| version 1.1, 2005/11/12 01:23:06 |
version 1.2, 2006/03/20 14:14:04 |
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| %#!platex |
%#!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} |
\documentclass{jarticle} |
| %\usepackage{amsmath} |
%\usepackage{amsmath} |
| \title{Yang Tutorial} |
\title{Yang Tutorial} |
| \author{Katsuyoshi OHARA} |
\author{¶âÂôÂç³ØÍý³ØÉô\ \ \ ¾®¸¶¸ùǤ} |
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\date{} |
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\topmargin -1.5cm |
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\textheight 23.5cm |
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\oddsidemargin 0cm |
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\evensidemargin 0cm |
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\textwidth 16.5cm |
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| \begin{document} |
\begin{document} |
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|
| \maketitle |
\maketitle |
| |
|
| \section{yang ¤È¤Ï} |
\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 |
¤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. |
\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, |
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 |
°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¤¬ |
\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. |
¤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. |
·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.} |
\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, |
¥°¥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. |
\verb|yang_D.rr| ¤ò»È¤Ã¤¿¤Û¤¦¤¬¹â®¤Ë¤Ê¤ê¤Þ¤¹. |
| |
|
| Line 31 yang ¤Ç¤Ç¤¤ë·×»»¤Ï,¥°¥ì¥Ö¥Ê´ðÄì, Àµµ¬·Á, 0¼¡¸µ¥¤¥Ç¥¢¥ |
|
| Line 38 yang ¤Ç¤Ç¤¤ë·×»»¤Ï,¥°¥ì¥Ö¥Ê´ðÄì, Àµµ¬·Á, 0¼¡¸µ¥¤¥Ç¥¢¥ |
|
| ohara:~> asir |
ohara:~> asir |
| [1] load("yang.rr"); |
[1] load("yang.rr"); |
| [2] yang.define_ring([x1,x2]); |
[2] yang.define_ring([x1,x2]); |
| |
{[euler,[x1,x2]],[x1,x2],[0,0],[0,0],[dx1,dx2]} |
| \end{verbatim} |
\end{verbatim} |
| |
|
| ´Ä¤È¤·¤Æ, $R=K(x_1,x_2)\langle \theta_1, \theta_2\rangle$ ¤¬ÄêµÁ |
´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} |
\begin{verbatim} |
| [3] S1=yang.operator(x1); |
[3] S=dx1+dx2; |
| <<1,0>> |
|
| [4] S2=yang.operator(x2); |
|
| <<0,1>> |
|
| \end{verbatim} |
\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. |
|
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| ¤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} |
\begin{verbatim} |
| [5] S=S1+S2; |
[4] L1 = yang.mul(dx1,S+c-1) - x1*yang.mul(dx1+b1,S+a); |
| <<0,1>>+<<1,0>> |
((-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} |
