| version 1.3, 2010/02/06 00:50:32 |
version 1.6, 2019/08/31 06:36:28 |
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| /*$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 1.5 2012/11/28 05:07:31 takayama Exp $ */ |
| |
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| /*&C |
/*&C |
| @c DO NOT EDIT THIS FILE |
@c DO NOT EDIT THIS FILE |
|
|
| @findex sm1.gb |
@findex sm1.gb |
| @findex sm1.gb_d |
@findex sm1.gb_d |
| @table @t |
@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 |
:: computes the Grobner basis of @var{f} in the ring of differential |
| operators with the variable @var{v}. |
operators with the variable @var{v}. |
| @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p}) |
@item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p}) |
|
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| Each polynomial is expressed as a string temporally for now. |
Each polynomial is expressed as a string temporally for now. |
| When the optional variable @var{r} is set to one, |
When the optional variable @var{r} is set to one, |
| the polynomials are dehomogenized (,i.e., h is set to 1). |
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, |
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execute sm1.auto_reduce(1) before executing this function. |
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@item When the needBack option @var{n} is 1, it returns the answer is a different format as [groebner basis,[gb,1,all,[groebner basis, backward transformation]]] |
| @end itemize |
@end itemize |
| */ |
*/ |
| /*&ja |
/*&ja |
|
|
| @findex sm1.gb |
@findex sm1.gb |
| @findex sm1.gb_d |
@findex sm1.gb_d |
| @table @t |
@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} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. |
:: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. |
| @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p}) |
@item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p}) |
| :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. $B7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B. |
:: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. $B7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B. |
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| $B$$$^$N$H$3$m$3$NB?9`<0$O(B, $BJ8;zNs$GI=8=$5$l$k(B. |
$B$$$^$N$H$3$m$3$NB?9`<0$O(B, $BJ8;zNs$GI=8=$5$l$k(B. |
| $B%*%W%7%g%J%kJQ?t(B @var{r} $B$,%;%C%H$5$l$F$$$k$H$-$O(B, |
$B%*%W%7%g%J%kJQ?t(B @var{r} $B$,%;%C%H$5$l$F$$$k$H$-$O(B, |
| $BLa$jB?9`<0$O(B dehomogenize $B$5$l$k(B ($B$9$J$o$A(B h $B$K(B 1 $B$,BeF~$5$l$k(B). |
$BLa$jB?9`<0$O(B dehomogenize $B$5$l$k(B ($B$9$J$o$A(B h $B$K(B 1 $B$,BeF~$5$l$k(B). |
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@item Reduced $B%0%l%V%J!<4pDl$^$?$O(B in_w $B$r7W;;$7$?$$$H$-$O(B, $B$3$N4X?t$N<B9T$NA0$K(B |
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sm1.auto_reduce(1) $B$r<B9T$7$F$*$/$3$H(B. |
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@item needBack $B%*%W%7%g%s$,(B 1 $B$N;~$O(B, $BB>$N>l9g$H$O0[$J$k7A<0(B |
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[groebner basis, [gb,1,all, [groebner basis, backward transformation]]] |
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$B$GEz$($rLa$9(B. (sm1 $B$N(B getAttribute $B$r;2>H(B) |
| @end itemize |
@end itemize |
| */ |
*/ |
| /*&C |
/*&C |
| Line 706 $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$ |
|
| Line 714 $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>>]]] |
,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]] |
| @end example |
@end example |
| */ |
*/ |
| |
/*&C |
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@example |
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[1834] sm1.gb([[dx^2-x,dx],[x]] | needBack=1); |
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[[[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]]]]] |
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@end example |
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*/ |
| |
|
| /*&en |
/*&en |
| @table @t |
@table @t |
| @item Reference |
@item Reference |
| @code{sm1.reduction}, @code{sm1.rat_to_p} |
@code{sm1.auto_reduce}, @code{sm1.reduction}, @code{sm1.rat_to_p} |
| @end table |
@end table |
| */ |
*/ |
| /*&ja |
/*&ja |
| @table @t |
@table @t |
| @item $B;2>H(B |
@item $B;2>H(B |
| @code{sm1.reduction}, @code{sm1.rat_to_p} |
@code{sm1.auto_reduce}, @code{sm1.reduction}, @code{sm1.rat_to_p} |
| @end table |
@end table |
| */ |
*/ |
| |
|
| Line 2075 not bihomogeneous. |
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| Line 2089 not bihomogeneous. |
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| Algorithm: |
Algorithm: |
| see "A.Assi, F.J.Castro-Jimenez and J.M.Granger, |
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" |
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. |
of the slopes are returned. |
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In other words, when pF+qV is the gap, -s'=q/p is returned. |
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Note that s=1-1/s' is called the slope in recent literatures. Solutions belongs to O(s). |
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The number s satisfies 1<= s. |
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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) |
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in the Borel and Laplace transformations respectively. |
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| */ |
*/ |
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| "A.Assi, F.J.Castro-Jimenez and J.M.Granger, |
"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" |
How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996" |
| $B$r$_$h(B. |
$B$r$_$h(B. |
| Slope $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B, $B$3$N%W%m%0%i%`$O(B, |
Slope s' $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B, $B$3$N%W%m%0%i%`$O(B, |
| Slope $B$N@dBPCM$rLa$9(B. |
Slope $B$N@dBPCM(B -s' $B$rLa$9(B. |
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$B$D$^$j(B pF+qV $B$,(Bmicro$BFC@-B?MMBN$N(Bgap$B$G$"$k$H$-(B, -s'=q/p $B$rLa$9(B. |
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$B:G6a$NJ88%$G$O(B s=1-1/s' $B$r(B slope $B$H8F$s$G$$$k(B. $B2r$O(B O(s) $B$KB0$9$k(B. |
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$B?t(B s $B$O(B 1<= s $B$rK~$9(B. |
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r=s-1=-1/s' $B$*$h$S(B kappa=1/r=-s' $B$G$"$k(B. |
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$B$3$l$i$N?t$O(BBorel and Laplace $BJQ49$K$*$$$F$=$l$>$l(B 1/Gamma(1+m*r) factor, |
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exp(-tau^kappa) $B9`$H$7$F;H$o$l$k(B. |
| */ |
*/ |
| |
|
| /*&C |
/*&C |
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