===================================================================
RCS file: /home/cvs/OpenXM/src/asir-contrib/packages/doc/yang/yang_tutorial-ja.tex,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -p -r1.1 -r1.2
--- OpenXM/src/asir-contrib/packages/doc/yang/yang_tutorial-ja.tex	2005/11/12 01:23:06	1.1
+++ OpenXM/src/asir-contrib/packages/doc/yang/yang_tutorial-ja.tex	2006/03/20 14:14:04	1.2
@@ -1,29 +1,36 @@
 %#!platex
-% $OpenXM$
+% $OpenXM: OpenXM/src/asir-contrib/packages/doc/yang/yang_tutorial-ja.tex,v 1.1 2005/11/12 01:23:06 ohara Exp $
 \documentclass{jarticle}
 %\usepackage{amsmath}
 \title{Yang Tutorial}
-\author{Katsuyoshi OHARA}
+\author{¶âÂôÂç³ØÍý³ØÉô\ \ \ ¾®¸¶¸ùǤ}
+\date{}
+\topmargin -1.5cm
+\textheight 23.5cm
+\oddsidemargin  0cm
+\evensidemargin 0cm
+\textwidth  16.5cm
+
 \begin{document}
 
 \maketitle
 
 \section{yang ¤È¤Ï}
 
-yang ¤Ç¤Ï¥ª¥¤¥é¡¼Èùʬ±é»»»Ò, shift operator, q-shift operator ¤«¤é¤Ê¤ë´Ä
+yang ¤Ç¤Ï¥ª¥¤¥é¡¼Èùʬ±é»»»Ò, shift operator, $q$-shift operator ¤«¤é¤Ê¤ë´Ä
 ¤Ç¤Î·×»»¤ò¹Ô¤¦ Risa/Asir ¤Î¥Ñ¥Ã¥±¡¼¥¸¤Ç¤¹.  ·×»»¤¹¤ëÁ°¤Ë
-\verb|yang.define_ring| ¤¢¤ë¤¤¤Ï¤½¤ÎÊѼï¤òÍѤ¤¤Æ, ɬ¤º´Ä¤òÄêµÁ¤·¤Þ¤¹. 
-Ʊ»þ¤Ë°·¤¨¤ë´Ä¤Ï¤Ò¤È¤Ä¤À¤±¤Ç¤¹¤¬, \verb|yang.define_ring| ¤ò¸Æ¤Ó½Ð¤¹¤È, 
-°ÊÁ°¤Î´Ä¤ÎÄêµÁ¤Ï¥¹¥¿¥Ã¥¯¤Ë¥×¥Ã¥·¥å¤µ¤ì¤ë¤¿¤á, \verb|yang.define_ring| ¤È 
+\verb|yang.define_ring| ¤¢¤ë¤¤¤Ï¤½¤ÎÊѼï¤òÍѤ¤¤Æ, ɬ¤º´Ä¤òÄêµÁ¤·¤Þ¤¹.
+Ʊ»þ¤Ë°·¤¨¤ë´Ä¤Ï¤Ò¤È¤Ä¤À¤±¤Ç¤¹¤¬, \verb|yang.define_ring| ¤ò¸Æ¤Ó½Ð¤¹¤È,
+°ÊÁ°¤Î´Ä¤ÎÄêµÁ¤Ï¥¹¥¿¥Ã¥¯¤Ë¥×¥Ã¥·¥å¤µ¤ì¤ë¤¿¤á, \verb|yang.define_ring| ¤È
 \verb|yang.pop_ring| ¤Ç¶´¤à¤³¤È¤Ç, ¥µ¥Ö¥ë¡¼¥Á¥óŪ¤Ê·×»»¤ò¼Â¸½¤¹¤ë¤³¤È¤¬
 ¤Ç¤­¤Þ¤¹.
 
