===================================================================
RCS file: /home/cvs/OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v
retrieving revision 1.3
retrieving revision 1.7
diff -u -p -r1.3 -r1.7
--- OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw	2010/02/06 00:50:32	1.3
+++ OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw	2019/09/09 23:39:52	1.7
@@ -1,4 +1,4 @@
-/*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v 1.2 2008/06/04 01:46:52 takayama Exp $ */
+/*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v 1.6 2019/08/31 06:36:28 takayama Exp $ */
 
 /*&C
 @c DO NOT EDIT THIS FILE   
@@ -511,7 +511,7 @@ x*dx+1
 @findex sm1.gb
 @findex sm1.gb_d
 @table @t
-@item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
+@item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r},needBack=@var{n})
 ::  computes the Grobner basis of @var{f} in the ring of differential
 operators with the variable @var{v}.
 @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
@@ -559,6 +559,9 @@ List
    Each polynomial is expressed as a string temporally for now.
    When the optional variable @var{r} is set to one,
    the polynomials are dehomogenized (,i.e., h is set to 1).
+@item If you want to have a reduced basis or compute the initial form ideal exactly, 
+execute sm1.auto_reduce(1) before executing this function.
+@item When the needBack option @var{n} is 1, it returns the answer is a different format as [groebner basis,initial, gb,1,all,[groebner basis, backward transformation]]
 @end itemize
 */
 /*&ja
@@ -568,7 +571,7 @@ List
 @findex sm1.gb
 @findex sm1.gb_d
 @table @t
-@item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
+@item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r},needBack=@var{n})
 ::  @var{v} 上の微分作用素環において @var{f} のグレブナ基底を計算する.
 @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
 ::  @var{v} 上の微分作用素環において @var{f} のグレブナ基底を計算する. 結果を分散多項式のリストで戻す.
@@ -609,6 +612,11 @@ List
     いまのところこの多項式は, 文字列で表現される.
     オプショナル変数 @var{r} がセットされているときは,
     戻り多項式は dehomogenize される (すなわち h に 1 が代入される).
+@item Reduced グレブナー基底または in_w を計算したいときは, この関数の実行の前に
+sm1.auto_reduce(1) を実行しておくこと.
+@item needBack オプションが 1 の時は, 他の場合とは異なる形式
+[groebner basis, initial, gb,1,all, [groebner basis, backward transformation]]
+で答えを戻す. (sm1 の getAttribute を参照)
 @end itemize
 */
 /*&C
@@ -706,17 +714,23 @@ $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
 ,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]]
 @end example
 */
+/*&C
+@example
+[1834] sm1.gb([[dx^2-x,dx],[x]] | needBack=1);
+[[dx,dx^2-x,1],[dx,dx^2,1],gb,1,all,[[dx,dx^2-x,1],[[0,1],[1,0],[-dx,dx^2-x]]]]
+@end example
+*/
 
 /*&en
 @table @t
 @item Reference
-    @code{sm1.reduction}, @code{sm1.rat_to_p}
+    @code{sm1.auto_reduce}, @code{sm1.reduction}, @code{sm1.rat_to_p}
 @end table
 */
 /*&ja
 @table @t
 @item 参照
-    @code{sm1.reduction}, @code{sm1.rat_to_p}
+    @code{sm1.auto_reduce}, @code{sm1.reduction}, @code{sm1.rat_to_p}
 @end table
 */
 
@@ -2075,8 +2089,13 @@ not bihomogeneous.
 Algorithm:
 see "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
 How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
-Note that the signs of the slopes are negative, but the absolute values
+Note that the signs of the slopes s' are negative, but the absolute values -s'
 of the slopes are returned.
+In other words, when pF+qV is the gap, -s'=q/p is returned.
+Note that s=1-1/s' is called the slope in recent literatures. Solutions belongs to O(s).
+The number s satisfies 1<= s.  
+We have r=s-1=-1/s', and kappa=1/r=-s', which is used 1/Gamma(1+m*r) factor and exp(-tau^kappa)
+in the Borel and Laplace transformations respectively.
 
 */
 
@@ -2119,8 +2138,14 @@ Algorithm:
 "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
 How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
 をみよ.
-Slope の本来の定義では, 符号が負となるが, このプログラムは,
-Slope の絶対値を戻す.
+Slope s' の本来の定義では, 符号が負となるが, このプログラムは,
+Slope の絶対値 -s' を戻す.
+つまり pF+qV がmicro特性多様体のgapであるとき, -s'=q/p を戻す.
+最近の文献では s=1-1/s' を slope と呼んでいる. 解は O(s) に属する.
+数 s は 1<= s を満す.  
+r=s-1=-1/s' および kappa=1/r=-s' である.
+これらの数はBorel and Laplace 変換においてそれぞれ 1/Gamma(1+m*r) factor,
+exp(-tau^kappa) 項として使われる.
 */
 
 /*&C