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version 1.14, 2004/03/05 15:30:50

version 1.18, 2004/05/28 01:22:13

Line 1

/*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1.oxweave,v 1.13 2003/07/28 01:36:36 takayama Exp $ */

/*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1.oxweave,v 1.17 2004/05/14 01:25:03 takayama Exp $ */

/*&C-texi

/*&C

@c DO NOT EDIT THIS FILE oxphc.texi

/*&C-texi

/*&C

@node SM1 Functions,,, Top

Line 68 Hence, the dimension of the first de Rham cohomology g

cohomology groups.

@end tex

/*&C-texi

/*&C

@example

@include opening.texi

Line 77 cohomology groups.

[1,2]

@end example

/*&C-texi

/*&C

@noindent

The author of @code{sm1} : Nobuki Takayama, @code{takayama@@math.sci.kobe-u.ac.jp} @*

The author of sm1 packages : Toshinori Oaku, @code{oaku@@twcu.ac.jp} @*

Line 87 Grobner Deformations of Hypergeometric Differential Eq

See the appendix.

/*&C-texi

/*&C

@menu

* ox_sm1_forAsir::

* sm1.start::

Line 279 The descriptor can be obtained by the function

$B$3$N<1JLHV9f$O4X?t(B @code{sm1.get_Sm1_proc()} $B$G$H$j$@$9$3$H$,$G$-$k(B.

@end itemize

/*&C-texi

/*&C

@example

[260] ord([da,a,db,b]);

[da,a,db,b,dx,dy,dz,x,y,z,dt,ds,t,s,u,v,w,

Line 361 to execute the command string @var{s}.

($B<!$NNc$G$O(B, $B<1JLHV9f(B 0)

@end itemize

/*&C-texi

/*&C

@example

[261] sm1.sm1(0," ( (x-1)^2 ) . ");

Line 604 List

$BLa$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).

@end itemize

/*&C-texi

/*&C

@example

[293] sm1.gb([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);

[[x*dx+y*dy-1,y^2*dy^2+2],[x*dx,y^2*dy^2]]

Line 633 graded reverse lexicographic order $B$K4X$9$k%0%l%V%J

$BBP$9$k(B leading monomial (initial monomial) $B$G$"$k(B.

@end tex

/*&C-texi

/*&C

@example

[294] sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]);

[[dx+dy^3-4*dy,-dy^4+4*dy^2-1],[dx,-dy^4]]

Line 666 $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$

$B$5$l$k(B).

@end tex

/*&C-texi

/*&C

@example

[294] F=sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);

map(print,F[2][0])$

map(print,F[2][1])$

@end example

/*&C-texi

/*&C

@example

[595]

sm1.gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],

Line 804 mode. So, it is strongly recommended to execute the co

$B$r0l;~(B shutdown $B$7$F%j%9%?!<%H$7$?J}$,0BA4$G$"$k(B.

@end itemize

/*&C-texi

/*&C

@example

[332] sm1.deRham([x^3-y^2,[x,y]]);

[1,1,0]

Line 917 List

@end itemize

/*&C-texi

/*&C

@example

[346] load("katsura")$

Line 1020 List

@var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.

@end itemize

/*&C-texi

/*&C

@example

[595] sm1.genericAnn([x^3+y^3+z^3,[s,x,y,z]]);

[-x*dx-y*dy-z*dz+3*s,z^2*dy-y^2*dz,z^2*dx-x^2*dz,y^2*dx-x^2*dy]

Line 1122 the inputs @var{f} and @var{g} are left ideals of D.

$B0lHL$K(B, $B=PNO$O<+M32C72(B D^r $B$NItJ,2C72$G$"$k(B.

@end itemize

/*&C-texi

/*&C

@example

[258] sm1.wTensor0([[x*dx -1, y*dy -4],[dx+dy,dx-dy^2],[x,y],[1,2]]);

[[-y*x*dx-y*x*dy+4*x+y],[5*x*dx^2+5*x*dx+2*y*dy^2+(-2*y-6)*dy+3],

Line 1210 sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_

$B$O(B, $BJ,;6B?9`<0MQ$G$"$k(B.

@end itemize

/*&C-texi

/*&C

@example

[259] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y]]);

[x^2+y^2-4,1,[0,0],[y^4-4*y^2+1,x+y^3-4*y]]

Line 1287 String

($B$?$H$($P(B, /usr/local/jdk1.1.8/bin $B$r%3%^%s%I%5!<%A%Q%9$KF~$l$k$J$I(B.)

@end itemize

/*&C-texi

/*&C

@example

[263] load("om");

Line 1405 syzygy $B$G$"$k(B.

$BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.

@end itemize

/*&C-texi

/*&C

@example

[293] sm1.syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);

[[[y*x*dy*dx-2,-x*dx-y*dy+1]], generators of the syzygy

Line 1414 syzygy $B$G$"$k(B.

