ntilde;&euml;&egrave; &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&aring;&iacute;&ucirc; &auml;&acirc;&agrave; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml;:
1) x ≥ a ∀ a ∈ A, &egrave;
2) &aring;&ntilde;&euml;&egrave; y ≥ a ∀ a ∈ A, &ograve;&icirc; y ≥ x.
15
&Agrave;&iacute;&agrave;&euml;&icirc;&atilde;&egrave;&divide;&iacute;&icirc; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &ograve;&icirc;&divide;&iacute;&agrave;&yuml; &iacute;&egrave;&aelig;&iacute;&yuml;&yuml; &atilde;&eth;&agrave;&iacute;&uuml; inf .
&Icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring;. &Oacute;&iuml;&icirc;&eth;&yuml;&auml;&icirc;&divide;&aring;&iacute;&iacute;&icirc;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; (X, ≥) &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &iuml;&icirc;&euml;&iacute;&icirc;&eacute; &eth;&aring;&oslash;&aring;&ograve;&ecirc;&icirc;&eacute;, &aring;&ntilde;&euml;&egrave; &auml;&euml;&yuml;
&euml;&thorn;&aacute;&icirc;&atilde;&icirc; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; A ⊂ X &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ecirc;&agrave;&ecirc; sup(A), &ograve;&agrave;&ecirc; &egrave; inf(A).
&Iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;, &euml;&thorn;&aacute;&icirc;&eacute; &icirc;&ograve;&eth;&aring;&ccedil;&icirc;&ecirc; [a, b] &acirc; R &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &iuml;&icirc;&euml;&iacute;&icirc;&eacute; &eth;&aring;&oslash;&aring;&ograve;&ecirc;&icirc;&eacute;, &egrave;&euml;&egrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; 2Z &acirc;&ntilde;&aring;&otilde;
&iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; Z. &Igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; R &acirc;&ntilde;&aring;&otilde; &divide;&egrave;&ntilde;&aring;&euml; &iuml;&icirc;&euml;&iacute;&icirc;&eacute; &eth;&aring;&oslash;&aring;&ograve;&ecirc;&icirc;&eacute; &iacute;&aring;
&yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml;.
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; (&Ograve;&agrave;&eth;&ntilde;&ecirc;&egrave;&eacute;). &Iuml;&oacute;&ntilde;&ograve;&uuml; f : X → X &igrave;&icirc;&iacute;&icirc;&ograve;&icirc;&iacute;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &iuml;&icirc;&euml;&iacute;&icirc;&eacute; &eth;&aring;&oslash;&aring;&ograve;&ecirc;&egrave; X &acirc;
&ntilde;&aring;&aacute;&yuml;. &Ograve;&icirc;&atilde;&auml;&agrave; &icirc;&iacute;&icirc; &egrave;&igrave;&aring;&aring;&ograve; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute;.
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;. &Ntilde;&iacute;&icirc;&acirc;&agrave; &egrave;&auml;&aring;&yuml; &iuml;&eth;&icirc;&ntilde;&ograve;&agrave;: &acirc;&icirc;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&icirc;&acirc;&agrave;&ograve;&uuml;&ntilde;&yuml; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&igrave;&egrave; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&egrave;&yuml;
&igrave;&egrave;, &iacute;&agrave;&divide;&egrave;&iacute;&agrave;&yuml; &ntilde; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave; 0 = inf(X). &Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&icirc;, &divide;&ograve;&icirc; x0 = 0 ≤ f (0) = x1 ,
&ograve;&icirc; &acirc; &ntilde;&egrave;&euml;&oacute; &igrave;&icirc;&iacute;&icirc;&ograve;&icirc;&iacute;&iacute;&icirc;&ntilde;&ograve;&egrave; x2 = f (x1 ) ≥ f (x0 ) = x1 , &egrave; &ograve;. &auml;. &Iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc;&icirc;&ccedil;&eth;&agrave;&ntilde;&ograve;&agrave;&thorn;&ugrave;&agrave;&yuml; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;
&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&uuml; &ograve;&icirc;&divide;&aring;&ecirc;
x0 ≤ x1 ≤ &middot; &middot; &middot; ≤ xn ≤ . . .
&Iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave; x∞ = sup(xi , i = 1, . . . , n, . . . ). &Ntilde;&iacute;&icirc;&acirc;&agrave; &egrave;&ccedil; &igrave;&icirc;&iacute;&icirc;&ograve;&icirc;&iacute;&iacute;&icirc;&ntilde;&ograve;&egrave; f (x∞ ) ≥ x∞ .
&Ecirc; &ntilde;&icirc;&aelig;&agrave;&euml;&aring;&iacute;&egrave;&thorn;, &ograve;&oacute;&ograve; &igrave;&icirc;&aelig;&aring;&ograve; &aacute;&ucirc;&ograve;&uuml; &ntilde;&ograve;&eth;&icirc;&atilde;&icirc;&aring; &iacute;&aring;&eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&icirc;, &egrave; &iuml;&eth;&icirc;&ouml;&aring;&ntilde;&ntilde; &iacute;&oacute;&aelig;&iacute;&icirc; &iuml;&eth;&icirc;&auml;&icirc;&euml;&aelig;&egrave;&ograve;&uuml;, &iacute;&agrave;&divide;&egrave;
&iacute;&agrave;&yuml; &ntilde; x∞ . &times;&ograve;&icirc;&aacute;&ucirc; &iacute;&aring; &iuml;&eth;&egrave;&acirc;&euml;&aring;&ecirc;&agrave;&ograve;&uuml; &ograve;&eth;&agrave;&iacute;&ntilde;&ocirc;&egrave;&iacute;&egrave;&ograve;&iacute;&oacute;&thorn; &egrave;&iacute;&auml;&oacute;&ecirc;&ouml;&egrave;&thorn;, &iacute;&oacute;&aelig;&iacute;&icirc; &iuml;&icirc;-&auml;&eth;&oacute;&atilde;&icirc;&igrave;&oacute; &eth;&aring;&agrave;&euml;&egrave;&ccedil;&icirc;&acirc;&agrave;&ograve;&uuml;
&yacute;&ograve;&oacute; &egrave;&auml;&aring;&thorn;.
&ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; A &ograve;&agrave;&ecirc;&egrave;&otilde; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&acirc; x ∈ X , &divide;&ograve;&icirc; x ≤ f (x). &Auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&icirc; &yuml;&ntilde;&iacute;&icirc;, &divide;&ograve;&icirc;
&aring;&ntilde;&euml;&egrave; x ∈ A, &ograve;&icirc; f (x) ∈ A. &Iuml;&oacute;&ntilde;&ograve;&uuml; a = sup(A). &Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; a ≥ x ∀x ∈ A, &ograve;&icirc; &iuml;&icirc; &igrave;&icirc;&iacute;&icirc;&ograve;&icirc;&iacute;&iacute;&icirc;&ntilde;&ograve;&egrave;
f (a) ≥ f (x) ≥ x, &iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; f (a) &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &acirc;&aring;&eth;&otilde;&iacute;&aring;&eacute; &atilde;&eth;&agrave;&iacute;&uuml;&thorn; &auml;&euml;&yuml; A, &egrave; &ccedil;&iacute;&agrave;&divide;&egrave;&ograve; f (a) ≥ a &iuml;&icirc;
&icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&thorn; &ograve;&icirc;&divide;&iacute;&icirc;&eacute; &acirc;&aring;&eth;&otilde;&iacute;&aring;&eacute; &atilde;&eth;&agrave;&iacute;&egrave;. &Ccedil;&iacute;&agrave;&divide;&egrave;&ograve; a &ograve;&icirc;&aelig;&aring; &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; A, &ecirc;&agrave;&ecirc; &egrave; f (a). &Iacute;&icirc; &ograve;&icirc;&atilde;&auml;&agrave;
a ≥ f (a), &egrave; &acirc; &ntilde;&egrave;&euml;&oacute; &agrave;&iacute;&ograve;&egrave;&ntilde;&egrave;&igrave;&igrave;&aring;&ograve;&eth;&egrave;&divide;&iacute;&icirc;&ntilde;&ograve;&egrave; a = f (a).
&Ecirc;&agrave;&ecirc; &oacute;&aelig;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&egrave;&euml;&icirc;&ntilde;&uuml;, &iacute;&agrave; &iuml;&aring;&eth;&acirc;&ucirc;&eacute; &acirc;&ccedil;&atilde;&euml;&yuml;&auml; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Ograve;&agrave;&eth;&ntilde;&ecirc;&icirc;&atilde;&icirc; &egrave;&igrave;&aring;&aring;&ograve; &igrave;&agrave;&euml;&icirc; &icirc;&aacute;&ugrave;&aring;&atilde;&icirc; &ntilde; &ograve;&aring;&icirc;&eth;&aring;
&igrave;&icirc;&eacute; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;. &Iacute;&icirc; &yacute;&ograve;&icirc; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &iacute;&agrave; &iuml;&aring;&eth;&acirc;&ucirc;&eacute; &acirc;&ccedil;&atilde;&euml;&yuml;&auml;. &Egrave;&igrave;&aring;&aring;&ograve;&ntilde;&yuml; &aacute;&icirc;&euml;&aring;&aring; &icirc;&aacute;&ugrave;&aring;&aring; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&aring; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&atilde;&icirc;
&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave;, &acirc;&ecirc;&euml;&thorn;&divide;&agrave;&thorn;&ugrave;&aring;&aring; &acirc; &ntilde;&aring;&aacute;&yuml; &ecirc;&agrave;&ecirc; &divide;&agrave;&ntilde;&ograve;&iacute;&ucirc;&aring; &ntilde;&euml;&oacute;&divide;&agrave;&egrave; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&aring; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; &egrave; &eth;&aring;&oslash;&aring;&ograve;&ecirc;&egrave;.
&Iuml;&eth;&egrave; &ograve;&agrave;&ecirc;&icirc;&igrave; &acirc;&ccedil;&atilde;&euml;&yuml;&auml;&aring; &igrave;&icirc;&iacute;&icirc;&ograve;&icirc;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml; &iuml;&eth;&aring;&acirc;&eth;&agrave;&ugrave;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&ntilde;&ograve;&uuml;, &agrave; &ograve;&eth;&aring;&aacute;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &iuml;&icirc;&euml;&iacute;&icirc;&ograve;&ucirc; &eth;&aring;
&oslash;&aring;&ograve;&ecirc;&egrave; &ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&egrave;&ograve;&ntilde;&yuml; &agrave;&iacute;&agrave;&euml;&icirc;&atilde;&icirc;&igrave; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave;. &Ograve;&agrave;&ecirc; &divide;&ograve;&icirc; &icirc;&aacute;&aring; &yacute;&ograve;&egrave; &ccedil;&iacute;&agrave;&igrave;&aring;&iacute;&egrave;&ograve;&ucirc;&aring; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&yuml;&ograve;&ntilde;&yuml;
&divide;&agrave;&ntilde;&ograve;&iacute;&ucirc;&igrave;&egrave; &ntilde;&euml;&oacute;&divide;&agrave;&yuml;&igrave;&egrave; &aacute;&icirc;&euml;&aring;&aring; &icirc;&aacute;&ugrave;&aring;&atilde;&icirc; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&aring;&iacute;&egrave;&yuml;.
&Oacute;&iuml;&eth;&agrave;&aelig;&iacute;&aring;&iacute;&egrave;&yuml;
2.1. &Iuml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml; &acirc;&ntilde;&aring;&otilde; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&eacute; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;: &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&ntilde;&ograve;&uuml;, &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;&ntilde;&ograve;&uuml; &egrave;
&icirc;&atilde;&eth;&agrave;&iacute;&egrave;&divide;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml; X , &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&ntilde;&ograve;&uuml; f .
2.2. &Iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &yacute;&ecirc;&acirc;&egrave;&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&agrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&eacute; &ograve;&aring;&icirc;&eth;&aring;&igrave;&aring; &icirc; &iacute;&aring;&eth;&aring;&ograve;&eth;&agrave;&atilde;&egrave;&eth;&oacute;&aring;&igrave;&icirc;&ntilde;&ograve;&egrave;:
&iacute;&aring; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&eacute; &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&egrave; n-&igrave;&aring;&eth;&iacute;&icirc;&atilde;&icirc; &auml;&egrave;&ntilde;&ecirc;&agrave; D &iacute;&agrave; &aring;&atilde;&icirc; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&oacute; ∂D = S .
2.3. &Iuml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &yacute;&ecirc;&acirc;&egrave;&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&agrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&eacute; &ograve;&aring;&icirc;&eth;&aring;&igrave;&aring; &icirc; &ntilde;&thorn;&eth;&uacute;&aring;&ecirc;&ograve;&egrave;&acirc;&iacute;&icirc;&ntilde;&ograve;&egrave;:
&aring;&ntilde;&euml;&egrave; f : D → D &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &auml;&egrave;&ntilde;&ecirc;&agrave; &acirc; &ntilde;&aring;&aacute;&yuml;, &icirc;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&thorn;&ugrave;&aring;&aring; &iacute;&agrave; &igrave;&aring;&ntilde;&ograve;&aring; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&oacute; ∂D,
&ograve;&icirc; f &ntilde;&thorn;&eth;&uacute;&aring;&ecirc;&ograve;&egrave;&acirc;&iacute;&icirc; (&ograve;. &aring;. &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &frac34;&iacute;&agrave;&iquest;).
2.4. &Iuml;&oacute;&ntilde;&ograve;&uuml; D n-&igrave;&aring;&eth;&iacute;&ucirc;&eacute; &oslash;&agrave;&eth; &acirc; Rn , &egrave; &iuml;&oacute;&ntilde;&ograve;&uuml; f &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; D &acirc; Rn , &ecirc;&icirc;
&ograve;&icirc;&eth;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&agrave;&aring;&ograve; &atilde;&eth;&agrave;&iacute;&egrave;&divide;&iacute;&oacute;&thorn; &ntilde;&ocirc;&aring;&eth;&oacute; ∂D &acirc; D. &Iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave;.
(&Oacute;&ecirc;&agrave;&ccedil;&agrave;&iacute;&egrave;&aring;: &egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&icirc;&acirc;&agrave;&ograve;&uuml; &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&thorn; Rn &iacute;&agrave; D.)
2.5. &Ccedil;&agrave;&acirc;&aring;&eth;&oslash;&egrave;&ograve;&aring; &iacute;&agrave;&igrave;&aring;&divide;&aring;&iacute;&iacute;&icirc;&aring; &acirc;&ucirc;&oslash;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc; &icirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&aring; &acirc; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &agrave;&ocirc;&ocirc;&egrave;&iacute;&iacute;&icirc;&atilde;&icirc;
&icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&atilde;&icirc; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&agrave; &acirc; &ntilde;&aring;&aacute;&yuml;
16
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 2.
&Ograve;&Aring;&Icirc;&ETH;&Aring;&Igrave;&Agrave; &Aacute;&ETH;&Agrave;&Oacute;&Yacute;&ETH;&Agrave;: &Ocirc;&Icirc;&ETH;&Igrave;&Oacute;&Euml;&Egrave;&ETH;&Icirc;&Acirc;&Ecirc;&Agrave; &Egrave; &Icirc;&Aacute;&Ntilde;&Oacute;&AElig;&Auml;&Aring;&Iacute;&Egrave;&Aring;
2.6. &Iuml;&eth;&egrave;&acirc;&aring;&ntilde;&ograve;&egrave; &iuml;&eth;&egrave;&igrave;&aring;&eth;&ucirc; &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&ucirc;&otilde;, &iacute;&icirc; &iacute;&aring; &iuml;&icirc;&euml;&oacute;&iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&otilde; &ntilde;&acirc;&aring;&eth;&otilde;&oacute; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&eacute;, &agrave; &ograve;&agrave;&ecirc;&aelig;&aring;
&iuml;&icirc;&euml;&oacute;&iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&otilde; &ntilde;&acirc;&aring;&eth;&otilde;&oacute;, &iacute;&icirc; &iacute;&aring; &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&ucirc;&otilde; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&eacute;.
2.7. &Iuml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc; &aring;&ntilde;&euml;&egrave; Y &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;, &ograve;&icirc; &euml;&thorn;&aacute;&icirc;&aring; &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;&aring; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; &egrave;&ccedil; X &acirc; Y &iuml;&icirc;&euml;&oacute;&iacute;&aring;
&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc; &ntilde;&acirc;&aring;&eth;&otilde;&oacute;.
2.8. &Icirc;&aacute;&eth;&agrave;&ograve;&iacute;&icirc;, &aring;&ntilde;&euml;&egrave; &acirc;&ntilde;&aring; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&yuml; &iuml;&icirc;&euml;&oacute;&iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&atilde;&icirc; &ntilde;&acirc;&aring;&eth;&otilde;&oacute; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&yuml; F &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&ucirc;, &ograve;&icirc; F
&iuml;&icirc;&euml;&oacute;&iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc; &ntilde;&acirc;&aring;&eth;&otilde;&oacute;.
2.9. &Iuml;&oacute;&ntilde;&ograve;&uuml; u : X &times; T → R &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&agrave;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml;. &ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; F : T ⇒ X ,
&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &ograve;&icirc;&divide;&ecirc;&aring; t ∈ T &ntilde;&icirc;&iuml;&icirc;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&aring;&ograve; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&oacute;&igrave;&icirc;&acirc; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; u(&middot;, t) &iacute;&agrave; X , &ograve;. &aring;.
F (t) = {x ∈ X, u(x, t) ≥ u(x0 , t) ∀ x0 ∈ X}.
&Iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; F &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;.
2.10. &Auml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&aring; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&aring;&iacute;&egrave;&aring;: &iuml;&oacute;&ntilde;&ograve;&uuml; X &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&eacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;, &agrave; F : X ⇒ X &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;&aring; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; &ntilde; &iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&ucirc;&igrave;&egrave; &icirc;&aacute;&eth;&agrave;&ccedil;&agrave;&igrave;&egrave;. &Ograve;&icirc;&atilde;&auml;&agrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ograve;&agrave;&ecirc;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave; x∗ , &ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml;
&iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; conv(F (x∗ )) &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&eacute; &icirc;&aacute;&icirc;&euml;&icirc;&divide;&ecirc;&aring; F (x∗ ).
2.11. &Iuml;&oacute;&ntilde;&ograve;&uuml; f &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &icirc;&ograve;&eth;&aring;&ccedil;&ecirc;&agrave; &acirc; &ntilde;&aring;&aacute;&yuml;, &icirc;&aacute;&euml;&agrave;&auml;&agrave;&thorn;&ugrave;&aring;&aring; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&igrave; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc;&igrave;, &icirc;&ntilde;&euml;&agrave;&aacute;&euml;&yuml;
&thorn;&ugrave;&egrave;&igrave; &ecirc;&agrave;&ecirc; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&ntilde;&ograve;&uuml;, &ograve;&agrave;&ecirc; &egrave; &igrave;&icirc;&iacute;&icirc;&ograve;&icirc;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml;: &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave; x &icirc;&ograve;&eth;&aring;&ccedil;&ecirc;&agrave;
lim f (x) ≤ f (x) ≤ lim f (x).
x→x−
x→x+
&Iuml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc; f &icirc;&aacute;&euml;&agrave;&auml;&agrave;&aring;&ograve; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&icirc;&eacute;.
2.12. &Iuml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&eacute; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Ecirc;&agrave;&ecirc;&oacute;&ograve;&agrave;&iacute;&egrave;.
&ETH;&aring;&ecirc;&icirc;&igrave;&aring;&iacute;&auml;&oacute;&aring;&igrave;&agrave;&yuml; &euml;&egrave;&ograve;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave;: [6, 8, 11].
&Euml;&aring;&ecirc;&ouml;&egrave;&yuml; 3
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;: &iuml;&eth;&egrave;&igrave;&aring;&iacute;&aring;&iacute;&egrave;&yuml;
&Ccedil;&auml;&aring;&ntilde;&uuml; &igrave;&ucirc; &eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &divide;&aring;&ograve;&ucirc;&eth;&aring; &iuml;&eth;&egrave;&igrave;&aring;&iacute;&aring;&iacute;&egrave;&yuml; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; (&egrave;&euml;&egrave; &aring;&aring; &acirc;&agrave;&eth;&egrave;&agrave;&iacute;&ograve;&icirc;&acirc;): &ecirc; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;
&acirc;&agrave;&iacute;&egrave;&thorn; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;, &ecirc; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&thorn; &Iacute;&yacute;&oslash;&agrave; &acirc; &egrave;&atilde;&eth;&agrave;&otilde;, &ecirc; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&igrave;&oacute; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;
&ntilde;&egrave;&thorn;, &egrave; &ecirc; &yuml;&auml;&eth;&oacute;.
&Igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&ucirc;
&Iuml;&oacute;&ntilde;&ograve;&uuml; &lt; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &aacute;&egrave;&iacute;&agrave;&eth;&iacute;&icirc;&aring; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; &iacute;&agrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&aring; X , &ograve;. &aring;. &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &acirc; X &times; X .
&Ntilde;&icirc;&icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; x &lt; y &egrave;&iacute;&ograve;&aring;&eth;&iuml;&eth;&aring;&ograve;&egrave;&eth;&oacute;&aring;&ograve;&ntilde;&yuml; &ecirc;&agrave;&ecirc; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&aring;&iacute;&egrave;&aring;, &divide;&ograve;&icirc; y &frac34;&euml;&oacute;&divide;&oslash;&aring;&iquest;, &egrave;&euml;&egrave; &frac34;&iuml;&eth;&aring;&auml;&iuml;&icirc;&divide;&ograve;&egrave;
&ograve;&aring;&euml;&uuml;&iacute;&aring;&aring;&iquest; x. &Iuml;&icirc; &yacute;&ograve;&icirc;&eacute; &iuml;&eth;&egrave;&divide;&egrave;&iacute;&aring; &lt; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&ograve;&ntilde;&yuml; &ograve;&agrave;&ecirc;&aelig;&aring; &ecirc;&agrave;&ecirc; P . &Icirc;&aacute;&ucirc;&divide;&iacute;&icirc; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; &lt; &icirc;&aacute;&euml;&agrave;&auml;&agrave;&aring;&ograve;
&eth;&yuml;&auml;&icirc;&igrave; &auml;&icirc;&iuml;&icirc;&euml;&iacute;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&otilde; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;, &ograve;&agrave;&ecirc;&egrave;&otilde; &ecirc;&agrave;&ecirc; &ograve;&eth;&agrave;&iacute;&ccedil;&egrave;&ograve;&egrave;&acirc;&iacute;&icirc;&ntilde;&ograve;&uuml;, &egrave;&eth;&eth;&aring;&ocirc;&euml;&aring;&ecirc;&ntilde;&egrave;&acirc;&iacute;&icirc;&ntilde;&ograve;&uuml;, &icirc;&ograve;&ecirc;&eth;&ucirc;&ograve;&icirc;&ntilde;&ograve;&uuml;
&egrave; &ograve;. &iuml;. &Igrave;&ucirc; &aacute;&oacute;&auml;&aring;&igrave; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&ograve;&uuml; &auml;&agrave;&euml;&aring;&aring;, &divide;&ograve;&icirc; &lt; &egrave;&eth;&eth;&aring;&ocirc;&euml;&aring;&ecirc;&ntilde;&egrave;&acirc;&iacute;&icirc;: &euml;&thorn;&aacute;&icirc;&eacute; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve; x &iacute;&aring; &euml;&oacute;&divide;&oslash;&aring;
&ntilde;&agrave;&igrave;&icirc;&atilde;&icirc; &ntilde;&aring;&aacute;&yuml;, &ograve;. &aring;. &iacute;&aring;&acirc;&aring;&eth;&iacute;&icirc;, &divide;&ograve;&icirc; x &lt; x.
&Acirc;&icirc;&ograve; &ntilde;&agrave;&igrave;&ucirc;&eacute; &iuml;&eth;&icirc;&ntilde;&ograve;&icirc;&eacute; &iuml;&eth;&egrave;&igrave;&aring;&eth; &acirc;&icirc;&ccedil;&iacute;&egrave;&ecirc;&iacute;&icirc;&acirc;&aring;&iacute;&egrave;&yuml; &ograve;&agrave;&ecirc;&icirc;&atilde;&icirc; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&yuml; &iuml;&eth;&aring;&auml;&iuml;&icirc;&divide;&ograve;&aring;&iacute;&egrave;&yuml; &lt;. &Iuml;&eth;&aring;&auml;&iuml;&icirc;
&euml;&icirc;&aelig;&egrave;&igrave;, &divide;&ograve;&icirc; &iacute;&agrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&aring; X &ccedil;&agrave;&auml;&agrave;&iacute;&agrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; &frac34;&iuml;&icirc;&euml;&aring;&ccedil;&iacute;&icirc;&ntilde;&ograve;&egrave;&iquest; u : X → R. &Ograve;&icirc;&atilde;&auml;&agrave; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&egrave;&igrave;
&icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; &lt;u , &iuml;&icirc;&euml;&agrave;&atilde;&agrave;&yuml; x &lt;u y ⇔ u(x) &lt; u(y).
&Ograve;&agrave;&ecirc;&icirc;&aring; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; &lt;u &icirc;&aacute;&euml;&agrave;&auml;&agrave;&aring;&ograve; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc;&igrave; &ograve;&eth;&agrave;&iacute;&ccedil;&egrave;&ograve;&egrave;&acirc;&iacute;&icirc;&ntilde;&ograve;&egrave; &egrave; &aring;&ugrave;&aring; &eth;&yuml;&auml;&icirc;&igrave; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;. &Ecirc;&euml;&agrave;&ntilde;&ntilde;&egrave;
&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &icirc;&aacute;&eth;&agrave;&ograve;&iacute;&agrave;&yuml; &ccedil;&agrave;&auml;&agrave;&divide;&agrave; &iuml;&eth;&egrave; &ecirc;&agrave;&ecirc;&egrave;&otilde; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml;&otilde; &iacute;&agrave; &lt; &icirc;&iacute;&icirc; &iuml;&eth;&icirc;&egrave;&ntilde;&otilde;&icirc;&auml;&egrave;&ograve; &egrave;&ccedil; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&eacute;
&ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; u, &ecirc;&icirc;&atilde;&auml;&agrave; u &igrave;&icirc;&aelig;&iacute;&icirc; &ntilde;&divide;&egrave;&ograve;&agrave;&ograve;&uuml; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&eacute; &egrave; &ograve;. &iuml;.
&Acirc;&icirc;&ccedil;&igrave;&icirc;&aelig;&aring;&iacute; &egrave; &aacute;&icirc;&euml;&aring;&aring; &ntilde;&euml;&icirc;&aelig;&iacute;&ucirc;&eacute; &igrave;&aring;&otilde;&agrave;&iacute;&egrave;&ccedil;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&yuml;. &Iuml;&oacute;&ntilde;&ograve;&uuml; &egrave;&igrave;&aring;&aring;&ograve;&ntilde;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml;
I : X &times; X → R; &divide;&egrave;&ntilde;&euml;&icirc; I(x, y) &egrave;&iacute;&ograve;&aring;&eth;&iuml;&eth;&aring;&ograve;&egrave;&eth;&oacute;&aring;&ograve;&ntilde;&yuml; &icirc;&aacute;&ucirc;&divide;&iacute;&icirc; &ecirc;&agrave;&ecirc; &igrave;&aring;&eth;&agrave; &iuml;&eth;&aring;&auml;&iuml;&icirc;&divide;&ograve;&aring;&iacute;&egrave;&yuml; x &iuml;&icirc; &ntilde;&eth;&agrave;&acirc;&iacute;&aring;
&iacute;&egrave;&thorn; &ntilde; y . &Ograve;&icirc;&atilde;&auml;&agrave; &igrave;&icirc;&aelig;&iacute;&icirc; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&acirc;&agrave;&ograve;&uuml; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; &gt;, &iuml;&icirc;&euml;&agrave;&atilde;&agrave;&yuml; x &gt; y ⇔ I(x, y) &gt; 0.
&Icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring;. &Yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve; x &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; X &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&igrave; (&icirc;&ograve;&iacute;&icirc;&ntilde;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc; &lt;), &aring;&ntilde;&euml;&egrave;
&iacute;&aring; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &aacute;&icirc;&euml;&aring;&aring; &euml;&oacute;&divide;&oslash;&egrave;&otilde; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;, &ograve;. &aring;. &aring;&ntilde;&euml;&egrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; {y ∈ X, x &lt; y} &iuml;&oacute;&ntilde;&ograve;&icirc;.
&Ntilde;&ograve;&icirc;&egrave;&ograve; &iuml;&eth;&aring;&auml;&icirc;&ntilde;&ograve;&aring;&eth;&aring;&divide;&uuml;, &divide;&ograve;&icirc; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&eacute; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve; &igrave;&icirc;&aelig;&aring;&ograve; &iacute;&aring; &aacute;&ucirc;&ograve;&uuml; &iacute;&agrave;&egrave;&aacute;&icirc;&euml;&uuml;&oslash;&egrave;&igrave; (&yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;
a &iacute;&agrave;&egrave;&aacute;&icirc;&euml;&uuml;&oslash;&egrave;&eacute;, &aring;&ntilde;&euml;&egrave; a &gt; x ∀ x ∈ X \ {a}). &Igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&acirc; &igrave;&icirc;&aelig;&aring;&ograve; &aacute;&ucirc;&ograve;&uuml; &igrave;&iacute;&icirc;&atilde;&icirc;,
&agrave; &igrave;&icirc;&aelig;&aring;&ograve; &iacute;&aring; &aacute;&ucirc;&ograve;&uuml; &iacute;&egrave; &icirc;&auml;&iacute;&icirc;&atilde;&icirc;. &Iacute;&egrave;&aelig;&aring; &igrave;&ucirc; &ccedil;&agrave;&eacute;&igrave;&aring;&igrave;&ntilde;&yuml; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml;&igrave;&egrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde;
&yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;. &Egrave;&ccedil; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Acirc;&aring;&eacute;&aring;&eth;&oslash;&ograve;&eth;&agrave;&ntilde;&ntilde;&agrave; &egrave;&ccedil;&acirc;&aring;&ntilde;&ograve;&iacute;&icirc;, &divide;&ograve;&icirc; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&agrave;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; &iacute;&agrave; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&aring;
&auml;&icirc;&ntilde;&ograve;&egrave;&atilde;&agrave;&aring;&ograve; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&oacute;&igrave;&agrave;. &Icirc;&aacute;&icirc;&aacute;&ugrave;&aring;&iacute;&egrave;&aring; &yacute;&ograve;&icirc;&atilde;&icirc; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&aring;&iacute;&egrave;&yuml; &iacute;&agrave; &ograve;&eth;&agrave;&iacute;&ccedil;&egrave;&ograve;&egrave;&acirc;&iacute;&ucirc;&aring; &iuml;&eth;&aring;&auml;&iuml;&icirc;&divide;&ograve;&aring;&iacute;&egrave;&yuml; &iuml;&eth;&egrave;
&acirc;&aring;&auml;&aring;&iacute;&icirc; &acirc; &oacute;&iuml;&eth;&agrave;&aelig;&iacute;&aring;&iacute;&egrave;&egrave; 3.1.
&Icirc;&ecirc;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml;, &divide;&ograve;&icirc; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc; &ograve;&eth;&agrave;&iacute;&ccedil;&egrave;&ograve;&egrave;&acirc;&iacute;&icirc;&ntilde;&ograve;&egrave; (&egrave; &ograve;&aring;&igrave; &aacute;&icirc;&euml;&aring;&aring; &iacute;&agrave;&euml;&egrave;&divide;&egrave;&aring; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; &frac34;&iuml;&icirc;&euml;&aring;&ccedil;&iacute;&icirc;&ntilde;&ograve;&egrave;&iquest;)
&igrave;&icirc;&aelig;&iacute;&icirc; &ccedil;&agrave;&igrave;&aring;&iacute;&egrave;&ograve;&uuml; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&igrave; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc;&igrave; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&ntilde;&ograve;&egrave;. &Ccedil;&auml;&aring;&ntilde;&uuml; &iacute;&agrave;&igrave; &oacute;&auml;&icirc;&aacute;&iacute;&aring;&aring; &aacute;&oacute;&auml;&aring;&ograve; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&ograve;&uuml;
&icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; &lt; &ntilde;&egrave;&igrave;&acirc;&icirc;&euml;&icirc;&igrave; P , &agrave; &ograve;&agrave;&ecirc;&aelig;&aring; &iuml;&icirc;&euml;&uuml;&ccedil;&icirc;&acirc;&agrave;&ograve;&uuml;&ntilde;&yuml; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&igrave;&egrave; &auml;&acirc;&oacute;&igrave;&yuml; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&yuml;&igrave;&egrave;:
P (x) = {y ∈ X, x &lt; y}
&egrave;
17
P −1 (x) = {y ∈ X, y &lt; x}.
