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Diff for /OpenXM/src/asir-contrib/packages/doc/Attic/sm1.oxweave between version 1.10 and 1.18

version 1.10, 2003/05/20 23:25:28 version 1.18, 2004/05/28 01:22:13
Line 1 
Line 1 
 /*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1.oxweave,v 1.9 2003/05/19 05:15:52 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 DO NOT EDIT THIS FILE   oxphc.texi
 */  */
 /*&C-texi  /*&C
 @node SM1 Functions,,, Top  @node SM1 Functions,,, Top
   
 */  */
 /*&jp-texi  /*&ja
 @chapter SM1 $BH!?t(B  @chapter SM1 $BH!?t(B
   
 $B$3$N@a$G$O(B sm1 $B$N(B ox $B%5!<%P(B @code{ox_sm1_forAsir}  $B$3$N@a$G$O(B sm1 $B$N(B ox $B%5!<%P(B @code{ox_sm1_forAsir}
Line 32  $X$ $B$OJ?LL$KFs$D$N7j$r$"$1$?6u4V$G$"$k$N$G(B, $BE
Line 33  $X$ $B$OJ?LL$KFs$D$N7j$r$"$1$?6u4V$G$"$k$N$G(B, $BE
 $B<!85$rEz$($k(B.  $B<!85$rEz$($k(B.
 @end tex  @end tex
 */  */
 /*&eg-texi  /*&en
 @chapter SM1 Functions  @chapter SM1 Functions
   
 This chapter describes  interface functions for  This chapter describes  interface functions for
Line 67  Hence, the dimension of the first de Rham cohomology g
Line 68  Hence, the dimension of the first de Rham cohomology g
 cohomology groups.  cohomology groups.
 @end tex  @end tex
 */  */
 /*&C-texi  /*&C
 @example  @example
   
 @include opening.texi  @include opening.texi
Line 76  cohomology groups.
Line 77  cohomology groups.
 [1,2]  [1,2]
 @end example  @end example
 */  */
 /*&C-texi  /*&C
 @noindent  @noindent
 The author of @code{sm1} : Nobuki Takayama, @code{takayama@@math.sci.kobe-u.ac.jp} @*  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} @*  The author of sm1 packages : Toshinori Oaku, @code{oaku@@twcu.ac.jp} @*
Line 86  Grobner Deformations of Hypergeometric Differential Eq
Line 87  Grobner Deformations of Hypergeometric Differential Eq
 See the appendix.  See the appendix.
 */  */
   
 /*  /*&C
 @menu  @menu
 * ox_sm1_forAsir::  * ox_sm1_forAsir::
 * sm1.start::  * sm1.start::
Line 95  See the appendix.
Line 96  See the appendix.
 * sm1.gb::  * sm1.gb::
 * sm1.deRham::  * sm1.deRham::
 * sm1.hilbert::  * sm1.hilbert::
 * hilbert_polynomial::  
 * sm1.genericAnn::  * sm1.genericAnn::
 * sm1.wTensor0::  * sm1.wTensor0::
 * sm1.reduction::  * sm1.reduction::
Line 109  See the appendix.
Line 109  See the appendix.
 * sm1.rank::  * sm1.rank::
 * sm1.auto_reduce::  * sm1.auto_reduce::
 * sm1.slope::  * sm1.slope::
   * sm1.ahg::
   * sm1.bfunction::
   * sm1.generalized_bfunction::
   * sm1.restriction::
   * sm1.saturation::
 @end menu  @end menu
 */  */
   
 /*&jp-texi  /*&ja
 @section @code{ox_sm1_forAsir} $B%5!<%P(B  @section @code{ox_sm1_forAsir} $B%5!<%P(B
 */  */
 /*&eg-texi  /*&en
 @section @code{ox_sm1_forAsir} Server  @section @code{ox_sm1_forAsir} Server
 */  */
   
 /*&eg-texi  /*&en
 @node ox_sm1_forAsir,,, Top  @node ox_sm1_forAsir,,, SM1 Functions
 @subsection @code{ox_sm1_forAsir}  @subsection @code{ox_sm1_forAsir}
 @findex ox_sm1_forAsir  @findex ox_sm1_forAsir
 @table @t  @table @t
Line 150  See the appendix.
Line 155  See the appendix.
 to build your own server by reading @code{sm1} macros.  to build your own server by reading @code{sm1} macros.
 @end itemize  @end itemize
 */  */
 /*&jp-texi  /*&ja
 @node ox_sm1_forAsir,,, Top  @node ox_sm1_forAsir,,, SM1 Functions
 @subsection @code{ox_sm1_forAsir}  @subsection @code{ox_sm1_forAsir}
 @findex ox_sm1_forAsir  @findex ox_sm1_forAsir
 @table @t  @table @t
Line 184  to build your own server by reading @code{sm1} macros.
Line 189  to build your own server by reading @code{sm1} macros.
 */  */
   
   
 /*&jp-texi  /*&ja
 @section $BH!?t0lMw(B  @section $BH!?t0lMw(B
 */  */
 /*&eg-texi  /*&en
 @section Functions  @section Functions
 */  */
   