-yang ¤Ç¤Ç¤­¤ë·×»»¤Ï,¥°¥ì¥Ö¥Ê´ðÄì, Àµµ¬·Á, 0¼¡¸µ¥¤¥Ç¥¢¥ë¤Î¥é¥ó¥¯, Pfaff 
+yang ¤Ç¤Ç¤­¤ë·×»»¤Ï,¥°¥ì¥Ö¥Ê´ðÄì, Àµµ¬·Á, 0¼¡¸µ¥¤¥Ç¥¢¥ë¤Î¥é¥ó¥¯, Pfaff
 ·Á¼°¤Ê¤É¤Ç¤¹.  ¤Þ¤¿¥°¥ì¥Ö¥Ê´ðÄì¤ÏÍ­Íý´Ø¿ôÂη¸¿ô¤Ç·×»»¤·¤Þ¤¹.
 
 \section{Appell's $F_1$ ¤ò·×»»¤·¤Æ¤ß¤ë.}
 
-¤³¤³¤Ç¤Ï, ¥ª¥¤¥é¡¼Èùʬ±é»»»Ò¤«¤é¤Ê¤ë´Ä¤òÄêµÁ¤·, Ķ´ö²¿ÊýÄø¼°·Ï $F_1$ ¤Î 
+¤³¤³¤Ç¤Ï, ¥ª¥¤¥é¡¼Èùʬ±é»»»Ò¤«¤é¤Ê¤ë´Ä¤òÄêµÁ¤·, Ķ´ö²¿ÊýÄø¼°·Ï $F_1$ ¤Î
 ¥°¥ì¥Ö¥Ê´ðÄì¤ò·×»»¤·¤Æ¤ß¤Þ¤¹.  ¼Â¤Ï¥ª¥¤¥é¡¼Èùʬ±é»»»Ò¤Î¤ß¤ò´Þ¤à¾ì¹ç¤Ë¤Ï,
 \verb|yang_D.rr| ¤ò»È¤Ã¤¿¤Û¤¦¤¬¹â®¤Ë¤Ê¤ê¤Þ¤¹.
 
@@ -31,85 +38,93 @@ yang ¤Ç¤Ç¤­¤ë·×»»¤Ï,¥°¥ì¥Ö¥Ê´ðÄì, Àµµ¬·Á, 0¼¡¸µ¥¤¥Ç¥¢¥
 ohara:~> asir
 [1] load("yang.rr");
 [2] yang.define_ring([x1,x2]);
+{[euler,[x1,x2]],[x1,x2],[0,0],[0,0],[dx1,dx2]}
 \end{verbatim}
 
 ´Ä¤È¤·¤Æ, $R=K(x_1,x_2)\langle \theta_1, \theta_2\rangle$ ¤¬ÄêµÁ
-¤µ¤ì¤Þ¤·¤¿($\theta_i = x_i\partial_i$).  ´Ä¤Î¸µ¤òÄêµÁ¤·¤Þ¤·¤ç¤¦.
+¤µ¤ì¤Þ¤·¤¿($\theta_i = x_i\partial_i$).
+\verb|yang.define_ring| ¤Î½ÐÎϤ˼¨¤µ¤ì¤Æ¤¤¤ë¤è¤¦¤Ë,
+\verb|x1| ¤ËÂбþ¤¹¤ë¥ª¥¤¥é¡¼Èùʬ±é»»»Ò $\theta_1$ ¤Ï \verb|dx1| ¤Çɽ¤µ¤ì¤Þ¤¹.
+´Ä¤Î¸µ¤òÄêµÁ¤·¤Þ¤·¤ç¤¦.
 
 \begin{verbatim}
-[3] S1=yang.operator(x1);
-<<1,0>>
-[4] S2=yang.operator(x2);
-<<0,1>>
+[3] S=dx1+dx2;
 \end{verbatim}
 
-$S_1 = \theta_1, S_2 = \theta_2$ ¤È¤Ê¤ê¤Þ¤¹.  
+$S = \theta_1 + \theta_2$ ¤Ç¤¹.
+$R$ ¤Ë¤ª¤±¤ëϤÏ, Ä̾ï¤Î $+$ ¤Ç½ñ¤¯¤³¤È¤¬¤Ç¤­¤Þ¤¹.
+¤Þ¤¿, $K(x_1,x_2)$ ¤Î¸µ¤Î³Ý¤±»»¤Ï, Ä̾ï¤Î $*$ ¤Ç½ñ¤¯¤³¤È¤¬¤Ç¤­¤Þ¤¹.
 