18
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 3.
&Ograve;&Aring;&Icirc;&ETH;&Aring;&Igrave;&Agrave; &Aacute;&ETH;&Agrave;&Oacute;&Yacute;&ETH;&Agrave;: &Iuml;&ETH;&Egrave;&Igrave;&Aring;&Iacute;&Aring;&Iacute;&Egrave;&szlig;
&Iuml;&aring;&eth;&acirc;&icirc;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &egrave;&ccedil; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&acirc;, &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &euml;&oacute;&divide;&oslash;&aring; x, &agrave; &acirc;&ograve;&icirc;&eth;&icirc;&aring; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &otilde;&oacute;&aelig;&aring; x.
&Icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; &iuml;&eth;&aring;&auml;&iuml;&icirc;&divide;&ograve;&aring;&iacute;&egrave;&yuml; P &iacute;&agrave; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&igrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&aring; X &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&igrave;, &aring;&ntilde;&euml;&egrave; &igrave;&iacute;&icirc;
&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; P (x) &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc; &iuml;&eth;&egrave; &euml;&thorn;&aacute;&icirc;&igrave; x ∈ X . &Icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; P &iacute;&agrave; &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; X
&iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &icirc;&ograve;&ecirc;&eth;&ucirc;&ograve;&ucirc;&igrave;, &aring;&ntilde;&euml;&egrave; P &icirc;&ograve;&ecirc;&eth;&ucirc;&ograve;&icirc; &ecirc;&agrave;&ecirc; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &auml;&aring;&ecirc;&agrave;&eth;&ograve;&icirc;&acirc;&agrave; &ecirc;&acirc;&agrave;&auml;&eth;&agrave;&ograve;&agrave; X &times; X .
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave;. &Iuml;&oacute;&ntilde;&ograve;&uuml; X &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&eacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;, &agrave; P &icirc;&ograve;&ecirc;&eth;&ucirc;&ograve;&icirc;&aring; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&aring; &egrave;&eth;&eth;&aring;&ocirc;&euml;&aring;&ecirc;&ntilde;&egrave;&acirc;&iacute;&icirc;&aring;
&icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring;. &Ograve;&icirc;&atilde;&auml;&agrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&eacute; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;.
&Iacute;&agrave; &ntilde;&agrave;&igrave;&icirc;&igrave; &auml;&aring;&euml;&aring;, &ograve;&eth;&aring;&aacute;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &ecirc; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&thorn; P &igrave;&icirc;&aelig;&iacute;&icirc; &ntilde;&euml;&aring;&atilde;&ecirc;&agrave; &icirc;&ntilde;&euml;&agrave;&aacute;&egrave;&ograve;&uuml;, &iacute;&aring; &oacute;&ntilde;&euml;&icirc;&aelig;&iacute;&yuml;&yuml; &auml;&icirc;&ecirc;&agrave;
&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave;. &Agrave; &egrave;&igrave;&aring;&iacute;&iacute;&icirc;, &acirc;&aring;&eth;&iacute;&agrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&agrave;&yuml;
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; (&Ecirc;&egrave; &Ocirc;&agrave;&iacute;, &Ccedil;&icirc;&iacute;&iacute;&aring;&iacute;&oslash;&agrave;&eacute;&iacute;, &Aacute;&aring;&eth;&atilde;&ntilde;&ograve;&eth;&aring;&igrave;). &Iuml;&oacute;&ntilde;&ograve;&uuml; X &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&eacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;, &agrave; P &aacute;&egrave;&iacute;&agrave;&eth;&iacute;&icirc;&aring; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; &iacute;&agrave; X , &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &icirc;&aacute;&euml;&agrave;&auml;&agrave;&aring;&ograve; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&igrave;&egrave; &auml;&acirc;&oacute;&igrave;&yuml; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&agrave;&igrave;&egrave;:
1) &auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&atilde;&icirc; x ∈ X &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; P −1 (x) &icirc;&ograve;&ecirc;&eth;&ucirc;&ograve;&icirc;;
2) &auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&atilde;&icirc; x ∈ X &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&agrave;&yuml; &icirc;&aacute;&icirc;&euml;&icirc;&divide;&ecirc;&agrave; P (x) &iacute;&aring; &ntilde;&icirc;&auml;&aring;&eth;&aelig;&egrave;&ograve; x.
&Ograve;&icirc;&atilde;&auml;&agrave; P &icirc;&aacute;&euml;&agrave;&auml;&agrave;&aring;&ograve; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&igrave;&egrave; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&igrave;&egrave;.
&Acirc;&egrave;&auml;&iacute;&icirc;, &divide;&ograve;&icirc; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&aring; 1) &icirc;&aacute;&icirc;&aacute;&ugrave;&agrave;&aring;&ograve; &icirc;&ograve;&ecirc;&eth;&ucirc;&ograve;&icirc;&ntilde;&ograve;&uuml;, &agrave; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&aring; 2) &icirc;&auml;&iacute;&icirc;&acirc;&eth;&aring;&igrave;&aring;&iacute;&iacute;&icirc; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&ntilde;&ograve;&uuml;
&egrave; &egrave;&eth;&eth;&aring;&ocirc;&euml;&aring;&ecirc;&ntilde;&egrave;&acirc;&iacute;&icirc;&ntilde;&ograve;&uuml;.
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;. &Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc; &ntilde;&iacute;&icirc;&acirc;&agrave; &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &acirc; &ntilde;&acirc;&aring;&auml;&aring;&iacute;&egrave;&egrave; &ecirc; &ograve;&aring;&icirc;&eth;&aring;&igrave;&aring; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; (&ecirc;&icirc;&ograve;&icirc;&eth;&oacute;&thorn;,
&acirc; &ntilde;&acirc;&icirc;&thorn; &icirc;&divide;&aring;&eth;&aring;&auml;&uuml;, &euml;&aring;&atilde;&ecirc;&icirc; &acirc;&ucirc;&acirc;&aring;&ntilde;&ograve;&egrave; &egrave;&ccedil; &yacute;&ograve;&icirc;&eacute; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc;), &egrave; &divide;&agrave;&ntilde;&ograve;&egrave;&divide;&iacute;&icirc; &iuml;&icirc;&acirc;&ograve;&icirc;&eth;&yuml;&aring;&ograve; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;
&ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Ecirc;&agrave;&ecirc;&oacute;&ograve;&agrave;&iacute;&egrave;. &Iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&iacute;&icirc;&aring;, &ograve;. &aring;. &divide;&ograve;&icirc; &auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&atilde;&icirc; y ∈ X &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; x,
&ograve;&agrave;&ecirc;&icirc;&eacute; &divide;&ograve;&icirc; yP x, &egrave;&euml;&egrave; y ∈ P −1 (x). &Egrave;&iacute;&agrave;&divide;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&yuml;, &ntilde;&aring;&igrave;&aring;&eacute;&ntilde;&ograve;&acirc;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc; (P −1 (x), x ∈ X ) &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&ograve;
&iuml;&icirc;&ecirc;&eth;&ucirc;&ograve;&egrave;&aring; X . &Agrave; &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &iuml;&icirc; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&thorn; 1) &yacute;&ograve;&icirc; &icirc;&ograve;&ecirc;&eth;&ucirc;&ograve;&icirc;&aring; &iuml;&icirc;&ecirc;&eth;&ucirc;&ograve;&egrave;&aring;, &ograve;&icirc; &acirc; &ntilde;&egrave;&euml;&oacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave; X
&ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&aring; &divide;&egrave;&ntilde;&euml;&icirc; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&acirc; x1 , . . . , xn &egrave;&ccedil; X , &ograve;&agrave;&ecirc;&egrave;&otilde; &divide;&ograve;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; Ui = P −1 (xi )
&iuml;&icirc;&ecirc;&eth;&ucirc;&acirc;&agrave;&thorn;&ograve; X .
&Acirc;&icirc;&ccedil;&uuml;&igrave;&aring;&igrave; &ograve;&aring;&iuml;&aring;&eth;&uuml; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&aring; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; ui &iacute;&agrave; X , &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &eth;&agrave;&acirc;&iacute;&ucirc; 0 &acirc;&iacute;&aring; Ui &egrave; &ntilde;&ograve;&eth;&icirc;&atilde;&icirc; &iuml;&icirc;&euml;&icirc;
&aelig;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc; &iacute;&agrave; Ui (&acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring; ui &igrave;&icirc;&aelig;&iacute;&icirc; &acirc;&ccedil;&yuml;&ograve;&uuml; &eth;&agrave;&ntilde;&ntilde;&ograve;&icirc;&yuml;&iacute;&egrave;&aring; &icirc;&ograve; &ograve;&icirc;&divide;&ecirc;&egrave; &auml;&icirc; &auml;&icirc;&iuml;&icirc;&euml;&iacute;&aring;&iacute;&egrave;&yuml; &ecirc; Ui &acirc; X ).
&Iuml;&icirc;&ntilde;&euml;&aring; &yacute;&ograve;&icirc;&atilde;&icirc; &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&igrave; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f : X → X &iuml;&icirc; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&aring;:
P
ui (x)xi
f (x) = Pi
.
i ui (x)
P
&Egrave;&iacute;&agrave;&divide;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&yuml;, f (x) &aring;&ntilde;&ograve;&uuml; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&agrave;&yuml; &ecirc;&icirc;&igrave;&aacute;&egrave;&iacute;&agrave;&ouml;&egrave;&yuml; &ograve;&icirc;&divide;&aring;&ecirc; xi &ntilde; &acirc;&aring;&ntilde;&agrave;&igrave;&egrave; ui (x)/( j uj (x)). &Iuml;&icirc; &ograve;&aring;&icirc;
&eth;&aring;&igrave;&aring; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave; x∗ &auml;&euml;&yuml; f . &Yacute;&ograve;&icirc; &ccedil;&iacute;&agrave;&divide;&egrave;&ograve;, &divide;&ograve;&icirc; x∗ &aring;&ntilde;&ograve;&uuml; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&agrave;&yuml;
&ecirc;&icirc;&igrave;&aacute;&egrave;&iacute;&agrave;&ouml;&egrave;&yuml; &ograve;&aring;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; xi , &auml;&euml;&yuml; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; ui (x∗ ) &gt; 0, &ograve;. &aring;. &auml;&euml;&yuml; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; x∗ ∈ P −1 (xi ), &ograve;. &aring;.
&auml;&euml;&yuml; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; xi ∈ P (x∗ ). &Iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&igrave; &icirc;&ecirc;&icirc;&iacute;&divide;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;, &divide;&ograve;&icirc; x∗ &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&eacute; &icirc;&aacute;&icirc;&euml;&icirc;&divide;&ecirc;&aring;
P (x∗ ), &divide;&ograve;&icirc; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&ograve; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&thorn; 2).
&ETH;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&yuml; &Iacute;&yacute;&oslash;&agrave;
&Egrave;&atilde;&eth;&icirc;&eacute; n &euml;&egrave;&ouml; (&acirc; &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&aring;) &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &iacute;&agrave;&aacute;&icirc;&eth; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc; S1 , . . . , Sn &egrave; &iacute;&agrave;&aacute;&icirc;&eth; &ocirc;&oacute;&iacute;&ecirc;
&ouml;&egrave;&eacute; u1 , . . . , un &iacute;&agrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&egrave; X = S1 &times; &middot; &middot; &middot; &times; Sn . &Igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; Si &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;&igrave;
&ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&eacute; &egrave;&atilde;&eth;&icirc;&ecirc;&agrave; i, &agrave; ui &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&aring;&eacute; &acirc;&ucirc;&egrave;&atilde;&eth;&ucirc;&oslash;&agrave; &egrave;&atilde;&eth;&icirc;&ecirc;&agrave; i. &Ecirc;&agrave;&aelig;&auml;&ucirc;&eacute; &egrave;&atilde;&eth;&icirc;&ecirc; &ntilde;&ograve;&eth;&aring;&igrave;&egrave;&ograve;
&ntilde;&yuml; &acirc;&ucirc;&aacute;&eth;&agrave;&ograve;&uuml; &ograve;&agrave;&ecirc;&oacute;&thorn; &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&thorn; si ∈ Si , &ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &auml;&agrave;&acirc;&agrave;&euml;&agrave; &aacute;&ucirc; &aring;&igrave;&oacute; &iacute;&agrave;&egrave;&aacute;&icirc;&euml;&uuml;&oslash;&egrave;&eacute; &acirc;&ucirc;&egrave;&atilde;&eth;&ucirc;&oslash;. &Icirc;&auml;&iacute;&agrave;&ecirc;&icirc;
&yacute;&ograve;&icirc;&ograve; &acirc;&ucirc;&egrave;&atilde;&eth;&ucirc;&oslash; &ccedil;&agrave;&acirc;&egrave;&ntilde;&egrave;&ograve; &ograve;&agrave;&ecirc;&aelig;&aring; &icirc;&ograve; &acirc;&ucirc;&aacute;&icirc;&eth;&agrave; &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&eacute; &icirc;&ntilde;&ograve;&agrave;&euml;&uuml;&iacute;&ucirc;&igrave;&egrave; &egrave;&atilde;&eth;&icirc;&ecirc;&agrave;&igrave;&egrave;, &iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &acirc; &icirc;&aacute;&ugrave;&aring;&igrave;
&ntilde;&euml;&oacute;&divide;&agrave;&aring; &iacute;&aring;&ograve; &aring;&ntilde;&ograve;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&atilde;&icirc; &egrave; &aacute;&aring;&ntilde;&ntilde;&iuml;&icirc;&eth;&iacute;&icirc;&atilde;&icirc; &iuml;&eth;&agrave;&acirc;&egrave;&euml;&agrave; &auml;&euml;&yuml; &acirc;&ucirc;&aacute;&icirc;&eth;&agrave; &ecirc;&agrave;&aelig;&auml;&ucirc;&igrave; &oacute;&divide;&agrave;&ntilde;&ograve;&iacute;&egrave;&ecirc;&icirc;&igrave; &iacute;&agrave;&egrave;&aacute;&icirc;&euml;&aring;&aring;
&icirc;&iuml;&ograve;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&icirc;&eacute; &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&egrave;.
19
&Iacute;&yacute;&oslash; (1950) &iuml;&eth;&aring;&auml;&euml;&icirc;&aelig;&egrave;&euml; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&aring; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&yuml; &acirc; &ograve;&agrave;&ecirc;&icirc;&eacute; &egrave;&atilde;&eth;&aring;, &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &ntilde; &ograve;&aring;&otilde; &iuml;&icirc;&eth; &iacute;&icirc;&ntilde;&egrave;&ograve; &aring;&atilde;&icirc;
&egrave;&igrave;&yuml;. &Agrave; &egrave;&igrave;&aring;&iacute;&iacute;&icirc;, &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&aring;&igrave; &Iacute;&yacute;&oslash;&agrave; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ograve;&agrave;&ecirc;&icirc;&eacute; &iacute;&agrave;&aacute;&icirc;&eth; &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&eacute; s∗ = (s∗1 , . . . , s∗n ) ∈ S1 &times;
&middot; &middot; &middot; &times; Sn , &divide;&ograve;&icirc; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; &egrave;&atilde;&eth;&icirc;&ecirc;&agrave; i &aring;&atilde;&icirc; &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&yuml; s∗i &auml;&icirc;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&aring;&ograve; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&oacute;&igrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; ui (&middot;, s∗−i ).
&Ograve;&icirc; &aring;&ntilde;&ograve;&uuml; &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&egrave; s∗−i &icirc;&ntilde;&ograve;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &egrave;&atilde;&eth;&icirc;&ecirc;&icirc;&acirc; &ntilde;&divide;&egrave;&ograve;&agrave;&thorn;&ograve;&ntilde;&yuml; &ocirc;&egrave;&ecirc;&ntilde;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&ucirc;&igrave;&egrave;, &agrave; &igrave;&aring;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc;
i-&agrave;&yuml; &iuml;&aring;&eth;&aring;&igrave;&aring;&iacute;&iacute;&agrave;&yuml;, &ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &iacute;&agrave;&otilde;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &acirc; &eth;&agrave;&ntilde;&iuml;&icirc;&eth;&yuml;&aelig;&aring;&iacute;&egrave;&egrave; &egrave;&atilde;&eth;&icirc;&ecirc;&agrave; i. &Egrave;&iacute;&agrave;&divide;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&yuml;, &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&yuml; s∗i
&egrave;&atilde;&eth;&icirc;&ecirc;&agrave; i &icirc;&iuml;&ograve;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&agrave; &iuml;&eth;&egrave; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&egrave;, &divide;&ograve;&icirc; &icirc;&ntilde;&ograve;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &ntilde;&icirc;&otilde;&eth;&agrave;&iacute;&yuml;&thorn;&ograve; &ntilde;&acirc;&icirc;&egrave; &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&egrave; s∗j , j 6= i.
&ETH;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&yuml; &Iacute;&yacute;&oslash;&agrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&thorn;&ograve; &auml;&agrave;&euml;&aring;&ecirc;&icirc; &iacute;&aring; &acirc;&ntilde;&aring;&atilde;&auml;&agrave;. &Icirc;&auml;&iacute;&agrave;&ecirc;&icirc; &oslash;&agrave;&iacute;&ntilde;&ucirc; &iacute;&agrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &ntilde;&egrave;&euml;&uuml;&iacute;&icirc;
&oacute;&acirc;&aring;&euml;&egrave;&divide;&egrave;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml;, &aring;&ntilde;&euml;&egrave; &icirc;&ograve; &egrave;&ntilde;&otilde;&icirc;&auml;&iacute;&ucirc;&otilde; (&divide;&egrave;&ntilde;&ograve;&ucirc;&otilde;) &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&eacute; &iuml;&aring;&eth;&aring;&eacute;&ograve;&egrave; &ecirc; &ntilde;&igrave;&aring;&oslash;&agrave;&iacute;&iacute;&ucirc;&igrave; (&eth;&agrave;&iacute;&auml;&icirc;&igrave;&egrave;&ccedil;&egrave;&eth;&icirc;
&acirc;&agrave;&iacute;&iacute;&ucirc;&igrave;) &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&yuml;&igrave;. &Iacute;&aring; &acirc;&auml;&agrave;&acirc;&agrave;&yuml;&ntilde;&uuml; &acirc; &iuml;&icirc;&auml;&eth;&icirc;&aacute;&iacute;&ucirc;&aring; &icirc;&aacute;&uacute;&yuml;&ntilde;&iacute;&aring;&iacute;&egrave;&yuml; (&ccedil;&agrave; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&igrave;&egrave; &igrave;&ucirc; &icirc;&ograve;&ntilde;&ucirc;&euml;&agrave;&aring;&igrave; &ecirc;
&ograve;&aring;&icirc;&eth;&egrave;&egrave; &egrave;&atilde;&eth;), &ntilde;&ecirc;&agrave;&aelig;&aring;&igrave; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc;, &divide;&ograve;&icirc; &yacute;&ograve;&icirc; &iuml;&eth;&egrave;&acirc;&icirc;&auml;&egrave;&ograve; &ecirc; &ograve;&icirc;&igrave;&oacute;, &divide;&ograve;&icirc; &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; Si &ntilde;&ograve;&agrave;
&iacute;&icirc;&acirc;&yuml;&ograve;&ntilde;&yuml; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&igrave;&egrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave;&igrave;&egrave;, &agrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; &acirc;&ucirc;&egrave;&atilde;&eth;&ucirc;&oslash;&agrave; &agrave;&ocirc;&ocirc;&egrave;&iacute;&iacute;&ucirc;&igrave;&egrave; &iuml;&icirc; &iuml;&aring;&eth;&aring;&igrave;&aring;&iacute;&iacute;&icirc;&eacute; si .
&Iacute;&agrave; &ntilde;&agrave;&igrave;&icirc;&igrave; &auml;&aring;&euml;&aring;, &acirc;&igrave;&aring;&ntilde;&ograve;&icirc; &agrave;&ocirc;&ocirc;&egrave;&iacute;&iacute;&icirc;&ntilde;&ograve;&egrave; &auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc; &ograve;&eth;&aring;&aacute;&icirc;&acirc;&agrave;&ograve;&uuml; &acirc;&icirc;&atilde;&iacute;&oacute;&ograve;&icirc;&ntilde;&ograve;&uuml;. &Ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; u : X → R
&iacute;&agrave; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&igrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&aring; X &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc;&icirc;&atilde;&iacute;&oacute;&ograve;&icirc;&eacute; (&egrave;&euml;&egrave; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&eacute; &acirc;&acirc;&aring;&eth;&otilde;), &aring;&ntilde;&euml;&egrave;
u(αx + (1 − α)y) ≥ αu(x) + (1 − α)u(y)
&auml;&euml;&yuml; &euml;&thorn;&aacute;&ucirc;&otilde; x, y ∈ X &egrave; 0 ≤ α ≤ 1.
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; (&Iacute;&yacute;&oslash;, 1950). &Iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave;, &divide;&ograve;&icirc; &ecirc;&agrave;&aelig;&auml;&icirc;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; Si &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&eacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;,
&agrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; &acirc;&ucirc;&egrave;&atilde;&eth;&ucirc;&oslash;&agrave; ui &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc; &iuml;&icirc; &acirc;&ntilde;&aring;&igrave; &iuml;&aring;&eth;&aring;&igrave;&aring;&iacute;&iacute;&ucirc;&igrave; &egrave; &acirc;&icirc;&atilde;&iacute;&oacute;&ograve;&ucirc; &iuml;&icirc; si . &Ograve;&icirc;&atilde;&auml;&agrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;
&aring;&ograve; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&aring; &Iacute;&yacute;&oslash;&agrave;.
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;. &Auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&atilde;&icirc; &egrave;&atilde;&eth;&icirc;&ecirc;&agrave; i &egrave; &iacute;&agrave;&aacute;&icirc;&eth;&agrave; s−i = (sj , j 6= i) &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&eacute; &icirc;&ntilde;&ograve;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde;
&icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&egrave;&igrave; &divide;&aring;&eth;&aring;&ccedil; Hi (s−i ) &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &iacute;&agrave;&egrave;&euml;&oacute;&divide;&oslash;&egrave;&otilde; &icirc;&ograve;&acirc;&aring;&ograve;&icirc;&acirc; i, &ograve;. &aring;.
Hi (s−i ) = Argmax(ui (&middot;, s−i )).
&Iuml;&eth;&egrave; &iuml;&aring;&eth;&aring;&igrave;&aring;&iacute;&iacute;&icirc;&igrave; s−i &igrave;&ucirc; &iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&igrave; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; Hi : &times;j6=i Sj ⇒ Si , &ograve;. &aring;. &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &acirc; X .
&Acirc; &ntilde;&egrave;&euml;&oacute; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&ntilde;&ograve;&egrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; ui &iuml;&icirc; &acirc;&ntilde;&aring;&igrave; &iuml;&aring;&eth;&aring;&igrave;&aring;&iacute;&iacute;&ucirc;&igrave; &yacute;&ograve;&icirc; &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;&aring; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &acirc; X
(&ntilde;&igrave;. &oacute;&iuml;&eth;. 2.9); &acirc; &ntilde;&egrave;&euml;&oacute; &acirc;&icirc;&atilde;&iacute;&oacute;&ograve;&icirc;&ntilde;&ograve;&egrave; ui &iuml;&icirc; &iuml;&aring;&eth;&aring;&igrave;&aring;&iacute;&iacute;&icirc;&eacute; si &icirc;&aacute;&eth;&agrave;&ccedil;&ucirc; Hi (s−i ) &iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&ucirc;&aring; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&aring;
&iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; Si .
&Iacute;&agrave;&aacute;&icirc;&eth; &ntilde;&ograve;&eth;&agrave;&ograve;&aring;&atilde;&egrave;&eacute; s = (si ) ∈ X &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&aring;&igrave; &Iacute;&yacute;&oslash;&agrave; &ograve;&icirc;&atilde;&auml;&agrave; &egrave; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &ograve;&icirc;&atilde;&auml;&agrave;, &ecirc;&icirc;
&atilde;&auml;&agrave; si &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; Hi (s−i ) &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; i. &Icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&igrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&aring; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; F : X ⇒ X ,
&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &ograve;&icirc;&divide;&ecirc;&aring; s ∈ X &ntilde;&icirc;&iuml;&icirc;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&aring;&ograve; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; F (s) = &times;i Hi (s−i ). &Ecirc;&agrave;&ecirc; &euml;&aring;&atilde;&ecirc;&icirc; &iuml;&icirc;&iacute;&yuml;&ograve;&uuml;, &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;
&ntilde;&ograve;&acirc;&egrave;&aring; F &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;, &agrave; &aring;&atilde;&icirc; &icirc;&aacute;&eth;&agrave;&ccedil;&ucirc; &iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&ucirc;&aring; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&aring; &ecirc;&agrave;&ecirc; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&aring;&iacute;&egrave;&yuml; &iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&ucirc;&otilde; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&otilde;
&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc; Hi (s−i ). &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &ecirc; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&thorn; F &iuml;&eth;&egrave;&igrave;&aring;&iacute;&egrave;&igrave;&agrave; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Ecirc;&agrave;&ecirc;&oacute;&ograve;&agrave;&iacute;&egrave;, &ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &auml;&agrave;&aring;&ograve;
&ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &ograve;&icirc;&divide;&ecirc;&egrave; s∗ ∈ F (s∗ ). &Iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&aring; &ecirc;&agrave;&ecirc; &eth;&agrave;&ccedil; &egrave; &icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&ograve;, &divide;&ograve;&icirc; s∗i ∈ Hi (s∗−i ) &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc;
i, &ograve;. &aring;. &divide;&ograve;&icirc; s∗ &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&aring; &Iacute;&yacute;&oslash;&agrave;.
&Ecirc;&icirc;&iacute;&ecirc;&oacute;&eth;&aring;&iacute;&ograve;&iacute;&ucirc;&aring; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&yuml;
&Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &acirc; &iacute;&agrave;&oslash;&oacute; &ccedil;&agrave;&auml;&agrave;&divide;&oacute; &iacute;&aring; &acirc;&otilde;&icirc;&auml;&egrave;&ograve; &auml;&aring;&ograve;&agrave;&euml;&uuml;&iacute;&icirc;&aring; &egrave;&ccedil;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring; &ograve;&aring;&icirc;&eth;&egrave;&egrave; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&atilde;&icirc; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;
&ntilde;&egrave;&yuml;, &iacute;&icirc; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &auml;&aring;&igrave;&icirc;&iacute;&ntilde;&ograve;&eth;&agrave;&ouml;&egrave;&yuml; &iuml;&eth;&egrave;&igrave;&aring;&iacute;&aring;&iacute;&egrave;&yuml; &ograve;&aring;&icirc;&eth;&aring;&igrave; &icirc; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&ecirc;&agrave;&otilde;, &igrave;&ucirc; &euml;&egrave;&oslash;&uuml; &aacute;&aring;&atilde;&euml;&icirc;
&icirc;&aacute;&eth;&egrave;&ntilde;&oacute;&aring;&igrave; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&oacute;&thorn; &divide;&agrave;&ntilde;&ograve;&uuml;, &iacute;&aring; &atilde;&icirc;&iacute;&yuml;&ntilde;&uuml; &ccedil;&agrave; &icirc;&aacute;&ugrave;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn;. &Iuml;&oacute;&ntilde;&ograve;&uuml; Rl &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; &ograve;&icirc;&acirc;&agrave;&eth;&icirc;&acirc;.
&Egrave;&igrave;&aring;&aring;&ograve;&ntilde;&yuml; n &iuml;&icirc;&ograve;&eth;&aring;&aacute;&egrave;&ograve;&aring;&euml;&aring;&eacute;, &ecirc;&agrave;&aelig;&auml;&ucirc;&eacute; &iuml;&icirc;&ograve;&eth;&aring;&aacute;&egrave;&ograve;&aring;&euml;&uuml; &ccedil;&agrave;&auml;&agrave;&aring;&ograve;&ntilde;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute; &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&icirc;&ntilde;&ograve;&egrave; ui &iacute;&agrave; &ntilde;&acirc;&icirc;&aring;&igrave; &iuml;&icirc;
&ograve;&eth;&aring;&aacute;&egrave;&ograve;&aring;&euml;&uuml;&ntilde;&ecirc;&icirc;&igrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&aring; Xi , &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &igrave;&ucirc; &auml;&euml;&yuml; &iuml;&eth;&icirc;&ntilde;&ograve;&icirc;&ograve;&ucirc; &ntilde;&divide;&egrave;&ograve;&agrave;&aring;&igrave; &eth;&agrave;&acirc;&iacute;&ucirc;&igrave; &iuml;&icirc;&euml;&icirc;&aelig;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&igrave;&oacute;
&icirc;&eth;&ograve;&agrave;&iacute;&ograve;&oacute; Rl+ . &Auml;&agrave;&euml;&aring;&aring;, &egrave;&igrave;&aring;&aring;&ograve;&ntilde;&yuml; &ograve;&aring;&otilde;&iacute;&icirc;&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; Y ⊂ Rl . &Iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave;, &divide;&ograve;&icirc; &acirc; &yacute;&ecirc;&icirc;
&iacute;&icirc;&igrave;&egrave;&ecirc;&aring; &auml;&aring;&eacute;&ntilde;&ograve;&acirc;&oacute;&thorn;&ograve; &ouml;&aring;&iacute;&ucirc; p, &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &iuml;&icirc;&ccedil;&acirc;&icirc;&euml;&yuml;&aring;&ograve; &ntilde; &ecirc;&agrave;&aelig;&auml;&ucirc;&igrave; &ograve;&icirc;&acirc;&agrave;&eth;&iacute;&ucirc;&igrave; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&icirc;&igrave; x ∈ Rl &ntilde;&acirc;&yuml;&ccedil;&agrave;&ograve;&uuml;
&aring;&atilde;&icirc; &ntilde;&ograve;&icirc;&egrave;&igrave;&icirc;&ntilde;&ograve;&uuml; px. &Ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&icirc;&iacute;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&ecirc;&egrave; &iuml;&eth;&icirc;&egrave;&ntilde;&otilde;&icirc;&auml;&egrave;&ograve; &ograve;&agrave;&ecirc;. &Iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&auml;&egrave;&ograve;&aring;&euml;&uuml; &iuml;&eth;&egrave; &auml;&aring;&eacute;
&ntilde;&ograve;&acirc;&oacute;&thorn;&ugrave;&egrave;&otilde; &ouml;&aring;&iacute;&agrave;&otilde; p &acirc;&ucirc;&aacute;&egrave;&eth;&agrave;&aring;&ograve; &ograve;&icirc;&ograve; &ograve;&aring;&otilde;&iacute;&icirc;&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&eacute; &ntilde;&iuml;&icirc;&ntilde;&icirc;&aacute;, &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&eacute; &auml;&agrave;&aring;&ograve; &aring;&igrave;&oacute; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&oacute;&thorn;
&iuml;&eth;&egrave;&aacute;&ucirc;&euml;&uuml; maxy∈Y py . &Iuml;&icirc;&euml;&oacute;&divide;&aring;&iacute;&iacute;&agrave;&yuml; &iuml;&eth;&egrave;&aacute;&ucirc;&euml;&uuml; &ecirc;&agrave;&ecirc;-&ograve;&icirc; &eth;&agrave;&ntilde;&iuml;&eth;&aring;&auml;&aring;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &igrave;&aring;&aelig;&auml;&oacute; &iuml;&icirc;&ograve;&eth;&aring;&aacute;&egrave;&ograve;&aring;&euml;&yuml;&igrave;&egrave;,
20
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 3.