 /*&eg-texi  /*&en
 @c sort-sm1.start  @c sort-sm1.start
 @node sm1.start,,, SM1 Functions  @node sm1.start,,, SM1 Functions
 @subsection @code{sm1.start}  @subsection @code{sm1.start}
Line 233  The descriptor can be obtained by the function
Line 238  The descriptor can be obtained by the function
 @code{sm1.get_Sm1_proc()}.  @code{sm1.get_Sm1_proc()}.
 @end itemize  @end itemize
 */  */
 /*&jp-texi  /*&ja
 @c sort-sm1.start  @c sort-sm1.start
 @node sm1.start,,, SM1 Functions  @node sm1.start,,, SM1 Functions
 @subsection @code{sm1.start}  @subsection @code{sm1.start}
Line 274  The descriptor can be obtained by the function
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.  $B$3$N<1JLHV9f$O4X?t(B @code{sm1.get_Sm1_proc()} $B$G$H$j$@$9$3$H$,$G$-$k(B.
 @end itemize  @end itemize
 */  */
 /*&C-texi  /*&C
 @example  @example
 [260] ord([da,a,db,b]);  [260] ord([da,a,db,b]);
 [da,a,db,b,dx,dy,dz,x,y,z,dt,ds,t,s,u,v,w,  [da,a,db,b,dx,dy,dz,x,y,z,dt,ds,t,s,u,v,w,
Line 290  a*da+1                  
Line 295  a*da+1                  
 a*da  a*da
 @end example  @end example
 */  */
 /*&eg-texi  /*&en
 @table @t  @table @t
 @item Reference  @item Reference
     @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},      @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
     @code{ord}      @code{ord}
 @end table  @end table
 */  */
 /*&jp-texi  /*&ja
 @table @t  @table @t
 @item $B;2>H(B  @item $B;2>H(B
     @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},      @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
Line 307  a*da
Line 312  a*da
   
   
   
 /*&eg-texi  /*&en
 @c sort-sm1  @c sort-sm1
 @node sm1.sm1,,, SM1 Functions  @node sm1.sm1,,, SM1 Functions
 @subsection @code{sm1.sm1}  @subsection @code{sm1.sm1}
Line 332  to execute the command string @var{s}.
Line 337  to execute the command string @var{s}.
 (In the next example, the descriptor number is 0.)  (In the next example, the descriptor number is 0.)
 @end itemize  @end itemize
 */  */
 /*&jp-texi  /*&ja
 @node sm1.sm1,,, SM1 Functions  @node sm1.sm1,,, SM1 Functions
 @subsection @code{sm1.sm1}  @subsection @code{sm1.sm1}
 @findex sm1.sm1  @findex sm1.sm1
Line 356  to execute the command string @var{s}.
Line 361  to execute the command string @var{s}.
  ($B<!$NNc$G$O(B, $B<1JLHV9f(B 0)   ($B<!$NNc$G$O(B, $B<1JLHV9f(B 0)
 @end itemize  @end itemize
 */  */
 /*&C-texi  /*&C
 @example  @example
 [261] sm1.sm1(0," ( (x-1)^2 ) . ");  [261] sm1.sm1(0," ( (x-1)^2 ) . ");
 0  0
Line 369  x^2-2*x+1
Line 374  x^2-2*x+1
 @end example  @end example
 */  */
   
 /*&jp-texi  /*&ja
 @table @t  @table @t
 @item $B;2>H(B  @item $B;2>H(B
     @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.      @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
 @end table  @end table
 */  */
 /*&eg-texi  /*&en
 @table @t  @table @t
 @item Reference  @item Reference
     @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.      @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
Line 383  x^2-2*x+1
Line 388  x^2-2*x+1
 */  */
   
   
 /*&eg-texi  /*&en
 @c sort-sm1.push_int0  @c sort-sm1.push_int0
 @node sm1.push_int0,,, SM1 Functions  @node sm1.push_int0,,, SM1 Functions
 @subsection @code{sm1.push_int0}  @subsection @code{sm1.push_int0}
Line 421  Note that @code{ox_push_cmo(@var{p},1234)} send the bi
Line 426  Note that @code{ox_push_cmo(@var{p},1234)} send the bi
 @item In other cases,  @code{ox_push_cmo} is called without data conversion.  @item In other cases,  @code{ox_push_cmo} is called without data conversion.
 @end itemize  @end itemize
 */  */
 /*&jp-texi  /*&ja
 @c sort-sm1.push_int0  @c sort-sm1.push_int0
 @node sm1.push_int0,,, SM1 Functions  @node sm1.push_int0,,, SM1 Functions
 @subsection @code{sm1.push_int0}  @subsection @code{sm1.push_int0}
Line 477  x*dx+1
Line 482  x*dx+1
 [1,2]  [1,2]
 @end example  @end example
 */  */
 /*&eg-texi  /*&en
 @table @t  @table @t
 @item Reference  @item Reference
     @code{ox_push_cmo}      @code{ox_push_cmo}
 @end table  @end table
 */  */
 /*&jp-texi  /*&ja
 @table @t  @table @t
 @item Reference  @item Reference
     @code{ox_push_cmo}      @code{ox_push_cmo}
Line 492  x*dx+1
Line 497  x*dx+1
   