-±é»»»Ò¤ÎÆâÉôɽ¸½¤Ïʬ»¶Â¿¹à¼°¤Ë¤Ê¤Ã¤Æ¤¤¤Þ¤¹¤Î¤Ç, ¤½¤Î¤Þ¤Þ¤Ç¤Ï
-$R'=K[x_1,x_2]\langle \theta_1, \theta_2\rangle$ ¤Î¸µ¤·¤«É½¤»¤º, $R$ ¤Î
-¸µ¤Ïɽ¸½¤Ç¤­¤Þ¤»¤ó.  ¤·¤¿¤¬¤Ã¤Æ, $f \in R'$ ¤È $q \in K[x_1,x_2]$ ¤ÎÂÐ
-\verb|[F,Q]| ¤Ç $R$ ¤Î¸µ $(1/q)f$ ¤òɽ¤·¤Þ¤¹.
-
-¤µ¤é¤Ë, $R$ ¤Ë¤ª¤±¤ë¹à½ç½ø¤Ï asir ¤Îʬ»¶Â¿¹à¼°¤Î¹à½ç½ø¤Ë°ìÃפ·¤Þ¤¹.  ´û
-ÄêÃͤÏ, Á´¼¡¿ôµÕ¼­½ñ¼°½ç½ø¤Ç¤¹.  ½ç½ø¤òÊѹ¹¤·¤Æ Groeber ´ðÄì¤ò·×»»¤·¤¿¤¤
-¾ì¹ç¤Ë¤Ï, ´Ä¤Î¸µ¤òÄêµÁ¤¹¤ëÁ°¤Ë dp\_ord ¤Ç½ç½ø¤òÊѹ¹¤·¤Æ¤ª¤¯¤Ù¤­¤Ç¤¹.
-
 \begin{verbatim}
-[5] S=S1+S2;
-<<0,1>>+<<1,0>>
+[4] L1 = yang.mul(dx1,S+c-1) - x1*yang.mul(dx1+b1,S+a);
+((-x1+1)*dx1-b1*x1)*dx2+(-x1+1)*dx1^2+((-a-b1)*x1+c-1)*dx1-b1*a*x1
+[5] L2 = yang.mul(dx2,S+c-1) - x2*yang.mul(dx2+b2,S+a);
+(-x2+1)*dx2^2+((-x2+1)*dx1+(-a-b2)*x2+c-1)*dx2-b2*x2*dx1-b2*a*x2
 \end{verbatim}
 
