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Diff for /OpenXM/src/k097/lib/minimal/minimal-note-ja.txt between version 1.11 and 1.12

version 1.11, 2000/08/02 05:14:30 version 1.12, 2000/08/09 03:45:27
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 $OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.10 2000/08/01 08:51:02 takayama Exp $  $OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.11 2000/08/02 05:14:30 takayama Exp $
   
 SpairAndReduction() :  SpairAndReduction() :
    $BM?$($i$l$?(B pair $B$r(B reduction $B$9$k(B.     $BM?$($i$l$?(B pair $B$r(B reduction $B$9$k(B.
Line 744  Sminimal $B$O(B [(Homogenize_vec) 0] system_variable
Line 744  Sminimal $B$O(B [(Homogenize_vec) 0] system_variable
 test17b(), test18() $B$O@5>oF0:n(B.  test17b(), test18() $B$O@5>oF0:n(B.
   
   
   August 7, Mon 13:00JST ( 5:00 St.Andrews, Scotland, 4039 $B9f<<(B)
   example-ja.tex $B$r=q$/$?$a$N=PNO(B.
   
   % k0
   sm1>macro package : dr.sm1,   9/26,1995 --- Version 6/15, 2000.
   sm1>macro package : module1.sm1, 1994 -- Nov 8, 1998
   This is kan/k0 Version 1998,12/15
   WARNING: This is an EXPERIMENTAL version
   sm1>var.sm1 : Version 3/7, 1997
   
   
   In(1)=Loading startup files (startup.k)   1997, 3/11.
   sm1 version = 3.000726
   Default ring is Z[x,h].
   WARNING(sm): You rewrited the protected symbol pushVariables.
   WARNING(sm): You rewrited the protected symbol popVariables.
   In(2)=load["minimal-test.k"];;
   cpp: -lang-c++: linker input file unused since linking not done
   cpp: -lang-c++: linker input file unused since linking not done
   cohom.sm1 is the top of an experimental package to compute restrictions
   of all degrees based on restall.sm1 and restall_s.sm1
   See, http://www.math.kobe-u.ac.jp to get these files of the latest version.
   Note that the package b-function.sm1 cannot be used with this package.
   r-interface.sm1 (C) N.Takayama,  restriction, deRham
   
   oxasir.sm1, --- open asir protocol module 3/1 1998, 6/5 1999
      asirconnect, asir, fctr, primadec, (C) M.Noro, N.Takayama
   ox.sm1, --- open sm1 protocol module 11/11,1999  (C) N.Takayama. oxhelp for help
   hol.sm1, basic package for holonomic systems (C) N.Takayama, 2000, 06/08
   rank characteristic ch rrank gb pgb syz  genericAnn  annfs gb_h syz_h isSameIdeal isSameIdeal_h
   sm1>gkz.sm1 generates gkz systems (C) N.Takayama, 1998, 11/8, cf. rrank in hol.sm1
   gkz
   sm1>appell.sm1 generates Appell hypergeometric differential equations (C) N.Takayama, 1998, 11/8, cf. rank in hol.sm1
   appell1 appell4
   sm1>resol0.sm1, package to construct schreyer resolutions -- not minimal
                                  (C) N.Takayama, 1999, 5/18. resol0, resol1
   complex.sm1 : 1999, 9/28, res-div, res-solv, res-kernel-image, res-dual
                 2000, 6/8,  isExact_h, isExact
   In this package, complex is expressed in terms of matrices.
   restall.sm1 ... compute all the cohomology groups of the restriction
                   of a D-module to tt = (t_1,...,t_d) = (0,...,0).
   non-Schreyer Version: 19980415 by T.Oaku
   usage: [(P1)...] [(t1)...] bfm --> the b-function
          [(P1)...] [(t1)...] k0 k1 deg restall --> cohomologies of restriction
          [(P1)...] [(t1)...] intbfm --> the b-function for integration
          [(P1)...] [(t1)...] k0 k1 deg intall --> cohomologies of integration
   restall_s.sm1...compute all the cohomology groups of the restriction
                   of a D-module to tt = (t_1,...,t_d) = (0,...,0).
   Schreyer Version: 19990521 by N.Takayama & T.Oaku
   usage: [(P1)...] [(t1)...] k0 k1 deg restall_s -> cohomologies of restriction
          [(P1)...] [(t1)...] k0 k1 deg intall_s --> cohomologies of integration
   No truncation from below in restall
   The variable Schreyer is set to 2.
   Loading tower.sm1 in the standard context. You cannot use Schyrer 1. It is controlled from cohom.sm1
   