   
   
 /*&eg-texi  /*&en
 @c sort-sm1.gb  @c sort-sm1.gb
 @node sm1.gb,,, SM1 Functions  @node sm1.gb,,, SM1 Functions
 @node sm1.gb_d,,, SM1 Functions  
 @subsection @code{sm1.gb}  @subsection @code{sm1.gb}
 @findex sm1.gb  @findex sm1.gb
 @findex sm1.gb_d  @findex sm1.gb_d
Line 550  List
Line 554  List
    the polynomials are dehomogenized (,i.e., h is set to 1).     the polynomials are dehomogenized (,i.e., h is set to 1).
 @end itemize  @end itemize
 */  */
 /*&jp-texi  /*&ja
 @c sort-sm1.gb  @c sort-sm1.gb
 @node sm1.gb,,, SM1 Functions  @node sm1.gb,,, SM1 Functions
 @node sm1.gb_d,,, SM1 Functions  
 @subsection @code{sm1.gb}  @subsection @code{sm1.gb}
 @findex sm1.gb  @findex sm1.gb
 @findex sm1.gb_d  @findex sm1.gb_d
Line 601  List
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).      $BLa$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  @end itemize
 */  */
 /*&C-texi  /*&C
 @example  @example
 [293] sm1.gb([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);  [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]]  [[x*dx+y*dy-1,y^2*dy^2+2],[x*dx,y^2*dy^2]]
 @end example  @end example
 */  */
 /*&eg-texi  /*&en
 In the example above,  In the example above,
 @tex the set $\{ x \partial_x + y \partial_y -1,  @tex the set $\{ x \partial_x + y \partial_y -1,
                  y^2 \partial_y^2+2\}$                   y^2 \partial_y^2+2\}$
Line 618  The set $\{x \partial_x, y^2 \partial_y\}$ is the lead
Line 621  The set $\{x \partial_x, y^2 \partial_y\}$ is the lead
 (the initial monominals) of the Gr\"obner basis.  (the initial monominals) of the Gr\"obner basis.
 @end tex  @end tex
 */  */
 /*&jp-texi  /*&ja
 $B>e$NNc$K$*$$$F(B,  $B>e$NNc$K$*$$$F(B,
 @tex $B=89g(B $\{ x \partial_x + y \partial_y -1,  @tex $B=89g(B $\{ x \partial_x + y \partial_y -1,
                  y^2 \partial_y^2+2\}$                   y^2 \partial_y^2+2\}$
Line 630  graded reverse lexicographic order $B$K4X$9$k%0%l%V%J
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.  $BBP$9$k(B leading monomial (initial monomial) $B$G$"$k(B.
 @end tex  @end tex
 */  */
 /*&C-texi  /*&C
 @example  @example
 [294] sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]);  [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]]  [[dx+dy^3-4*dy,-dy^4+4*dy^2-1],[dx,-dy^4]]
 @end example  @end example
 */  */
 /*&eg-texi  /*&en
 In the example above, two monomials  In the example above, two monomials
 @tex  @tex
 $m = x^a y^b \partial_x^c \partial_y^d$ and  $m = x^a y^b \partial_x^c \partial_y^d$ and
Line 649  compared by the reverse lexicographic order
Line 652  compared by the reverse lexicographic order
 (i.e., if $50c+2d+a = 50c'+2d'+a'$, then use the reverse lexicogrpahic order).  (i.e., if $50c+2d+a = 50c'+2d'+a'$, then use the reverse lexicogrpahic order).
 @end tex  @end tex
 */  */
 /*&jp-texi  /*&ja
 $B>e$NNc$K$*$$$FFs$D$N%b%N%_%"%k(B  $B>e$NNc$K$*$$$FFs$D$N%b%N%_%"%k(B
 @tex  @tex
 $m = x^a y^b \partial_x^c \partial_y^d$ $B$*$h$S(B  $m = x^a y^b \partial_x^c \partial_y^d$ $B$*$h$S(B
Line 663  $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
Line 666  $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
 $B$5$l$k(B).  $B$5$l$k(B).
 @end tex  @end tex
 */  */
 /*&C-texi  /*&C
 @example  @example
 [294] F=sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);  [294] F=sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);
       map(print,F[2][0])$        map(print,F[2][0])$
       map(print,F[2][1])$        map(print,F[2][1])$
 @end example  @end example
 */  */
 /*&C-texi  /*&C
 @example  @example
 [595]  [595]
    sm1.gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],     sm1.gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],
Line 697  $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
Line 700  $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
 @end example  @end example
 */  */
   