-$S = S_1 + S_2$ ¤Ç¤¹.  $R'$ ¤Ë¤ª¤±¤ëϤÏ, Ä̾ï¤Î $+$ ¤Ç½ñ¤¯¤³¤È¤¬¤Ç¤­¤Þ¤¹.
-
-\begin{verbatim}
-[6] L1 = yang.multi(S1,S+c-1) - x1*yang.multi(S1+b1,S+a);
-[7] L2 = yang.multi(S2,S+c-1) - x2*yang.multi(S2+b2,S+a);
-\end{verbatim}
-
-$L_1 = S_1 (S + c-1) - x_1 (S_1 + b_1)(S+a)$ ¤Ç¤¹.  $R'$ ¤Ë¤ª¤±¤ë±é»»»Ò
-¤ÎÀÑ¤Ï \verb|yang.multi| ¤Ç·×»»¤·¤Þ¤¹.  ¤¿¤À¤·, $K[x_1,x_2]$ ¤Î¸µ¤Ï¤½¤Î¤Þ
-¤Þ¤«¤±¤Æ¤â¹½¤¤¤Þ¤»¤ó.  ¤¤¤Þ¤Î¤È¤³¤í, \verb|yang.multi| ¤Î°ú¿ô¤Ë»È¤¨¤ë¤Î
+$L_1 = \theta_1 (S + c-1) - x_1 (\theta_1 + b_1)(S+a)$ ¤Ç¤¹.
+$R'=K[x_1,x_2]\langle \theta_1, \theta_2\rangle$ ¤Ë¤ª¤±¤ë±é»»»Ò
+¤ÎÀÑ¤Ï \verb|yang.mul| ¤Ç·×»»¤·¤Þ¤¹.  ¤¿¤À¤·, $K[x_1,x_2]$ ¤Î¸µ¤Ï¤½¤Î¤Þ
+¤Þ¤«¤±¤Æ¤â¹½¤¤¤Þ¤»¤ó.  ¤¤¤Þ¤Î¤È¤³¤í, \verb|yang.mul| ¤Î°ú¿ô¤Ë»È¤¨¤ë¤Î
 ¤Ï$R'$ ¤Î¸µ¤Î¤ß¤Ç¤¹.
 
 \begin{verbatim}
-[8] G = yang.buchberger([L1,L2]);
-[[(-x2^2+(x1+1)*x2-x1)*<<0,2>>+((b2*x1-b2)*x2)*<<1,0>>
-+((-a-b2)*x2^2+((a-b1+b2)*x1+c-1)*x2+(-c+b1+1)*x1)*<<0,1>>
-+(-b2*a*x2^2+b2*a*x1*x2)*<<0,0>>,-x2^2+(x1+1)*x2-x1],
-[(-x2+x1)*<<1,1>>+(-b2*x2)*<<1,0>>+(b1*x1)*<<0,1>>,-x2+x1],
-[((-x1+1)*x2+x1^2-x1)*<<2,0>>+
-(((-a-b1+b2)*x1+c-b2-1)*x2+(a+b1)*x1^2+(-c+1)*x1)*<<1,0>>
-+(-b1*x1*x2+b1*x1)*<<0,1>>
-+(-b1*a*x1*x2+b1*a*x1^2)*<<0,0>>,(-x1+1)*x2+x1^2-x1]]
+[6] G = yang.gr([L1,L2]);
+[((-x2^2+(x1+1)*x2-x1)*dx2^2+((-a-b2)*x2^2+((a-b1+b2)*x1+c-1)*x2+(-c+b1+1)*x1)*dx2
++(b2*x1-b2)*x2*dx1-b2*a*x2^2+b2*a*x1*x2)/(-x2^2+(x1+1)*x2-x1),
+(((-x2+x1)*dx1+b1*x1)*dx2-b2*x2*dx1)/(-x2+x1),
+((-b1*x1*x2+b1*x1)*dx2+((-x1+1)*x2+x1^2-x1)*dx1^2+(((-a-b1+b2)*x1+c-b2-1)*x2
++(a+b1)*x1^2+(-c+1)*x1)*dx1-b1*a*x1*x2+b1*a*x1^2)/((-x1+1)*x2+x1^2-x1)]
 \end{verbatim}
 