\end{verbatim} |
| |
|
| $S = S_1 + S_2$ ¤Ç¤¹. $R'$ ¤Ë¤ª¤±¤ëϤÏ, Ä̾ï¤Î $+$ ¤Ç½ñ¤¯¤³¤È¤¬¤Ç¤¤Þ¤¹. |
$L_1 = \theta_1 (S + c-1) - x_1 (\theta_1 + b_1)(S+a)$ ¤Ç¤¹. |
| |
$R'=K[x_1,x_2]\langle \theta_1, \theta_2\rangle$ ¤Ë¤ª¤±¤ë±é»»»Ò |
| \begin{verbatim} |
¤ÎÀÑ¤Ï \verb|yang.mul| ¤Ç·×»»¤·¤Þ¤¹. ¤¿¤À¤·, $K[x_1,x_2]$ ¤Î¸µ¤Ï¤½¤Î¤Þ |
| [6] L1 = yang.multi(S1,S+c-1) - x1*yang.multi(S1+b1,S+a); |
¤Þ¤«¤±¤Æ¤â¹½¤¤¤Þ¤»¤ó. ¤¤¤Þ¤Î¤È¤³¤í, \verb|yang.mul| ¤Î°ú¿ô¤Ë»È¤¨¤ë¤Î |
| [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 |
|
| ¤Ï$R'$ ¤Î¸µ¤Î¤ß¤Ç¤¹. |
¤Ï$R'$ ¤Î¸µ¤Î¤ß¤Ç¤¹. |
| |
|
| \begin{verbatim} |
\begin{verbatim} |
| [8] G = yang.buchberger([L1,L2]); |
[6] G = yang.gr([L1,L2]); |
| [[(-x2^2+(x1+1)*x2-x1)*<<0,2>>+((b2*x1-b2)*x2)*<<1,0>> |
[((-x2^2+(x1+1)*x2-x1)*dx2^2+((-a-b2)*x2^2+((a-b1+b2)*x1+c-1)*x2+(-c+b1+1)*x1)*dx2 |
| +((-a-b2)*x2^2+((a-b1+b2)*x1+c-1)*x2+(-c+b1+1)*x1)*<<0,1>> |
+(b2*x1-b2)*x2*dx1-b2*a*x2^2+b2*a*x1*x2)/(-x2^2+(x1+1)*x2-x1), |
| +(-b2*a*x2^2+b2*a*x1*x2)*<<0,0>>,-x2^2+(x1+1)*x2-x1], |
(((-x2+x1)*dx1+b1*x1)*dx2-b2*x2*dx1)/(-x2+x1), |
| [(-x2+x1)*<<1,1>>+(-b2*x2)*<<1,0>>+(b1*x1)*<<0,1>>,-x2+x1], |
((-b1*x1*x2+b1*x1)*dx2+((-x1+1)*x2+x1^2-x1)*dx1^2+(((-a-b1+b2)*x1+c-b2-1)*x2 |
| [((-x1+1)*x2+x1^2-x1)*<<2,0>>+ |
+(a+b1)*x1^2+(-c+1)*x1)*dx1-b1*a*x1*x2+b1*a*x1^2)/((-x1+1)*x2+x1^2-x1)] |
| (((-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]] |
|
| \end{verbatim} |
\end{verbatim} |
| |
|
| $R$ ¤Î¥¤¥Ç¥¢¥ë $I=\{ L_1, L_2 \}$ ¤Î¥°¥ì¥Ö¥Ê´ðÄì $G$ ¤ò·×»»¤·¤Þ¤¹. |
$R$ ¤Î¥¤¥Ç¥¢¥ë $I=\langle L_1, L_2 \rangle$ ¤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 |
·×»»·ë²Ì¤Ï |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| G = &\biggl\{ |
G = &\biggl\{ |
| & |
& |
| \frac{t_1 S_2^2+(b_2 x_1-b_2) x_2 S_1 |
\frac{t_1 \theta_2^2+(b_2 x_1-b_2) x_2 \theta_1 |
| + t_2 S_2 |
+ 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},\\ |
+(-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{(-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 S_1^2 +t_4 S_1 |
&&\frac{t_3 \theta_1^2 +t_4 \theta_1 |
| + (-b_1 x_1 x_2+b_1 x_1 ) S_2 |
+ (-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} |
+ (-b_1 a x_1 x_2+b_1 a x_1^2)}{(-x_1+1) x_2+x_1^2-x_1} |
| \biggr\} |
\biggr\} |
| \end{eqnarray*} |
\end{eqnarray*} |
| ¤ò°ÕÌ£¤·¤Þ¤¹. |
¤ò°ÕÌ£¤·¤Þ¤¹. |
| ¤³¤³¤Ç, |
¤³¤³¤Ç, |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| t_1 &=& -x_2^2+(x_1+1) x_2-x_1 \\ |
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_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_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*} |
\end{eqnarray*} |
| ¤Ä¤Þ¤ê, ·×»»·ë²Ì¤Ï $R$ ¤Î¸µ¤Î¥ê¥¹¥È¤Ç¤¹. |
¤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. |