 /*&eg-texi  /*&en
 @table @t  @table @t
 @item Reference  @item Reference
     @code{sm1.reduction}, @code{sm1.rat_to_p}      @code{sm1.reduction}, @code{sm1.rat_to_p}
 @end table  @end table
 */  */
 /*&jp-texi  /*&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.reduction}, @code{sm1.rat_to_p}
Line 712  $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
Line 715  $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
   
   
   
 /*&eg-texi  /*&en
 @c sort-sm1.deRham  @c sort-sm1.deRham
 @node sm1.deRham,,, SM1 Functions  @node sm1.deRham,,, SM1 Functions
 @subsection @code{sm1.deRham}  @subsection @code{sm1.deRham}
Line 757  mode. So, it is strongly recommended to execute the co
Line 760  mode. So, it is strongly recommended to execute the co
 @code{ox_shutdown(sm1.get_Sm1_proc());} to interrupt and restart the server.  @code{ox_shutdown(sm1.get_Sm1_proc());} to interrupt and restart the server.
 @end itemize  @end itemize
 */  */
 /*&jp-texi  /*&ja
 @c sort-sm1.deRham  @c sort-sm1.deRham
 @node sm1.deRham,,, SM1 Functions  @node sm1.deRham,,, SM1 Functions
 @subsection @code{sm1.deRham}  @subsection @code{sm1.deRham}
Line 801  mode. So, it is strongly recommended to execute the co
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.    $B$r0l;~(B shutdown $B$7$F%j%9%?!<%H$7$?J}$,0BA4$G$"$k(B.
 @end itemize  @end itemize
 */  */
 /*&C-texi  /*&C
 @example  @example
 [332] sm1.deRham([x^3-y^2,[x,y]]);  [332] sm1.deRham([x^3-y^2,[x,y]]);
 [1,1,0]  [1,1,0]
Line 809  mode. So, it is strongly recommended to execute the co
Line 812  mode. So, it is strongly recommended to execute the co
 [1,2]  [1,2]
 @end example  @end example
 */  */
 /*&eg-texi  /*&en
 @table @t  @table @t
 @item Reference  @item Reference
     @code{sm1.start}, @code{deRham} (sm1 command)      @code{sm1.start}, @code{deRham} (sm1 command)
Line 819  mode. So, it is strongly recommended to execute the co
Line 822  mode. So, it is strongly recommended to execute the co
     Journal of pure and applied algebra 139 (1999), 201--233.      Journal of pure and applied algebra 139 (1999), 201--233.
 @end table  @end table
 */  */
 /*&jp-texi  /*&ja
 @table @t  @table @t
 @item $B;2>H(B  @item $B;2>H(B
     @code{sm1.start}, @code{deRham} (sm1 command)      @code{sm1.start}, @code{deRham} (sm1 command)
Line 833  mode. So, it is strongly recommended to execute the co
Line 836  mode. So, it is strongly recommended to execute the co
   
   
   
 /*&eg-texi  /*&en
 @c sort-sm1.hilbert  @c sort-sm1.hilbert
 @node sm1.hilbert,,, SM1 Functions  @node sm1.hilbert,,, SM1 Functions
 @subsection @code{sm1.hilbert}  @subsection @code{sm1.hilbert}
Line 875  List
Line 878  List
    polynomials in @code{sm1} is  slower than in @code{asir}.     polynomials in @code{sm1} is  slower than in @code{asir}.
 @end itemize  @end itemize
 */  */
 /*&jp-texi  /*&ja
 @c sort-sm1.hilbert  @c sort-sm1.hilbert
 @node sm1.hilbert,,, SM1 Functions  @node sm1.hilbert,,, SM1 Functions
 @subsection @code{sm1.hilbert}  @subsection @code{sm1.hilbert}
Line 914  List
Line 917  List
 @end itemize  @end itemize
 */  */
   
 /*&C-texi  /*&C
 @example  @example
   
 [346] load("katsura")$  [346] load("katsura")$
Line 944  List
Line 947  List
 @end example  @end example
 */  */
   
 /*&eg-texi  /*&en
 @table @t  @table @t
 @item Reference  @item Reference
     @code{sm1.start}, @code{sm1.gb}, @code{longname}      @code{sm1.start}, @code{sm1.gb}, @code{longname}
 @end table  @end table
 */  */
 /*&jp-texi  /*&ja
 @table @t  @table @t
 @item $B;2>H(B  @item $B;2>H(B
     @code{sm1.start}, @code{sm1.gb}, @code{longname}      @code{sm1.start}, @code{sm1.gb}, @code{longname}
Line 958  List
Line 961  List
 */  */
   