&Ograve;&Aring;&Icirc;&ETH;&Aring;&Igrave;&Agrave; &Aacute;&ETH;&Agrave;&Oacute;&Yacute;&ETH;&Agrave;: &Iuml;&ETH;&Egrave;&Igrave;&Aring;&Iacute;&Aring;&Iacute;&Egrave;&szlig;
&acirc; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve;&aring; &divide;&aring;&atilde;&icirc; &ecirc;&agrave;&aelig;&auml;&ucirc;&eacute; &iuml;&icirc;&ograve;&eth;&aring;&aacute;&egrave;&ograve;&aring;&euml;&uuml; i &iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&ograve; &auml;&icirc;&otilde;&icirc;&auml; βi (p). &Yacute;&ograve;&icirc;&ograve; &auml;&icirc;&otilde;&icirc;&auml; &icirc;&iacute; &ograve;&eth;&agrave;&ograve;&egrave;&ograve; &iacute;&agrave;
&iuml;&eth;&egrave;&icirc;&aacute;&eth;&aring;&ograve;&aring;&iacute;&egrave;&aring; &iuml;&icirc;&ograve;&eth;&aring;&aacute;&egrave;&ograve;&aring;&euml;&uuml;&ntilde;&ecirc;&icirc;&atilde;&icirc;P&iacute;&agrave;&aacute;&icirc;&eth;&agrave; xi , &ograve;&agrave;&ecirc; &divide;&ograve;&icirc; pxi ≤ βi (p). &Ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;,P
&aring;&ntilde;&euml;&egrave; &iacute;&aring; &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&aring;&iacute;
&iacute;&agrave;&ograve;&oacute;&eth;&agrave;&euml;&uuml;&iacute;&ucirc;&eacute; &aacute;&agrave;&euml;&agrave;&iacute;&ntilde; &acirc; &ocirc;&icirc;&eth;&igrave;&aring; i xi = y (&egrave;&euml;&egrave; &otilde;&icirc;&ograve;&yuml; &aacute;&ucirc; &ecirc;&agrave;&ecirc; &iacute;&aring;&eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&icirc; i xi ≤ y ), &ouml;&aring;&iacute;&agrave; p
&egrave;&igrave;&aring;&aring;&ograve; &igrave;&agrave;&euml;&icirc; &ntilde;&igrave;&ucirc;&ntilde;&euml;&agrave;. &Iacute;&agrave;&iuml;&eth;&icirc;&ograve;&egrave;&acirc;, &aring;&ntilde;&euml;&egrave; &iacute;&agrave;&ograve;&oacute;&eth;&agrave;&euml;&uuml;&iacute;&ucirc;&eacute; &aacute;&agrave;&euml;&agrave;&iacute;&ntilde; &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&aring;&iacute;, &atilde;&icirc;&acirc;&icirc;&eth;&yuml;&ograve;, &divide;&ograve;&icirc; &iacute;&agrave;&aacute;&icirc;&eth; (p,
y , (xi )) &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &ecirc;&icirc;&iacute;&ecirc;&oacute;&eth;&aring;&iacute;&ograve;&iacute;&ucirc;&igrave; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&aring;&igrave;.
&Igrave;&ucirc; &aacute;&oacute;&auml;&aring;&igrave; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&ograve;&uuml; &auml;&agrave;&euml;&aring;&aring;, &divide;&ograve;&icirc; &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &ccedil;&agrave;&ecirc;&icirc;&iacute; &Acirc;&agrave;&euml;&uuml;&eth;&agrave;&ntilde;&agrave;, &ograve;. &aring;. &divide;&ograve;&icirc; &iuml;&eth;&egrave; &euml;&thorn;&aacute;&icirc;&eacute;
&ouml;&aring;&iacute;&aring; &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &ntilde;&icirc;&icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring;
X
βi (p) = max py.
y∈Y
i
&Ecirc;&eth;&icirc;&igrave;&aring; &ograve;&icirc;&atilde;&icirc;, &igrave;&ucirc; &aacute;&oacute;&auml;&aring;&igrave; &ntilde;&divide;&egrave;&ograve;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&aring;&iacute;&ucirc; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&aring; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&yuml;:
1) &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&icirc;&ntilde;&ograve;&egrave; ui &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc; &egrave; &ecirc;&acirc;&agrave;&ccedil;&egrave;&acirc;&icirc;&atilde;&iacute;&oacute;&ograve;&ucirc;;
2) &ograve;&aring;&otilde;&iacute;&icirc;&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; Y &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&eacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;;
3) &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; &auml;&icirc;&otilde;&icirc;&auml;&agrave; βi &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc; &egrave; &iuml;&icirc;&euml;&icirc;&aelig;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc; &iuml;&eth;&egrave; p &gt; 0.
&Iuml;&eth;&egrave; &yacute;&ograve;&egrave;&otilde; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&yuml;&otilde; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&aring;. &Aacute;&icirc;&euml;&aring;&aring; &ograve;&icirc;&atilde;&icirc;, &igrave;&icirc;&aelig;&iacute;&icirc; &ntilde;&divide;&egrave;&ograve;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &eth;&agrave;&acirc;
&iacute;&icirc;&acirc;&aring;&ntilde;&iacute;&agrave;&yuml; &ouml;&aring;&iacute;&agrave; p &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; &aring;&auml;&egrave;&iacute;&egrave;&divide;&iacute;&icirc;&igrave;&oacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&oacute; ∆l , &ograve;. &aring;. p1l = 1l .
&Auml;&euml;&yuml; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave; &acirc;&acirc;&aring;&auml;&aring;&igrave; &iacute;&aring;&ntilde;&ecirc;&icirc;&euml;&uuml;&ecirc;&icirc; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&eacute;. &Iuml;&aring;&eth;&acirc;&icirc;&aring; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&auml;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&aring;. &Auml;&euml;&yuml;
&ecirc;&agrave;&aelig;&auml;&icirc;&eacute; &ouml;&aring;&iacute;&ucirc; p &egrave;&ccedil; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; ∆l &divide;&aring;&eth;&aring;&ccedil; Y (p) &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&egrave;&igrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; y ∈ Y , &auml;&agrave;&thorn;&ugrave;&egrave;&otilde; &igrave;&agrave;&ecirc;&ntilde;&egrave;
&igrave;&agrave;&euml;&uuml;&iacute;&oacute;&thorn; &iuml;&eth;&egrave;&aacute;&ucirc;&euml;&uuml; &iuml;&eth;&egrave; &ouml;&aring;&iacute;&aring; p. &Agrave;&iacute;&agrave;&euml;&icirc;&atilde;&egrave;&divide;&iacute;&icirc;, &auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&atilde;&icirc; &iuml;&icirc;&ograve;&eth;&aring;&aacute;&egrave;&ograve;&aring;&euml;&yuml; i &acirc;&acirc;&aring;&auml;&aring;&igrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;
Xi (p) &aring;&atilde;&icirc; &icirc;&iuml;&ograve;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &iuml;&icirc;&ograve;&eth;&aring;&aacute;&euml;&aring;&iacute;&egrave;&eacute;, &ograve;. &aring;. Xi (p) &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &egrave;&ccedil; &eth;&aring;&oslash;&aring;&iacute;&egrave;&eacute; &ccedil;&agrave;&auml;&agrave;&divide;&egrave; ui (x) → max &iuml;&eth;&egrave;
&icirc;&atilde;&eth;&agrave;&iacute;&egrave;&divide;&aring;&iacute;&egrave;&egrave; x ∈ Xi &egrave; px ≤ βi (p). &Ograve;&icirc;&divide;&iacute;&aring;&aring;, &acirc; &yacute;&ograve;&icirc;&igrave; &igrave;&aring;&ntilde;&ograve;&aring; &iuml;&icirc;&euml;&icirc;&aelig;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&eacute; &icirc;&eth;&ograve;&agrave;&iacute;&ograve; Xi &iacute;&oacute;&aelig;&iacute;&icirc;
&ccedil;&agrave;&igrave;&aring;&iacute;&egrave;&ograve;&uuml; &auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc; &aacute;&icirc;&euml;&uuml;&oslash;&egrave;&igrave; &ecirc;&oacute;&aacute;&icirc;&igrave; K , &iacute;&icirc; &yacute;&ograve;&icirc; &oacute;&aelig;&aring; &ograve;&aring;&otilde;&iacute;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &igrave;&aring;&ntilde;&ograve;&icirc;, &iacute;&agrave; &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&igrave; &igrave;&ucirc; &iacute;&aring;
&otilde;&icirc;&ograve;&aring;&euml;&egrave; &aacute;&ucirc; &ntilde;&aring;&eacute;&divide;&agrave;&ntilde; &icirc;&ntilde;&ograve;&agrave;&iacute;&agrave;&acirc;&euml;&egrave;&acirc;&agrave;&ograve;&uuml;&ntilde;&yuml;. &Ograve;&icirc;&divide;&iacute;&icirc; &ograve;&agrave;&ecirc; &aelig;&aring; &igrave;&ucirc; &icirc;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&aring;&igrave; &divide;&egrave;&ograve;&agrave;&ograve;&aring;&euml;&thorn; &iuml;&eth;&icirc;&acirc;&aring;&eth;&ecirc;&oacute; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc;
&ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&yuml; Y &egrave; Xi &egrave;&ccedil; ∆l &acirc; Rl &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&ucirc; (&ntilde;&igrave;. &oacute;&iuml;&eth;. 2.9). &Icirc;&ograve;&igrave;&aring;&ograve;&egrave;&igrave; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc;, &divide;&ograve;&icirc; &egrave;&igrave;&aring;&iacute;&iacute;&icirc; &ccedil;&auml;&aring;&ntilde;&uuml;
&iacute;&oacute;&aelig;&iacute;&icirc; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring; &icirc; &ntilde;&ograve;&eth;&icirc;&atilde;&icirc;&eacute; &iuml;&icirc;&euml;&icirc;&aelig;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&egrave; &auml;&icirc;&otilde;&icirc;&auml;&icirc;&acirc; βi (&ograve;. &iacute;. &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&aring; &Ntilde;&euml;&aring;&eacute;&ograve;&aring;&eth;&agrave; ); &iuml;&eth;&egrave;
&aring;&atilde;&icirc; &iacute;&agrave;&eth;&oacute;&oslash;&aring;&iacute;&egrave;&egrave; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&yuml; &igrave;&icirc;&aelig;&aring;&ograve; &egrave; &iacute;&aring; &aacute;&ucirc;&ograve;&uuml;. &Ecirc;&eth;&icirc;&igrave;&aring; &ograve;&icirc;&atilde;&icirc;, &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; Y (p) &egrave; Xi (p) &iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&ucirc;
&egrave; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;. &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &aring;&ntilde;&euml;&egrave; &igrave;&ucirc; &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&igrave; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; &egrave;&ccedil;&aacute;&ucirc;&ograve;&icirc;&divide;&iacute;&icirc;&atilde;&icirc; &ntilde;&iuml;&eth;&icirc;&ntilde;&agrave; E : ∆l ⇒ Rl &iuml;&icirc;
&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&aring;:
nX
o
X
E(p) =
Xi (p) − Y (p) =
xi − y, xi ∈ Xi (p), y ∈ Y (p) ,
i
i
&ograve;&icirc; &yacute;&ograve;&icirc; &ograve;&agrave;&ecirc;&aelig;&aring; &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;&aring; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; &ntilde; &iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&ucirc;&igrave;&egrave; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&igrave;&egrave; &icirc;&aacute;&eth;&agrave;&ccedil;&agrave;&igrave;&egrave;. &Icirc;&ograve;&igrave;&aring;&ograve;&egrave;&igrave; &aring;&ugrave;&aring; &icirc;&auml;&iacute;&icirc;
&acirc;&agrave;&aelig;&iacute;&icirc;&aring; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&yuml; E . &Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; pxi ≤ βi (p) &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; i, &ograve;&icirc;
X
X
p
xi − y &lt;
βi (p) − py = 0
i
i
&iuml;&icirc; &ccedil;&agrave;&ecirc;&icirc;&iacute;&oacute; &Acirc;&agrave;&euml;&uuml;&eth;&agrave;&ntilde;&agrave;. &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; p ∈ ∆l &egrave; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; z ∈ E(p) &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&aring;&iacute;&icirc; &ntilde;&icirc;&icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring;
pz ≤ 0. &Icirc;&ecirc;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml;, &divide;&ograve;&icirc; &yacute;&ograve;&egrave;&otilde; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc; &oacute;&aelig;&aring; &auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc; &auml;&euml;&yuml; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave; &ouml;&aring;&iacute; p∗ &egrave;
&acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave; z ∗ ∈ E(p∗ ), &ograve;&agrave;&ecirc;&egrave;&otilde; &divide;&ograve;&icirc; z ∗ ≤ 0, &divide;&ograve;&icirc; &egrave; &auml;&agrave;&aring;&ograve; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&aring;. &Acirc; &ntilde;&agrave;&igrave;&icirc;&igrave; &auml;&aring;&euml;&aring;, &acirc;&aring;&eth;&iacute;&agrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&agrave;&yuml;
&ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; (&ograve;&icirc;&aelig;&aring; &yacute;&ecirc;&acirc;&egrave;&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&agrave;&yuml; &ograve;&aring;&icirc;&eth;&aring;&igrave;&aring; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;):
&Euml;&aring;&igrave;&igrave;&agrave; (&Atilde;&aring;&eacute;&euml;, 1955; &Iacute;&egrave;&ecirc;&agrave;&eacute;&auml;&icirc;, 1956). &Iuml;&oacute;&ntilde;&ograve;&uuml; ∆l &aring;&auml;&egrave;&iacute;&egrave;&divide;&iacute;&ucirc;&eacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;, &egrave; E &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;
&ograve;&icirc;&aring; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; &egrave;&ccedil; ∆l &acirc; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&iacute;&icirc;&aring; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&aring; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; K ⊂ Rl &ntilde; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&igrave;&egrave;
&iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&ucirc;&igrave;&egrave; &icirc;&aacute;&eth;&agrave;&ccedil;&agrave;&igrave;&egrave;. &Iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave;, &divide;&ograve;&icirc; pz ≤ 0 &auml;&euml;&yuml; &euml;&thorn;&aacute;&ucirc;&otilde; p ∈ ∆l &egrave; z ∈ E(p). &Ograve;&icirc;&atilde;&auml;&agrave;
&ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&thorn;&ograve; p∗ ∈ ∆l &egrave; z ∗ ∈ E(p∗ ), &ograve;&agrave;&ecirc;&egrave;&aring; &divide;&ograve;&icirc; z ∗ ≤ 0.
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;. &Auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&atilde;&icirc; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave; z ∈ K &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&egrave;&igrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&aring; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &acirc; ∆l :
G(z) = Argmax(z) = {p ∈ ∆l , pz = max(qz)}.
q∈∆l
21
&Icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&icirc; (&egrave;&euml;&egrave; &euml;&aring;&atilde;&ecirc;&icirc; &iuml;&eth;&icirc;&acirc;&aring;&eth;&egrave;&ograve;&uuml; &acirc; &auml;&oacute;&otilde;&aring; &oacute;&iuml;&eth;. 2.9), &divide;&ograve;&icirc; &yacute;&ograve;&icirc; &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;&aring; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring;; &icirc;&aacute;&eth;&agrave;&ccedil;&ucirc;
&aring;&atilde;&icirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &atilde;&eth;&agrave;&iacute;&egrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; (&ograve;&icirc;&divide;&ecirc;&egrave; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&oacute;&igrave;&agrave; &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; z ), &ograve;&agrave;&ecirc; &divide;&ograve;&icirc; &icirc;&iacute;&egrave;
&iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&ucirc;&aring; &egrave; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&aring;.
&Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &aring;&ntilde;&euml;&egrave; &igrave;&ucirc; &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&aring;&igrave; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; F &egrave;&ccedil; ∆l &times; K &acirc; &ntilde;&aring;&aacute;&yuml; &iuml;&icirc; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&aring;:
F (p, z) = G(z) &times; E(p),
&ograve;&icirc; &ecirc; &iacute;&aring;&igrave;&oacute; &iuml;&eth;&egrave;&igrave;&aring;&iacute;&egrave;&igrave;&agrave; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Ecirc;&agrave;&ecirc;&oacute;&ograve;&agrave;&iacute;&egrave; &egrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ograve;&agrave;&ecirc;&agrave;&yuml; &iuml;&agrave;&eth;&agrave; (p∗ , z ∗ ) ∈ ∆l &times; K , &divide;&ograve;&icirc;
p∗ ∈ G(z ∗ ) &egrave; z ∗ ∈ E(p∗ ).
&Igrave;&ucirc; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&agrave;&aring;&igrave;, &divide;&ograve;&icirc; z ∗ ≤ 0. &Auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;, &auml;&icirc;&iuml;&oacute;&ntilde;&ograve;&egrave;&igrave;, &divide;&ograve;&icirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&agrave; zj∗ &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave;
z ∗ &ntilde;&ograve;&eth;&icirc;&atilde;&icirc; &iuml;&icirc;&euml;&icirc;&aelig;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&agrave;. &Ograve;&icirc;&atilde;&auml;&agrave; max(z ∗ |∆l ) &gt; 0 (&acirc;&ccedil;&yuml;&ograve;&uuml; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&aring; z ∗ &iacute;&agrave; j -&igrave; &aacute;&agrave;&ccedil;&egrave;&ntilde;&iacute;&icirc;&igrave; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&aring;
ej = (0, . . . , 1, . . . , 0) ∈ ∆l ), &ccedil;&iacute;&agrave;&divide;&egrave;&ograve; &egrave; p∗ z ∗ = max(z ∗ |∆l ) &gt; 0 (&ntilde;&igrave;. &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; G), &divide;&ograve;&icirc;
&iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&ograve; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&thorn; p∗ z ∗ &lt; 0.
&Ccedil;&agrave;&igrave;&aring;&divide;&agrave;&iacute;&egrave;&aring; 1. &Ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;, &igrave;&icirc;&auml;&aring;&euml;&uuml; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&yuml; &egrave; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml; &igrave;&icirc;&atilde;&oacute;&ograve; &igrave;&aring;&iacute;&yuml;&ograve;&uuml;&ntilde;&yuml;, &iacute;&icirc; &icirc;&aacute;&ugrave;&egrave;&eacute; &igrave;&aring;&ograve;&icirc;&auml; &auml;&icirc;
&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave; &icirc;&ntilde;&ograve;&agrave;&aring;&ograve;&ntilde;&yuml;. &Iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;, &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring;, &divide;&ograve;&icirc; Y &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&eacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;, &divide;&agrave;&ntilde;&ograve;&icirc; &ccedil;&agrave;&igrave;&aring;
&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring;&igrave;, &divide;&ograve;&icirc; Y &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&eacute; &ecirc;&icirc;&iacute;&oacute;&ntilde;; &iacute;&icirc; &acirc; &yacute;&ograve;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &iuml;&eth;&egrave;&otilde;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &auml;&icirc;&iuml;&icirc;&euml;&iacute;&egrave;
&ograve;&aring;&euml;&uuml;&iacute;&icirc; &iuml;&icirc;&ccedil;&agrave;&aacute;&icirc;&ograve;&egrave;&ograve;&uuml;&ntilde;&yuml; &icirc;&aacute; &icirc;&atilde;&eth;&agrave;&iacute;&egrave;&divide;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&egrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; &auml;&icirc;&iuml;&oacute;&ntilde;&ograve;&egrave;&igrave;&ucirc;&otilde; &eth;&agrave;&ntilde;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&eacute;.
&Ccedil;&agrave;&igrave;&aring;&divide;&agrave;&iacute;&egrave;&aring; 2. &Egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&icirc;&acirc;&agrave;&iacute;&iacute;&ucirc;&eacute; &acirc; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&aring; &euml;&aring;&igrave;&igrave;&ucirc; &Atilde;&aring;&eacute;&euml;&agrave;-&Iacute;&egrave;&ecirc;&agrave;&eacute;&auml;&icirc; &ntilde;&iuml;&icirc;&ntilde;&icirc;&aacute; &iacute;&agrave;&ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&yuml;
&iacute;&icirc;&acirc;&ucirc;&otilde; &ouml;&aring;&iacute; &iacute;&agrave;&acirc;&icirc;&auml;&egrave;&ograve; &iacute;&agrave; &igrave;&ucirc;&ntilde;&euml;&uuml; &acirc;&acirc;&aring;&ntilde;&ograve;&egrave; &aring;&ugrave;&aring; &icirc;&auml;&iacute;&icirc;&atilde;&icirc; &oacute;&divide;&agrave;&ntilde;&ograve;&iacute;&egrave;&ecirc;&agrave;, &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&eacute; &icirc;&ograve;&acirc;&aring;&divide;&agrave;&aring;&ograve; &ccedil;&agrave; &ouml;&aring;&iacute;&ucirc;. &Ograve;&icirc;&divide;
&iacute;&aring;&aring;, &icirc;&iacute; &ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&ograve; &iacute;&agrave; &yacute;&ecirc;&ntilde;&ouml;&aring;&ntilde;&ntilde; &ntilde;&iuml;&eth;&icirc;&ntilde;&agrave; z &egrave; &iacute;&agrave;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&ograve; &ouml;&aring;&iacute;&oacute; p &ograve;&agrave;&ecirc;, &divide;&ograve;&icirc;&aacute;&ucirc; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&egrave;&ccedil;&egrave;&eth;&icirc;&acirc;&agrave;&ograve;&uuml; pz &iacute;&agrave;
&ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&aring;. &Iuml;&eth;&egrave; &ograve;&agrave;&ecirc;&icirc;&igrave; &acirc;&ccedil;&atilde;&euml;&yuml;&auml;&aring; &ecirc;&icirc;&iacute;&ecirc;&oacute;&eth;&aring;&iacute;&ograve;&iacute;&icirc;&aring; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&aring; &iuml;&eth;&aring;&acirc;&eth;&agrave;&ugrave;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&aring; &Iacute;&yacute;&oslash;&agrave; &acirc;
&iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&iacute;&icirc;&eacute; &egrave;&atilde;&eth;&aring;. &Egrave; &auml;&euml;&yuml; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &igrave;&icirc;&aelig;&iacute;&icirc; &iuml;&eth;&egrave;&acirc;&euml;&aring;&ecirc;&agrave;&ograve;&uuml; &auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&icirc; &igrave;&icirc;&ugrave;&iacute;&ucirc;&aring; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &ntilde;&oacute;
&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml;. &Acirc; &divide;&agrave;&ntilde;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave;, &acirc;&igrave;&aring;&ntilde;&ograve;&icirc; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute; &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&icirc;&ntilde;&ograve;&egrave; &iuml;&icirc;&ograve;&eth;&aring;&aacute;&egrave;&ograve;&aring;&euml;&aring;&eacute; &igrave;&icirc;&aelig;&iacute;&icirc; &eth;&agrave;&ntilde;&ntilde;&igrave;&agrave;&ograve;&eth;&egrave;&acirc;&agrave;&ograve;&uuml;
&icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&yuml; &iuml;&eth;&aring;&auml;&iuml;&icirc;&divide;&ograve;&aring;&iacute;&egrave;&yuml; P &ograve;&egrave;&iuml;&agrave; &eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&aring;&iacute;&iacute;&ucirc;&otilde; &acirc;&ucirc;&oslash;&aring;.
&szlig;&auml;&eth;&icirc;
&Ecirc;&icirc;&iacute;&ecirc;&oacute;&eth;&aring;&iacute;&ograve;&iacute;&ucirc;&aring; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&yuml; &icirc;&aacute;&euml;&agrave;&auml;&agrave;&thorn;&ograve; &icirc;&auml;&iacute;&egrave;&igrave; &ccedil;&agrave;&igrave;&aring;&divide;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&igrave; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc;&igrave;, &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &igrave;&ucirc; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc;
&icirc;&ograve;&igrave;&aring;&ograve;&egrave;&igrave;, &iacute;&icirc; &iacute;&aring; &aacute;&oacute;&auml;&aring;&igrave; &iuml;&icirc;&auml;&eth;&icirc;&aacute;&iacute;&icirc; &icirc;&aacute;&ntilde;&oacute;&aelig;&auml;&agrave;&ograve;&uuml;. &Ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc; &yacute;&ograve;&icirc; &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &acirc; &ograve;&icirc;&igrave;, &divide;&ograve;&icirc; &iacute;&egrave;&ecirc;&agrave;&ecirc;&agrave;&yuml; &ecirc;&icirc;&agrave;&euml;&egrave;
&ouml;&egrave;&yuml; (&ograve;. &aring;. &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;) &iuml;&icirc;&ograve;&eth;&aring;&aacute;&egrave;&ograve;&aring;&euml;&aring;&eacute; &iacute;&aring; &igrave;&icirc;&aelig;&aring;&ograve; &oacute;&euml;&oacute;&divide;&oslash;&egrave;&ograve;&uuml; &aacute;&euml;&agrave;&atilde;&icirc;&ntilde;&icirc;&ntilde;&ograve;&icirc;&yuml;&iacute;&egrave;&aring; &ntilde;&acirc;&icirc;&egrave;&otilde; &divide;&euml;&aring;&iacute;&icirc;&acirc; &iuml;&icirc;
&ntilde;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&thorn; &ntilde; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&iacute;&ucirc;&igrave; &eth;&agrave;&ntilde;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring;&igrave;, &iuml;&icirc;&euml;&uuml;&ccedil;&oacute;&yuml;&ntilde;&uuml; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &ntilde;&acirc;&icirc;&egrave;&igrave;&egrave; &eth;&aring;&ntilde;&oacute;&eth;&ntilde;&agrave;&igrave;&egrave;. &Acirc; &divide;&agrave;&ntilde;&ograve;&iacute;&icirc;
&ntilde;&ograve;&egrave;, &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&iacute;&icirc;&aring; &eth;&agrave;&ntilde;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; &icirc;&iuml;&ograve;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&icirc; &iuml;&icirc; &Iuml;&agrave;&eth;&aring;&ograve;&icirc; &egrave; &egrave;&iacute;&auml;&egrave;&acirc;&egrave;&auml;&oacute;&agrave;&euml;&uuml;&iacute;&icirc; &eth;&agrave;&ouml;&egrave;&icirc;&iacute;&agrave;&euml;&uuml;&iacute;&icirc;.
&Yacute;&ograve;&icirc; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&icirc;&iacute;&iacute;&icirc;&eacute; &iacute;&aring;&oacute;&euml;&oacute;&divide;&oslash;&agrave;&aring;&igrave;&icirc;&ntilde;&ograve;&egrave; &igrave;&icirc;&aelig;&iacute;&icirc; &ntilde;&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&egrave;&eth;&icirc;&acirc;&agrave;&ograve;&uuml; &agrave;&aacute;&ntilde;&ograve;&eth;&agrave;&ecirc;&ograve;&iacute;&icirc;. &Iuml;&oacute;&ntilde;&ograve;&uuml;
&egrave;&igrave;&aring;&thorn;&ograve;&ntilde;&yuml; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &agrave;&euml;&uuml;&ograve;&aring;&eth;&iacute;&agrave;&ograve;&egrave;&acirc; X &egrave; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&agrave;&yuml; &atilde;&eth;&oacute;&iuml;&iuml;&agrave; &oacute;&divide;&agrave;&ntilde;&ograve;&iacute;&egrave;&ecirc;&icirc;&acirc; I .
&Ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&aring;&eacute; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&icirc;&aring; &iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&icirc;&aring; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &acirc; I . &Auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&eacute; &agrave;&euml;&uuml;&ograve;&aring;&eth;&iacute;&agrave;&ograve;&egrave;&acirc;&ucirc;
x ∈ X &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&egrave;&igrave; &divide;&aring;&eth;&aring;&ccedil; H(x) &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &ograve;&aring;&otilde; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&eacute;, &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &igrave;&icirc;&atilde;&oacute;&ograve; &oacute;&euml;&oacute;&divide;&oslash;&egrave;&ograve;&uuml; &egrave;&ntilde;&otilde;&icirc;&auml; x.
&Iuml;&eth;&egrave; &iacute;&agrave;&oslash;&aring;&igrave; &iuml;&icirc;&auml;&otilde;&icirc;&auml;&aring; &frac34;&oacute;&euml;&oacute;&divide;&oslash;&egrave;&ograve;&uuml;&iquest; &yacute;&ograve;&icirc; &iuml;&aring;&eth;&acirc;&egrave;&divide;&iacute;&icirc;&aring;, &iacute;&aring;&icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&yuml;&aring;&igrave;&icirc;&aring; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&aring;, &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &auml;&icirc;&euml;&aelig;&iacute;&icirc;
&oacute;&ograve;&icirc;&divide;&iacute;&yuml;&ograve;&uuml;&ntilde;&yuml; &acirc; &aacute;&icirc;&euml;&aring;&aring; &ecirc;&icirc;&iacute;&ecirc;&eth;&aring;&ograve;&iacute;&ucirc;&otilde; &ntilde;&egrave;&ograve;&oacute;&agrave;&ouml;&egrave;&yuml;&otilde;. &Atilde;&icirc;&acirc;&icirc;&eth;&yuml;&ograve;, &divide;&ograve;&icirc; &ntilde;&icirc;&ntilde;&ograve;&icirc;&yuml;&iacute;&egrave;&aring; x∗ &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; &yuml;&auml;&eth;&oacute;,
&aring;&ntilde;&euml;&egrave; H(x∗ ) &iuml;&oacute;&ntilde;&ograve;&icirc;. &Iuml;&eth;&egrave;&acirc;&aring;&auml;&aring;&igrave; &icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&oacute;&thorn; &igrave;&icirc;&auml;&aring;&euml;&uuml; (&egrave;&euml;&egrave; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&oacute;), &atilde;&auml;&aring; &iuml;&icirc;&yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &yuml;&auml;&eth;&icirc;, &egrave;
&oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml;, &atilde;&agrave;&eth;&agrave;&iacute;&ograve;&egrave;&eth;&oacute;&thorn;&ugrave;&egrave;&aring; &iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&icirc;&ograve;&oacute; &yuml;&auml;&eth;&agrave;. &Yacute;&ograve;&agrave; &igrave;&icirc;&auml;&aring;&euml;&uuml; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&icirc;&iacute;&iacute;&icirc;&eacute; &egrave;&atilde;&eth;&icirc;&eacute; &aacute;&aring;&ccedil;
&iuml;&icirc;&aacute;&icirc;&divide;&iacute;&ucirc;&otilde; &iuml;&euml;&agrave;&ograve;&aring;&aelig;&aring;&eacute;.
&Iuml;&oacute;&ntilde;&ograve;&uuml; I &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &egrave;&atilde;&eth;&icirc;&ecirc;&icirc;&acirc; (&oacute;&divide;&agrave;&ntilde;&ograve;&iacute;&egrave;&ecirc;&icirc;&acirc;). &Egrave;&ntilde;&otilde;&icirc;&auml; &egrave;&atilde;&eth;&ucirc; &icirc;&iuml;&egrave;&ntilde;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&icirc;&igrave; u =
(ui , i ∈ I) ∈ RI , &ecirc;&icirc;&iacute;&ecirc;&eth;&aring;&ograve;&egrave;&ccedil;&egrave;&eth;&oacute;&thorn;&ugrave;&egrave;&igrave;, &ecirc;&agrave;&ecirc;&oacute;&thorn; &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&icirc;&ntilde;&ograve;&uuml; &iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&ograve; &ograve;&icirc;&ograve; &egrave;&euml;&egrave; &egrave;&iacute;&icirc;&eacute; &egrave;&atilde;&eth;&icirc;&ecirc;.