   oxpath.oxlog.xterm is set to /home/nobuki/OpenXM/lib/sm1/bin/oxlog
   In(3)=a=Sannfs2("x^3-y^2");
   Starting ox_asir server.
   Hello from open. serverName is localhost and portnumber is 0
   Done the initialization. port =1024
   Hello from open. serverName is localhost and portnumber is 0
   Done the initialization. port =1025
   [    7 , 1025 , 6 , 1024 ]
   [1] 250
   Trying to accept from localhost... len= 16
    4  2  7f  0  0  1  0  0  0  0  0  0  0  0  8  0
   Authentification: localhost is allowed to be accepted.
   Accepted.
   Trying to accept from localhost... len= 16
    4  3  7f  0  0  1  0  0  0  0  0  0  0  0  6  0
   Authentification: localhost is allowed to be accepted.
   Accepted.
   
   Control port 1024 : Connected.
   
   Stream port 1025 : Connected.
   Byte order for control process is network byte order.
   Byte order for engine process is network byte order.
   WeightOfSweyl=[    x , -1 , y , -1 , Dx , 1 , Dy , 1 ]
   Automatic homogenization.
   [    -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ]
   Warning: Homogenization and ReduceLowerTerms options are automatically turned off.
   ....Done. betti=4
   Warning: Homogenization and ReduceLowerTerms options are automatically turned ON.
   Warning: Homogenization and ReduceLowerTerms options are automatically turned off.
   .Done. betti=1
   Warning: Homogenization and ReduceLowerTerms options are automatically turned ON.
   Warning: Homogenization and ReduceLowerTerms options are automatically turned off.
   Done. betti=0
   Warning: Homogenization and ReduceLowerTerms options are automatically turned ON.
   rf= [
     [
      [
        [    -9*y^2*Dy , 0 , 2*x , 0 ]
        [    0 , 0 , -3*y*Dy , Dx ]
        [    0 , -3*y*Dy , Dx , 0 ]
        [    -3*y*Dx , 2*x , 0 , 0 ]
      ]
      [
        [    -Dx , 0 , 0 , 3*y*Dy ]
      ]
       [   ]
     ]
     [
       [    -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ]
       [    -9*y^2*Dy , -3*es^2*y*Dy , -3*es*y*Dy , -3*y*Dx ]
       [    -Dx ]
     ]
     [
       [   ]
      [
       [
         [    0 , 2 ]
         [    -9*y^2*Dy , 2*x ]
       ]
       [
         [    2 , 3 ]
         [    -3*y*Dy , Dx ]
       ]
       [
         [    1 , 2 ]
         [    -3*y*Dy , Dx ]
       ]
       [
         [    0 , 1 ]
         [    -3*y*Dx , 2*x ]
       ]
      ]
      [
       [
         [    0 , 3 ]
         [    -Dx , 3*y*Dy ]
       ]
      ]
       [   ]
     ]
     [
       [    -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , -9*y^2*Dx*Dy-3*y*Dx*h^2-4*x^2*Dy*h , -27*y^3*Dy^2-27*y^2*Dy*h^2+3*y*h^4+8*x^3*Dy*h ]
     ]
    ]
   Generating reduction table which gives an order of reduction.
   WeghtOfSweyl=[    x , -1 , y , -1 , Dx , 1 , Dy , 1 ]
   tower[    [    -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ]  , [    -9*y^2*Dy , -3*es^2*y*Dy , -3*es*y*Dy , -3*y*Dx ]  , [    -Dx ]  ]
   reductionTable= [
      [    1 , 2 , 3 , 4 ]
      [    3 , 4 , 3 , 2 ]
      [    3 ]
    ]
   [    0 , 0 ]
   Processing [level,i]= [    0 , 0 ]    Strategy = 1