-$R$ ¤Î¥¤¥Ç¥¢¥ë $I=\{ L_1, L_2 \}$ ¤Î¥°¥ì¥Ö¥Ê´ðÄì $G$ ¤ò·×»»¤·¤Þ¤¹.
-$I$ ¤Ï \verb|[[L1,1],[L2,1]]| ¤Î¤è¤¦¤Ëɽ¤¹¤³¤È¤â¤Ç¤­¤Þ¤¹.  ·×»»·ë²Ì¤Ï
+$R$ ¤Î¥¤¥Ç¥¢¥ë $I=\langle L_1, L_2 \rangle$ ¤Î¥°¥ì¥Ö¥Ê´ðÄì $G$ ¤ò·×»»¤·¤Þ¤¹.
+·×»»·ë²Ì¤Ï
 \begin{eqnarray*}
 G = &\biggl\{
 &
-\frac{t_1 S_2^2+(b_2 x_1-b_2) x_2 S_1
-+ t_2 S_2
+\frac{t_1 \theta_2^2+(b_2 x_1-b_2) x_2 \theta_1
++ t_2 \theta_2
 +(-b_2 a x_2^2+b_2 a x_1 x_2)}{-x_2^2+(x_1+1) x_2-x_1},\\
-&&\frac{(-x_2+x_1) S_1 S_2 + (-b_2 x_2) S_1 + b_1 x_1 S_2}{-x_2+x_1},\\
-&&\frac{t_3 S_1^2 +t_4 S_1 
-+ (-b_1 x_1 x_2+b_1 x_1 ) S_2 
+&&\frac{(-x_2+x_1) \theta_1 \theta_2 + (-b_2 x_2) \theta_1 + b_1 x_1 \theta_2}{-x_2+x_1},\\
+&&\frac{t_3 \theta_1^2 +t_4 \theta_1
++ (-b_1 x_1 x_2+b_1 x_1 ) \theta_2
 + (-b_1 a x_1 x_2+b_1 a x_1^2)}{(-x_1+1) x_2+x_1^2-x_1}
 \biggr\}
 \end{eqnarray*}
 ¤ò°ÕÌ£¤·¤Þ¤¹.
-¤³¤³¤Ç, 
+¤³¤³¤Ç,
 \begin{eqnarray*}
 t_1 &=& -x_2^2+(x_1+1) x_2-x_1 \\
 t_2 &=& (-a-b_2) x_2^2+((a-b_1+b_2) x_1+c-1) x_2+(-c+1+b_1) x_1 \\
 t_3 &=& (-x_1+1) x_2+x_1^2-x_1 \\
-t_4 &=& ((-a-b_1+b_2) x_1+c-b_2-1) x_2+(a+b_1) x_1^2+(-c+1) x_1 
+t_4 &=& ((-a-b_1+b_2) x_1+c-b_2-1) x_2+(a+b_1) x_1^2+(-c+1) x_1
 \end{eqnarray*}
 ¤Ä¤Þ¤ê, ·×»»·ë²Ì¤Ï $R$ ¤Î¸µ¤Î¥ê¥¹¥È¤Ç¤¹.
-ɸ½àñ¹à¼°¤Ï
+
+% ±é»»»Ò¤ÎÆâÉôɽ¸½¤òÃΤë¤Ë¤Ï \verb|yang.op| ´Ø¿ô¤ò,
+% ÆâÉôɽ¸½¤«¤é¿¹à¼°É½¸½¤òÃΤë¤Ë¤Ï \verb|yang.opr| ´Ø¿ô¤ò»È¤¤¤Þ¤¹.
+
+% \begin{verbatim}
+% [5] yang.op(dx1);
+% (1)*<<1,0>>
+% [6] yang.opr(<<1,0>>);
+% dx1
+% \end{verbatim}
+
+% ¤³¤Î¤è¤¦¤Ë±é»»»Ò¤ÎÆâÉôɽ¸½¤Ïʬ»¶Â¿¹à¼°¤Ë¤Ê¤Ã¤Æ¤¤¤Þ¤¹.
+% ¤½¤Î¤Þ¤Þ¤Ç¤Ï $R'$ ¤Î¸µ¤·¤«É½¤»¤º, $R$ ¤Î
+% ¸µ¤Ïɽ¸½¤Ç¤­¤Þ¤»¤ó.  ¤·¤¿¤¬¤Ã¤Æ, $f \in R'$ ¤È $q \in K[x_1,x_2]$ ¤ÎÂÐ
+% \verb|[F,Q]| ¤Ç $R$ ¤Î¸µ $(1/q)f$ ¤òɽ¤·¤Þ¤¹.
+
+% ¤µ¤é¤Ë, $R$ ¤Ë¤ª¤±¤ë¹à½ç½ø¤Ï asir ¤Îʬ»¶Â¿¹à¼°¤Î¹à½ç½ø¤Ë°ìÃפ·¤Þ¤¹.  ´û
+% ÄêÃͤÏ, Á´¼¡¿ôµÕ¼­½ñ¼°½ç½ø¤Ç¤¹.  ½ç½ø¤òÊѹ¹¤·¤Æ Groeber ´ðÄì¤ò·×»»¤·¤¿¤¤
+% ¾ì¹ç¤Ë¤Ï, ´Ä¤Î¸µ¤òÄêµÁ¤¹¤ëÁ°¤Ë dp\_ord ¤Ç½ç½ø¤òÊѹ¹¤·¤Æ¤ª¤¯¤Ù¤­¤Ç¤¹.
+
+¥¤¥Ç¥¢¥ë $I$ ¤Î¥é¥ó¥¯¤òÄ´¤Ù¤Þ¤·¤ç¤¦.
+$G$ ¤Îɸ½àñ¹à¼°¤Ï
 \begin{verbatim}
-[9] yang.stdmon(G);
-[(1)*<<1,0>>,(1)*<<0,1>>,(1)*<<0,0>>]		
+[7] yang.stdmon(G);
+[dx1,dx2,1]
 \end{verbatim}
 ¤Çµá¤Þ¤ê¤Þ¤¹.
 ¤è¤Ã¤Æ $I$ ¤Î¥é¥ó¥¯¤Ï 3 ¤Ç¤¹.
@@ -118,11 +133,9 @@ t_4 &=& ((-a-b_1+b_2) x_1+c-b_2-1) x_2+(a+b_1) x_1^2+(
 Àµµ¬·Á¤òµá¤á¤Þ¤·¤ç¤¦.
 ¤³¤ì¤Ë¤Ï \verb|yang.nf| ¤òÍѤ¤¤Þ¤¹.
 \begin{verbatim}
-[10] T=(x2-x1)*S1*S1;
-(x2-x1)*<<2,0>>
-[11] yang.nf(T,G);
-[(((-a-b1+b2)*x1+c-b2-1)*x2+(a+b1)*x1^2+(-c+1)*x1)*<<1,0>>
-+(-b1*x1*x2+b1*x1)*<<0,1>>+(-b1*a*x1*x2+b1*a*x1^2)*<<0,0>>,x1-1]
+[8] yang.nf((x2-x1)*dx1^2,G);
+((-b1*x1*x2+b1*x1)*dx2+(((-a-b1+b2)*x1+c-b2-1)*x2+(a+b1)*x1^2+(-c+1)*x1)*dx1
+-b1*a*x1*x2+b1*a*x1^2)/(x1-1)
 \end{verbatim}
 ¤Ä¤Þ¤ê·×»»·ë²Ì¤Ï $\mathrm{mod}\ I$ ¤Ç
 \begin{eqnarray*}
@@ -136,62 +149,59 @@ t &\equiv&
 ¼¡¤Ë $F_1$ ¤Î Pfaff ·Á¼°¤ò·×»»¤·¤Þ¤·¤ç¤¦.
 