   
 /*&eg-texi  /*&en
 @c sort-sm1.genericAnn  @c sort-sm1.genericAnn
 @node sm1.genericAnn,,, SM1 Functions  @node sm1.genericAnn,,, SM1 Functions
 @subsection @code{sm1.genericAnn}  @subsection @code{sm1.genericAnn}
Line 987  List
Line 990  List
     @var{f} is a polynomial in the variables @code{rest}(@var{v}).      @var{f} is a polynomial in the variables @code{rest}(@var{v}).
 @end itemize  @end itemize
 */  */
 /*&jp-texi  /*&ja
 @c sort-sm1.genericAnn  @c sort-sm1.genericAnn
 @node sm1.genericAnn,,, SM1 Functions  @node sm1.genericAnn,,, SM1 Functions
 @subsection @code{sm1.genericAnn}  @subsection @code{sm1.genericAnn}
Line 1017  List
Line 1020  List
     @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.      @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
 @end itemize  @end itemize
 */  */
 /*&C-texi  /*&C
 @example  @example
 [595] sm1.genericAnn([x^3+y^3+z^3,[s,x,y,z]]);  [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]  [-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]
 @end example  @end example
 */  */
 /*&eg-texi  /*&en
 @table @t  @table @t
 @item Reference  @item Reference
     @code{sm1.start}      @code{sm1.start}
 @end table  @end table
 */  */
 /*&jp-texi  /*&ja
 @table @t  @table @t
 @item $B;2>H(B  @item $B;2>H(B
     @code{sm1.start}      @code{sm1.start}
Line 1038  List
Line 1041  List
   
   
   
 /*&eg-texi  /*&en
 @c sort-sm1.wTensor0  @c sort-sm1.wTensor0
 @node sm1.wTensor0,,, SM1 Functions  @node sm1.wTensor0,,, SM1 Functions
 @subsection @code{sm1.wTensor0}  @subsection @code{sm1.wTensor0}
Line 1079  the inputs @var{f} and @var{g} are left ideals of D.
Line 1082  the inputs @var{f} and @var{g} are left ideals of D.
 @end itemize  @end itemize
 */  */
   
 /*&jp-texi  /*&ja
 @c sort-sm1.wTensor0  @c sort-sm1.wTensor0
 @node sm1.wTensor0,,, SM1 Functions  @node sm1.wTensor0,,, SM1 Functions
 @subsection @code{sm1.wTensor0}  @subsection @code{sm1.wTensor0}
Line 1119  the inputs @var{f} and @var{g} are left ideals of D.
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.  $B0lHL$K(B, $B=PNO$O<+M32C72(B D^r $B$NItJ,2C72$G$"$k(B.
 @end itemize  @end itemize
 */  */
 /*&C-texi  /*&C
 @example  @example
 [258]  sm1.wTensor0([[x*dx -1, y*dy -4],[dx+dy,dx-dy^2],[x,y],[1,2]]);  [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],  [[-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 1130  the inputs @var{f} and @var{g} are left ideals of D.
Line 1133  the inputs @var{f} and @var{g} are left ideals of D.
   
   
   
 /*&eg-texi  /*&en
 @c sort-sm1.reduction  @c sort-sm1.reduction
 @node sm1.reduction,,, SM1 Functions  @node sm1.reduction,,, SM1 Functions
 @subsection @code{sm1.reduction}  @subsection @code{sm1.reduction}
Line 1168  sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G)
Line 1171  sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G)
 are for distributed polynomials.  are for distributed polynomials.
 @end itemize  @end itemize
 */  */
 /*&jp-texi  /*&ja
 @node sm1.reduction,,, SM1 Functions  @node sm1.reduction,,, SM1 Functions
 @subsection @code{sm1.reduction}  @subsection @code{sm1.reduction}
 @findex sm1.reduction  @findex sm1.reduction
Line 1207  sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_
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.  $B$O(B, $BJ,;6B?9`<0MQ$G$"$k(B.
 @end itemize  @end itemize
 */  */
 /*&C-texi  /*&C
 @example  @example
 [259] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y]]);  [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]]  [x^2+y^2-4,1,[0,0],[y^4-4*y^2+1,x+y^3-4*y]]
Line 1215  sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_
Line 1218  sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_
 [0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]]  [0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]]
 @end example  @end example
 */  */
 /*&eg-texi  /*&en
 @table @t  @table @t
 @item Reference  @item Reference
     @code{sm1.start}, @code{d_true_nf}      @code{sm1.start}, @code{d_true_nf}
 @end table  @end table
 */  */
 /*&jp-texi  /*&ja
 @table @t  @table @t
 @item $B;2>H(B  @item $B;2>H(B
     @code{sm1.start}, @code{d_true_nf}      @code{sm1.start}, @code{d_true_nf}
Line 1229  sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_
Line 1232  sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_
 */  */
   