| |
|
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% \begin{verbatim} |
| |
% [5] yang.op(dx1); |
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% (1)*<<1,0>> |
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% [6] yang.opr(<<1,0>>); |
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% dx1 |
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% \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} |
\begin{verbatim} |
| [9] yang.stdmon(G); |
[7] yang.stdmon(G); |
| [(1)*<<1,0>>,(1)*<<0,1>>,(1)*<<0,0>>] |
[dx1,dx2,1] |
| \end{verbatim} |
\end{verbatim} |
| ¤Çµá¤Þ¤ê¤Þ¤¹. |
¤Çµá¤Þ¤ê¤Þ¤¹. |
| ¤è¤Ã¤Æ $I$ ¤Î¥é¥ó¥¯¤Ï 3 ¤Ç¤¹. |
¤è¤Ã¤Æ $I$ ¤Î¥é¥ó¥¯¤Ï 3 ¤Ç¤¹. |
| Line 118 t_4 &=& ((-a-b_1+b_2) x_1+c-b_2-1) x_2+(a+b_1) x_1^2+( |
|
| Line 133 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¤¦. |
Àµµ¬·Á¤òµá¤á¤Þ¤·¤ç¤¦. |
| ¤³¤ì¤Ë¤Ï \verb|yang.nf| ¤òÍѤ¤¤Þ¤¹. |
¤³¤ì¤Ë¤Ï \verb|yang.nf| ¤òÍѤ¤¤Þ¤¹. |
| \begin{verbatim} |
\begin{verbatim} |
| [10] T=(x2-x1)*S1*S1; |
[8] yang.nf((x2-x1)*dx1^2,G); |
| (x2-x1)*<<2,0>> |
((-b1*x1*x2+b1*x1)*dx2+(((-a-b1+b2)*x1+c-b2-1)*x2+(a+b1)*x1^2+(-c+1)*x1)*dx1 |
| [11] yang.nf(T,G); |
-b1*a*x1*x2+b1*a*x1^2)/(x1-1) |
| [(((-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] |
|
| \end{verbatim} |
\end{verbatim} |
| ¤Ä¤Þ¤ê·×»»·ë²Ì¤Ï $\mathrm{mod}\ I$ ¤Ç |
¤Ä¤Þ¤ê·×»»·ë²Ì¤Ï $\mathrm{mod}\ I$ ¤Ç |
| \begin{eqnarray*} |
\begin{eqnarray*} |
|
|
| ¼¡¤Ë $F_1$ ¤Î Pfaff ·Á¼°¤ò·×»»¤·¤Þ¤·¤ç¤¦. |
¼¡¤Ë $F_1$ ¤Î Pfaff ·Á¼°¤ò·×»»¤·¤Þ¤·¤ç¤¦. |
| |
|
| \begin{verbatim} |
\begin{verbatim} |
| [10] S0 = yang.constant(1); |
[9] Base=[1,dx1,dx2]; |
| (1)*<<0,0>> |
[10] Pf=yang.pf(Base,G); |
| [11] Base=[S0,S1,S2]; |
|
| [(1)*<<0,0>>,(1)*<<1,0>>,(1)*<<0,1>>] |
|
| [12] Pf=yang.pfaffian(Base,G); |
|
| [ [ 0 (1)/(x1) 0 ] |
[ [ 0 (1)/(x1) 0 ] |
| [ (-b1*a)/(x1-1) |
[ (-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) |
(((-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) ] |
(-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 0 (1)/(x2) ] |
| [ 0 (-b2)/(x2-x1) (b1*x1)/(x2^2-x1*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) ] ] |
((-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 |
2 |
| [14] P1 = Pf[0]; |
[12] P1 = Pf[0]; |
| [ 0 (1)/(x1) 0 ] |
[ 0 (1)/(x1) 0 ] |
| [ (-b1*a)/(x1-1) |
[ (-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) |
(((-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) ] |
(-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) ] |
| [15] P2 = Pf[1]; |
[13] P2 = Pf[1]; |
| [ 0 0 (1)/(x2) ] |
[ 0 0 (1)/(x2) ] |
| [ 0 (-b2)/(x2-x1) (b1*x1)/(x2^2-x1*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) ] |