 \begin{verbatim}
-[10] S0 = yang.constant(1);
-(1)*<<0,0>>
-[11] Base=[S0,S1,S2];
-[(1)*<<0,0>>,(1)*<<1,0>>,(1)*<<0,1>>]
-[12] Pf=yang.pfaffian(Base,G);
+[9] Base=[1,dx1,dx2];
+[10] Pf=yang.pf(Base,G);
 [ [ 0 (1)/(x1) 0 ]
-[ (-b1*a)/(x1-1) 
-(((-a-b1+b2)*x1+c-b2-1)*x2+(a+b1)*x1^2+(-c+1)*x1)/((x1^2-x1)*x2-x1^3+x1^2) 
+[ (-b1*a)/(x1-1)
+(((-a-b1+b2)*x1+c-b2-1)*x2+(a+b1)*x1^2+(-c+1)*x1)/((x1^2-x1)*x2-x1^3+x1^2)
 (-b1*x2+b1)/((x1-1)*x2-x1^2+x1) ]
-[ 0 (-b2*x2)/(x1*x2-x1^2) (b1)/(x2-x1) ] 
+[ 0 (-b2*x2)/(x1*x2-x1^2) (b1)/(x2-x1) ]
 [ 0 0 (1)/(x2) ]
 [ 0 (-b2)/(x2-x1) (b1*x1)/(x2^2-x1*x2) ]
-[ (-b2*a)/(x2-1) (b2*x1-b2)/(x2^2+(-x1-1)*x2+x1) 
+[ (-b2*a)/(x2-1) (b2*x1-b2)/(x2^2+(-x1-1)*x2+x1)
 ((-a-b2)*x2^2+((a-b1+b2)*x1+c-1)*x2+(-c+b1+1)*x1)/(x2^3+(-x1-1)*x2^2+x1*x2) ] ]
-[13] length(Pf);
+[11] length(Pf);
 2
-[14] P1 = Pf[0];
+[12] P1 = Pf[0];
 [ 0 (1)/(x1) 0 ]
-[ (-b1*a)/(x1-1) 
-(((-a-b1+b2)*x1+c-b2-1)*x2+(a+b1)*x1^2+(-c+1)*x1)/((x1^2-x1)*x2-x1^3+x1^2) 
+[ (-b1*a)/(x1-1)
+(((-a-b1+b2)*x1+c-b2-1)*x2+(a+b1)*x1^2+(-c+1)*x1)/((x1^2-x1)*x2-x1^3+x1^2)
 (-b1*x2+b1)/((x1-1)*x2-x1^2+x1) ]
 [ 0 (-b2*x2)/(x1*x2-x1^2) (b1)/(x2-x1) ]
-[15] P2 = Pf[1];
+[13] P2 = Pf[1];
 [ 0 0 (1)/(x2) ]
 [ 0 (-b2)/(x2-x1) (b1*x1)/(x2^2-x1*x2) ]
-[ (-b2*a)/(x2-1) (b2*x1-b2)/(x2^2+(-x1-1)*x2+x1) 
+[ (-b2*a)/(x2-1) (b2*x1-b2)/(x2^2+(-x1-1)*x2+x1)
 ((-a-b2)*x2^2+((a-b1+b2)*x1+c-1)*x2+(-c+b1+1)*x1)/(x2^3+(-x1-1)*x2^2+x1*x2) ]
 \end{verbatim}
 