   
 /*&eg-texi  /*&en
 @node sm1.xml_tree_to_prefix_string,,, SM1 Functions  @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
 @subsection @code{sm1.xml_tree_to_prefix_string}  @subsection @code{sm1.xml_tree_to_prefix_string}
 @findex sm1.xml_tree_to_prefix_string  @findex sm1.xml_tree_to_prefix_string
Line 1257  asir has not yet understood this CMO.
Line 1260  asir has not yet understood this CMO.
 command search path.)  command search path.)
 @end itemize  @end itemize
 */  */
 /*&jp-texi  /*&ja
 @node sm1.xml_tree_to_prefix_string,,, SM1 Functions  @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
 @subsection @code{sm1.xml_tree_to_prefix_string}  @subsection @code{sm1.xml_tree_to_prefix_string}
 @findex sm1.xml_tree_to_prefix_string  @findex sm1.xml_tree_to_prefix_string
Line 1284  String
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$I(B.)  ($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  @end itemize
 */  */
 /*&C-texi  /*&C
 @example  @example
 [263] load("om");  [263] load("om");
 1  1
Line 1301  Trying to connect to the server... Done.
Line 1304  Trying to connect to the server... Done.
 basic_plus(basic_times(basic_power(x,4),1),basic_times(basic_power(x,0),-1))  basic_plus(basic_times(basic_power(x,4),1),basic_times(basic_power(x,0),-1))
 @end example  @end example
 */  */
 /*&eg-texi  /*&en
 @table @t  @table @t
 @item Reference  @item Reference
     @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}      @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
 @end table  @end table
 */  */
 /*&jp-texi  /*&ja
 @table @t  @table @t
 @item $B;2>H(B  @item $B;2>H(B
     @code{om_*}, @code{OpenXM/src/OpenMath},  @code{eval_str}      @code{om_*}, @code{OpenXM/src/OpenMath},  @code{eval_str}
Line 1317  basic_plus(basic_times(basic_power(x,4),1),basic_times
Line 1320  basic_plus(basic_times(basic_power(x,4),1),basic_times
   
   
   
 /*&eg-texi  /*&en
 @c sort-sm1.syz  @c sort-sm1.syz
 @node sm1.syz,,, SM1 Functions  @node sm1.syz,,, SM1 Functions
 @node sm1.syz_d,,, SM1 Functions  
 @subsection @code{sm1.syz}  @subsection @code{sm1.syz}
 @findex sm1.syz  @findex sm1.syz
 @findex sm1.syz_d  @findex sm1.syz_d
Line 1347  Here @var{s} is the syzygy of @var{f} in the ring of d
Line 1349  Here @var{s} is the syzygy of @var{f} in the ring of d
 operators with the variable @var{v}.  operators with the variable @var{v}.
 @var{g} is a Groebner basis of @var{f} with the weight vector @var{w},  @var{g} is a Groebner basis of @var{f} with the weight vector @var{w},
 and @var{m} is a matrix that translates the input matrix @var{f} to the Gr\"obner  and @var{m} is a matrix that translates the input matrix @var{f} to the Gr\"obner
 basis @var {g}.  basis @var{g}.
 @var{t} is the syzygy of the Gr\"obner basis @var{g}.  @var{t} is the syzygy of the Gr\"obner basis @var{g}.
 In summary, @var{g} = @var{m} @var{f} and  In summary, @var{g} = @var{m} @var{f} and
 @var{s} @var{f} = 0 hold as matrices.  @var{s} @var{f} = 0 hold as matrices.
Line 1361  In summary, @var{g} = @var{m} @var{f} and
Line 1363  In summary, @var{g} = @var{m} @var{f} and
    The homogenization variable h is automatically added.     The homogenization variable h is automatically added.
 @end itemize  @end itemize
 */  */
 /*&jp-texi  /*&ja
 @c sort-sm1.syz  @c sort-sm1.syz
 @node sm1.syz,,, SM1 Functions  @node sm1.syz,,, SM1 Functions
 @node sm1.syz_d,,, SM1 Functions  
 @subsection @code{sm1.syz}  @subsection @code{sm1.syz}
 @findex sm1.syz  @findex sm1.syz
 @findex sm1.syz_d  @findex sm1.syz_d
Line 1404  syzygy $B$G$"$k(B.
Line 1405  syzygy $B$G$"$k(B.
 $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.  $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
 @end itemize  @end itemize
 */  */
 /*&C-texi  /*&C
 @example  @example
 [293] sm1.syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);  [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  [[[y*x*dy*dx-2,-x*dx-y*dy+1]],    generators of the syzygy
Line 1413  syzygy $B$G$"$k(B.
Line 1414  syzygy $B$G$"$k(B.
  [[y*x*dy*dx-2,-x*dx-y*dy+1]]]]   [[y*x*dy*dx-2,-x*dx-y*dy+1]]]]
 @end example  @end example
 */  */
 /*&C-texi  /*&C
 @example  @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]]]);  [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  [[[y*x*dy*dx-1,-x^2*dx^2-x*dx-y^2*dy^2-y*dy+4]], generators of the syzygy
Line 1426  syzygy $B$G$"$k(B.
Line 1427  syzygy $B$G$"$k(B.
   