((-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} |
\end{verbatim} |
| |
|
| ·×»»¤òÀâÌÀ¤·¤Þ¤¹. |
·×»»¤òÀâÌÀ¤·¤Þ¤¹. |
| ¥é¥ó¥¯¤¬ 3 ¤Ç¤¢¤ë¤³¤È¤ËÃí°Õ¤·¤Þ¤¹. |
¥é¥ó¥¯¤¬ 3 ¤Ç¤¢¤ë¤³¤È¤ËÃí°Õ¤·¤Þ¤¹. |
| ´ðÄì¤ò |
´ðÄì¤ò |
| \[ |
\[ |
| F = |
F = |
| \left( |
\left( |
| \begin{array}{c} |
\begin{array}{c} |
| f \\ S_1f \\ S_2 f |
f \\ S_1f \\ S_2 f |
| \end{array} |
\end{array} |
| \right) |
\right) |
| = |
= |
| \left( |
\left( |
| \begin{array}{c} |
\begin{array}{c} |
| f_1 \\ f_2 \\ f_3 |
f_1 \\ f_2 \\ f_3 |
| \end{array} |
\end{array} |
| \right) |
\right) |
| \] |
\] |
| ¤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. |
\verb|Pf| ¤Ï$3\times 3$-¹ÔÎó¤Î¥ê¥¹¥È¤ÇŤµ¤Ï $2$ ¤Ç¤¹. |
| ·ë²Ì¤Ï, |
·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) |
\left(\begin{array}{c} f_1 \\ f_2 \\ f_3\end{array}\right) |
| = P_1 |
= P_1 |
| \left(\begin{array}{c} f_1 \\ f_2 \\ f_3\end{array}\right), |
\left(\begin{array}{c} f_1 \\ f_2 \\ f_3\end{array}\right), |
| \qquad |
\qquad |
| \frac{\partial}{\partial x_2} |
\frac{\partial}{\partial x_2} |
| \left(\begin{array}{c} f_1 \\ f_2 \\ f_3\end{array}\right) |
\left(\begin{array}{c} f_1 \\ f_2 \\ f_3\end{array}\right) |
| = P_2 |
= P_2 |
| \left(\begin{array}{c} f_1 \\ f_2 \\ f_3\end{array}\right) |
\left(\begin{array}{c} f_1 \\ f_2 \\ f_3\end{array}\right) |
|
|
| 0& \frac{-b_2}{x_2-x_1}& \frac{b_1 x_1}{x_2^2 -x_1 x_2} \cr |
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- |
\frac{-b_2 a}{x_2-1}& \frac{b_2 x_1- |
| b_2}{x_2^2 + (-x_1-1) x_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 |
\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 |
+ (-c+ b_1+ 1) x_1}{x_2^3 + (-x_1-1) x_2^2 + x_1 x_2} \cr |
| } |
} |
| Line 222 b_2}{x_2^2 + (-x_1-1) x_2+ x_1} |
|
| Line 232 b_2}{x_2^2 + (-x_1-1) x_2+ x_1} |
|
| |
|
| \section{$\mathcal{A}$-Ķ´ö²¿Èùʬº¹Ê¬·Ï¤ò·×»»¤·¤Æ¤ß¤ë} |
\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 |
¤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. |
ʬ±é»»»Ò¤Î·Á¤ÇÊýÄø¼°·Ï¤òµá¤á¤ëɬÍפ¬¤¢¤ê¤Þ¤¹. |
| |
|
|
|
| Ring := '[' ( Vars | RingDef ) ']' |
Ring := '[' ( Vars | RingDef ) ']' |
| Vars := Variable [ , Variable ]* |
Vars := Variable [ , Variable ]* |
| RingDef := RingEl [ , RingEl ]* |
RingDef := RingEl [ , RingEl ]* |
| RingEl := Keyword , '[' ( Vars | Pairs ) ']' |
RingEl := Keyword , '[' ( Vars | Pairs ) ']' |
| Keyword := "euler" | "differential" | "difference" |
Keyword := "euler" | "differential" | "difference" |
| Pairs := Pair [ , Pair ]* |
Pairs := Pair [ , Pair ]* |
| Pair := '[' Variable , ( Number | Variable ) ']' |
Pair := '[' Variable , ( Number | Variable ) ']' |
| \end{verbatim} |
\end{verbatim} |
| |
|
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