-·×»»¤òÀâÌÀ¤·¤Þ¤¹.  
+·×»»¤òÀâÌÀ¤·¤Þ¤¹.
 ¥é¥ó¥¯¤¬ 3 ¤Ç¤¢¤ë¤³¤È¤ËÃí°Õ¤·¤Þ¤¹.
-´ðÄì¤ò 
+´ðÄì¤ò
 \[
-F = 
+F =
 \left(
 \begin{array}{c}
 f \\ S_1f \\ S_2 f
 \end{array}
 \right)
-= 
+=
 \left(
 \begin{array}{c}
 f_1 \\ f_2 \\ f_3
 \end{array}
 \right)
 \]
-¤È¤È¤ê, $S_i f_j$ ¤ÎÀµµ¬·Á¤ò·×»»¤¹¤ë¤³¤È¤Ç Pfaff ·Á¼°¤òµá¤á¤Æ¤¤¤Þ¤¹. 
+¤È¤È¤ê, $S_i f_j$ ¤ÎÀµµ¬·Á¤ò·×»»¤¹¤ë¤³¤È¤Ç Pfaff ·Á¼°¤òµá¤á¤Æ¤¤¤Þ¤¹.
 \verb|Pf| ¤Ï$3\times 3$-¹ÔÎó¤Î¥ê¥¹¥È¤ÇŤµ¤Ï $2$ ¤Ç¤¹.
 ·ë²Ì¤Ï,
 \[
-\frac{\partial}{\partial x_1} 
+\frac{\partial}{\partial x_1}
 \left(\begin{array}{c} f_1 \\ f_2 \\ f_3\end{array}\right)
-= P_1 
-\left(\begin{array}{c} f_1 \\ f_2 \\ f_3\end{array}\right), 
+= P_1
+\left(\begin{array}{c} f_1 \\ f_2 \\ f_3\end{array}\right),
 \qquad
-\frac{\partial}{\partial x_2} 
+\frac{\partial}{\partial x_2}
 \left(\begin{array}{c} f_1 \\ f_2 \\ f_3\end{array}\right)
 = P_2
 \left(\begin{array}{c} f_1 \\ f_2 \\ f_3\end{array}\right)
@@ -214,7 +224,7 @@ P_2 &=&
 0& \frac{-b_2}{x_2-x_1}& \frac{b_1 x_1}{x_2^2 -x_1 x_2} \cr
 \frac{-b_2 a}{x_2-1}& \frac{b_2 x_1-
 b_2}{x_2^2 + (-x_1-1) x_2+ x_1}
-& 
+&
 \frac{(-a-b_2) x_2^2 + ((a -b_1+b_2) x_1+ c-1) x_2
 + (-c+ b_1+ 1) x_1}{x_2^3 + (-x_1-1) x_2^2 + x_1 x_2} \cr
 }
@@ -222,7 +232,7 @@ b_2}{x_2^2 + (-x_1-1) x_2+ x_1}
 