   
   
 /*&eg-texi  /*&en
 @node sm1.mul,,, SM1 Functions  @node sm1.mul,,, SM1 Functions
 @subsection @code{sm1.mul}  @subsection @code{sm1.mul}
 @findex sm1.mul  @findex sm1.mul
Line 1449  List
Line 1450  List
 @itemize @bullet  @itemize @bullet
 @item Ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.  @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 @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  @end itemize
 */  */
   
 /*&jp-texi  /*&ja
 @node sm1.mul,,, SM1 Functions  @node sm1.mul,,, SM1 Functions
 @subsection @code{sm1.mul}  @subsection @code{sm1.mul}
 @findex sm1.mul  @findex sm1.mul
Line 1477  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}  @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.  $B>e$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 @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  @end itemize
 */  */
   
 /*&C-texi  /*&C
   
 @example  @example
 [277] sm1.mul(dx,x,[x]);  [277] sm1.mul(dx,x,[x]);
Line 1496  x+2*y
Line 1504  x+2*y
   
   
   
 /*&eg-texi  /*&en
 @node sm1.distraction,,, SM1 Functions  @node sm1.distraction,,, SM1 Functions
 @subsection @code{sm1.distraction}  @subsection @code{sm1.distraction}
 @findex sm1.distraction  @findex sm1.distraction
Line 1527  See Saito, Sturmfels, Takayama : Grobner Deformations 
Line 1535  See Saito, Sturmfels, Takayama : Grobner Deformations 
 @end itemize  @end itemize
 */  */
   
 /*&jp-texi  /*&ja
 @node sm1.distraction,,, SM1 Functions  @node sm1.distraction,,, SM1 Functions
   
 @subsection @code{sm1.distraction}  @subsection @code{sm1.distraction}
Line 1558  See Saito, Sturmfels, Takayama : Grobner Deformations 
Line 1566  See Saito, Sturmfels, Takayama : Grobner Deformations 
 @end itemize  @end itemize
 */  */
   
 /*&C-texi  /*&C
   
 @example  @example
 [280] sm1.distraction([x*dx,[x],[x],[dx],[x]]);  [280] sm1.distraction([x*dx,[x],[x],[dx],[x]]);
Line 1574  x^2+3*x+2
Line 1582  x^2+3*x+2
 @end example  @end example
 */  */
   
 /*&eg-texi  /*&en
 @table @t  @table @t
 @item Reference  @item Reference
     @code{distraction2(sm1)},      @code{distraction2(sm1)},
 @end table  @end table
 */  */
   
 /*&jp-texi  /*&ja
 @table @t  @table @t
 @item $B;2>H(B  @item $B;2>H(B
     @code{distraction2(sm1)},      @code{distraction2(sm1)},
Line 1590  x^2+3*x+2
Line 1598  x^2+3*x+2
   
   
   
 /*&eg-texi  /*&en
 @node sm1.gkz,,, SM1 Functions  @node sm1.gkz,,, SM1 Functions
 @subsection @code{sm1.gkz}  @subsection @code{sm1.gkz}
 @findex sm1.gkz  @findex sm1.gkz
Line 1615  List
Line 1623  List
 @end itemize  @end itemize
 */  */
   
 /*&jp-texi  /*&ja
 @node sm1.gkz,,, SM1 Functions  @node sm1.gkz,,, SM1 Functions
 @subsection @code{sm1.gkz}  @subsection @code{sm1.gkz}
 @findex sm1.gkz  @findex sm1.gkz
Line 1638  List
Line 1646  List
 @end itemize  @end itemize
 */  */
   
 /*&C-texi  /*&C
   
 @example  @example
   
Line 1654  List
Line 1662  List
   
   
   
 /*&eg-texi  /*&en
 @node sm1.appell1,,, SM1 Functions  @node sm1.appell1,,, SM1 Functions
 @subsection @code{sm1.appell1}  @subsection @code{sm1.appell1}
 @findex sm1.appell1  @findex sm1.appell1
Line 1678  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).  where @var{a} =(a,c,b1,...,bn).
 When n=2, the Lauricella function is called the Appell function F_1.  When n=2, the Lauricella function is called the Appell function F_1.
 The parameters a, c, b1, ..., bn may be rational numbers.  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  @end itemize
 */  */
   
 /*&jp-texi  /*&ja
 @node sm1.appell1,,, SM1 Functions  @node sm1.appell1,,, SM1 Functions
 @subsection @code{sm1.appell1}  @subsection @code{sm1.appell1}
 @findex sm1.appell1  @findex sm1.appell1
Line 1705  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,  $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
 @var{a} =(a,c,b1,...,bn).  @var{a} =(a,c,b1,...,bn).
 $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.  $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(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  @end itemize
 */  */
   
 /*&C-texi  /*&C
   
 @example  @example
   
Line 1727  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]]   [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]));  [283] sm1.rank(sm1.appell1([1/2,3,5,-1/3]));
 1  3
   
 [285] Mu=2$ Beta = 1/3$  [285] Mu=2$ Beta = 1/3$
 [287] sm1.rank(sm1.appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));  [287] sm1.rank(sm1.appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));
Line 1738  F_D(a,b1,b2,...,bn,c;x1,...,xn)
Line 1751  F_D(a,b1,b2,...,bn,c;x1,...,xn)
   
 */  */
   
 /*&eg-texi  /*&en
 @node sm1.appell4,,, SM1 Functions  @node sm1.appell4,,, SM1 Functions
 @subsection @code{sm1.appell4}  @subsection @code{sm1.appell4}
 @findex sm1.appell4  @findex sm1.appell4
Line 1762  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).  where @var{a} =(a,b,c1,...,cn).
 When n=2, the Lauricella function is called the Appell function F_4.  When n=2, the Lauricella function is called the Appell function F_4.
 The parameters a, b, c1, ..., cn may be rational numbers.  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  @end itemize
 */  */
   