&Acirc;&icirc;&ccedil;&igrave;&icirc;&aelig;&iacute;&icirc;&ntilde;&ograve;&egrave; &ecirc;&agrave;&aelig;&auml;&icirc;&eacute; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&egrave; K ⊂ I &ccedil;&agrave;&auml;&agrave;&thorn;&ograve;&ntilde;&yuml; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;&igrave; V (K) ⊂ RK ; &aring;&ntilde;&euml;&egrave; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;
xK = (xi , i ∈ K) &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; V (K), &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&yuml; K &igrave;&icirc;&aelig;&aring;&ograve; &icirc;&aacute;&aring;&ntilde;&iuml;&aring;&divide;&egrave;&ograve;&uuml; &ecirc;&agrave;&aelig;&auml;&icirc;&igrave;&oacute; &ntilde;&acirc;&icirc;&aring;&igrave;&oacute;
&oacute;&divide;&agrave;&ntilde;&ograve;&iacute;&egrave;&ecirc;&oacute; i ∈ K &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&icirc;&ntilde;&ograve;&uuml; &iacute;&aring; &igrave;&aring;&iacute;&uuml;&oslash;&aring;, &divide;&aring;&igrave; xi . &Ograve;&agrave;&ecirc;&egrave;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;, &egrave;&atilde;&eth;&agrave; &aacute;&aring;&ccedil; &iuml;&icirc;&aacute;&icirc;&divide;&iacute;&ucirc;&otilde; &iuml;&euml;&agrave;
&ograve;&aring;&aelig;&aring;&eacute; &ccedil;&agrave;&auml;&agrave;&aring;&ograve;&ntilde;&yuml; &ntilde;&aring;&igrave;&aring;&eacute;&ntilde;&ograve;&acirc;&icirc;&igrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc; V = (V (K), K ⊂ I). &Ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&yuml; K &igrave;&icirc;&aelig;&aring;&ograve; &oacute;&euml;&oacute;&divide;&oslash;&egrave;&ograve;&uuml;
22
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 3.
&Ograve;&Aring;&Icirc;&ETH;&Aring;&Igrave;&Agrave; &Aacute;&ETH;&Agrave;&Oacute;&Yacute;&ETH;&Agrave;: &Iuml;&ETH;&Egrave;&Igrave;&Aring;&Iacute;&Aring;&Iacute;&Egrave;&szlig;
&egrave;&ntilde;&otilde;&icirc;&auml; u ∈ RI , &aring;&ntilde;&euml;&egrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ograve;&agrave;&ecirc;&icirc;&eacute; &acirc;&aring;&ecirc;&ograve;&icirc;&eth; xK = (xi , i ∈ K) &egrave;&ccedil; V (K), &divide;&ograve;&icirc; xi &gt; ui &auml;&euml;&yuml; &acirc;&ntilde;&aring;&otilde;
i ∈ K . &Acirc; &yacute;&ograve;&egrave;&otilde; &ograve;&aring;&eth;&igrave;&egrave;&iacute;&agrave;&otilde; &yuml;&auml;&eth;&icirc; &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &egrave;&ccedil; &ograve;&agrave;&ecirc;&egrave;&otilde; &iacute;&agrave;&aacute;&icirc;&eth;&icirc;&acirc; u ∈ RI , &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &ntilde; &icirc;&auml;&iacute;&icirc;&eacute; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute;&ucirc;
&auml;&icirc;&ntilde;&ograve;&egrave;&aelig;&egrave;&igrave;&ucirc;, &ograve;. &aring;. &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&agrave;&ograve; V (I), &agrave; &ntilde; &auml;&eth;&oacute;&atilde;&icirc;&eacute; &iacute;&aring;&oacute;&euml;&oacute;&divide;&oslash;&agrave;&aring;&igrave;&ucirc; &iacute;&egrave;&ecirc;&agrave;&ecirc;&icirc;&eacute; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&aring;&eacute;.
&Icirc;&aacute;&ucirc;&divide;&iacute;&icirc; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&aring;&ograve;&ntilde;&yuml;, &divide;&ograve;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; V (K) &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&ucirc;, &icirc;&atilde;&eth;&agrave;&iacute;&egrave;&divide;&aring;&iacute;&ucirc; &ntilde;&acirc;&aring;&eth;&otilde;&oacute; &egrave; &iacute;&icirc;&eth;&igrave;&agrave;&euml;&uuml;
&iacute;&ucirc; (&ograve;. &aring;. &ntilde; &ecirc;&agrave;&aelig;&auml;&ucirc;&igrave; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&igrave; &ntilde;&icirc;&auml;&aring;&eth;&aelig;&agrave;&ograve; &egrave; &igrave;&aring;&iacute;&uuml;&oslash;&egrave;&aring;). &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&aring; &iacute;&aring;&oacute;&euml;&oacute;&divide;&oslash;&agrave;&aring;&igrave;&icirc;&ntilde;&ograve;&egrave;
&igrave;&icirc;&aelig;&iacute;&icirc; &iuml;&aring;&eth;&aring;&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&egrave;&eth;&icirc;&acirc;&agrave;&ograve;&uuml; &ograve;&agrave;&ecirc;: &iuml;&eth;&icirc;&aring;&ecirc;&ouml;&egrave;&yuml; uK &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave; u &iacute;&agrave; &euml;&thorn;&aacute;&icirc;&aring; &iuml;&icirc;&auml;&iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; RK &iacute;&aring;
&iuml;&icirc;&iuml;&agrave;&auml;&agrave;&aring;&ograve; &ntilde;&ograve;&eth;&icirc;&atilde;&icirc; &acirc;&iacute;&oacute;&ograve;&eth;&uuml; V (K).
&Egrave;&iacute;&ograve;&oacute;&egrave;&ograve;&egrave;&acirc;&iacute;&icirc; &yuml;&ntilde;&iacute;&icirc;, &divide;&ograve;&icirc; &aring;&ntilde;&euml;&egrave; &frac34;&ccedil;&agrave;&iuml;&eth;&icirc;&ntilde;&ucirc;&iquest; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&eacute; (&ograve;. &aring;. &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; V (K)) &acirc;&aring;&euml;&egrave;&ecirc;&egrave; &iuml;&icirc; &ntilde;&eth;&agrave;&acirc;
&iacute;&aring;&iacute;&egrave;&thorn; &ntilde; &acirc;&icirc;&ccedil;&igrave;&icirc;&aelig;&iacute;&icirc;&ntilde;&ograve;&yuml;&igrave;&egrave; &acirc;&ntilde;&aring;&atilde;&icirc; &icirc;&aacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&agrave; (&ograve;. &aring;. &ntilde; V (I)), &ograve;&icirc; &yuml;&auml;&eth;&icirc; &iuml;&oacute;&ntilde;&ograve;&icirc;. &Acirc; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&iacute;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring;
&igrave;&icirc;&aelig;&iacute;&icirc; &eth;&agrave;&ntilde;&ntilde;&divide;&egrave;&ograve;&ucirc;&acirc;&agrave;&ograve;&uuml; &iacute;&agrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &yuml;&auml;&aring;&eth;&iacute;&ucirc;&otilde; &egrave;&ntilde;&otilde;&icirc;&auml;&icirc;&acirc;. &Iacute;&icirc; &ecirc;&agrave;&ecirc; &yacute;&ograve;&icirc; &ograve;&icirc;&divide;&iacute;&icirc; &ntilde;&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&egrave;&eth;&icirc;
&acirc;&agrave;&ograve;&uuml;? &Icirc;&auml;&iacute;&agrave; &egrave;&ccedil; &acirc;&icirc;&ccedil;&igrave;&icirc;&aelig;&iacute;&icirc;&ntilde;&ograve;&aring;&eacute; &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &acirc; &iuml;&eth;&egrave;&acirc;&euml;&aring;&divide;&aring;&iacute;&egrave;&egrave; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&yuml; &ntilde;&aacute;&agrave;&euml;&agrave;&iacute;&ntilde;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&icirc;&ntilde;&ograve;&egrave;.
&Iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave;, &divide;&ograve;&icirc; &ecirc;&agrave;&aelig;&auml;&icirc;&eacute; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&egrave; K &iuml;&eth;&aring;&auml;&iuml;&egrave;&ntilde;&agrave;&iacute;&icirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &iacute;&aring;&icirc;&ograve;&eth;&egrave;&ouml;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&aring; &divide;&egrave;&ntilde;&euml;&icirc;
λK . &Iacute;&agrave;&aacute;&icirc;&eth; λ = (λK , K ⊂ I) &ograve;&agrave;&ecirc;&egrave;&otilde; &divide;&egrave;&ntilde;&aring;&euml; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ntilde;&aacute;&agrave;&euml;&agrave;&iacute;&ntilde;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&ucirc;&igrave;, &aring;&ntilde;&euml;&egrave;
X
λK = 1 &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; i ∈ I.
(3.1)
i∈K
&Ograve;&agrave;&ecirc; &acirc;&icirc;&ograve;, &egrave;&atilde;&eth;&agrave; V &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ntilde;&aacute;&agrave;&euml;&agrave;&iacute;&ntilde;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&icirc;&eacute;, &aring;&ntilde;&euml;&egrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; V (I) &frac34;&auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc; &acirc;&aring;&euml;&egrave;&ecirc;&icirc;&iquest;
&acirc; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&igrave; &ntilde;&igrave;&ucirc;&ntilde;&euml;&aring;. &Iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave;, &divide;&ograve;&icirc; &acirc;&aring;&ecirc;&ograve;&icirc;&eth; &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&icirc;&ntilde;&ograve;&aring;&eacute; u ∈ RI &ograve;&agrave;&ecirc;&icirc;&acirc;, &divide;&ograve;&icirc; &iacute;&agrave;&eacute;&auml;&aring;&ograve;&ntilde;&yuml;
&ntilde;&aacute;&agrave;&euml;&agrave;&iacute;&ntilde;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&ucirc;&eacute; &iacute;&agrave;&aacute;&icirc;&eth; &ecirc;&icirc;&yacute;&ocirc;&ocirc;&egrave;&ouml;&egrave;&aring;&iacute;&ograve;&icirc;&acirc; λ = (λK ), &ograve;&agrave;&ecirc;&icirc;&eacute; &divide;&ograve;&icirc; &aring;&ntilde;&euml;&egrave; λK 6= 0, &ograve;&icirc; &iuml;&eth;&icirc;&aring;&ecirc;&ouml;&egrave;&yuml;
uK ∈ V (K); &ograve;&icirc;&atilde;&auml;&agrave; &acirc;&aring;&ecirc;&ograve;&icirc;&eth; u ∈ V (I).
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; (&Ntilde;&ecirc;&agrave;&eth;&ocirc;, 1967). &Aring;&ntilde;&euml;&egrave; &egrave;&atilde;&eth;&agrave; V &ntilde;&aacute;&agrave;&euml;&agrave;&iacute;&ntilde;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&agrave;, &ograve;&icirc; &aring;&aring; &yuml;&auml;&eth;&icirc; &iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&icirc;.
&Ntilde;&ograve;&agrave;&iacute;&auml;&agrave;&eth;&ograve;&iacute;&ucirc;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Ntilde;&ecirc;&agrave;&eth;&ocirc;&agrave; &agrave;&iuml;&aring;&euml;&euml;&egrave;&eth;&oacute;&thorn;&ograve; &icirc;&aacute;&ucirc;&divide;&iacute;&icirc; &ecirc; &ograve;&icirc;&igrave;&oacute; &egrave;&euml;&egrave; &egrave;&iacute;&icirc;&igrave;&oacute;
&acirc;&agrave;&eth;&egrave;&agrave;&iacute;&ograve;&oacute; &euml;&aring;&igrave;&igrave;&ucirc; &Ecirc;&oacute;&eth;&agrave;&ograve;&icirc;&acirc;&ntilde;&ecirc;&icirc;&atilde;&icirc;&Ecirc;&iacute;&agrave;&ntilde;&ograve;&aring;&eth;&agrave;&Igrave;&agrave;&ccedil;&oacute;&eth;&ecirc;&aring;&acirc;&egrave;&divide;&agrave;&Oslash;&aring;&iuml;&euml;&egrave; (&ntilde;&igrave;. &iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;, &ecirc;&iacute;&egrave;&atilde;&oacute; &Yacute;&ecirc;
&euml;&agrave;&iacute;&auml;&agrave; [10]). &szlig; &iuml;&eth;&egrave;&acirc;&aring;&auml;&oacute; &ccedil;&auml;&aring;&ntilde;&uuml; &auml;&eth;&oacute;&atilde;&icirc;&aring;, &aacute;&icirc;&euml;&aring;&aring; &iuml;&eth;&yuml;&igrave;&icirc;&aring; &egrave; &ecirc;&icirc;&eth;&icirc;&ograve;&ecirc;&icirc;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;. &Icirc;&iacute;&icirc; &icirc;&ntilde;&iacute;&icirc;&acirc;&agrave;&iacute;&icirc;
&iacute;&agrave; &egrave;&iacute;&ograve;&aring;&eth;&iuml;&eth;&aring;&ograve;&agrave;&ouml;&egrave;&egrave; &egrave;&atilde;&eth;&ucirc; &aacute;&aring;&ccedil; &iuml;&icirc;&aacute;&icirc;&divide;&iacute;&ucirc;&otilde; &iuml;&euml;&agrave;&ograve;&aring;&aelig;&aring;&eacute; &ecirc;&agrave;&ecirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&eacute; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&ecirc;&egrave;, &acirc; &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&eacute; &ecirc;&icirc;
&agrave;&euml;&egrave;&ouml;&egrave;&egrave; &egrave;&atilde;&eth;&agrave;&thorn;&ograve; &eth;&icirc;&euml;&uuml; &ocirc;&egrave;&eth;&igrave;, &agrave; &acirc;&ucirc;&iuml;&euml;&agrave;&ograve;&ucirc; &oacute;&divide;&agrave;&ntilde;&ograve;&iacute;&egrave;&ecirc;&agrave;&igrave; &iuml;&icirc;&iacute;&egrave;&igrave;&agrave;&thorn;&ograve;&ntilde;&yuml; &ecirc;&agrave;&ecirc; &ccedil;&agrave;&eth;&iuml;&euml;&agrave;&ograve;&ucirc;. &szlig;&auml;&aring;&eth;&iacute;&icirc;&aring;
&eth;&agrave;&ntilde;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; &ograve;&icirc;&atilde;&auml;&agrave; &ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&egrave;&ograve;&ntilde;&yuml; &iuml;&eth;&icirc;&ntilde;&ograve;&icirc; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&iacute;&ucirc;&igrave; &eth;&agrave;&ntilde;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring;&igrave; &acirc; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&oacute;&thorn;&ugrave;&aring;&eacute;
&yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&ecirc;&aring;.
&Icirc;&auml;&iacute;&icirc; &iacute;&agrave;&acirc;&icirc;&auml;&yuml;&ugrave;&aring;&aring; &ntilde;&icirc;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring;. &ETH;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve;&icirc;&igrave; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&icirc;&iacute;&iacute;&icirc;&atilde;&icirc; &acirc;&ccedil;&agrave;&egrave;&igrave;&icirc;&auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&yuml; &aring;&ntilde;&ograve;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;
&ntilde;&divide;&egrave;&ograve;&agrave;&ograve;&uuml; &iuml;&agrave;&eth;&oacute; (u, λ), &atilde;&auml;&aring; u = (ui , i ∈ I) &eth;&agrave;&ntilde;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; &acirc;&ucirc;&egrave;&atilde;&eth;&ucirc;&oslash;&agrave;, &agrave; λ = (λK , K ⊂ I)
&icirc;&ograve;&eth;&agrave;&aelig;&agrave;&aring;&ograve; &egrave;&iacute;&ograve;&aring;&iacute;&ntilde;&egrave;&acirc;&iacute;&icirc;&ntilde;&ograve;&egrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&icirc;&iacute;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&eacute;. &Iuml;&eth;&icirc;&ugrave;&aring; &acirc;&ntilde;&aring;&atilde;&icirc; &ntilde;&divide;&egrave;&ograve;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; λK &iuml;&eth;&egrave;
&iacute;&egrave;&igrave;&agrave;&thorn;&ograve; &ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&yuml; 1 &egrave;&euml;&egrave; 0 &acirc; &ccedil;&agrave;&acirc;&egrave;&ntilde;&egrave;&igrave;&icirc;&ntilde;&ograve;&egrave; &icirc;&ograve; &ograve;&icirc;&atilde;&icirc;, &eth;&aring;&agrave;&euml;&uuml;&iacute;&icirc; &euml;&egrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&acirc;&agrave;&euml;&agrave;&ntilde;&uuml; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&yuml; K &egrave;&euml;&egrave;
&iacute;&aring;&ograve;. &Icirc;&auml;&iacute;&agrave;&ecirc;&icirc; &aacute;&icirc;&euml;&aring;&aring; &atilde;&egrave;&aacute;&ecirc;&egrave;&eacute; &iuml;&icirc;&auml;&otilde;&icirc;&auml; &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &acirc; &ograve;&icirc;&igrave;, &divide;&ograve;&icirc;&aacute;&ucirc; &auml;&icirc;&iuml;&oacute;&ntilde;&ograve;&egrave;&ograve;&uuml; &egrave; &iuml;&eth;&icirc;&igrave;&aring;&aelig;&oacute;&ograve;&icirc;&divide;&iacute;&ucirc;&aring; &ccedil;&iacute;&agrave;&divide;&aring;
&iacute;&egrave;&yuml; &auml;&euml;&yuml; λK . &Icirc;&auml;&iacute;&agrave;&ecirc;&icirc; &iacute;&oacute;&aelig;&iacute;&icirc; &acirc; &euml;&thorn;&aacute;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &iuml;&icirc;&ograve;&eth;&aring;&aacute;&icirc;&acirc;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc;&aacute;&ucirc; &egrave;&igrave;&aring;&thorn;&ugrave;&egrave;&otilde;&ntilde;&yuml; &oacute;&divide;&agrave;&ntilde;&ograve;&iacute;&egrave;&ecirc;&icirc;&acirc; &acirc;
&ograve;&icirc;&divide;&iacute;&icirc;&ntilde;&ograve;&egrave; &otilde;&acirc;&agrave;&ograve;&egrave;&euml;&icirc; &auml;&euml;&yuml; &ograve;&agrave;&ecirc;&icirc;&atilde;&icirc; &ocirc;&icirc;&eth;&igrave;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml;, &ograve;. &aring;. &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&aring;&iacute;&egrave;&aring; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml; &ntilde;&aacute;&agrave;&euml;&agrave;&iacute;&ntilde;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&icirc;&ntilde;&ograve;&egrave;
(3.1). &Acirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&igrave; &ntilde;&igrave;&ucirc;&ntilde;&euml;&aring; &yacute;&ograve;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&icirc; &aacute;&agrave;&euml;&agrave;&iacute;&ntilde; &iuml;&icirc; &ograve;&eth;&oacute;&auml;&oacute;.
&Acirc;&ograve;&icirc;&eth;&icirc;&aring; &aring;&ntilde;&ograve;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&aring; &ograve;&eth;&aring;&aacute;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &iacute;&agrave; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&egrave;&eth;&oacute;&thorn;&ugrave;&oacute;&thorn; &iuml;&agrave;&eth;&oacute; (u, λ) &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &acirc; &ograve;&icirc;&igrave;, &divide;&ograve;&icirc; &acirc;&aring;&ecirc;
&ograve;&icirc;&eth; u &auml;&icirc;&euml;&aelig;&aring;&iacute; &aacute;&ucirc;&ograve;&uuml; &auml;&icirc;&ntilde;&ograve;&egrave;&aelig;&egrave;&igrave; &oacute;&ntilde;&egrave;&euml;&egrave;&yuml;&igrave;&egrave; &ntilde;&ocirc;&icirc;&eth;&igrave;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&ucirc;&otilde; &ntilde;&icirc;&atilde;&euml;&agrave;&ntilde;&iacute;&icirc; λ &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&eacute;. &Iuml;&icirc;&auml; &yacute;&ograve;&egrave;&igrave;
&yuml; &iuml;&icirc;&iacute;&egrave;&igrave;&agrave;&thorn;, &divide;&ograve;&icirc; &aring;&ntilde;&euml;&egrave; λK &gt; 0 &auml;&euml;&yuml; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&eacute; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&egrave; K , &ograve;&icirc; uK ∈ V (K). &Aring;&ntilde;&euml;&egrave; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&egrave;&ograve;&uuml;
&divide;&aring;&eth;&aring;&ccedil; V (λ) &ograve;&agrave;&ecirc;&icirc;&eacute; &iacute;&agrave;&aacute;&icirc;&eth; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&icirc;&acirc; u, &ograve;&icirc; &yacute;&ograve;&icirc; &acirc;&ograve;&icirc;&eth;&icirc;&aring; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&aring; &igrave;&icirc;&aelig;&iacute;&icirc; &iuml;&aring;&eth;&aring;&iuml;&egrave;&ntilde;&agrave;&ograve;&uuml; &ecirc;&agrave;&ecirc; u ∈ V (λ).
&Iacute;&agrave;&ecirc;&icirc;&iacute;&aring;&ouml;, &aring;&ntilde;&euml;&egrave; &auml;&aring;&euml;&aring;&aelig; u &igrave;&icirc;&aelig;&aring;&ograve; &aacute;&ucirc;&ograve;&uuml; &oacute;&euml;&oacute;&divide;&oslash;&aring;&iacute; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&eacute; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&aring;&eacute; K (&ograve;. &aring;. &aring;&ntilde;&euml;&egrave; uK ∈
Int(V (K))), &ograve;&icirc; &ntilde;&euml;&icirc;&aelig;&egrave;&acirc;&oslash;&agrave;&yuml;&ntilde;&yuml; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&agrave; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&eacute; &aacute;&oacute;&auml;&aring;&ograve; &eth;&agrave;&ccedil;&eth;&oacute;&oslash;&aring;&iacute;&agrave;. &times;&ograve;&icirc;&aacute;&ucirc; &yacute;&ograve;&icirc; &egrave;&ntilde;&ecirc;&euml;&thorn;&divide;&egrave;&ograve;&uuml;,
&iuml;&icirc;&ograve;&eth;&aring;&aacute;&oacute;&aring;&igrave;, &divide;&ograve;&icirc;&aacute;&ucirc; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&eacute; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&egrave; K &acirc;&aring;&ecirc;&ograve;&icirc;&eth; uK &iacute;&aring; &iuml;&icirc;&iuml;&agrave;&auml;&agrave;&euml; &acirc;&icirc; &acirc;&iacute;&oacute;&ograve;&eth;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml; V (K).
&Iuml;&agrave;&eth;&oacute; (u, λ), &oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&eth;&yuml;&thorn;&ugrave;&oacute;&thorn; &yacute;&ograve;&egrave;&igrave; &ograve;&eth;&aring;&igrave; &ograve;&eth;&aring;&aacute;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml;&igrave; (&ntilde;&aacute;&agrave;&euml;&agrave;&iacute;&ntilde;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&icirc;&ntilde;&ograve;&egrave;, &auml;&icirc;&ntilde;&ograve;&egrave;&aelig;&egrave;
&igrave;&icirc;&ntilde;&ograve;&egrave; &egrave; &iacute;&aring;&auml;&icirc;&igrave;&egrave;&iacute;&egrave;&eth;&oacute;&aring;&igrave;&icirc;&ntilde;&ograve;&egrave;), &iacute;&agrave;&ccedil;&icirc;&acirc;&aring;&igrave; B -&yuml;&auml;&aring;&eth;&iacute;&icirc;&eacute;.
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave;. &Acirc; &euml;&thorn;&aacute;&icirc;&eacute; &egrave;&atilde;&eth;&aring; V &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&thorn;&ograve; B -&yuml;&auml;&aring;&eth;&iacute;&ucirc;&aring; &egrave;&ntilde;&otilde;&icirc;&auml;&ucirc;.
23
&Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &ntilde;&aacute;&agrave;&euml;&agrave;&iacute;&ntilde;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml; &egrave;&atilde;&eth;&ucirc; &icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&ograve;, &divide;&ograve;&icirc; V (λ) ⊂ V (I) &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; &ntilde;&aacute;&agrave;&euml;&agrave;&iacute;&ntilde;&egrave;
&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&icirc;&atilde;&icirc; &iacute;&agrave;&aacute;&icirc;&eth;&agrave; λ, &ograve;&icirc; &euml;&thorn;&aacute;&icirc;&eacute; B -&yuml;&auml;&aring;&eth;&iacute;&ucirc;&eacute; &egrave;&ntilde;&otilde;&icirc;&auml; &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; &yuml;&auml;&eth;&oacute;. &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&ucirc;&igrave;
&ntilde;&euml;&aring;&auml;&ntilde;&ograve;&acirc;&egrave;&aring;&igrave; &iacute;&agrave;&oslash;&aring;&eacute; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &oacute;&aelig;&aring; &oacute;&iuml;&icirc;&igrave;&yuml;&iacute;&oacute;&ograve;&agrave;&yuml; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Ntilde;&ecirc;&agrave;&eth;&ocirc;&agrave;.
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;. &Ecirc;&agrave;&ecirc; &oacute;&aelig;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&egrave;&euml;&icirc;&ntilde;&uuml;, &igrave;&ucirc; &ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &iacute;&agrave; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&icirc;&iacute;&iacute;&oacute;&thorn; &egrave;&atilde;&eth;&oacute; &ecirc;&agrave;&ecirc; &iacute;&agrave; &yacute;&ecirc;&icirc;&iacute;&icirc;
&igrave;&egrave;&ecirc;&oacute;, &acirc; &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&eacute; &ocirc;&egrave;&eth;&igrave;&ucirc; &auml;&euml;&yuml; &ntilde;&acirc;&icirc;&aring;&eacute; &eth;&agrave;&aacute;&icirc;&ograve;&ucirc; &iuml;&eth;&egrave;&atilde;&euml;&agrave;&oslash;&agrave;&thorn;&ograve; &oacute;&divide;&agrave;&ntilde;&ograve;&iacute;&egrave;&ecirc;&icirc;&acirc; &yacute;&ograve;&icirc;&eacute; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&egrave;. &ETH;&icirc;&euml;&uuml;
&ouml;&aring;&iacute; (&ccedil;&agrave;&eth;&iuml;&euml;&agrave;&ograve;) &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&yuml;&aring;&ograve; &acirc;&aring;&ecirc;&ograve;&icirc;&eth; u. &Acirc; &ccedil;&agrave;&acirc;&egrave;&ntilde;&egrave;&igrave;&icirc;&ntilde;&ograve;&egrave; &icirc;&ograve; &acirc;&aring;&euml;&egrave;&divide;&egrave;&iacute;&ucirc; u &ocirc;&egrave;&eth;&igrave;&agrave;-&ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&yuml; K &euml;&egrave;&aacute;&icirc;
&aacute;&aring;&ccedil;&auml;&aring;&eacute;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; (&aring;&ntilde;&euml;&egrave; &ccedil;&agrave;&iuml;&eth;&icirc;&ntilde;&ucirc; &oacute;&divide;&agrave;&ntilde;&ograve;&iacute;&egrave;&ecirc;&icirc;&acirc; &acirc;&aring;&euml;&egrave;&ecirc;&egrave; &egrave; uK &iacute;&aring; &iuml;&icirc;&iuml;&agrave;&auml;&agrave;&aring;&ograve; &acirc; V (K)), &euml;&egrave;&aacute;&icirc; &eth;&agrave;&ccedil;&acirc;&egrave;&acirc;&agrave;&aring;&ograve;
&aacute;&aring;&oslash;&aring;&iacute;&oacute;&thorn; &agrave;&ecirc;&ograve;&egrave;&acirc;&iacute;&icirc;&ntilde;&ograve;&uuml; (&aring;&ntilde;&euml;&egrave; &ccedil;&agrave;&iuml;&eth;&icirc;&ntilde;&ucirc; &oacute;&divide;&agrave;&ntilde;&ograve;&iacute;&egrave;&ecirc;&icirc;&acirc; &igrave;&agrave;&euml;&ucirc; &egrave; uK &iuml;&icirc;&iuml;&agrave;&auml;&agrave;&aring;&ograve; &acirc;&icirc; &acirc;&iacute;&oacute;&ograve;&eth;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml; V (K)),
&euml;&egrave;&aacute;&icirc; (&acirc; &iuml;&eth;&icirc;&igrave;&aring;&aelig;&oacute;&ograve;&icirc;&divide;&iacute;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring;, &ecirc;&icirc;&atilde;&auml;&agrave; uK &euml;&aring;&aelig;&egrave;&ograve; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; V (K)) &aacute;&aring;&ccedil;&eth;&agrave;&ccedil;&euml;&egrave;&divide;&iacute;&ucirc;
&ecirc; &oacute;&eth;&icirc;&acirc;&iacute;&thorn; &egrave;&iacute;&ograve;&aring;&iacute;&ntilde;&egrave;&acirc;&iacute;&icirc;&ntilde;&ograve;&egrave;. &Acirc; &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&igrave; &egrave;&ograve;&icirc;&atilde;&aring; &ocirc;&egrave;&eth;&igrave;&ucirc;-&ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&egrave; &iuml;&eth;&aring;&auml;&uacute;&yuml;&acirc;&euml;&yuml;&thorn;&ograve; &ntilde;&iuml;&eth;&icirc;&ntilde; &iacute;&agrave; &oacute;&divide;&agrave;&ntilde;&ograve;
&iacute;&egrave;&ecirc;&icirc;&acirc;. &Aring;&ntilde;&euml;&egrave; &ntilde;&iuml;&eth;&icirc;&ntilde; &iacute;&agrave; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&atilde;&icirc; &oacute;&divide;&agrave;&ntilde;&ograve;&iacute;&egrave;&ecirc;&agrave; i &iuml;&eth;&aring;&acirc;&ucirc;&oslash;&agrave;&aring;&ograve; 1, &ograve;&icirc; ui &oacute;&acirc;&aring;&euml;&egrave;&divide;&egrave;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml;, &aring;&ntilde;&euml;&egrave; &iacute;&aring;&ograve; &oacute;&igrave;&aring;&iacute;&uuml;&oslash;&agrave;&aring;&ograve;&ntilde;&yuml;. &Iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave; &auml;&agrave;&aring;&ograve; &yuml;&auml;&aring;&eth;&iacute;&ucirc;&eacute; &egrave;&ntilde;&otilde;&icirc;&auml;.
&Aacute;&icirc;&euml;&aring;&aring; &ograve;&icirc;&divide;&iacute;&icirc;, &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&egrave;&igrave; &divide;&aring;&eth;&aring;&ccedil; Λ &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; {λ = (λK ), 0 ≤ λK ≤ 2 ∀ K }, X &oslash;&agrave;&eth;
&aacute;&icirc;&euml;&uuml;&oslash;&icirc;&atilde;&icirc; &eth;&agrave;&auml;&egrave;&oacute;&ntilde;&agrave; &acirc;&icirc;&ecirc;&eth;&oacute;&atilde; 0 &acirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; RI . &Igrave;&ucirc; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&egrave;&igrave; &ntilde;&aring;&eacute;&divide;&agrave;&ntilde; (&igrave;&iacute;&icirc;&atilde;&icirc;&ccedil;&iacute;&agrave;&divide;&iacute;&icirc;&aring;) &icirc;&ograve;&icirc;&aacute;
&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; F &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; X &times; Λ &acirc; &ntilde;&aring;&aacute;&yuml;, &ograve;. &aring;. &iuml;&icirc; &iuml;&agrave;&eth;&aring; (u, λ) &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&egrave;&igrave; &iacute;&icirc;&acirc;&oacute;&thorn; &iuml;&agrave;&eth;&oacute; (u0 , λ0 ).
P
&Icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; u0 . &Ccedil;&agrave;&auml;&agrave;&auml;&egrave;&igrave; u0 &ecirc;&agrave;&ecirc; Argmax &iacute;&agrave; X &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&eacute; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&egrave; K λK 1K − 1I . &Yacute;&ograve;&icirc;
&aring;&auml;&egrave;&iacute;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&aring; &oslash;&agrave;&eth;&agrave; X &ecirc;&eth;&icirc;&igrave;&aring; &ograve;&icirc;&atilde;&icirc; &ntilde;&euml;&oacute;&divide;&agrave;&yuml;, &ecirc;&icirc;&atilde;&auml;&agrave; λ &ntilde;&aacute;&agrave;&euml;&agrave;&iacute;&ntilde;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&icirc;. &Acirc; &yacute;&ograve;&icirc;&igrave;
&ntilde;&euml;&oacute;&divide;&agrave;&aring; &acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring; u0 &igrave;&icirc;&aelig;&iacute;&icirc; &aacute;&eth;&agrave;&ograve;&uuml; &euml;&thorn;&aacute;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute; &egrave;&ccedil; &oslash;&agrave;&eth;&agrave; X .
&Icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; λ0 . &Ccedil;&agrave;&auml;&agrave;&auml;&egrave;&igrave; λ0 &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;. &Auml;&euml;&yuml; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&egrave; K &iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave;
λ0K = 0, &aring;&ntilde;&euml;&egrave; uK &iacute;&aring; &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; V (K);
λ0K &euml;&thorn;&aacute;&icirc;&aring; &divide;&egrave;&ntilde;&euml;&icirc; &igrave;&aring;&aelig;&auml;&oacute; 0 &egrave; 2, &aring;&ntilde;&euml;&egrave; uK &euml;&aring;&aelig;&egrave;&ograve; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&aring; V (K);
λ0K = 2, &aring;&ntilde;&euml;&egrave; uK &iuml;&icirc;&iuml;&agrave;&auml;&agrave;&aring;&ograve; &acirc;&icirc; &acirc;&iacute;&oacute;&ograve;&eth;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml; V (K).
&szlig;&ntilde;&iacute;&icirc;, &divide;&ograve;&icirc; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; (&ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring;) F &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc; &egrave; &egrave;&igrave;&aring;&aring;&ograve; &iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&ucirc;&aring; &acirc;&ucirc;
&iuml;&oacute;&ecirc;&euml;&ucirc;&aring; &icirc;&aacute;&eth;&agrave;&ccedil;&ucirc;. &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &iuml;&icirc; &ograve;&aring;&icirc;&eth;&aring;&igrave;&aring; &Ecirc;&agrave;&ecirc;&oacute;&ograve;&agrave;&iacute;&egrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&agrave;&yuml; &iuml;&agrave;&eth;&agrave; (u∗ , λ∗ ). &szlig;
&oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&agrave;&thorn;, &divide;&ograve;&icirc; &icirc;&iacute;&agrave; B -&yuml;&auml;&aring;&eth;&iacute;&agrave;&yuml;.
&Auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;, &aring;&ntilde;&euml;&egrave; u∗ &euml;&aring;&aelig;&egrave;&ograve; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&aring; &oslash;&agrave;&eth;&agrave; X , &ograve;&icirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&agrave; u∗i &frac34;&aacute;&icirc;&euml;&uuml;
&oslash;&agrave;&yuml;&iquest;. &Iacute;&icirc; &ograve;&icirc;&atilde;&auml;&agrave; &iacute;&egrave;&ecirc;&agrave;&ecirc;&agrave;&yuml; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&yuml; K , &ntilde;&icirc;&auml;&aring;&eth;&aelig;&agrave;&ugrave;&agrave;&yuml; i, &iacute;&aring; &igrave;&icirc;&aelig;&aring;&ograve; &icirc;&aacute;&aring;&ntilde;&iuml;&aring;&divide;&egrave;&ograve;&uuml; i &ograve;&agrave;&ecirc;&icirc;&eacute; &aacute;&icirc;&euml;&uuml;&oslash;&icirc;&eacute;
&auml;&icirc;&otilde;&icirc;&auml;, &egrave; &ccedil;&iacute;&agrave;&divide;&egrave;&ograve; λ∗K = 0, &ntilde;&iuml;&eth;&icirc;&ntilde; &iacute;&agrave; &yacute;&ograve;&icirc;&atilde;&icirc; &oacute;&divide;&agrave;&ntilde;&ograve;&iacute;&egrave;&ecirc;&agrave; i &eth;&agrave;&acirc;&aring;&iacute; 0, &egrave; u∗i &auml;&icirc;&euml;&aelig;&iacute;&icirc; &oacute;&iuml;&agrave;&ntilde;&ograve;&uuml;. &Agrave;&iacute;&agrave;&euml;&icirc;&atilde;&egrave;&divide;&iacute;&icirc;
&aring;&ntilde;&euml;&egrave; u∗i &frac34;&ntilde;&egrave;&euml;&uuml;&iacute;&icirc; &icirc;&ograve;&eth;&egrave;&ouml;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&iquest;.
&Ograve;&agrave;&ecirc;&egrave;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;, &ograve;&icirc;&divide;&ecirc;&agrave; u∗ &euml;&aring;&aelig;&egrave;&ograve; &ntilde;&ograve;&eth;&icirc;&atilde;&icirc; &acirc;&iacute;&oacute;&ograve;&eth;&egrave; X , &icirc;&ograve;&ecirc;&oacute;&auml;&agrave; λ &ntilde;&aacute;&agrave;&euml;&agrave;&iacute;&ntilde;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&icirc;.
&Aring;&ntilde;&euml;&egrave; λ∗K &gt; 0, &ograve;&icirc; &egrave;&ccedil; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&yuml; λ0 &acirc;&egrave;&auml;&iacute;&icirc;, &divide;&ograve;&icirc; u∗K ∈ V (K), &ograve;. &aring;. &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc;
&auml;&icirc;&ntilde;&ograve;&egrave;&aelig;&egrave;&igrave;&icirc;&ntilde;&ograve;&egrave;. &Iacute;&agrave;&ecirc;&icirc;&iacute;&aring;&ouml;, &acirc; &ntilde;&egrave;&euml;&oacute; &ograve;&icirc;&eacute; &aelig;&aring; &ntilde;&aacute;&agrave;&euml;&agrave;&iacute;&ntilde;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&icirc;&ntilde;&ograve;&egrave; λ∗K ≤ 1, &ograve;. &aring;. uK &iacute;&aring; &iuml;&icirc;&iuml;&agrave;&auml;&agrave;&aring;&ograve;
&ntilde;&ograve;&eth;&icirc;&atilde;&icirc; &acirc;&iacute;&oacute;&ograve;&eth;&uuml; V (K) &iacute;&egrave; &auml;&euml;&yuml; &ecirc;&agrave;&ecirc;&icirc;&eacute; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&egrave; K , &egrave; &igrave;&ucirc; &egrave;&igrave;&aring;&aring;&igrave; &iacute;&aring;&auml;&icirc;&igrave;&egrave;&iacute;&egrave;&eth;&oacute;&aring;&igrave;&icirc;&ntilde;&ograve;&uuml;.
&Oacute;&iuml;&eth;&agrave;&aelig;&iacute;&aring;&iacute;&egrave;&yuml;
3.1. &Auml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&aring; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&aring;&iacute;&egrave;&aring;: &iuml;&oacute;&ntilde;&ograve;&uuml; X &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&iacute;&icirc;&aring; &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;
&ntilde;&ograve;&acirc;&icirc;, &agrave; &lt; &egrave;&eth;&eth;&aring;&ocirc;&euml;&aring;&ecirc;&ntilde;&egrave;&acirc;&iacute;&icirc;&aring; &aacute;&egrave;&iacute;&agrave;&eth;&iacute;&icirc;&aring; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; &iacute;&agrave; &iacute;&aring;&igrave;, &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &icirc;&aacute;&euml;&agrave;&auml;&agrave;&aring;&ograve; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&igrave;&egrave;
&auml;&acirc;&oacute;&igrave;&yuml; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&agrave;&igrave;&egrave;: 1) &icirc;&iacute;&icirc; &icirc;&ograve;&ecirc;&eth;&ucirc;&ograve;&icirc; (&ecirc;&agrave;&ecirc; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; X &times; X ), &egrave; 2) &icirc;&iacute;&icirc; &ograve;&eth;&agrave;&iacute;&ccedil;&egrave;&ograve;&egrave;&acirc;&iacute;&icirc; (&ograve;. &aring;.
x &lt; y &lt; z &acirc;&euml;&aring;&divide;&aring;&ograve; x &lt; z ). &Ograve;&icirc;&atilde;&auml;&agrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &igrave;&agrave;&ecirc;&ntilde;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&eacute; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;.
3.2. &Ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; u &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ecirc;&acirc;&agrave;&ccedil;&egrave;&acirc;&icirc;&atilde;&iacute;&oacute;&ograve;&icirc;&eacute;, &aring;&ntilde;&euml;&egrave; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; &divide;&egrave;&ntilde;&euml;&agrave; r &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;
{x ∈ X, u(x) ≥ r}. &Iuml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc; &euml;&thorn;&aacute;&agrave;&yuml; &acirc;&icirc;&atilde;&iacute;&oacute;&ograve;&agrave;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml; &ecirc;&acirc;&agrave;&ccedil;&egrave;&acirc;&icirc;&atilde;&iacute;&oacute;&ograve;&agrave;. &Iuml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc;
&ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Iacute;&yacute;&oslash;&agrave; &icirc;&ntilde;&ograve;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc;&aring;&eth;&iacute;&icirc;&eacute;, &aring;&ntilde;&euml;&egrave; &acirc;&icirc;&atilde;&iacute;&oacute;&ograve;&icirc;&ntilde;&ograve;&uuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute; ui &ccedil;&agrave;&igrave;&aring;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &ecirc;&acirc;&agrave;&ccedil;&egrave;&acirc;&icirc;&atilde;&iacute;&oacute;&ograve;&icirc;&ntilde;&ograve;&uuml;&thorn;.
3.3. &Iuml;&oacute;&ntilde;&ograve;&uuml; &acirc; &egrave;&atilde;&eth;&aring; &oacute;&divide;&agrave;&ntilde;&ograve;&acirc;&oacute;&thorn;&ograve; 3 &egrave;&atilde;&eth;&icirc;&ecirc;&agrave;. &Egrave;&atilde;&eth;&icirc;&ecirc; &acirc; &icirc;&auml;&egrave;&iacute;&icirc;&divide;&ecirc;&oacute; &iacute;&aring; &igrave;&icirc;&aelig;&aring;&ograve; &acirc;&ucirc;&egrave;&atilde;&eth;&agrave;&ograve;&uuml; &iacute;&egrave;&divide;&aring;&atilde;&icirc; (&ograve;. &aring;.
≤ 0), &euml;&thorn;&aacute;&ucirc;&aring; &auml;&eth;&oacute;&atilde;&egrave;&aring; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&egrave; (&ntilde;&icirc;&auml;&aring;&eth;&aelig;&agrave;&ugrave;&egrave;&aring; &auml;&acirc;&oacute;&otilde; &egrave;&euml;&egrave; &ograve;&eth;&aring;&otilde; &divide;&euml;&aring;&iacute;&icirc;&acirc;) &igrave;&icirc;&atilde;&oacute;&ograve; &iuml;&icirc;&euml;&oacute;&divide;&egrave;&ograve;&uuml; 1 &eth;&oacute;&aacute;. &egrave;
&ecirc;&agrave;&ecirc; &oacute;&atilde;&icirc;&auml;&iacute;&icirc; &iuml;&aring;&eth;&aring;&eth;&agrave;&ntilde;&iuml;&eth;&aring;&auml;&aring;&euml;&egrave;&ograve;&uuml; &igrave;&aring;&aelig;&auml;&oacute; &ntilde;&acirc;&icirc;&egrave;&igrave;&egrave; &divide;&euml;&aring;&iacute;&agrave;&igrave;&egrave;. &Iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &yuml;&auml;&eth;&icirc; &iuml;&oacute;&ntilde;&ograve;&icirc;. &Iacute;&agrave;&ntilde;&ecirc;&icirc;&euml;&uuml;&ecirc;&icirc;
&iacute;&oacute;&aelig;&iacute;&icirc; &oacute;&acirc;&aring;&euml;&egrave;&divide;&egrave;&ograve;&uuml; &ouml;&aring;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml; &ecirc;&icirc;&agrave;&euml;&egrave;&ouml;&egrave;&egrave; &acirc;&ntilde;&aring;&otilde; &egrave;&atilde;&eth;&icirc;&ecirc;&icirc;&acirc;, &divide;&ograve;&icirc;&aacute;&ucirc; &iuml;&icirc;&yuml;&acirc;&egrave;&euml;&icirc;&ntilde;&uuml; &yuml;&auml;&eth;&icirc;?
24
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 3.
&Ograve;&Aring;&Icirc;&ETH;&Aring;&Igrave;&Agrave; &Aacute;&ETH;&Agrave;&Oacute;&Yacute;&ETH;&Agrave;: &Iuml;&ETH;&Egrave;&Igrave;&Aring;&Iacute;&Aring;&Iacute;&Egrave;&szlig;
&ETH;&aring;&ecirc;&icirc;&igrave;&aring;&iacute;&auml;&oacute;&aring;&igrave;&agrave;&yuml; &euml;&egrave;&ograve;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave;: [2, 4, 6, 7, 8, 10, 11].
&Euml;&aring;&ecirc;&ouml;&egrave;&yuml; 4
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;: &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave; &egrave;
&agrave;&euml;&atilde;&icirc;&eth;&egrave;&ograve;&igrave;&ucirc;
&Yacute;&ograve;&agrave; &euml;&aring;&ecirc;&ouml;&egrave;&yuml; &iuml;&icirc;&ntilde;&acirc;&yuml;&ugrave;&aring;&iacute;&agrave; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&oacute; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;. &Iacute;&agrave;&iuml;&icirc;&igrave;&iacute;&egrave;&igrave; &aring;&aring; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&egrave;&eth;&icirc;&acirc;&ecirc;&oacute;: &aring;&ntilde;&euml;&egrave;
f : X → X &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&atilde;&icirc; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&agrave; X &acirc; &ntilde;&aring;&aacute;&yuml;, &ograve;&icirc; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &iacute;&aring;&iuml;&icirc;
&auml;&acirc;&egrave;&aelig;&iacute;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave;. &Igrave;&ucirc; &iuml;&eth;&egrave;&acirc;&aring;&auml;&aring;&igrave; (&ntilde; &iacute;&aring;&aacute;&icirc;&euml;&uuml;&oslash;&egrave;&igrave;&egrave; &iuml;&eth;&icirc;&iuml;&oacute;&ntilde;&ecirc;&agrave;&igrave;&egrave;) &ecirc;&icirc;&igrave;&aacute;&egrave;&iacute;&agrave;&ograve;&icirc;&eth;&iacute;&icirc;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;,
&agrave; &ccedil;&agrave;&ograve;&aring;&igrave; &aacute;&aring;&atilde;&euml;&icirc; &icirc;&aacute;&ntilde;&oacute;&auml;&egrave;&igrave; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &agrave;&euml;&uuml;&ograve;&aring;&eth;&iacute;&agrave;&ograve;&egrave;&acirc;&iacute;&ucirc;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave;.
&ETH;&aring;&auml;&oacute;&ecirc;&ouml;&egrave;&yuml; &ecirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave;&igrave;
&Iuml;&eth;&aring;&aelig;&auml;&aring; &acirc;&ntilde;&aring;&atilde;&icirc; &ccedil;&agrave;&igrave;&aring;&ograve;&egrave;&igrave;, &divide;&ograve;&icirc; &yacute;&ograve;&agrave; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &egrave;&igrave;&aring;&aring;&ograve; &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&oacute;&thorn; &iuml;&eth;&egrave;&eth;&icirc;&auml;&oacute;: &aring;&ntilde;&euml;&egrave; &icirc;&iacute;&agrave; &acirc;&aring;&eth;&iacute;&agrave; &auml;&euml;&yuml;
X , &ograve;&icirc; &acirc;&aring;&eth;&iacute;&agrave; &egrave; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&atilde;&icirc; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&agrave;, &atilde;&icirc;&igrave;&aring;&icirc;&igrave;&icirc;&eth;&ocirc;&iacute;&icirc;&atilde;&icirc; X .
&Igrave;&ucirc; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&agrave;&aring;&igrave;, &divide;&ograve;&icirc; &euml;&thorn;&aacute;&icirc;&eacute; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&eacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave; n &atilde;&icirc;&igrave;&aring;&icirc;&igrave;&icirc;&eth;&ocirc;&aring;&iacute; &aring;&auml;&egrave;&iacute;&egrave;&divide;&iacute;&icirc;&igrave;&oacute;
&oslash;&agrave;&eth;&oacute; Dn &acirc; Rn , &atilde;&auml;&aring;
Dn = {x ∈ Rn , |x| ≤ 1},
&agrave; | &middot; | &aring;&acirc;&ecirc;&euml;&egrave;&auml;&icirc;&acirc;&agrave; &iacute;&icirc;&eth;&igrave;&agrave; &acirc; Rn , &ograve;. &aring;. |(x1 , . . . , xn )| = (x21 + &middot; &middot; &middot; + x2n )1/2 . &Acirc; &ntilde;&agrave;&igrave;&icirc;&igrave; &auml;&aring;&euml;&aring;, &iuml;&oacute;&ntilde;&ograve;&uuml;
X n-&igrave;&aring;&eth;&iacute;&ucirc;&eacute; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&eacute; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;. &Ograve;&icirc;&atilde;&auml;&agrave; &oacute; &iacute;&aring;&atilde;&icirc; &egrave;&igrave;&aring;&thorn;&ograve;&ntilde;&yuml; &acirc;&iacute;&oacute;&ograve;&eth;&aring;&iacute;&iacute;&egrave;&aring; &ograve;&icirc;&divide;&ecirc;&egrave;; &aacute;&aring;&ccedil; &icirc;&atilde;&eth;&agrave;&iacute;&egrave;&divide;&aring;&iacute;&egrave;&yuml;
&icirc;&aacute;&ugrave;&iacute;&icirc;&ntilde;&ograve;&egrave; &igrave;&icirc;&aelig;&iacute;&icirc; &ntilde;&divide;&egrave;&ograve;&agrave;&ograve;&uuml; &yacute;&ograve;&oacute; &acirc;&iacute;&oacute;&ograve;&eth;&aring;&iacute;&iacute;&thorn;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute; &eth;&agrave;&acirc;&iacute;&icirc;&eacute; 0. &Auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&atilde;&icirc; &aring;&auml;&egrave;&iacute;&egrave;&divide;&iacute;&icirc;&atilde;&icirc; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave;
v &iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave; k(v) = max(r ∈ R, rv ∈ X). &Auml;&icirc;&acirc;&icirc;&euml;&uuml;&iacute;&icirc; &yuml;&ntilde;&iacute;&icirc;, &divide;&ograve;&icirc; 0 &lt; k(v) &egrave; k(v) &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;
&ccedil;&agrave;&acirc;&egrave;&ntilde;&egrave;&ograve; &icirc;&ograve; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave; v .
&Ograve;&aring;&iuml;&aring;&eth;&uuml; &egrave;&ntilde;&ecirc;&icirc;&igrave;&ucirc;&eacute; &atilde;&icirc;&igrave;&aring;&icirc;&igrave;&icirc;&eth;&ocirc;&egrave;&ccedil;&igrave; g : Dn → X &ntilde;&ograve;&eth;&icirc;&egrave;&ograve;&ntilde;&yuml; &aacute;&aring;&ccedil; &ograve;&eth;&oacute;&auml;&agrave;. &Icirc;&iacute; &iuml;&aring;&eth;&aring;&acirc;&icirc;&auml;&egrave;&ograve; 0 &acirc; 0,
&agrave; &auml;&euml;&yuml; &iacute;&aring;&iacute;&oacute;&euml;&aring;&acirc;&icirc;&atilde;&icirc; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&agrave; y ∈ Dn &igrave;&ucirc; &iuml;&icirc;&euml;&agrave;&atilde;&agrave;&aring;&igrave; g(y) = k(y/|y|)y . &Egrave;&iacute;&agrave;&divide;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&yuml;, &acirc;&aring;&ecirc;&ograve;&icirc;&eth; y
&oacute;&auml;&euml;&egrave;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &acirc; k(y/|y|) &eth;&agrave;&ccedil;. &szlig;&ntilde;&iacute;&icirc;, &divide;&ograve;&icirc; g &icirc;&aacute;&eth;&agrave;&ograve;&egrave;&igrave;: &icirc;&aacute;&eth;&agrave;&ograve;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; g −1 &ccedil;&agrave;&auml;&agrave;&aring;&ograve;&ntilde;&yuml; &ograve;&agrave;&ecirc;&aelig;&aring;
&yuml;&acirc;&iacute;&icirc;: &auml;&euml;&yuml; x ∈ X g −1 (x) = x/k(x/|x|). &Igrave;&ucirc; &icirc;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&aring;&igrave; &divide;&egrave;&ograve;&agrave;&ograve;&aring;&euml;&thorn; &iuml;&eth;&icirc;&acirc;&aring;&eth;&ecirc;&oacute; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&ntilde;&ograve;&egrave; g
&egrave; g −1 .
&Ograve;&agrave;&ecirc;&egrave;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;, &auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml; &ograve;&aring;&icirc;&eth;&aring;&igrave;&oacute; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; &oslash;&agrave;&eth;&agrave; Dn &egrave;&euml;&egrave; &euml;&thorn;&aacute;&icirc;&atilde;&icirc;
(&ntilde;&ograve;&agrave;&iacute;&auml;&agrave;&eth;&ograve;&iacute;&icirc;&atilde;&icirc;) &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; ∆n . &Ntilde;&ograve;&icirc;&egrave;&ograve;, &icirc;&auml;&iacute;&agrave;&ecirc;&icirc;, &iacute;&agrave;&iuml;&icirc;&igrave;&iacute;&egrave;&ograve;&uuml;, &divide;&ograve;&icirc; &ograve;&agrave;&ecirc;&icirc;&aring; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;.
&Iuml;&oacute;&ntilde;&ograve;&uuml; S &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&aring; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;. &Ntilde;&ograve;&agrave;&iacute;&auml;&agrave;&eth;&ograve;&iacute;&ucirc;&igrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&icirc;&igrave; &ntilde; &acirc;&aring;&eth;&oslash;&egrave;&iacute;&agrave;&igrave;&egrave; &acirc; S &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml;
P
&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &ograve;&agrave;&ecirc;&egrave;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; x = (xs ) &acirc; RS , &divide;&ograve;&icirc; &acirc;&ntilde;&aring; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&ucirc; xs ≥ 0 &egrave; &egrave;&otilde; &ntilde;&oacute;&igrave;&igrave;&agrave; s xs = 1.
&Ograve;&agrave;&ecirc;&icirc;&eacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&ograve;&ntilde;&yuml; &ecirc;&agrave;&ecirc; ∆S (&icirc;&aacute;&ucirc;&divide;&iacute;&icirc; S = {1, . . . , n}, &egrave; &ograve;&icirc;&atilde;&auml;&agrave; &igrave;&ucirc; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&igrave; &aring;&atilde;&icirc;
&ecirc;&agrave;&ecirc; ∆n ). &Iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;, ∆1 &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &egrave;&ccedil; &icirc;&auml;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave;, ∆2 &icirc;&ograve;&eth;&aring;&ccedil;&icirc;&ecirc;, ∆3 &ograve;&eth;&aring;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;, ∆4 &ograve;&aring;&ograve;&eth;&agrave;&yacute;&auml;&eth;, &egrave; &ograve;. &auml;.
&Ecirc;&agrave;&aelig;&auml;&ucirc;&eacute; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve; s ∈ S &eth;&aring;&agrave;&euml;&egrave;&ccedil;&oacute;&aring;&ograve;&ntilde;&yuml; &ecirc;&agrave;&ecirc; &acirc;&aring;&eth;&oslash;&egrave;&iacute;&agrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; ∆S , &agrave; &egrave;&igrave;&aring;&iacute;&iacute;&icirc; &ecirc;&agrave;&ecirc; &ograve;&agrave;&ecirc;&agrave;&yuml;
&ograve;&icirc;&divide;&ecirc;&agrave;, &divide;&ograve;&icirc; &aring;&atilde;&icirc; s-&yuml; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&agrave; &eth;&agrave;&acirc;&iacute;&agrave; 1, &agrave; &icirc;&ntilde;&ograve;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &eth;&agrave;&acirc;&iacute;&ucirc; 0. &Egrave; &acirc;&icirc;&icirc;&aacute;&ugrave;&aring;, &aring;&ntilde;&euml;&egrave; T &iuml;&icirc;&auml;&igrave;&iacute;&icirc;
&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; S , &ograve;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde; ∆T &aring;&ntilde;&ograve;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc; &eth;&aring;&agrave;&euml;&egrave;&ccedil;&oacute;&aring;&ograve;&ntilde;&yuml; &ecirc;&agrave;&ecirc; &atilde;&eth;&agrave;&iacute;&uuml; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; ∆S . &Iuml;&eth;&egrave; &yacute;&ograve;&icirc;&igrave;
25
26
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 4.
&Ograve;&Aring;&Icirc;&ETH;&Aring;&Igrave;&Agrave; &Aacute;&ETH;&Agrave;&Oacute;&Yacute;&ETH;&Agrave;: &Auml;&Icirc;&Ecirc;&Agrave;&Ccedil;&Agrave;&Ograve;&Aring;&Euml;&Uuml;&Ntilde;&Ograve;&Acirc;&Agrave; &Egrave; &Agrave;&Euml;&Atilde;&Icirc;&ETH;&Egrave;&Ograve;&Igrave;&Ucirc;
T
∆T ∆U = ∆T T U &auml;&euml;&yuml; T, U ⊂ S . &Atilde;&eth;&agrave;&iacute;&egrave; &acirc;&egrave;&auml;&agrave; ∆S−{s} , &atilde;&auml;&aring; s ∈ S , &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &ntilde;&ograve;&aring;&iacute;&ecirc;&agrave;&igrave;&egrave; ∆S ,
&iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&iuml;&icirc;&euml;&icirc;&aelig;&iacute;&ucirc;&igrave;&egrave; &acirc;&aring;&eth;&oslash;&egrave;&iacute;&aring; s; &icirc;&iacute;&egrave; &ccedil;&agrave;&auml;&agrave;&thorn;&ograve;&ntilde;&yuml; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&aring;&igrave; xs = 0.
&Icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&icirc;, &divide;&ograve;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde; ∆S &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&igrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;&igrave; (&eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave; n − 1), &egrave;
&aacute;&icirc;&euml;&aring;&aring; &ograve;&icirc;&atilde;&icirc;, &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&eacute; &icirc;&aacute;&icirc;&euml;&icirc;&divide;&ecirc;&icirc;&eacute; &ntilde;&acirc;&icirc;&egrave;&otilde; &acirc;&aring;&eth;&oslash;&egrave;&iacute; s ∈ S . &Yacute;&ograve;&egrave; &acirc;&aring;&eth;&oslash;&egrave;&iacute;&ucirc; &agrave;&ocirc;&ocirc;&egrave;&iacute;&iacute;&icirc; &iacute;&aring;&ccedil;&agrave;&acirc;&egrave;&ntilde;&egrave;&igrave;&ucirc;
&acirc; &ograve;&icirc;&igrave; &ntilde;&igrave;&ucirc;&ntilde;&euml;&aring;, &divide;&ograve;&icirc; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&atilde;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; Y &egrave; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&yuml; f : S → Y
&ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &aring;&auml;&egrave;&iacute;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&aring; &agrave;&ocirc;&ocirc;&egrave;&iacute;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; fˆ : ∆S → Y , &iuml;&eth;&icirc;&auml;&icirc;&euml;&aelig;&agrave;&thorn;&ugrave;&aring;&aring; f (&egrave;&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml;,
&egrave;&euml;&egrave; &ntilde;&eth;&aring;&auml;&iacute;&aring;&aring;). &times;&agrave;&ntilde;&ograve;&icirc; &iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&icirc;&igrave; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve; &euml;&thorn;&aacute;&icirc;&aring; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &acirc; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&iacute;&icirc;&igrave; &iuml;&eth;&icirc;
&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&aring; V , &yuml;&acirc;&euml;&yuml;&thorn;&ugrave;&aring;&aring;&ntilde;&yuml; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&eacute; &icirc;&aacute;&icirc;&euml;&icirc;&divide;&ecirc;&icirc;&eacute; &agrave;&ocirc;&ocirc;&egrave;&iacute;&iacute;&icirc; &iacute;&aring;&ccedil;&agrave;&acirc;&egrave;&ntilde;&egrave;&igrave;&icirc;&atilde;&icirc; &iacute;&agrave;&aacute;&icirc;&eth;&agrave; &ograve;&icirc;&divide;&aring;&ecirc;. &Ograve;&agrave;&ecirc;&egrave;&aring;
&ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&ucirc; &agrave;&ocirc;&ocirc;&egrave;&iacute;&iacute;&icirc; &egrave;&ccedil;&icirc;&igrave;&icirc;&eth;&ocirc;&iacute;&ucirc; &ntilde;&ograve;&agrave;&iacute;&auml;&agrave;&eth;&ograve;&iacute;&ucirc;&igrave;.
&Euml;&aring;&igrave;&igrave;&agrave; &Oslash;&iuml;&aring;&eth;&iacute;&aring;&eth;&agrave;
&Ouml;&aring;&iacute;&ograve;&eth;&agrave;&euml;&uuml;&iacute;&oacute;&thorn; &eth;&icirc;&euml;&uuml; &acirc; &iuml;&eth;&egrave;&acirc;&icirc;&auml;&egrave;&igrave;&icirc;&igrave; &iacute;&egrave;&aelig;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&aring; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &aacute;&oacute;&auml;&aring;&ograve; &egrave;&atilde;&eth;&agrave;&ograve;&uuml; &acirc;&ntilde;&iuml;&icirc;
&igrave;&icirc;&atilde;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&agrave;&yuml; &ecirc;&icirc;&igrave;&aacute;&egrave;&iacute;&agrave;&ograve;&icirc;&eth;&iacute;&agrave;&yuml; &euml;&aring;&igrave;&igrave;&agrave;, &iuml;&icirc;&euml;&oacute;&divide;&aring;&iacute;&iacute;&agrave;&yuml; &Oslash;&iuml;&aring;&eth;&iacute;&aring;&eth;&icirc;&igrave; &acirc; 1928 &atilde;. &times;&ograve;&icirc;&aacute;&ucirc; &icirc;&iacute;&agrave; &iacute;&aring; &acirc;&ucirc;
&ntilde;&ecirc;&agrave;&ecirc;&egrave;&acirc;&agrave;&euml;&agrave;, &ecirc;&agrave;&ecirc; &divide;&aring;&eth;&ograve; &egrave;&ccedil; &ograve;&agrave;&aacute;&agrave;&ecirc;&aring;&eth;&ecirc;&egrave;, &ntilde;&auml;&aring;&euml;&agrave;&aring;&igrave; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &iuml;&icirc;&yuml;&ntilde;&iacute;&aring;&iacute;&egrave;&yuml;. &Iuml;&oacute;&ntilde;&ograve;&uuml; f : ∆S → ∆S &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &ntilde;&ograve;&agrave;&iacute;&auml;&agrave;&eth;&ograve;&iacute;&icirc;&atilde;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; &acirc; &ntilde;&aring;&aacute;&yuml;. &Auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&atilde;&icirc; s ∈ S &eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave;
&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;
Fs = {x = (xt ) ∈ ∆S , f (x)s ≤ xs }.
&Ograve;&icirc; &aring;&ntilde;&ograve;&uuml; &yacute;&ograve;&icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &ograve;&agrave;&ecirc;&egrave;&otilde; &ograve;&icirc;&divide;&aring;&ecirc;, &icirc;&aacute;&eth;&agrave;&ccedil; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; &iacute;&aring; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&agrave;&aring;&ograve;&ntilde;&yuml; &ecirc; &acirc;&aring;&eth;&oslash;&egrave;&iacute;&aring; s. &Iacute;&agrave;&oslash;&agrave;
&ccedil;&agrave;&auml;&agrave;&divide;&agrave; &iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &acirc;&ntilde;&aring; &yacute;&ograve;&egrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; Fs &egrave;&igrave;&aring;&thorn;&ograve; &icirc;&aacute;&ugrave;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute;. &Acirc; &ntilde;&agrave;&igrave;&icirc;&igrave; &auml;&aring;&euml;&aring;, &aring;&ntilde;&euml;&egrave; &ograve;&icirc;&divide;&ecirc;&agrave;
&ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; x∗ &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; &acirc;&ntilde;&aring;&igrave; Fs , &icirc;&iacute;&agrave; (&iuml;&icirc;&auml; &auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&aring;&igrave; f ) &iacute;&aring; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&agrave;&aring;&ograve;&ntilde;&yuml; &iacute;&egrave; &ecirc; &ecirc;&agrave;&ecirc;&icirc;&eacute;
&acirc;&aring;&eth;&oslash;&egrave;&iacute;&aring;, &agrave; &ccedil;&iacute;&agrave;&divide;&egrave;&ograve;, &icirc;&ntilde;&ograve;&agrave;&aring;&ograve;&ntilde;&yuml; &iacute;&agrave; &igrave;&aring;&ntilde;&ograve;&aring;.