 \section{$\mathcal{A}$-Ķ´ö²¿Èùʬº¹Ê¬·Ï¤ò·×»»¤·¤Æ¤ß¤ë}
 
-$\mathcal{A}$-Ķ´ö²¿Èùʬº¹Ê¬·Ï¤Ë¤Ä¤¤¤Æ¤ÏÀìÍѤδؿô¤¬ÍÑ°Õ¤µ¤ì¤Æ¤¤¤Þ¤¹. 
+$\mathcal{A}$-Ķ´ö²¿Èùʬº¹Ê¬·Ï¤Ë¤Ä¤¤¤Æ¤ÏÀìÍѤδؿô¤¬ÍÑ°Õ¤µ¤ì¤Æ¤¤¤Þ¤¹.
 ¤Þ¤º¹ÔÎó $A$ ¤È ¥Ñ¥é¥á¡¼¥¿¥Ù¥¯¥È¥ë $\beta$ ¤¬Í¿¤¨¤é¤ì¤¿¤È¤­, ¥ª¥¤¥é¡¼Èù
 ʬ±é»»»Ò¤Î·Á¤ÇÊýÄø¼°·Ï¤òµá¤á¤ëɬÍפ¬¤¢¤ê¤Þ¤¹.
 
@@ -278,9 +288,9 @@ Ring ¤Î ÄêµÁ
 Ring := '[' ( Vars | RingDef )  ']'
 Vars := Variable [ , Variable ]*
 RingDef := RingEl [ , RingEl ]*
-RingEl := Keyword , '[' ( Vars | Pairs ) ']' 
+RingEl := Keyword , '[' ( Vars | Pairs ) ']'
 Keyword := "euler" | "differential" | "difference"
-Pairs := Pair [ , Pair ]* 
+Pairs := Pair [ , Pair ]*
 Pair := '[' Variable , ( Number | Variable ) ']'
 \end{verbatim}