 /*&jp-texi  /*&ja
 @node sm1.appell4,,, SM1 Functions  @node sm1.appell4,,, SM1 Functions
 @subsection @code{sm1.appell4}  @subsection @code{sm1.appell4}
 @findex sm1.appell4  @findex sm1.appell4
Line 1789  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,  $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
 @var{a} =(a,b,c1,...,cn).  @var{a} =(a,b,c1,...,cn).
 $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.  $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(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  @end itemize
 */  */
   
 /*&C-texi  /*&C
   
 @example  @example
   
Line 1812  F_C(a,b,c1,c2,...,cn;x1,...,xn)
Line 1830  F_C(a,b,c1,c2,...,cn;x1,...,xn)
   
   
   
 /*&eg-texi  /*&en
 @node sm1.rank,,, SM1 Functions  @node sm1.rank,,, SM1 Functions
 @subsection @code{sm1.rank}  @subsection @code{sm1.rank}
 @findex sm1.rank  @findex sm1.rank
Line 1841  holonomic. It is generally faster than @code{sm1.rank}
Line 1859  holonomic. It is generally faster than @code{sm1.rank}
 @end itemize  @end itemize
 */  */
   
 /*&jp-texi  /*&ja
 @node sm1.rank,,, SM1 Functions  @node sm1.rank,,, SM1 Functions
 @subsection @code{sm1.rank}  @subsection @code{sm1.rank}
 @findex sm1.rank  @findex sm1.rank
Line 1869  holonomic. It is generally faster than @code{sm1.rank}
Line 1887  holonomic. It is generally faster than @code{sm1.rank}
 @end itemize  @end itemize
 */  */
   
 /*&C-texi  /*&C
   
 @example  @example
   
Line 1892  holonomic. It is generally faster than @code{sm1.rank}
Line 1910  holonomic. It is generally faster than @code{sm1.rank}
 */  */
   
   
 /*&eg-texi  /*&en
 @node sm1.auto_reduce,,, SM1 Functions  @node sm1.auto_reduce,,, SM1 Functions
 @subsection @code{sm1.auto_reduce}  @subsection @code{sm1.auto_reduce}
 @findex sm1.auto_reduce  @findex sm1.auto_reduce
Line 1918  Grobner bases.  This is the default.
Line 1936  Grobner bases.  This is the default.
 @end itemize  @end itemize
 */  */
   
 /*&jp-texi  /*&ja
 @node sm1.auto_reduce,,, SM1 Functions  @node sm1.auto_reduce,,, SM1 Functions
 @subsection @code{sm1.auto_reduce}  @subsection @code{sm1.auto_reduce}
 @findex sm1.auto_reduce  @findex sm1.auto_reduce
Line 1946  reduced $B%0%l%V%J4pDl$H$O$+$.$i$J$$(B. $B$3$A$i$,%
Line 1964  reduced $B%0%l%V%J4pDl$H$O$+$.$i$J$$(B. $B$3$A$i$,%
   
   
   
 /*&eg-texi  /*&en
 @node sm1.slope,,, SM1 Functions  @node sm1.slope,,, SM1 Functions
 @subsection @code{sm1.slope}  @subsection @code{sm1.slope}
 @findex sm1.slope  @findex sm1.slope
Line 1991  of the slopes are returned.
Line 2009  of the slopes are returned.
   
 */  */
   
 /*&jp-texi  /*&ja
 @node sm1.slope,,, SM1 Functions  @node sm1.slope,,, SM1 Functions
 @subsection @code{sm1.slope}  @subsection @code{sm1.slope}
 @findex sm1.slope  @findex sm1.slope
Line 2034  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.  Slope $B$N@dBPCM$rLa$9(B.
 */  */
   
 /*&C-texi  /*&C
   
 @example  @example
   
Line 2053  Slope $B$N@dBPCM$rLa$9(B.
Line 2071  Slope $B$N@dBPCM$rLa$9(B.
 @end example  @end example
   
 */  */
 /*&eg-texi  /*&en
 @table @t  @table @t
 @item Reference  @item Reference
     @code{sm.gb}      @code{sm.gb}
 @end table  @end table
 */  */
 /*&jp-texi  /*&ja
 @table @t  @table @t
 @item $B;2>H(B  @item $B;2>H(B
     @code{sm.gb}      @code{sm.gb}
Line 2067  Slope $B$N@dBPCM$rLa$9(B.
Line 2085  Slope $B$N@dBPCM$rLa$9(B.
 */  */
   
   
 /*&eg-texi  /*&en
 @include sm1-auto-en.texi  @include sm1-auto.en
 */  */
   
 /*&jp-texi  /*&ja
 @include sm1-auto-ja.texi  @include sm1-auto.ja
 */  */
   
   
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