&Agrave; &divide;&ograve;&icirc; &igrave;&ucirc; &ccedil;&iacute;&agrave;&aring;&igrave; &icirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave;&otilde; Fs ? &szlig;&ntilde;&iacute;&icirc;, &divide;&ograve;&icirc; &acirc;&ntilde;&aring; &icirc;&iacute;&egrave; &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&ucirc;. &Auml;&agrave;&euml;&aring;&aring;, &acirc; &ntilde;&icirc;&acirc;&icirc;&ecirc;&oacute;&iuml;&iacute;&icirc;&ntilde;&ograve;&egrave;
&icirc;&iacute;&egrave; &iuml;&icirc;&ecirc;&eth;&ucirc;&acirc;&agrave;&thorn;&ograve; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde; ∆S ; &auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;, &iacute;&aring; &igrave;&icirc;&aelig;&aring;&ograve; &aelig;&aring; &ograve;&icirc;&divide;&ecirc;&agrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&agrave;&ograve;&uuml;&ntilde;&yuml;
&ecirc;&icirc; &acirc;&ntilde;&aring;&igrave; &ntilde;&acirc;&icirc;&egrave;&igrave; &acirc;&aring;&eth;&oslash;&egrave;&iacute;&agrave;&igrave;! &Iacute;&agrave; &ntilde;&agrave;&igrave;&icirc;&igrave; &auml;&aring;&euml;&aring;, &yacute;&ograve;&icirc; &ntilde;&icirc;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &atilde;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&eacute; &atilde;&eth;&agrave;&iacute;&egrave;, &egrave; &igrave;&ucirc;
&egrave;&igrave;&aring;&aring;&igrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&aring; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc;: &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; T ⊂ S &atilde;&eth;&agrave;&iacute;&uuml; ∆T &iuml;&icirc;&ecirc;&eth;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave;&igrave;&egrave; Ft , &atilde;&auml;&aring;
t ∈ T.
&Ccedil;&agrave;&igrave;&aring;&divide;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;, &divide;&ograve;&icirc;S&ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &yacute;&ograve;&egrave;&otilde; &auml;&acirc;&oacute;&otilde; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc; &ntilde;&aring;&igrave;&aring;&eacute;&ntilde;&ograve;&acirc;&agrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc; (Fs , s ∈ S ) (&ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;
&ntilde;&ograve;&egrave; &egrave; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; ∆T ⊂ t∈T Ft &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; T ⊂ S ) &auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc;, &divide;&ograve;&icirc;&aacute;&ucirc; &atilde;&agrave;&eth;&agrave;&iacute;&ograve;&egrave;&eth;&icirc;&acirc;&agrave;&ograve;&uuml; &iacute;&agrave;&euml;&egrave;&divide;&egrave;&aring;
&icirc;&aacute;&ugrave;&aring;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave; &oacute; Fs . &Yacute;&ograve;&icirc; &ntilde;&icirc;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&aring;&ograve; &ntilde;&icirc;&auml;&aring;&eth;&aelig;&agrave;&iacute;&egrave;&aring; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Ecirc;&iacute;&agrave;&ntilde;&ograve;&aring;&eth;&agrave;&Ecirc;&oacute;&eth;&agrave;&ograve;&icirc;&acirc;&ntilde;&ecirc;&icirc;&atilde;&icirc;&Igrave;&agrave;&ccedil;&oacute;&eth;&ecirc;&aring;
&acirc;&egrave;&divide;&agrave; (1929), &ograve;&agrave;&ecirc;&aelig;&aring; &yacute;&ecirc;&acirc;&egrave;&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&icirc;&eacute; &ograve;&aring;&icirc;&eth;&aring;&igrave;&aring; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;. &Igrave;&ucirc;, &icirc;&auml;&iacute;&agrave;&ecirc;&icirc;, &iacute;&aring; &aacute;&oacute;&auml;&aring;&igrave; &icirc;&ntilde;&ograve;&agrave;&iacute;&agrave;&acirc;&euml;&egrave;&acirc;&agrave;&ograve;&uuml;&ntilde;&yuml;
&iacute;&agrave; &Ecirc;&Ecirc;&Igrave;, &agrave; &auml;&acirc;&egrave;&iacute;&aring;&igrave;&ntilde;&yuml; &ecirc; &Oslash;&iuml;&aring;&eth;&iacute;&aring;&eth;&oacute;. &Ograve;&oacute;&ograve; &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&icirc; &divide;&oacute;&ograve;&uuml; &egrave;&ccedil;&igrave;&aring;&iacute;&egrave;&ograve;&uuml; &ograve;&aring;&eth;&igrave;&egrave;&iacute;&icirc;&euml;&icirc;&atilde;&egrave;&thorn;. &Acirc;&igrave;&aring;&ntilde;&ograve;&icirc; &ograve;&icirc;&atilde;&icirc;,
&divide;&ograve;&icirc;&aacute;&ucirc; &atilde;&icirc;&acirc;&icirc;&eth;&egrave;&ograve;&uuml;, &divide;&ograve;&icirc; &ograve;&icirc;&divide;&ecirc;&agrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; x &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; Fs , &ntilde;&ecirc;&agrave;&aelig;&aring;&igrave;, &divide;&ograve;&icirc; &ograve;&icirc;&divide;&ecirc;&agrave; x &egrave;&igrave;&aring;&aring;&ograve; &igrave;&aring;&ograve;&ecirc;&oacute;
s. &Ecirc;&Ecirc;&Igrave; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&agrave;&aring;&ograve;, &divide;&ograve;&icirc; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ograve;&icirc;&divide;&ecirc;&agrave; x∗ , &egrave;&igrave;&aring;&thorn;&ugrave;&agrave;&yuml; &acirc;&ntilde;&aring; &igrave;&aring;&ograve;&ecirc;&egrave;.
&Ograve;&agrave;&ecirc; &acirc;&icirc;&ograve;, &euml;&aring;&igrave;&igrave;&agrave; &Oslash;&iuml;&aring;&eth;&iacute;&aring;&eth;&agrave; &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&aring;&ograve; &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&oacute;&thorn; &agrave;&iuml;&iuml;&eth;&icirc;&ecirc;&ntilde;&egrave;&igrave;&agrave;&ouml;&egrave;&thorn; &Ecirc;&Ecirc;&Igrave;. &Iuml;&oacute;&ntilde;&ograve;&uuml; V &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&aring;, &iacute;&icirc; &auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc; &iuml;&euml;&icirc;&ograve;&iacute;&icirc;&aring; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &acirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&aring; ∆S , &egrave; &ecirc;&agrave;&aelig;&auml;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&aring; v ∈ V &iuml;&eth;&egrave;
&iuml;&egrave;&ntilde;&agrave;&iacute;&agrave; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &igrave;&aring;&ograve;&ecirc;&agrave; &egrave;&ccedil; S . &Ograve;&icirc;&atilde;&auml;&agrave; (&iuml;&eth;&egrave; &auml;&icirc;&iuml;&icirc;&euml;&iacute;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&igrave; &atilde;&eth;&agrave;&iacute;&egrave;&divide;&iacute;&icirc;&igrave; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&egrave;) &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve;
&atilde;&eth;&oacute;&iuml;&iuml;&agrave; &aacute;&euml;&egrave;&ccedil;&ecirc;&icirc;&eth;&agrave;&ntilde;&iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc;, &egrave;&igrave;&aring;&thorn;&ugrave;&egrave;&otilde; &acirc; &ntilde;&icirc;&acirc;&icirc;&ecirc;&oacute;&iuml;&iacute;&icirc;&ntilde;&ograve;&egrave; &acirc;&ntilde;&aring; &igrave;&aring;&ograve;&ecirc;&egrave;.
&Aacute;&icirc;&euml;&aring;&aring; &ograve;&icirc;&divide;&iacute;&icirc;, &acirc; &euml;&aring;&igrave;&igrave;&aring; &Oslash;&iuml;&aring;&eth;&iacute;&aring;&eth;&agrave; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&aring;&ograve;&ntilde;&yuml; &ccedil;&agrave;&auml;&agrave;&iacute;&iacute;&icirc;&eacute; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&yuml; Σ
&ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; ∆S ; &acirc;&aring;&eth;&oslash;&egrave;&iacute;&ucirc; &yacute;&ograve;&icirc;&eacute; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave; &egrave; &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&thorn;&ograve; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; V . &Acirc;&aring;&eth;&oslash;&egrave;&iacute;&ucirc; &ntilde;&divide;&egrave;&ograve;&agrave;&thorn;&ograve;&ntilde;&yuml;
&aacute;&euml;&egrave;&ccedil;&ecirc;&egrave;&igrave;&egrave;, &aring;&ntilde;&euml;&egrave; &icirc;&iacute;&egrave; &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&agrave;&ograve; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&igrave;&oacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&oacute; &egrave;&ccedil; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave; Σ. &Igrave;&ucirc; &iacute;&aring; &iuml;&eth;&egrave;&acirc;&icirc;
&auml;&egrave;&igrave; &ocirc;&icirc;&eth;&igrave;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&yuml; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave;; &yacute;&ograve;&icirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&aring; &iuml;&icirc;&ecirc;&eth;&ucirc;&ograve;&egrave;&aring; ∆S &aacute;&icirc;&euml;&aring;&aring;
&igrave;&aring;&euml;&ecirc;&egrave;&igrave;&egrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave;&igrave;&egrave; (&icirc;&iacute;&egrave; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&thorn;&ograve;&ntilde;&yuml; σ , τ &egrave; &ograve;. &iuml;.), &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&eth;&yuml;&aring;&ograve; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&igrave;&oacute;
&atilde;&euml;&agrave;&acirc;&iacute;&icirc;&igrave;&oacute; &ograve;&eth;&aring;&aacute;&icirc;&acirc;&agrave;&iacute;&egrave;&thorn;: &auml;&euml;&yuml; &euml;&thorn;&aacute;&ucirc;&otilde; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&icirc;&acirc; σ &egrave; τ &egrave;&otilde; &iuml;&aring;&eth;&aring;&ntilde;&aring;&divide;&aring;&iacute;&egrave;&aring; σ ∩ τ &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &atilde;&eth;&agrave;&iacute;&uuml;&thorn;
&ecirc;&agrave;&ecirc; &acirc; σ , &ograve;&agrave;&ecirc; &egrave; &acirc; τ (&aacute;&ucirc;&ograve;&uuml; &igrave;&icirc;&aelig;&aring;&ograve;, &iuml;&oacute;&ntilde;&ograve;&icirc;&eacute;). &Ecirc;&agrave;&aelig;&auml;&agrave;&yuml; &acirc;&aring;&eth;&oslash;&egrave;&iacute;&agrave; v &iacute;&agrave;&oslash;&aring;&eacute; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave; &egrave;&igrave;&aring;&aring;&ograve;
&igrave;&aring;&ograve;&ecirc;&oacute; l(v) ∈ S . &Atilde;&eth;&agrave;&iacute;&egrave;&divide;&iacute;&icirc;&aring; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&aring; &Ecirc;&Ecirc;&Igrave; &iuml;&eth;&egrave;&icirc;&aacute;&eth;&aring;&ograve;&agrave;&aring;&ograve; &ograve;&agrave;&ecirc;&icirc;&eacute; &acirc;&egrave;&auml;: &aring;&ntilde;&euml;&egrave; &acirc;&aring;&eth;&oslash;&egrave;&iacute;&agrave; v ∈ V &euml;&aring;
&aelig;&egrave;&ograve; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave; ∆T , T ⊂ S , &ograve;&icirc; &aring;&aring; &igrave;&aring;&ograve;&ecirc;&agrave; &ograve;&agrave;&ecirc;&aelig;&aring; &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve; T . &Iuml;&icirc;&ntilde;&euml;&aring; &yacute;&ograve;&egrave;&otilde; &iuml;&eth;&aring;&auml;&acirc;&agrave;&eth;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&otilde;
&iuml;&icirc;&yuml;&ntilde;&iacute;&aring;&iacute;&egrave;&eacute; &igrave;&icirc;&aelig;&iacute;&icirc; &auml;&agrave;&ograve;&uuml; &icirc;&ecirc;&icirc;&iacute;&divide;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&oacute;&thorn; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&egrave;&eth;&icirc;&acirc;&ecirc;&oacute; &euml;&aring;&igrave;&igrave;&ucirc;.
27
&Euml;&aring;&igrave;&igrave;&agrave; (&Oslash;&iuml;&aring;&eth;&iacute;&aring;&eth;, 1928). &Iuml;&oacute;&ntilde;&ograve;&uuml; &auml;&agrave;&iacute;&agrave; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&yuml; Σ &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; ∆S &ntilde; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;&igrave;
&acirc;&aring;&eth;&oslash;&egrave;&iacute; V . &Iuml;&oacute;&ntilde;&ograve;&uuml; &ecirc;&agrave;&aelig;&auml;&agrave;&yuml; &acirc;&aring;&eth;&oslash;&egrave;&iacute;&agrave; v ∈ V &iuml;&icirc;&igrave;&aring;&divide;&aring;&iacute;&agrave; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&igrave; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&igrave; &egrave;&ccedil; S , &ograve;. &aring;. &auml;&agrave;&iacute;&icirc;
&icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; l : V → S , &iuml;&eth;&egrave;&divide;&aring;&igrave; &acirc;&aring;&eth;&oslash;&egrave;&iacute;&ucirc;, &euml;&aring;&aelig;&agrave;&ugrave;&egrave;&aring; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave; ∆T , T ⊂ S , &egrave;&igrave;&aring;&thorn;&ograve; &igrave;&aring;&ograve;&ecirc;&egrave;
&egrave;&ccedil; T . &Ograve;&icirc;&atilde;&auml;&agrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde; σ &iacute;&agrave;&oslash;&aring;&eacute; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave;, &acirc;&aring;&eth;&oslash;&egrave;&iacute;&ucirc; &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&atilde;&icirc; &iacute;&aring;&ntilde;&oacute;&ograve; &acirc;&ntilde;&aring;
&igrave;&aring;&ograve;&ecirc;&egrave; &egrave;&ccedil; S .
&Ograve;&agrave;&ecirc;&icirc;&eacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &auml;&agrave;&euml;&aring;&aring; &iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&igrave;. &Acirc;&ucirc;&aacute;&egrave;&eth;&agrave;&yuml; &iuml;&eth;&aring;&auml;&aring;&euml;&uuml;&iacute;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute; &acirc;&ntilde;&aring; &aacute;&icirc;&euml;&aring;&aring; &igrave;&aring;&euml;
&ecirc;&egrave;&otilde; &iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&otilde; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&icirc;&acirc;, &igrave;&ucirc; &acirc;&egrave;&auml;&egrave;&igrave;, &divide;&ograve;&icirc; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; (&egrave;&euml;&egrave; &Ecirc;&Ecirc;&Igrave;) &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &ntilde;&euml;&aring;&auml;&ntilde;&ograve;&acirc;&egrave;&aring;&igrave;
&euml;&aring;&igrave;&igrave;&ucirc; &Oslash;&iuml;&aring;&eth;&iacute;&aring;&eth;&agrave; &egrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &ntilde;&ecirc;&icirc;&euml;&uuml; &oacute;&atilde;&icirc;&auml;&iacute;&icirc; &igrave;&aring;&euml;&ecirc;&icirc;&eacute; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave;.
&Ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave;
&Ntilde;&ecirc;&agrave;&aelig;&aring;&igrave; &ecirc;&eth;&agrave;&ograve;&ecirc;&icirc; &icirc; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&egrave; &igrave;&aring;&euml;&ecirc;&egrave;&otilde; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&eacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; ∆S . &Icirc;&auml;&egrave;&iacute; &egrave;&ccedil; &ntilde;&iuml;&icirc;&ntilde;&icirc;&aacute;&icirc;&acirc;
&iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&egrave;&yuml; &egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&oacute;&aring;&ograve; &aacute;&agrave;&eth;&egrave;&ouml;&aring;&iacute;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&aring; &eth;&agrave;&ccedil;&aacute;&egrave;&aring;&iacute;&egrave;&yuml;. &Icirc;&auml;&iacute;&agrave;&ecirc;&icirc; &aacute;&icirc;&euml;&aring;&aring; &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&ucirc;&igrave; &divide;&agrave;&ntilde;&ograve;&icirc; &aacute;&ucirc;&acirc;&agrave;
&aring;&ograve; &auml;&eth;&oacute;&atilde;&icirc;&eacute; &eth;&aring;&atilde;&oacute;&euml;&yuml;&eth;&iacute;&ucirc;&eacute; &ntilde;&iuml;&icirc;&ntilde;&icirc;&aacute; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&egrave;&yuml; &ntilde;&ecirc;&icirc;&euml;&uuml; &oacute;&atilde;&icirc;&auml;&iacute;&icirc; &igrave;&aring;&euml;&ecirc;&icirc;&eacute; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave;. &Iuml;&oacute;&ograve;&uuml; m (&aacute;&icirc;&euml;&uuml;&oslash;&icirc;&aring;) &ouml;&aring;&euml;&icirc;&aring; &divide;&egrave;&ntilde;&euml;&icirc;. &Igrave;&ucirc; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&egrave;&igrave; &ntilde;&aring;&eacute;&divide;&agrave;&ntilde; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&thorn; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; ∆n , &acirc;&aring;&eth;&oslash;&egrave;&iacute;&agrave;&igrave;&egrave; &ecirc;&icirc;
&ograve;&icirc;&eth;&icirc;&eacute; &aacute;&oacute;&auml;&oacute;&ograve; &ograve;&icirc;&divide;&ecirc;&egrave; &egrave;&ccedil; ∆n , &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&ucirc; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&agrave;&ograve; (1/m)Z. &Auml;&euml;&yuml; &yacute;&ograve;&icirc;&atilde;&icirc; &oacute;&auml;&icirc;&aacute;&iacute;&aring;&aring;
&ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&egrave;&eth;&icirc;&acirc;&agrave;&ograve;&uuml; &auml;&eth;&oacute;&atilde;&icirc;&eacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;, &ccedil;&agrave;&auml;&agrave;&iacute;&iacute;&ucirc;&eacute; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml;&igrave;&egrave;
∆ = {x = (xi ) ∈ Rn , m ≥ x1 ≥ &middot; &middot; &middot; ≥ xn ≥ 0},
&ograve;&agrave;&ecirc; &divide;&ograve;&icirc; &acirc;&aring;&eth;&oslash;&egrave;&iacute;&ucirc; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave; &aacute;&oacute;&auml;&oacute;&ograve; &iacute;&agrave;&otilde;&icirc;&auml;&egrave;&ograve;&uuml;&ntilde;&yuml; &acirc; &ouml;&aring;&euml;&icirc;&divide;&egrave;&ntilde;&euml;&aring;&iacute;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&ecirc;&agrave;&otilde;. &Ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&ucirc; &yacute;&ograve;&icirc;&eacute;
&ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave; &oacute;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&ucirc; &ograve;&agrave;&ecirc;. &Ccedil;&agrave;&ocirc;&egrave;&ecirc;&ntilde;&egrave;&eth;&oacute;&aring;&igrave; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&oacute;&thorn; &iuml;&aring;&eth;&aring;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&ecirc;&oacute; π &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; {1, . . . , n}
&egrave; &ouml;&aring;&euml;&icirc;&divide;&egrave;&ntilde;&euml;&aring;&iacute;&iacute;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute; y ∈ Zn .
&Ograve;&icirc;&atilde;&auml;&agrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde; σy,π &iacute;&agrave;&ograve;&yuml;&atilde;&egrave;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &iacute;&agrave; &ograve;&icirc;&divide;&ecirc;&egrave;
x0 = y,
x1 = x0 + eπ(1) ,
...,
xn = xn−1 + eπ(n) .
&Ccedil;&auml;&aring;&ntilde;&uuml; ei i-&eacute; &aacute;&agrave;&ccedil;&egrave;&ntilde;&iacute;&ucirc;&eacute; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;.
&Iacute;&oacute;&aelig;&iacute;&icirc; &iuml;&eth;&icirc;&acirc;&aring;&eth;&egrave;&ograve;&uuml;, &divide;&ograve;&icirc; &euml;&thorn;&aacute;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; ∆ = {x = (xi ) ∈ Rn , m ≥ x1 ≥ &middot; &middot; &middot; ≥ xn ≥
0}, &iuml;&icirc;&iuml;&agrave;&auml;&agrave;&aring;&ograve; &acirc; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&eacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave;. &Icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&egrave;&igrave; &divide;&aring;&eth;&aring;&ccedil; [a] &ouml;&aring;&euml;&oacute;&thorn; &divide;&agrave;&ntilde;&ograve;&uuml; &divide;&egrave;&ntilde;&euml;&agrave;
a. &Ograve;&icirc;&atilde;&auml;&agrave; &acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring; y &auml;&euml;&yuml; &ograve;&icirc;&divide;&ecirc;&egrave; x &acirc;&icirc;&ccedil;&uuml;&igrave;&aring;&igrave; &ograve;&icirc;&divide;&ecirc;&oacute; [x] &ntilde; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&agrave;&igrave;&egrave; [xi ], &agrave; &iuml;&aring;&eth;&aring;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&ecirc;&oacute; π
&acirc;&icirc;&ccedil;&uuml;&igrave;&aring;&igrave; &acirc; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&egrave;&egrave; &ntilde; &oacute;&aacute;&ucirc;&acirc;&agrave;&iacute;&egrave;&aring;&igrave; &divide;&egrave;&ntilde;&aring;&euml; xi − [xi ]. &Ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;, &iuml;&eth;&egrave; &yacute;&ograve;&icirc;&igrave; &iacute;&oacute;&aelig;&iacute;&icirc; &oacute;&aacute;&aring;&auml;&egrave;&ograve;&uuml;&ntilde;&yuml;,
&divide;&ograve;&icirc; &acirc;&aring;&eth;&oslash;&egrave;&iacute;&ucirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; σy,π &ograve;&agrave;&ecirc;&aelig;&aring; &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&agrave;&ograve; ∆, &aring;&ntilde;&euml;&egrave; &ograve;&icirc;&divide;&ecirc;&agrave; x &aacute;&ucirc;&euml;&agrave; &egrave;&ccedil; ∆. &Icirc;&ntilde;&ograve;&agrave;&acirc;&egrave;&igrave; &yacute;&ograve;&icirc;
&acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring; &oacute;&iuml;&eth;&agrave;&aelig;&iacute;&aring;&iacute;&egrave;&yuml;.
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc; &euml;&aring;&igrave;&igrave;&ucirc; &Oslash;&iuml;&aring;&eth;&iacute;&aring;&eth;&agrave;
&Ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave;, &acirc;&aring;&eth;&oslash;&egrave;&iacute;&ucirc; &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&atilde;&icirc; &egrave;&igrave;&aring;&thorn;&ograve; &acirc;&ntilde;&aring; &igrave;&aring;&ograve;&ecirc;&egrave;, &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&igrave;. &Ograve;&agrave;&ecirc;&egrave;&igrave;
&icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;, &euml;&aring;&igrave;&igrave;&agrave; &Oslash;&iuml;&aring;&eth;&iacute;&aring;&eth;&agrave; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&agrave;&aring;&ograve; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &iuml;&aring;&ntilde;&ograve;&eth;&icirc;&atilde;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave;. &Ecirc;&agrave;&ecirc; &yacute;&ograve;&icirc; &egrave;&iacute;&icirc;&atilde;&auml;&agrave;
&aacute;&ucirc;&acirc;&agrave;&aring;&ograve;, &euml;&aring;&atilde;&divide;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml; &aacute;&icirc;&euml;&aring;&aring; &ntilde;&egrave;&euml;&uuml;&iacute;&icirc;&aring; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&aring;&iacute;&egrave;&aring;. &Agrave; &egrave;&igrave;&aring;&iacute;&iacute;&icirc;, &igrave;&ucirc; &iuml;&icirc;&ecirc;&agrave;&aelig;&aring;&igrave;, &divide;&ograve;&icirc; &iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&otilde;
&ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&icirc;&acirc; &iacute;&aring;&divide;&aring;&ograve;&iacute;&icirc;&aring; &divide;&egrave;&ntilde;&euml;&icirc;. &Iuml;&icirc;&divide;&aring;&igrave;&oacute; &aelig;&aring; &euml;&aring;&atilde;&divide;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&ucirc;&acirc;&agrave;&ograve;&uuml; &aacute;&icirc;&euml;&aring;&aring; &ntilde;&egrave;&euml;&uuml;&iacute;&icirc;&aring; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&aring;&iacute;&egrave;&aring;? &Auml;&aring;
&euml;&icirc; &acirc; &ograve;&icirc;&igrave;, &divide;&ograve;&icirc; &iuml;&eth;&egrave; &egrave;&iacute;&auml;&oacute;&ecirc;&ograve;&egrave;&acirc;&iacute;&icirc;&igrave; &eth;&agrave;&ntilde;&ntilde;&oacute;&aelig;&auml;&aring;&iacute;&egrave;&egrave; &igrave;&ucirc; &icirc;&iuml;&egrave;&eth;&agrave;&aring;&igrave;&ntilde;&yuml; &iacute;&agrave; &aacute;&icirc;&euml;&aring;&aring; &ntilde;&egrave;&euml;&uuml;&iacute;&oacute;&thorn; &iuml;&icirc;&ntilde;&ucirc;&euml;&ecirc;&oacute;, &egrave;
&yacute;&ograve;&icirc; &icirc;&aacute;&euml;&aring;&atilde;&divide;&agrave;&aring;&ograve; &egrave; &iuml;&icirc;&igrave;&icirc;&atilde;&agrave;&aring;&ograve; &iuml;&icirc;&euml;&oacute;&divide;&egrave;&ograve;&uuml; &egrave; &aacute;&icirc;&euml;&aring;&aring; &ntilde;&egrave;&euml;&uuml;&iacute;&icirc;&aring; &ccedil;&agrave;&ecirc;&euml;&thorn;&divide;&aring;&iacute;&egrave;&aring;. &Igrave;&ucirc; &oacute;&aacute;&aring;&auml;&egrave;&igrave;&ntilde;&yuml; &acirc; &yacute;&ograve;&icirc;&igrave; &iuml;&eth;&egrave;
&auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&aring; &euml;&aring;&igrave;&igrave;&ucirc; &Oslash;&iuml;&aring;&eth;&iacute;&aring;&eth;&agrave;.
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;. &Icirc;&iacute;&icirc; &iuml;&eth;&icirc;&acirc;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &iuml;&icirc; &egrave;&iacute;&auml;&oacute;&ecirc;&ouml;&egrave;&egrave; &egrave; &icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&icirc; &acirc;&aring;&eth;&iacute;&icirc; &auml;&euml;&yuml; 0-&igrave;&aring;&eth;&iacute;&icirc;&atilde;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave;
∆1 . &Acirc; &icirc;&aacute;&ugrave;&aring;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &icirc;&ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&egrave;&igrave; S &ntilde; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc;&igrave; {1, . . . , n}. &Ecirc;&eth;&icirc;&igrave;&aring; &iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&otilde; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&icirc;&acirc;,
&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &egrave;&igrave;&aring;&thorn;&ograve; &acirc;&ntilde;&aring; &igrave;&aring;&ograve;&ecirc;&egrave; &icirc;&ograve; 1 &auml;&icirc; n, &acirc; &eth;&agrave;&ntilde;&ntilde;&oacute;&aelig;&auml;&aring;&iacute;&egrave;&egrave; &acirc;&agrave;&aelig;&iacute;&oacute;&thorn; &eth;&icirc;&euml;&uuml; &aacute;&oacute;&auml;&oacute;&ograve; &egrave;&atilde;&eth;&agrave;&ograve;&uuml; &ograve;. &iacute;. &iuml;&icirc;&euml;&oacute;&iuml;&aring;
&ntilde;&ograve;&eth;&ucirc;&aring; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&ucirc;, &iuml;&icirc;&igrave;&aring;&divide;&aring;&iacute;&iacute;&ucirc;&aring; &igrave;&aring;&ograve;&ecirc;&agrave;&igrave;&egrave; &icirc;&ograve; 1 &auml;&icirc; n − 1. &szlig;&ntilde;&iacute;&icirc;, &divide;&ograve;&icirc; &iuml;&icirc;&euml;&oacute;&iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&igrave; &igrave;&icirc;&aelig;&aring;&ograve; &aacute;&ucirc;&ograve;&uuml;
&ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave; n − 1 &egrave;&euml;&egrave; n − 2. &ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &ograve;&eth;&egrave; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave;:
1) A &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&otilde; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&icirc;&acirc;,
28
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 4.
&Ograve;&Aring;&Icirc;&ETH;&Aring;&Igrave;&Agrave; &Aacute;&ETH;&Agrave;&Oacute;&Yacute;&ETH;&Agrave;: &Auml;&Icirc;&Ecirc;&Agrave;&Ccedil;&Agrave;&Ograve;&Aring;&Euml;&Uuml;&Ntilde;&Ograve;&Acirc;&Agrave; &Egrave; &Agrave;&Euml;&Atilde;&Icirc;&ETH;&Egrave;&Ograve;&Igrave;&Ucirc;
2) B &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &iuml;&icirc;&euml;&oacute;&iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&otilde; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&icirc;&acirc; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave;, &euml;&aring;&aelig;&agrave;&ugrave;&egrave;&otilde; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave; ∆{1,...,n−1} ,
3) I &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; &iuml;&agrave;&eth; (σ, τ ), &atilde;&auml;&aring; σ (n − 1)-&igrave;&aring;&eth;&iacute;&ucirc;&eacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;, &agrave; τ &aring;&atilde;&icirc; &iuml;&icirc;&euml;&oacute;&iuml;&aring;&ntilde;&ograve;&eth;&agrave;&yuml;
&atilde;&eth;&agrave;&iacute;&uuml;.
&Iuml;&icirc;&ntilde;&divide;&egrave;&ograve;&agrave;&aring;&igrave; &divide;&egrave;&ntilde;&euml;&icirc; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&icirc;&acirc; |I| &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; I &auml;&acirc;&oacute;&igrave;&yuml; &ntilde;&iuml;&icirc;&ntilde;&icirc;&aacute;&agrave;&igrave;&egrave;.
&Iuml;&aring;&eth;&acirc;&ucirc;&eacute; &ntilde;&iuml;&icirc;&ntilde;&icirc;&aacute;. &Aring;&ntilde;&euml;&egrave; (σ, τ ) &iuml;&agrave;&eth;&agrave; &egrave;&ccedil; I , &ograve;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde; σ &euml;&egrave;&aacute;&icirc; &iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&eacute; (&ograve;. &aring;. &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve;
A), &euml;&egrave;&aacute;&icirc; &iuml;&icirc;&euml;&oacute;&iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&eacute;. &Acirc; &iuml;&aring;&eth;&acirc;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &auml;&euml;&yuml; &ograve;&agrave;&ecirc;&icirc;&atilde;&icirc; σ &egrave;&igrave;&aring;&aring;&ograve;&ntilde;&yuml; &eth;&icirc;&acirc;&iacute;&icirc; &icirc;&auml;&iacute;&agrave; &iuml;&agrave;&eth;&agrave; (σ, &middot;) &egrave;&ccedil; I (&agrave;
&egrave;&igrave;&aring;&iacute;&iacute;&icirc;, (σ, τ )), &acirc;&icirc; &acirc;&ograve;&icirc;&eth;&icirc;&igrave; &eth;&icirc;&acirc;&iacute;&icirc; &auml;&acirc;&aring; &ograve;&agrave;&ecirc;&egrave;&otilde; &iuml;&agrave;&eth;&ucirc;. &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; |I| = |A| + &divide;&aring;&ograve;&iacute;&icirc;&aring; &divide;&egrave;&ntilde;&euml;&icirc;.