[[y*x*dy*dx-2,-x*dx-y*dy+1]]]]

@end example

/*&C-texi

/*&C

@example

[294]sm1.syz([[x^2*dx^2+x*dx+y^2*dy^2+y*dy-4,x*y*dx*dy-1],[x,y],[[dx,-1,x,1]]]);

[[[y*x*dy*dx-1,-x^2*dx^2-x*dx-y^2*dy^2-y*dy+4]], generators of the syzygy

Line 1450 List

@itemize @bullet

@item Ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.

@item @code{sm1.mul_h} is for homogenized Weyl algebra.

@item BUG: @code{sm1.mul(p0*dp0,1,[p0])} returns

@code{dp0*p0+1}.

A variable order such that d-variables come after non-d-variables

is necessary for the correct computation.

@end itemize

Line 1478 List

Line 1482 List

@item sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}

$B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.

@item @code{sm1.mul_h} $B$O(B homogenized Weyl $BBe?tMQ(B.

@item BUG: @code{sm1.mul(p0*dp0,1,[p0])} $B$O(B

@code{dp0*p0+1} $B$rLa$9(B.

d$BJQ?t$,8e$m$K$/$k$h$&$JJQ?t=g=x$,$O$$$C$F$$$J$$$H(B, $B$3$N4X?t$O@5$7$$Ez$($rLa$5$J$$(B.

@end itemize

/*&C-texi

/*&C

@example

[277] sm1.mul(dx,x,[x]);

Line 1559 See Saito, Sturmfels, Takayama : Grobner Deformations

Line 1566 See Saito, Sturmfels, Takayama : Grobner Deformations

@end itemize

/*&C-texi

/*&C

@example

[280] sm1.distraction([x*dx,[x],[x],[dx],[x]]);

Line 1639 List

Line 1646 List

@end itemize

/*&C-texi

/*&C

@example

Line 1679 F_D(a,b1,b2,...,bn,c;x1,...,xn)

Line 1686 F_D(a,b1,b2,...,bn,c;x1,...,xn)

where @var{a} =(a,c,b1,...,bn).

When n=2, the Lauricella function is called the Appell function F_1.

The parameters a, c, b1, ..., bn may be rational numbers.

@item It does not call sm1 function appell1. As a concequence,

when parameters are rational or symbolic, this function also works

as well as integral parameters.

@end itemize

Line 1706 F_D(a,b1,b2,...,bn,c;x1,...,xn)

Line 1716 F_D(a,b1,b2,...,bn,c;x1,...,xn)

$B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,

@var{a} =(a,c,b1,...,bn).

$B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.

@item sm1 $B$N4X?t(B appell1 $B$r$h$V$o$1$G$J$$$N$G(B, $B%Q%i%a!<%?$,M-M}?t$dJ8;z<0$N>l9g$b(B

$B@5$7$/F0$/(B.

@end itemize

/*&C-texi

/*&C

@example

Line 1728 F_D(a,b1,b2,...,bn,c;x1,...,xn)

Line 1740 F_D(a,b1,b2,...,bn,c;x1,...,xn)

[x1*dx1*dx2,-x1^2*dx1^2,-x2^2*dx1*dx2,-x1*x2^2*dx2^2]]

[283] sm1.rank(sm1.appell1([1/2,3,5,-1/3]));

[285] Mu=2$ Beta = 1/3$

[287] sm1.rank(sm1.appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));

Line 1763 F_4(a,b,c1,c2,...,cn;x1,...,xn)

Line 1775 F_4(a,b,c1,c2,...,cn;x1,...,xn)

where @var{a} =(a,b,c1,...,cn).

When n=2, the Lauricella function is called the Appell function F_4.

The parameters a, b, c1, ..., cn may be rational numbers.

@item @item It does not call sm1 function appell4. As a concequence,

when parameters are rational or symbolic, this function also works

as well as integral parameters.

@end itemize

Line 1790 F_C(a,b,c1,c2,...,cn;x1,...,xn)

Line 1805 F_C(a,b,c1,c2,...,cn;x1,...,xn)

$B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,

@var{a} =(a,b,c1,...,cn).

$B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.

@item sm1 $B$N4X?t(B appell1 $B$r$h$V$o$1$G$J$$$N$G(B, $B%Q%i%a!<%?$,M-M}?t$dJ8;z<0$N>l9g$b(B

$B@5$7$/F0$/(B.

@end itemize

/*&C-texi

/*&C

@example

Line 1870 holonomic. It is generally faster than @code{sm1.rank}

Line 1887 holonomic. It is generally faster than @code{sm1.rank}

@end itemize

/*&C-texi

/*&C

@example

Line 2035 Slope $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B,

Line 2052 Slope $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B,

Slope $B$N@dBPCM$rLa$9(B.

/*&C-texi

/*&C

@example

Line 2069 Slope $B$N@dBPCM$rLa$9(B.

Line 2086 Slope $B$N@dBPCM$rLa$9(B.

/*&en

@include sm1-auto-en.texi

@include sm1-auto.en

/*&ja

@include sm1-auto-ja.texi

@include sm1-auto.ja

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