&Acirc;&ograve;&icirc;&eth;&icirc;&eacute; &ntilde;&iuml;&icirc;&ntilde;&icirc;&aacute;. &Aring;&ntilde;&euml;&egrave; (σ, τ ) &iuml;&agrave;&eth;&agrave; &egrave;&ccedil; I , &ograve;&icirc; &euml;&egrave;&aacute;&icirc; σ &euml;&aring;&aelig;&egrave;&ograve; &frac34;&acirc;&iacute;&oacute;&ograve;&eth;&egrave;&iquest; ∆n , &euml;&egrave;&aacute;&icirc; &euml;&aring;&aelig;&egrave;&ograve; &iacute;&agrave;
&atilde;&eth;&agrave;&iacute;&egrave;&ouml;&aring;. &Aring;&ntilde;&euml;&egrave; τ &euml;&aring;&aelig;&egrave;&ograve; &acirc;&iacute;&oacute;&ograve;&eth;&egrave; ∆n , &ograve;&icirc; &icirc;&iacute; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &atilde;&eth;&agrave;&iacute;&uuml;&thorn; &auml;&acirc;&oacute;&otilde; (n − 1)-&igrave;&aring;&eth;&iacute;&ucirc;&otilde; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&icirc;&acirc;
&ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave; Σ, &egrave; &egrave;&igrave;&aring;&aring;&ograve;&ntilde;&yuml; &eth;&icirc;&acirc;&iacute;&icirc; &auml;&acirc;&aring; &iuml;&agrave;&eth;&ucirc; (&middot;, τ ) ∈ I . &Aring;&ntilde;&euml;&egrave; τ &euml;&aring;&aelig;&egrave;&ograve; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&aring;, &ograve;&icirc; &eth;&icirc;&acirc;&iacute;&icirc; &icirc;&auml;&iacute;&agrave;
&ograve;&agrave;&ecirc;&agrave;&yuml; &iuml;&agrave;&eth;&agrave;. &Iuml;&eth;&egrave; &yacute;&ograve;&icirc;&igrave; &acirc; &ntilde;&egrave;&euml;&oacute; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml; &euml;&aring;&igrave;&igrave;&ucirc; τ &euml;&aring;&aelig;&egrave;&ograve; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave; ∆{1,...,n−1} , &ograve;. &aring;. &iuml;&eth;&egrave;&iacute;&agrave;&auml;&euml;&aring;&aelig;&egrave;&ograve;
&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&oacute; B . &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; |I| = &divide;&aring;&ograve;&iacute;&icirc;&aring; &divide;&egrave;&ntilde;&euml;&icirc; + |B|.
&Ccedil;&agrave;&ecirc;&euml;&thorn;&divide;&agrave;&aring;&igrave;, &divide;&ograve;&icirc; &egrave;&iacute;&ograve;&aring;&eth;&aring;&ntilde;&oacute;&thorn;&ugrave;&aring;&aring; &iacute;&agrave;&ntilde; &divide;&egrave;&ntilde;&euml;&icirc; |A| &icirc;&ograve;&euml;&egrave;&divide;&agrave;&aring;&ograve;&ntilde;&yuml; &icirc;&ograve; &divide;&egrave;&ntilde;&euml;&agrave; |B| &iacute;&agrave; &divide;&aring;&ograve;&iacute;&icirc;&aring; &divide;&egrave;&ntilde;&euml;&icirc;.
&Iacute;&icirc; |B| &yacute;&ograve;&icirc; &acirc; &ograve;&icirc;&divide;&iacute;&icirc;&ntilde;&ograve;&egrave; &divide;&egrave;&ntilde;&euml;&icirc; &iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&otilde; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&icirc;&acirc; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave; &atilde;&eth;&agrave;&iacute;&egrave; ∆{1,...,n−1} , &egrave; &iuml;&icirc;
&egrave;&iacute;&auml;&oacute;&ecirc;&ouml;&egrave;&egrave; &yacute;&ograve;&icirc; &divide;&egrave;&ntilde;&euml;&icirc; &iacute;&aring;&divide;&aring;&ograve;&iacute;&icirc;&aring;. &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &iacute;&aring;&divide;&aring;&ograve;&iacute;&icirc; &egrave; |A|.
&Icirc;&aacute;&ntilde;&oacute;&aelig;&auml;&aring;&iacute;&egrave;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave;
&Iacute;&agrave; &eth;&egrave;&ntilde;. ?? &egrave;&ccedil;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&agrave; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&yuml; ∆3 . &Aring;&ntilde;&euml;&egrave; &igrave;&ucirc; &iuml;&icirc;&igrave;&aring;&ograve;&egrave;&igrave; &iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&aring; &egrave; &iuml;&icirc;&euml;&oacute;&iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&aring; &ntilde;&egrave;&igrave;
&iuml;&euml;&aring;&ecirc;&ntilde;&ucirc;, &ograve;&icirc; &oacute;&acirc;&egrave;&auml;&egrave;&igrave;, &divide;&ograve;&icirc; &icirc;&iacute;&egrave; &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&thorn;&ograve; &iuml;&oacute;&ograve;&egrave;. &Egrave; &yacute;&ograve;&egrave; &iuml;&oacute;&ograve;&egrave; &aacute;&ucirc;&acirc;&agrave;&thorn;&ograve; &divide;&aring;&ograve;&ucirc;&eth;&aring;&otilde; &ograve;&egrave;&iuml;&icirc;&acirc;:
&ETH;&egrave;&ntilde;. 1
1) &iacute;&agrave;&divide;&egrave;&iacute;&agrave;&thorn;&ograve;&ntilde;&yuml; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave; &egrave; &ccedil;&agrave;&ecirc;&agrave;&iacute;&divide;&egrave;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;;
2) &ouml;&egrave;&ecirc;&euml;&ucirc;;
3) &iacute;&agrave;&divide;&egrave;&iacute;&agrave;&thorn;&ograve;&ntilde;&yuml; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave; &egrave; &ccedil;&agrave;&ecirc;&agrave;&iacute;&divide;&egrave;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &acirc; &iuml;&aring;&ntilde;&ograve;&eth;&icirc;&igrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&aring;;
4) &iacute;&agrave;&divide;&egrave;&iacute;&agrave;&thorn;&ograve;&ntilde;&yuml; &egrave; &ccedil;&agrave;&ecirc;&agrave;&iacute;&divide;&egrave;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &acirc; &iuml;&aring;&ntilde;&ograve;&eth;&icirc;&igrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&aring;.
&Yacute;&ograve;&icirc; &iacute;&agrave;&acirc;&icirc;&auml;&egrave;&ograve; &iacute;&agrave; &igrave;&ucirc;&ntilde;&euml;&uuml; &icirc;&aacute; &agrave;&euml;&atilde;&icirc;&eth;&egrave;&ograve;&igrave;&aring; &iuml;&icirc;&egrave;&ntilde;&ecirc;&agrave; &iuml;&aring;&ntilde;&ograve;&eth;&icirc;&atilde;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave;. &Iacute;&agrave;&auml;&icirc; &iacute;&agrave;&divide;&agrave;&ograve;&uuml; &ntilde; &iuml;&icirc;&euml;&oacute;
&iuml;&aring;&ntilde;&ograve;&eth;&icirc;&atilde;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;, &egrave; &auml;&acirc;&egrave;&atilde;&agrave;&ograve;&uuml;&ntilde;&yuml; &iuml;&icirc; &iuml;&icirc;&euml;&oacute;&iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&igrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave;&igrave;, &iuml;&icirc;&ecirc;&agrave; &iacute;&aring; &iuml;&eth;&egrave;&auml;&aring;&igrave; &acirc;
&iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&eacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;. &Aacute;&icirc;&euml;&aring;&aring; &ograve;&icirc;&divide;&iacute;&icirc;, &igrave;&ucirc; &iacute;&agrave;&divide;&egrave;&iacute;&agrave;&aring;&igrave; &ntilde; &iuml;&icirc;&euml;&oacute;&iuml;&aring;&ntilde;&ograve;&eth;&icirc;&atilde;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave; n − 2,
&euml;&aring;&aelig;&agrave;&ugrave;&aring;&atilde;&icirc; &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;-&icirc;&ntilde;&iacute;&icirc;&acirc;&agrave;&iacute;&egrave;&egrave;. &Icirc;&iacute; &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &atilde;&eth;&agrave;&iacute;&uuml;&thorn; &aring;&auml;&egrave;&iacute;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&atilde;&icirc; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave;
n − 1. &Aring;&ntilde;&euml;&egrave; &icirc;&iacute; &iacute;&aring; &iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&eacute;, &ograve;&icirc; &iuml;&icirc;&euml;&oacute;&iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&eacute;, &egrave; &oacute; &iacute;&aring;&atilde;&icirc; &aring;&ntilde;&ograve;&uuml; &acirc;&ograve;&icirc;&eth;&agrave;&yuml; &iuml;&icirc;&euml;&oacute;&iuml;&aring;&ntilde;&ograve;&eth;&agrave;&yuml; &atilde;&eth;&agrave;&iacute;&uuml;, &iuml;&eth;&egrave;&divide;&aring;&igrave;
&aring;&auml;&egrave;&iacute;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&agrave;&yuml;. &Ecirc; &yacute;&ograve;&icirc;&eacute; &atilde;&eth;&agrave;&iacute;&egrave; &icirc;&iuml;&yuml;&ograve;&uuml; &iuml;&eth;&egrave;&igrave;&ucirc;&ecirc;&agrave;&aring;&ograve; &aring;&auml;&egrave;&iacute;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&ucirc;&eacute; &iacute;&icirc;&acirc;&ucirc;&eacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave;
n − 1, &egrave; &ograve;. &auml;.
29
&Iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &yuml;&acirc;&iacute;&icirc;&aring; &iacute;&aring;&oacute;&auml;&icirc;&aacute;&ntilde;&ograve;&acirc;&icirc; &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &acirc; &ograve;&icirc;&igrave;, &divide;&ograve;&icirc; &igrave;&ucirc; &igrave;&icirc;&aelig;&aring;&igrave; &iuml;&eth;&egrave;&eacute;&ograve;&egrave; &iacute;&aring; &acirc; &iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&eacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;,
&agrave; &acirc;&aring;&eth;&iacute;&oacute;&ograve;&uuml;&ntilde;&yuml; &iacute;&agrave; &ograve;&oacute; &aelig;&aring; &atilde;&eth;&agrave;&iacute;&uuml;. &Ccedil;&agrave;&igrave;&aring;&ograve;&egrave;&igrave;, &icirc;&auml;&iacute;&agrave;&ecirc;&icirc;, &divide;&ograve;&icirc; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&atilde;&icirc; &iacute;&aring; &ntilde;&euml;&oacute;&divide;&egrave;&ograve;&ntilde;&yuml;, &aring;&ntilde;&euml;&egrave; &iacute;&agrave; &icirc;&ntilde;&iacute;&icirc;&acirc;&agrave;&iacute;&egrave;&egrave;
&egrave;&igrave;&aring;&aring;&ograve;&ntilde;&yuml; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &icirc;&auml;&egrave;&iacute; &iuml;&icirc;&euml;&oacute;&iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&eacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;. &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &iuml;&eth;&aring;&auml;&acirc;&agrave;&eth;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc; &auml;&icirc;&aacute;&agrave;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &aring;&ugrave;&aring; &icirc;&auml;&egrave;&iacute;
&ntilde;&euml;&icirc;&eacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&icirc;&acirc; (&iuml;&icirc;&auml; &icirc;&ntilde;&iacute;&icirc;&acirc;&agrave;&iacute;&egrave;&aring;&igrave;), &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&eacute; &eth;&agrave;&ccedil;&igrave;&aring;&divide;&agrave;&aring;&ograve;&ntilde;&yuml; &ograve;&agrave;&ecirc;, &divide;&ograve;&icirc;&aacute;&ucirc; &iacute;&agrave; &iacute;&aring;&igrave; &egrave;&igrave;&aring;&euml;&ntilde;&yuml; &eth;&icirc;&acirc;&iacute;&icirc;
&icirc;&auml;&egrave;&iacute; &iuml;&icirc;&euml;&oacute;&iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&eacute; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;. &Ntilde;&ograve;&agrave;&eth;&ograve;&oacute;&yuml; &ntilde; &iacute;&aring;&atilde;&icirc;, &iuml;&oacute;&ograve;&uuml; &oacute;&aelig;&aring; &icirc;&aacute;&yuml;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc; &iuml;&eth;&egrave;&acirc;&aring;&auml;&aring;&ograve; &ecirc; &iuml;&aring;&ntilde;&ograve;&eth;&icirc;&igrave;&oacute;
&ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&oacute; (&ntilde;&igrave;. &eth;&egrave;&ntilde;. ??).
&ETH;&egrave;&ntilde;. 2
&Yacute;&ograve;&agrave; &egrave;&auml;&aring;&yuml; &igrave;&icirc;&aelig;&aring;&ograve; &icirc;&ocirc;&icirc;&eth;&igrave;&euml;&yuml;&ograve;&uuml;&ntilde;&yuml; &igrave;&iacute;&icirc;&atilde;&egrave;&igrave;&egrave; &eth;&agrave;&ccedil;&iacute;&ucirc;&igrave;&egrave; &ntilde;&iuml;&icirc;&ntilde;&icirc;&aacute;&agrave;&igrave;&egrave; &egrave; &iuml;&eth;&egrave;&acirc;&icirc;&auml;&egrave;&ograve; &ecirc; &eth;&agrave;&ccedil;&euml;&egrave;&divide;&iacute;&ucirc;&igrave; &agrave;&euml;
&atilde;&icirc;&eth;&egrave;&ograve;&igrave;&agrave;&igrave; &iuml;&eth;&egrave;&aacute;&euml;&egrave;&aelig;&aring;&iacute;&iacute;&icirc;&atilde;&icirc; &iacute;&agrave;&otilde;&icirc;&aelig;&auml;&aring;&iacute;&egrave;&yuml; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave;. &Ccedil;&auml;&aring;&ntilde;&uuml; &iacute;&aring;&ograve; &acirc;&icirc;&ccedil;&igrave;&icirc;&aelig;&iacute;&icirc;&ntilde;&ograve;&egrave; &acirc;&otilde;&icirc;&auml;&egrave;&ograve;&uuml;
&acirc; &auml;&aring;&ograve;&agrave;&euml;&egrave; &egrave; &eth;&agrave;&ccedil;&euml;&egrave;&divide;&iacute;&ucirc;&aring; &oacute;&otilde;&egrave;&ugrave;&eth;&aring;&iacute;&egrave;&yuml;; &iuml;&icirc;&auml;&eth;&icirc;&aacute;&iacute;&aring;&aring; &ntilde;&igrave;. &ecirc;&iacute;&egrave;&atilde;&oacute; &Ograve;&icirc;&auml;&auml;&agrave; [8].
&Auml;&eth;&oacute;&atilde;&egrave;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave;
&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;, &aacute;&oacute;&auml;&oacute;&divide;&egrave; &acirc;&aring;&ntilde;&uuml;&igrave;&agrave; &iacute;&aring;&ograve;&eth;&egrave;&acirc;&egrave;&agrave;&euml;&uuml;&iacute;&icirc;&eacute;, &egrave;&igrave;&aring;&aring;&ograve; &igrave;&iacute;&icirc;&atilde;&icirc; &eth;&agrave;&ccedil;&euml;&egrave;&divide;&iacute;&ucirc;&otilde; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;,
&egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&oacute;&thorn;&ugrave;&egrave;&otilde; &eth;&agrave;&ccedil;&iacute;&ucirc;&aring; &igrave;&aring;&ograve;&icirc;&auml;&ucirc; &egrave; &iuml;&icirc;&auml;&otilde;&icirc;&auml;&ucirc;. &Aacute;&aring;&ccedil; &iuml;&eth;&aring;&oacute;&acirc;&aring;&euml;&egrave;&divide;&aring;&iacute;&egrave;&yuml; &igrave;&icirc;&aelig;&iacute;&icirc; &ntilde;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &yacute;&ograve;&egrave; &auml;&icirc;
&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave; &ntilde;&euml;&oacute;&aelig;&agrave;&ograve; &otilde;&icirc;&eth;&icirc;&oslash;&egrave;&igrave; &acirc;&acirc;&aring;&auml;&aring;&iacute;&egrave;&aring;&igrave; &acirc;&icirc; &igrave;&iacute;&icirc;&atilde;&egrave;&aring; &icirc;&aacute;&euml;&agrave;&ntilde;&ograve;&egrave; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&ecirc;&egrave;. &Icirc;&eth;&egrave;&atilde;&egrave;&iacute;&agrave;&euml;&uuml;&iacute;&icirc;&aring;
&auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&icirc;&acirc;&agrave;&euml;&icirc; &ograve;&aring;&icirc;&eth;&egrave;&thorn; &egrave;&iacute;&auml;&aring;&ecirc;&ntilde;&icirc;&acirc; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&iacute;&ucirc;&otilde; &iuml;&icirc;&euml;&aring;&eacute;. &Iuml;&eth;&egrave;&acirc;&aring;&auml;&aring;&iacute;&iacute;&icirc;&aring; &acirc;&ucirc;
&oslash;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc; &icirc;&iuml;&egrave;&eth;&agrave;&euml;&icirc;&ntilde;&uuml; &iacute;&agrave; &ecirc;&icirc;&igrave;&aacute;&egrave;&iacute;&agrave;&ograve;&icirc;&eth;&iacute;&oacute;&thorn; &euml;&aring;&igrave;&igrave;&oacute; &Oslash;&iuml;&aring;&eth;&iacute;&aring;&eth;&agrave;. &Iacute;&egrave;&aelig;&aring; &igrave;&ucirc; &iacute;&agrave;&igrave;&aring;&ograve;&egrave;&igrave; &ograve;&eth;&egrave;
&auml;&eth;&oacute;&atilde;&egrave;&otilde; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave;. &Aacute;&icirc;&euml;&aring;&aring; &ograve;&icirc;&divide;&iacute;&icirc;, &igrave;&ucirc; &aacute;&oacute;&auml;&aring;&igrave; &icirc;&aacute;&ntilde;&oacute;&aelig;&auml;&agrave;&ograve;&uuml; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &icirc; &iacute;&aring;&eth;&aring;
&ograve;&eth;&agrave;&atilde;&egrave;&eth;&oacute;&aring;&igrave;&icirc;&ntilde;&ograve;&egrave; &auml;&egrave;&ntilde;&ecirc;&agrave; Dn &iacute;&agrave; &aring;&atilde;&icirc; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&oacute; ∂Dn = Sn−1 , &yacute;&ecirc;&acirc;&egrave;&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&icirc;&eacute; (&ntilde;&igrave;. &euml;&aring;&ecirc;&ouml;&egrave;&thorn; 2) &ograve;&aring;&icirc;&eth;&aring;&igrave;&aring;
&Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;.
&Agrave;&euml;&atilde;&aring;&aacute;&eth;&agrave;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml; &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&yuml;. &Igrave;&aring;&ograve;&icirc;&auml; &agrave;&euml;&atilde;&aring;&aacute;&eth;&agrave;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&egrave; &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &acirc; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&egrave;&egrave; &iuml;&icirc;&auml;
&otilde;&icirc;&auml;&yuml;&ugrave;&aring;&atilde;&icirc; &ocirc;&oacute;&iacute;&ecirc;&ograve;&icirc;&eth;&agrave; &egrave;&ccedil; &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&egrave; &acirc; &agrave;&euml;&atilde;&aring;&aacute;&eth;&oacute;. &Ntilde; &ecirc;&agrave;&aelig;&auml;&ucirc;&igrave; &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&igrave; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc;&igrave; X
&ntilde;&acirc;&yuml;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &agrave;&euml;&atilde;&aring;&aacute;&eth;&agrave;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&eacute; &icirc;&aacute;&uacute;&aring;&ecirc;&ograve; H(X) (&acirc; &iacute;&agrave;&oslash;&aring;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &yacute;&ograve;&icirc; &aacute;&oacute;&auml;&aring;&ograve; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;
&ntilde;&ograve;&acirc;&icirc;), &agrave; &ntilde; &ecirc;&agrave;&aelig;&auml;&ucirc;&igrave; (&iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&ucirc;&igrave;) &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring;&igrave; f : X → Y &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring;
H(f ) : H(X) → H(Y ) &yacute;&ograve;&egrave;&otilde; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&iacute;&ucirc;&otilde; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;. &Iuml;&eth;&egrave; &yacute;&ograve;&icirc;&igrave; &ecirc;&icirc;&igrave;&iuml;&icirc;&ccedil;&egrave;&ouml;&egrave;&yuml; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute;
&auml;&agrave;&aring;&ograve; &ecirc;&icirc;&igrave;&iuml;&icirc;&ccedil;&egrave;&ouml;&egrave;&thorn;, &agrave; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc;&aring;. &Auml;&aring;&ograve;&agrave;&euml;&egrave; &yacute;&ograve;&egrave;&otilde; &ecirc;&icirc;&iacute;&ntilde;&ograve;&eth;&oacute;&ecirc;
&ouml;&egrave;&eacute; &acirc;&ucirc;&otilde;&icirc;&auml;&yuml;&ograve; &ccedil;&agrave; &eth;&agrave;&igrave;&ecirc;&egrave; &iacute;&agrave;&ntilde;&ograve;&icirc;&yuml;&ugrave;&aring;&atilde;&icirc; &egrave;&ccedil;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&yuml;. &Iuml;&eth;&egrave;&igrave;&aring;&iacute;&egrave;&igrave; &yacute;&ograve;&icirc; &ecirc; &iacute;&agrave;&oslash;&aring;&eacute; &ntilde;&egrave;&ograve;&oacute;&agrave;&ouml;&egrave;&egrave;, &ograve;&icirc;&divide;&iacute;&aring;&aring;, &ecirc;
&iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&thorn; &icirc; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&egrave; &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&egrave; f : D → S &oslash;&agrave;&eth;&agrave; &iacute;&agrave; &aring;&atilde;&icirc; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&oacute;. &Acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring;
H &acirc;&icirc;&ccedil;&uuml;&igrave;&aring;&igrave; &ograve;&aring;&icirc;&eth;&egrave;&thorn; &atilde;&icirc;&igrave;&icirc;&euml;&icirc;&atilde;&egrave;&eacute; (&eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave; n − 1 &ntilde; &ecirc;&icirc;&yacute;&ocirc;&ocirc;&egrave;&ouml;&egrave;&aring;&iacute;&ograve;&agrave;&igrave;&egrave; &acirc; R). &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&aring;&iacute;&egrave;&yuml; &auml;&agrave;
&thorn;&ograve;, &divide;&ograve;&icirc; H(D) = {0}, &ograve;&icirc;&atilde;&auml;&agrave; &ecirc;&agrave;&ecirc; H(S) = R. &Ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&egrave; &auml;&agrave;&aring;&ograve; &atilde;&icirc;&igrave;&icirc;&igrave;&icirc;&eth;&ocirc;&egrave;&ccedil;&igrave;&ucirc;
R = H(S) → H(D) → H(S) = R, &ecirc;&icirc;&igrave;&iuml;&icirc;&ccedil;&egrave;&ouml;&egrave;&yuml; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; &ograve;&icirc;&aelig;&auml;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&agrave;, &egrave;&aacute;&icirc; &ograve;&agrave;&ecirc;&icirc;&acirc;&agrave; &ecirc;&icirc;&igrave;&iuml;&icirc;&ccedil;&egrave;
&ouml;&egrave;&yuml; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute; S → D → S (&ntilde;&igrave;. &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&egrave; &egrave;&ccedil; &euml;&aring;&ecirc;&ouml;&egrave;&egrave; 2). &Iacute;&icirc; &ntilde; &auml;&eth;&oacute;&atilde;&icirc;&eacute; &ntilde;&ograve;&icirc;
&eth;&icirc;&iacute;&ucirc;, &yacute;&ograve;&icirc;&ograve; &atilde;&icirc;&igrave;&icirc;&igrave;&icirc;&eth;&ocirc;&egrave;&ccedil;&igrave; &iacute;&oacute;&euml;&aring;&acirc;&icirc;&eacute;, &egrave;&aacute;&icirc; &icirc;&iacute; &iuml;&eth;&icirc;&iuml;&oacute;&ntilde;&ecirc;&agrave;&aring;&ograve;&ntilde;&yuml; &divide;&aring;&eth;&aring;&ccedil; &iacute;&oacute;&euml;&aring;&acirc;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc; H(D).
&Yacute;&ograve;&icirc; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&aring; &egrave; &auml;&icirc;&ecirc;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve; &iacute;&aring;&acirc;&icirc;&ccedil;&igrave;&icirc;&aelig;&iacute;&icirc;&ntilde;&ograve;&uuml; &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&egrave; D &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&oacute;.
&Iacute;&aring;&auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&ecirc; &yacute;&ograve;&icirc;&atilde;&icirc; &iuml;&icirc;&auml;&otilde;&icirc;&auml;&agrave; &acirc; &iacute;&aring;&icirc;&aacute;&otilde;&icirc;&auml;&egrave;&igrave;&icirc;&ntilde;&ograve;&egrave; &eth;&agrave;&ccedil;&acirc;&egrave;&ograve;&egrave;&yuml; &atilde;&icirc;&igrave;&icirc;&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&atilde;&icirc; &ocirc;&icirc;&eth;&igrave;&agrave;&euml;&egrave;&ccedil;&igrave;&agrave;,
30
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 4.
&Ograve;&Aring;&Icirc;&ETH;&Aring;&Igrave;&Agrave; &Aacute;&ETH;&Agrave;&Oacute;&Yacute;&ETH;&Agrave;: &Auml;&Icirc;&Ecirc;&Agrave;&Ccedil;&Agrave;&Ograve;&Aring;&Euml;&Uuml;&Ntilde;&Ograve;&Acirc;&Agrave; &Egrave; &Agrave;&Euml;&Atilde;&Icirc;&ETH;&Egrave;&Ograve;&Igrave;&Ucirc;
&iacute;&aring;&ntilde;&icirc;&igrave;&iacute;&aring;&iacute;&iacute;&icirc;, &iuml;&icirc;&euml;&aring;&ccedil;&iacute;&icirc;&atilde;&icirc; &auml;&euml;&yuml; &igrave;&iacute;&icirc;&atilde;&egrave;&otilde; &auml;&eth;&oacute;&atilde;&egrave;&otilde; &ccedil;&agrave;&auml;&agrave;&divide;, &iacute;&icirc; &yuml;&acirc;&iacute;&icirc; &divide;&eth;&aring;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&atilde;&icirc; &iuml;&eth;&egrave;&igrave;&aring;&iacute;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc; &ecirc;
&ograve;&aring;&icirc;&eth;&aring;&igrave;&aring; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;. &Icirc;&ograve;&igrave;&aring;&ograve;&egrave;&igrave;, &divide;&ograve;&icirc; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&egrave;&aring; &atilde;&eth;&oacute;&iuml;&iuml; &atilde;&icirc;&igrave;&icirc;&euml;&icirc;&atilde;&egrave;&eacute; &egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&oacute;&aring;&ograve; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave;, &iuml;&icirc;
&yuml;&acirc;&euml;&yuml;&thorn;&ugrave;&egrave;&aring;&ntilde;&yuml; &acirc; &euml;&aring;&igrave;&igrave;&aring; &Oslash;&iuml;&aring;&eth;&iacute;&aring;&eth;&agrave;.
&Auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&agrave;&yuml; &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&yuml;. &Ntilde;&iacute;&icirc;&acirc;&agrave; &iuml;&oacute;&ntilde;&ograve;&uuml; f : D → S &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&yuml; &oslash;&agrave;&eth;&agrave; &iacute;&agrave; &aring;&atilde;&icirc; &atilde;&eth;&agrave;
&iacute;&egrave;&ouml;&oacute;. &Auml;&euml;&yuml; &ograve;&icirc;&divide;&ecirc;&egrave; y ∈ S &eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &aring;&aring; &iuml;&eth;&icirc;&icirc;&aacute;&eth;&agrave;&ccedil; f −1 (y). &Egrave;&iacute;&ograve;&oacute;&egrave;&ograve;&egrave;&acirc;&iacute;&icirc; &ntilde;&euml;&aring;&auml;&oacute;&aring;&ograve; &icirc;&aelig;&egrave;&auml;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc;
f −1 (y) &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &icirc;&auml;&iacute;&icirc;&igrave;&aring;&eth;&iacute;&icirc;&eacute; &ecirc;&eth;&egrave;&acirc;&icirc;&eacute;, &ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &iacute;&agrave;&divide;&egrave;&iacute;&agrave;&aring;&ograve;&ntilde;&yuml; &acirc; &ograve;&icirc;&divide;&ecirc;&aring; y . &Iuml;&icirc; &egrave;&auml;&aring;&aring;, &oacute; &iacute;&aring;&aring; &auml;&icirc;&euml;&aelig;&aring;&iacute;
&aacute;&ucirc;&ograve;&uuml; &egrave; &auml;&eth;&oacute;&atilde;&icirc;&eacute; &ecirc;&icirc;&iacute;&aring;&ouml; (&iuml;&icirc;&divide;&aring;&igrave;&oacute;?). &Iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&aring; ∂D &icirc;&iacute; &iacute;&aring; &igrave;&icirc;&aelig;&aring;&ograve; &euml;&aring;&aelig;&agrave;&ograve;&uuml;, &acirc;&ntilde;&aring; &ograve;&icirc;&divide;&ecirc;&egrave; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&ucirc;
&ccedil;&agrave;&iacute;&yuml;&ograve;&ucirc; &auml;&eth;&oacute;&atilde;&egrave;&igrave;&egrave; &ecirc;&eth;&egrave;&acirc;&ucirc;&igrave;&egrave;. &Acirc;&iacute;&oacute;&ograve;&eth;&egrave; &yacute;&ograve;&agrave; &ecirc;&eth;&egrave;&acirc;&agrave;&yuml; &ograve;&icirc;&aelig;&aring; &iacute;&aring; &igrave;&icirc;&aelig;&aring;&ograve; &icirc;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&egrave;&ograve;&uuml;&ntilde;&yuml; (&iuml;&icirc;&divide;&aring;&igrave;&oacute;?). &Acirc;&icirc;&ograve;
&egrave; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&aring;.
&Ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;, &yacute;&ograve;&icirc; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &atilde;&eth;&oacute;&aacute;&agrave;&yuml; &egrave;&auml;&aring;&yuml;, &iacute;&icirc; &icirc;&iacute;&agrave; &icirc;&ecirc;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &iuml;&eth;&agrave;&acirc;&egrave;&euml;&uuml;&iacute;&icirc;&eacute;, &otilde;&icirc;&ograve;&yuml; &aring;&aring; &eth;&aring;&agrave;&euml;&egrave;&ccedil;&agrave;&ouml;&egrave;&yuml;
&ograve;&agrave;&ecirc;&aelig;&aring; &ograve;&eth;&aring;&aacute;&oacute;&aring;&ograve; &ccedil;&iacute;&agrave;&divide;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&eacute; &acirc;&icirc;&ccedil;&iacute;&egrave;. &Acirc;&icirc; &iuml;&aring;&eth;&acirc;&ucirc;&otilde;, &aacute;&aring;&ccedil; &icirc;&ntilde;&icirc;&aacute;&ucirc;&otilde; &ograve;&eth;&oacute;&auml;&iacute;&icirc;&ntilde;&ograve;&aring;&eacute; &igrave;&icirc;&aelig;&iacute;&icirc; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&ograve;&uuml;,
&divide;&ograve;&icirc; &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&yuml; f &yuml;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &atilde;&euml;&agrave;&auml;&ecirc;&egrave;&igrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring;&igrave;. &Acirc;&icirc; &acirc;&ograve;&icirc;&eth;&ucirc;&otilde;, &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; &Ntilde;&agrave;&eth;&auml;&agrave; (&egrave; &yacute;&ograve;&icirc; &ntilde;&agrave;&igrave;&icirc;&aring;
&ograve;&eth;&oacute;&auml;&iacute;&icirc;&aring; &egrave; &ograve;&aring;&otilde;&iacute;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &igrave;&aring;&ntilde;&ograve;&icirc;) &atilde;&agrave;&eth;&agrave;&iacute;&ograve;&egrave;&eth;&oacute;&aring;&ograve; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &ograve;&agrave;&ecirc;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave; y ∈ S , &divide;&ograve;&icirc; &auml;&egrave;&ocirc;&ocirc;&aring;
&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml; df &iacute;&aring;&acirc;&ucirc;&eth;&icirc;&aelig;&auml;&aring;&iacute; &acirc;&icirc; &acirc;&ntilde;&aring;&otilde; &ograve;&icirc;&divide;&ecirc;&agrave;&otilde; &frac34;&ecirc;&eth;&egrave;&acirc;&icirc;&eacute;&iquest; f −1 (y). &Agrave; &ograve;&icirc;&atilde;&auml;&agrave; f −1 (y) &auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;
&frac34;&otilde;&icirc;&eth;&icirc;&oslash;&agrave;&yuml; &ecirc;&eth;&egrave;&acirc;&agrave;&yuml;&iquest; &egrave; &iacute;&aring; &igrave;&icirc;&aelig;&aring;&ograve; &ecirc;&icirc;&iacute;&divide;&agrave;&ograve;&uuml;&ntilde;&yuml; &acirc;&iacute;&oacute;&ograve;&eth;&egrave; D. &Iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&ograve;&ntilde;&yuml;, &divide;&ograve;&icirc; &oacute; &iacute;&aring;&aring; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &icirc;&auml;&egrave;&iacute;
&ecirc;&icirc;&iacute;&aring;&ouml; &ograve;&icirc;&divide;&ecirc;&agrave; y , &divide;&aring;&atilde;&icirc; &ograve;&icirc;&aelig;&aring; &aacute;&ucirc;&ograve;&uuml; &iacute;&aring; &igrave;&icirc;&aelig;&aring;&ograve;.
&Icirc;&ograve;&igrave;&aring;&ograve;&egrave;&igrave;, &divide;&ograve;&icirc; &yacute;&ograve;&icirc; &aelig;&aring; &eth;&agrave;&ntilde;&ntilde;&oacute;&aelig;&auml;&aring;&iacute;&egrave;&aring; &iuml;&icirc;&ecirc;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;, &divide;&ograve;&icirc; &iacute;&aring; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&egrave; &igrave;&iacute;&icirc;&atilde;&icirc;&icirc;&aacute;
&eth;&agrave;&ccedil;&egrave;&yuml; &ntilde; &ecirc;&eth;&agrave;&aring;&igrave; X &iacute;&agrave; &aring;&atilde;&icirc; &ecirc;&eth;&agrave;&eacute; ∂X . &Ecirc;&ntilde;&ograve;&agrave;&ograve;&egrave;, &iuml;&oacute;&ograve;&uuml; &iuml;&icirc; &iuml;&icirc;&euml;&oacute;&iuml;&aring;&ntilde;&ograve;&eth;&ucirc;&igrave; &ntilde;&egrave;&igrave;&iuml;&euml;&aring;&ecirc;&ntilde;&agrave;&igrave; &yacute;&ograve;&icirc; &egrave; &aring;&ntilde;&ograve;&uuml;
&atilde;&eth;&oacute;&aacute;&ucirc;&eacute; &agrave;&iacute;&agrave;&euml;&icirc;&atilde; &auml;&acirc;&egrave;&aelig;&aring;&iacute;&egrave;&yuml; &iuml;&icirc; &ecirc;&eth;&egrave;&acirc;&icirc;&eacute; f −1 (y) &egrave;&ccedil; &ograve;&icirc;&divide;&ecirc;&egrave; y &acirc; &iuml;&icirc;&egrave;&ntilde;&ecirc;&agrave;&otilde; &frac34;&auml;&eth;&oacute;&atilde;&icirc;&atilde;&icirc; &ecirc;&icirc;&iacute;&ouml;&agrave;&iquest;.
&Ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave; &Ntilde;&ograve;&icirc;&ecirc;&ntilde;&agrave;. &Iuml;&eth;&egrave;&acirc;&aring;&auml;&aring;&igrave; &aring;&ugrave;&aring; &icirc;&auml;&iacute;&icirc; &frac34;&iuml;&icirc;&divide;&ograve;&egrave; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&icirc;&aring;&iquest; &egrave; &iacute;&aring;&ntilde;&icirc;&igrave;&iacute;&aring;&iacute;&iacute;&icirc; &ntilde;&agrave;&igrave;&icirc;&aring; &egrave;&ccedil;&yuml;&ugrave;
&iacute;&icirc;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;, &iuml;&eth;&egrave;&auml;&oacute;&igrave;&agrave;&iacute;&iacute;&icirc;&aring; &ntilde;&eth;&agrave;&acirc;&iacute;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc; &iacute;&aring;&auml;&agrave;&acirc;&iacute;&icirc; (&acirc; 1981 &atilde;.) &Ecirc;&agrave;&iacute;&iacute;&agrave;&egrave;. &Ntilde;&iacute;&icirc;&acirc;&agrave; &igrave;&ucirc; &aacute;&oacute;&auml;&aring;&igrave;
&oacute;&ntilde;&ograve;&agrave;&iacute;&agrave;&acirc;&euml;&egrave;&acirc;&agrave;&ograve;&uuml; &iacute;&aring;&ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&icirc;&acirc;&agrave;&iacute;&egrave;&aring; &atilde;&euml;&agrave;&auml;&ecirc;&icirc;&eacute; &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&egrave; f : Dn → S &oslash;&agrave;&eth;&agrave; D &iacute;&agrave; &aring;&atilde;&icirc; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&oacute;. &times;&ograve;&icirc;
&aacute;&ucirc; &euml;&oacute;&divide;&oslash;&aring; &iuml;&icirc;&iacute;&yuml;&ograve;&uuml; &egrave;&auml;&aring;&thorn;, &eth;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &ntilde;&iacute;&agrave;&divide;&agrave;&euml;&agrave; &frac34;&icirc;&divide;&aring;&acirc;&egrave;&auml;&iacute;&ucirc;&eacute;&iquest; &ntilde;&euml;&oacute;&divide;&agrave;&eacute; n = 1.
&Egrave;&ograve;&agrave;&ecirc;, &iuml;&oacute;&ntilde;&ograve;&uuml; f &atilde;&euml;&agrave;&auml;&ecirc;&agrave;&yuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml;, &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&agrave;&thorn;&ugrave;&agrave;&yuml; &icirc;&ograve;&eth;&aring;&ccedil;&icirc;&ecirc; [0, 1] &acirc; &igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; {0, 1},
&iuml;&eth;&egrave;&divide;&aring;&igrave; f (0) = 0 &egrave; f (1) = 1. &Iuml;&icirc; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&aring; &Iacute;&uuml;&thorn;&ograve;&icirc;&iacute;&agrave;&Euml;&aring;&eacute;&aacute;&iacute;&egrave;&ouml;&agrave;
Z
1
f 0 (x)dx = f (1) − f (0) = 1 − 0 = 1.
0
&Ntilde; &auml;&eth;&oacute;&atilde;&icirc;&eacute; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute;&ucirc;, f 0 (x) = 0 &auml;&euml;&yuml; &acirc;&ntilde;&aring;&otilde; x, &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &icirc;&aacute;&eth;&agrave;&ccedil; f &auml;&egrave;&ntilde;&ecirc;&eth;&aring;&ograve;&aring;&iacute;. &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &euml;&aring;&acirc;&agrave;&yuml;
&ntilde;&ograve;&icirc;&eth;&icirc;&iacute;&agrave; &eth;&agrave;&acirc;&iacute;&agrave; 0. &Iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&aring;!
&Ecirc;&agrave;&ecirc; &yacute;&ograve;&icirc; &iuml;&eth;&icirc;&ccedil;&eth;&agrave;&divide;&iacute;&icirc;&aring; &eth;&agrave;&ntilde;&ntilde;&oacute;&aelig;&auml;&aring;&iacute;&egrave;&aring; &iuml;&aring;&eth;&aring;&iacute;&aring;&ntilde;&ograve;&egrave; &iacute;&agrave; &ntilde;&euml;&oacute;&divide;&agrave;&eacute; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &divide;&egrave;&ntilde;&euml;&agrave; n? &ETH;&icirc;&euml;&uuml;
&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc; &Iacute;&uuml;&thorn;&ograve;&icirc;&iacute;&agrave;&Euml;&aring;&eacute;&aacute;&iacute;&egrave;&ouml;&agrave; &aacute;&oacute;&auml;&aring;&ograve; &egrave;&atilde;&eth;&agrave;&ograve;&uuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave; &Ntilde;&ograve;&icirc;&ecirc;&ntilde;&agrave;, &ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&agrave;&aring;&ograve;, &divide;&ograve;&icirc; &auml;&euml;&yuml;
&igrave;&iacute;&icirc;&atilde;&icirc;&icirc;&aacute;&eth;&agrave;&ccedil;&egrave;&yuml; &ntilde; &ecirc;&eth;&agrave;&aring;&igrave; X &egrave; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&ucirc; ω &iacute;&agrave; &iacute;&aring;&igrave;
Z
Z
dω =
ω.
X
∂X
&Iuml;&oacute;&ntilde;&ograve;&uuml; &ograve;&aring;&iuml;&aring;&eth;&uuml; f &atilde;&euml;&agrave;&auml;&ecirc;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &auml;&egrave;&ntilde;&ecirc;&agrave; D &acirc; &ntilde;&aring;&aacute;&yuml;, &icirc;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&thorn;&ugrave;&aring;&aring; &iacute;&agrave; &igrave;&aring;&ntilde;&ograve;&aring; &ograve;&icirc;&divide;&ecirc;&egrave;
&atilde;&eth;&agrave;&iacute;&egrave;&ouml;&ucirc; ∂D (&iuml;&icirc;&ecirc;&agrave; &iacute;&agrave;&igrave; &iacute;&aring;&acirc;&agrave;&aelig;&iacute;&icirc;, &divide;&ograve;&icirc; f &eth;&aring;&ograve;&eth;&agrave;&atilde;&egrave;&eth;&oacute;&aring;&ograve; D &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&oacute;). &Iuml;&oacute;&ntilde;&ograve;&uuml; x1 , . . . , xn &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&ucirc; &acirc; Rn ; &acirc; &yacute;&ograve;&egrave;&otilde; &ecirc;&icirc;&icirc;&eth;&auml;&egrave;&iacute;&agrave;&ograve;&agrave;&otilde; f &ccedil;&agrave;&auml;&agrave;&aring;&ograve;&ntilde;&yuml; n &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&yuml;&igrave;&egrave; f1 (x), . . . , fn (x). &ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave;
&auml;&acirc;&aring; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; (n − 1)-&ocirc;&icirc;&eth;&igrave;&ucirc; &iacute;&agrave; D:
ω = x1 dx2 ∧ &middot; &middot; &middot; ∧ dxn
&egrave; ω̃ = f1 df2 ∧ &middot; &middot; &middot; ∧ dfn .
&Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; fi = xi &iacute;&agrave; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&aring; ∂D, &ograve;&icirc; &icirc;&atilde;&eth;&agrave;&iacute;&egrave;&divide;&aring;&iacute;&egrave;&yuml; ω &egrave; ω̃ &iacute;&agrave; ∂D &ntilde;&icirc;&acirc;&iuml;&agrave;&auml;&agrave;&thorn;&ograve;. &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute; &egrave;&ccedil; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc;
&Ntilde;&ograve;&icirc;&ecirc;&ntilde;&agrave; &igrave;&ucirc; &egrave;&igrave;&aring;&aring;&igrave;:
Z
Z
Z
Z
dω =
ω=
ω̃ =
dω̃.
D
∂D
∂D
D
&Ntilde;&euml;&aring;&acirc;&agrave; &ntilde;&ograve;&icirc;&egrave;&ograve; &egrave;&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml; &ocirc;&icirc;&eth;&igrave;&ucirc; &icirc;&aacute;&uacute;&aring;&igrave;&agrave; dx1 ∧ dx2 ∧ &middot; &middot; &middot; ∧ dxn , &ograve;. &aring;. &icirc;&aacute;&uacute;&aring;&igrave; &oslash;&agrave;&eth;&agrave; Dn &divide;&egrave;&ntilde;&euml;&icirc;
&yuml;&acirc;&iacute;&icirc; &iacute;&aring;&iacute;&oacute;&euml;&aring;&acirc;&icirc;&aring;. &Ntilde;&iuml;&eth;&agrave;&acirc;&agrave; &ntilde;&ograve;&icirc;&egrave;&ograve; &egrave;&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml; &ocirc;&icirc;&eth;&igrave;&ucirc; dω̃ = df1 ∧ df2 ∧ &middot; &middot; &middot; ∧ dfn . &Aring;&ntilde;&euml;&egrave; &ograve;&aring;&iuml;&aring;&eth;&uuml; &igrave;&ucirc;
31
&iuml;&icirc;&auml;&ecirc;&euml;&thorn;&divide;&egrave;&igrave; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&aring; &icirc; &ograve;&icirc;&igrave;, &divide;&ograve;&icirc; f &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&yuml; &iacute;&agrave; &ecirc;&eth;&agrave;&eacute;, &ograve;&icirc; &acirc; &ntilde;&egrave;&euml;&oacute; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; &icirc;&aacute;&eth;&agrave;&ccedil;
f &egrave;&igrave;&aring;&aring;&ograve; &eth;&agrave;&ccedil;&igrave;&aring;&eth;&iacute;&icirc;&ntilde;&ograve;&uuml; &igrave;&aring;&iacute;&uuml;&oslash;&aring; n, &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&ucirc; df1 , . . . , dfn &euml;&egrave;&iacute;&aring;&eacute;&iacute;&icirc; &ccedil;&agrave;&acirc;&egrave;&ntilde;&egrave;&igrave;&ucirc;. &Iuml;&icirc;&yacute;&ograve;&icirc;&igrave;&oacute;
dω̃ ≡ 0 &egrave; &iuml;&eth;&agrave;&acirc;&ucirc;&eacute; &egrave;&iacute;&ograve;&aring;&atilde;&eth;&agrave;&euml; &eth;&agrave;&acirc;&aring;&iacute; 0. &Iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&aring;!
&Yacute;&ograve;&icirc; &aelig;&aring; &eth;&agrave;&ntilde;&ntilde;&oacute;&aelig;&auml;&aring;&iacute;&egrave;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve; &icirc;&ograve;&ntilde;&oacute;&ograve;&ntilde;&ograve;&acirc;&egrave;&aring; &atilde;&euml;&agrave;&auml;&ecirc;&icirc;&eacute; &eth;&aring;&ograve;&eth;&agrave;&ecirc;&ouml;&egrave;&egrave; &icirc;&atilde;&eth;&agrave;&iacute;&egrave;&divide;&aring;&iacute;&iacute;&icirc;&eacute; &icirc;&aacute;&euml;&agrave;&ntilde;&ograve;&egrave;
B ⊂ Rn &iacute;&agrave; &aring;&aring; &atilde;&eth;&agrave;&iacute;&egrave;&ouml;&oacute; ∂B .
&Egrave;&ograve;&agrave;&ecirc;, &igrave;&ucirc; &icirc;&ccedil;&iacute;&agrave;&ecirc;&icirc;&igrave;&egrave;&euml;&egrave;&ntilde;&uuml; &iuml;&icirc;&divide;&ograve;&egrave; &ntilde;&icirc; &acirc;&ntilde;&aring;&igrave;&egrave; &egrave;&ccedil;&acirc;&aring;&ntilde;&ograve;&iacute;&ucirc;&igrave;&egrave; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave;&igrave;&egrave; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;
&egrave; &acirc;&egrave;&auml;&egrave;&igrave;, &divide;&ograve;&icirc; &ntilde;&eth;&aring;&auml;&egrave; &iacute;&egrave;&otilde; &iacute;&aring;&ograve; &acirc;&iuml;&icirc;&euml;&iacute;&aring; &oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&eth;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&atilde;&icirc;. &Ecirc;&icirc;&igrave;&aacute;&egrave;&iacute;&agrave;&ograve;&icirc;&eth;&iacute;&icirc;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;
&yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&icirc; (&aring;&ntilde;&euml;&egrave; &icirc;&ntilde;&ograve;&agrave;&acirc;&egrave;&ograve;&uuml; &acirc; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute;&aring; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&egrave;), &icirc;&auml;&iacute;&agrave;&ecirc;&icirc; &yuml;&acirc;&iacute;&icirc; &ecirc;&agrave;&ecirc;&icirc;&aring;-&ograve;&icirc; &frac34;&oacute;&atilde;&euml;&icirc;&acirc;&agrave;&ograve;&icirc;&aring;&iquest; &egrave;
&iuml;&eth;&egrave;&acirc;&euml;&aring;&ecirc;&agrave;&aring;&ograve; &acirc;&ntilde;&iuml;&icirc;&igrave;&icirc;&atilde;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&oacute;&thorn; &iuml;&icirc;&ntilde;&ograve;&icirc;&eth;&icirc;&iacute;&iacute;&thorn;&thorn; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&oacute; &ograve;&eth;&egrave;&agrave;&iacute;&atilde;&oacute;&euml;&yuml;&ouml;&egrave;&thorn;. &Icirc;&ntilde;&ograve;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &ograve;&eth;&aring;&aacute;&oacute;&thorn;&ograve;
&egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &ograve;&icirc;&atilde;&icirc; &egrave;&euml;&egrave; &egrave;&iacute;&icirc;&atilde;&icirc; &auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc; &eth;&agrave;&ccedil;&acirc;&egrave;&ograve;&icirc;&atilde;&icirc; &ocirc;&icirc;&eth;&igrave;&agrave;&euml;&egrave;&ccedil;&igrave;&agrave;.
&Oacute;&iuml;&eth;&agrave;&aelig;&iacute;&aring;&iacute;&egrave;&yuml;
4.1. &Auml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&ntilde;&ograve;&uuml; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&eacute; k , g &egrave; g −1 &egrave;&ccedil; &iacute;&agrave;&divide;&agrave;&euml;&agrave; &euml;&aring;&ecirc;&ouml;&egrave;&egrave;.
4.2.* &Acirc;&ucirc;&acirc;&aring;&ntilde;&ograve;&egrave; &euml;&aring;&igrave;&igrave;&oacute; &Ecirc;&Ecirc;&Igrave; &egrave;&ccedil; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;. (&Oacute;&ecirc;&agrave;&ccedil;&agrave;&iacute;&egrave;&aring;. &Iuml;&oacute;&ntilde;&ograve;&uuml; Us = ∆\Fs &auml;&icirc;&iuml;&icirc;&euml;&iacute;&aring;&iacute;&egrave;&yuml;
&ecirc; Fs . &Aring;&ntilde;&euml;&egrave; &iuml;&aring;&eth;&aring;&ntilde;&aring;&divide;&aring;&iacute;&egrave;&aring; Fs &iuml;&oacute;&ntilde;&ograve;&icirc;, &ograve;&icirc; Us &icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&thorn;&ograve; &icirc;&ograve;&ecirc;&eth;&ucirc;&ograve;&icirc;&aring; &iuml;&icirc;&ecirc;&eth;&ucirc;&ograve;&egrave;&aring; ∆S . &Iuml;&oacute;&ntilde;&ograve;&uuml; (us , s ∈ S ) &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&aring; &eth;&agrave;&ccedil;&aacute;&egrave;&aring;&iacute;&egrave;&aring; &aring;&auml;&egrave;&iacute;&egrave;&ouml;&ucirc;, &iuml;&icirc;&auml;&divide;&egrave;&iacute;&aring;&iacute;&iacute;&icirc;&aring; &icirc;&ograve;&ecirc;&eth;&ucirc;&ograve;&icirc;&igrave;&oacute; &iuml;&icirc;&ecirc;&eth;&ucirc;&ograve;&egrave;&thorn; (Us ). &Yacute;&ograve;&icirc; &ntilde;&aring;&igrave;&aring;&eacute;&ntilde;&ograve;&acirc;&icirc;
&ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute; (us ) &ccedil;&agrave;&auml;&agrave;&aring;&ograve; &iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; u : ∆S → ∆S &iuml;&icirc; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&aring;: u(x) = (us (x)).
&Iuml;&eth;&egrave;&igrave;&aring;&iacute;&egrave;&ograve;&uuml; &ograve;&aring;&icirc;&eth;&aring;&igrave;&oacute; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave; &ecirc; u.)
4.3. &Auml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring; &ntilde; &iuml;&icirc;&igrave;&icirc;&ugrave;&uuml;&thorn; &euml;&aring;&igrave;&igrave;&ucirc; &Ecirc;&Ecirc;&Igrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&eacute; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve; (&Ecirc;&egrave; &Ocirc;&agrave;&iacute;). &Iuml;&oacute;&ntilde;&ograve;&uuml; V &ograve;&icirc;&iuml;&icirc;
&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &acirc;&aring;&ecirc;&ograve;&icirc;&eth;&iacute;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc;, S ⊂ V . &Iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave;, &divide;&ograve;&icirc; &auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&atilde;&icirc; s ∈ S &ccedil;&agrave;&auml;&agrave;&iacute;&icirc;
&ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&iacute;&icirc;&aring; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; F (s) &acirc; V , &oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&eth;&yuml;&thorn;&ugrave;&aring;&aring; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&igrave;&oacute; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&thorn;: &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc;
&ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&atilde;&icirc; &iacute;&agrave;&aacute;&icirc;&eth;&agrave; &ograve;&icirc;&divide;&aring;&ecirc; s1 , . . . , sn ∈ S &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&agrave;&yuml; &icirc;&aacute;&icirc;&euml;&icirc;&divide;&ecirc;&agrave; conv(s1 , . . . , sn ) &euml;&aring;&aelig;&egrave;&ograve; &acirc; &icirc;&aacute;&uacute;&aring;&auml;&egrave;&iacute;&aring;
&iacute;&egrave;&egrave; F (s1 ), . . . , F (sn ). &Iuml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ograve;&icirc;&divide;&ecirc;&agrave; x, &icirc;&aacute;&ugrave;&agrave;&yuml; &acirc;&ntilde;&aring;&igrave; F (s)T, s ∈ S . (&Oacute;&ecirc;&agrave;&ccedil;&agrave;&iacute;&egrave;&aring;:
&iuml;&eth;&icirc;&acirc;&aring;&eth;&egrave;&ograve;&uuml;, &divide;&ograve;&icirc; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&atilde;&icirc; &iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&agrave; T ⊂ S &iuml;&aring;&eth;&aring;&ntilde;&aring;&divide;&aring;&iacute;&egrave;&aring; t∈T F (t) &iacute;&aring;&iuml;&oacute;&ntilde;&ograve;&icirc;.)
4.4. &Iuml;&oacute;&ntilde;&ograve;&uuml; Y &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &iuml;&eth;&icirc;&ntilde;&ograve;&eth;&agrave;&iacute;&ntilde;&ograve;&acirc;&icirc;, &iuml;&icirc;&otilde;&icirc;&aelig;&aring;&aring; &iacute;&agrave; &aacute;&oacute;&ecirc;&acirc;&oacute; &frac34;Y&iquest;. &Iuml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc; &euml;&thorn;&aacute;&icirc;&aring;
&iacute;&aring;&iuml;&eth;&aring;&eth;&ucirc;&acirc;&iacute;&icirc;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; Y &acirc; &ntilde;&aring;&aacute;&yuml; &egrave;&igrave;&aring;&aring;&ograve; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&oacute;&thorn; &ograve;&icirc;&divide;&ecirc;&oacute;.
4.5.* &Iuml;&icirc;&auml;&igrave;&iacute;&icirc;&aelig;&aring;&ntilde;&ograve;&acirc;&icirc; X &acirc; Rn &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &ccedil;&acirc;&aring;&ccedil;&auml;&iacute;&ucirc;&igrave;, &aring;&ntilde;&euml;&egrave; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &ograve;&agrave;&ecirc;&agrave;&yuml; &ograve;&icirc;&divide;&ecirc;&agrave; x∗ ∈ X ,
&divide;&ograve;&icirc; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&atilde;&icirc; x ∈ X &icirc;&ograve;&eth;&aring;&ccedil;&icirc;&ecirc; [x, x∗ ] &ouml;&aring;&euml;&egrave;&ecirc;&icirc;&igrave; &euml;&aring;&aelig;&egrave;&ograve; &acirc; X . &Auml;&euml;&yuml; &ccedil;&acirc;&aring;&ccedil;&auml;&iacute;&ucirc;&otilde; &ecirc;&icirc;&igrave;&iuml;&agrave;&ecirc;&ograve;&icirc;&acirc; &acirc;&aring;&eth;&aring;&iacute;
&agrave;&iacute;&agrave;&euml;&icirc;&atilde; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &Aacute;&eth;&agrave;&oacute;&yacute;&eth;&agrave;. &Ecirc;&agrave;&ecirc;&oacute;&thorn; &egrave;&auml;&aring;&thorn; &acirc;&ucirc; &igrave;&icirc;&atilde;&euml;&egrave; &aacute;&ucirc; &iuml;&eth;&aring;&auml;&euml;&icirc;&aelig;&egrave;&ograve;&uuml; &auml;&euml;&yuml; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave;?
&ETH;&aring;&ecirc;&icirc;&igrave;&aring;&iacute;&auml;&oacute;&aring;&igrave;&agrave;&yuml; &euml;&egrave;&ograve;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave;: [5, 6, 8, 12].
32
&Euml;&Aring;&Ecirc;&Ouml;&Egrave;&szlig; 4.
&Ograve;&Aring;&Icirc;&ETH;&Aring;&Igrave;&Agrave; &Aacute;&ETH;&Agrave;&Oacute;&Yacute;&ETH;&Agrave;: &Auml;&Icirc;&Ecirc;&Agrave;&Ccedil;&Agrave;&Ograve;&Aring;&Euml;&Uuml;&Ntilde;&Ograve;&Acirc;&Agrave; &Egrave; &Agrave;&Euml;&Atilde;&Icirc;&ETH;&Egrave;&Ograve;&Igrave;&Ucirc;
&Euml;&egrave;&ograve;&aring;&eth;&agrave;&ograve;&oacute;&eth;&agrave;
[1] &Agrave;&eth;&iacute;&icirc;&euml;&uuml;&auml; &Acirc;.&Egrave;., &Icirc;&aacute;&ucirc;&ecirc;&iacute;&icirc;&acirc;&aring;&iacute;&iacute;&ucirc;&aring; &auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &oacute;&eth;&agrave;&acirc;&iacute;&aring;&iacute;&egrave;&yuml;. &Igrave;., 1974
[2] &Ecirc;&egrave;&eth;&oacute;&ograve;&agrave; &Agrave;.&szlig;., &ETH;&oacute;&aacute;&egrave;&iacute;&icirc;&acirc; &Agrave;.&Igrave;., &szlig;&iacute;&icirc;&acirc;&ntilde;&ecirc;&agrave;&yuml; &Aring;.&Aacute;., &Icirc;&iuml;&ograve;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&eacute; &acirc;&ucirc;&aacute;&icirc;&eth; &eth;&agrave;&ntilde;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&eacute; &acirc; &ntilde;&euml;&icirc;&aelig;
&iacute;&ucirc;&otilde; &ntilde;&icirc;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&icirc;-&yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&otilde; &ccedil;&agrave;&auml;&agrave;&divide;&agrave;&otilde;, &Euml;., 1980
[3] &Euml;&thorn;&ntilde;&ograve;&aring;&eth;&iacute;&egrave;&ecirc; &Euml;.&Agrave;., &Ntilde;&icirc;&aacute;&icirc;&euml;&aring;&acirc; &Acirc;.&Egrave;. &Yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&ucirc; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&icirc;&iacute;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &agrave;&iacute;&agrave;&euml;&egrave;&ccedil;&agrave;. &Igrave;. 1965
[4] &Igrave;&agrave;&ecirc;&agrave;&eth;&icirc;&acirc; &Acirc;.&Euml;., &ETH;&oacute;&aacute;&egrave;&iacute;&icirc;&acirc; &Agrave;.&Igrave;., &Igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml; &ograve;&aring;&icirc;&eth;&egrave;&yuml; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &auml;&egrave;&iacute;&agrave;&igrave;&egrave;&ecirc;&egrave; &egrave; &eth;&agrave;&acirc;
&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&yuml;, &Igrave;., 1973
[5] &Igrave;&egrave;&euml;&iacute;&icirc;&eth; &Auml;&aelig;., &Oacute;&icirc;&euml;&euml;&aring;&ntilde; &Agrave;., &Auml;&egrave;&ocirc;&ocirc;&aring;&eth;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&agrave;&yuml; &ograve;&icirc;&iuml;&icirc;&euml;&icirc;&atilde;&egrave;&yuml;, &Igrave;., 1972
[6] &Iacute;&egrave;&ecirc;&agrave;&eacute;&auml;&icirc; &Otilde;. &Acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&ucirc;&aring; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&ucirc; &egrave; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&agrave;&yuml; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&ecirc;&agrave;. &Igrave;. 1972
[7] &Iuml;&icirc;&euml;&ograve;&aring;&eth;&icirc;&acirc;&egrave;&divide; &Acirc;.&Igrave;., &Yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&ntilde;&egrave;&aring; &egrave; &otilde;&icirc;&ccedil;&yuml;&eacute;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&ucirc;&eacute; &igrave;&aring;&otilde;&agrave;&iacute;&egrave;&ccedil;&igrave;, &Igrave;., 1990
[8] &Ograve;&icirc;&auml;&auml; &Igrave;.&Auml;&aelig;. &Acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&aring;&iacute;&egrave;&aring; &iacute;&aring;&iuml;&icirc;&auml;&acirc;&egrave;&aelig;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; &egrave; &iuml;&eth;&egrave;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&yuml; &ecirc; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&ecirc;&aring;. &Igrave;. 1983
[9] &Oslash;&egrave;&euml;&icirc;&acirc; &Atilde;.&Aring;., &Igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&eacute; &agrave;&iacute;&agrave;&euml;&egrave;&ccedil;. &Ntilde;&iuml;&aring;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&eacute; &ecirc;&oacute;&eth;&ntilde;. &Igrave;., 1961
[10] &Yacute;&ecirc;&euml;&agrave;&iacute;&auml; &Egrave;., &Yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&ucirc; &igrave;&agrave;&ograve;&aring;&igrave;&agrave;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&ecirc;&egrave;, &Igrave;., 1983
[11] Border K.C., Fixed point theorems with applications to economics and game theory. Cambridge, 1985
[12] Kannai Y., An elementary proof of the no-retraction theorem. Amer. Math. Monthly, 88
(1981) 264-268
&Ntilde;&acirc;&aring;&auml;&aring;&iacute;&egrave;&yuml; &icirc;&aacute; &agrave;&acirc;&ograve;&icirc;&eth;&aring;: &Auml;&agrave;&iacute;&egrave;&euml;&icirc;&acirc; &Acirc;&euml;&agrave;&auml;&egrave;&igrave;&egrave;&eth; &Egrave;&acirc;&agrave;&iacute;&icirc;&acirc;&egrave;&divide;, &auml;. &ocirc;.-&igrave;. &iacute;., &atilde;&euml;&agrave;&acirc;&iacute;&ucirc;&eacute; &iacute;&agrave;&oacute;&divide;&iacute;&ucirc;&eacute; &ntilde;&icirc;&ograve;&eth;&oacute;&auml;
&iacute;&egrave;&ecirc; &Ouml;&Yacute;&Igrave;&Egrave; &ETH;&Agrave;&Iacute;, &iuml;&eth;&icirc;&ocirc;&aring;&ntilde;&ntilde;&icirc;&eth; &ETH;&icirc;&ntilde;&ntilde;&egrave;&eacute;&ntilde;&ecirc;&icirc;&eacute; &yacute;&ecirc;&icirc;&iacute;&icirc;&igrave;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&eacute; &oslash;&ecirc;&icirc;&euml;&ucirc;.
33
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