   [    0 , 1 ]
   Processing [level,i]= [    0 , 1 ]    Strategy = 2
   [    1 , 3 ]
   Processing [level,i]= [    1 , 3 ]    Strategy = 2
   SpairAndReduction:
   [    p and bases  , [    [    0 , 1 ]  , [    -3*y*Dx , 2*x ]  ]  , [    -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ]  ]
   [    level= , 1 ]
   [    tower2= , [    [   ]  ]  ]
   [    -3*y*Dx , 2*es*x ]
   [gi, gj] = [    -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h ]
   1
   Reduce the element 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h
   by  [    -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ]
   result is [    9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , 1 , [    0 , 0 , 0 , 0 ]  ]
   vdegree of the original = 0
   vdegree of the remainder = 0
   [    9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , [    -3*y*Dx , 2*x , 0 , 0 ]  , 3 , 2 , 0 , 0 ]
   [    0 , 2 ]
   Processing [level,i]= [    0 , 2 ]    Strategy = 3
   [    1 , 0 ]
   Processing [level,i]= [    1 , 0 ]    Strategy = 3
   SpairAndReduction:
   [    p and bases  , [    [    0 , 2 ]  , [    -9*y^2*Dy , 2*x ]  ]  , [    -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , %[null] ]  ]
   [    level= , 1 ]
   [    tower2= , [    [   ]  ]  ]
   [    9*y^2*Dy , 2*es^2*x ]
   [gi, gj] = [    -2*x*Dx-3*y*Dy+h^2 , 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
   1
   Reduce the element -27*y^3*Dy^2+6*x*y*Dx*h^2-18*y^2*Dy*h^2+8*x^3*Dy*h
   by  [    -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , %[null] ]
   result is [    27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h , -1 , [    -3*y*h^2 , 0 , 0 , 0 ]  ]
   vdegree of the original = -1
   vdegree of the remainder = -1
   [    27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h , [    -9*y^2*Dy-3*y*h^2 , 0 , -2*x , 0 ]  , 0 , 3 , -1 , -1 ]
   [    1 , 2 ]
   Processing [level,i]= [    1 , 2 ]    Strategy = 3
   SpairAndReduction:
   [    p and bases  , [    [    1 , 2 ]  , [    -3*y*Dy , Dx ]  ]  , [    -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , 27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]  ]
   [    level= , 1 ]
   [    tower2= , [    [   ]  ]  ]
   [    3*es*y*Dy , es^2*Dx ]
   [gi, gj] = [    -3*y*Dx^2+2*x*Dy*h , 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
   1
   Reduce the element -6*y*Dx^2*h^2+4*x^2*Dx*Dy*h+6*x*y*Dy^2*h+8*x*Dy*h^3
   by  [    -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , 27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]
   result is [    0 , 1 , [    2*x*Dy*h , -2*h^2 , 0 , 0 ]  ]
   vdegree of the original = 1
   vdegree of the remainder = %[null]
   [    0 , [    2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]  , 2 , -1 , 1 , %[null] ]
   [    2 , 0 ]
   Processing [level,i]= [    2 , 0 ]    Strategy = 3
   SpairAndReduction:
   [    p and bases  , [    [    0 , 3 ]  , [    -Dx , 3*y*Dy ]  ]  , [    [    9*y^2*Dy+3*y*h^2 , 0 , 2*x , 1 ]  , %[null] , [    2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]  , [    3*y*Dx , -2*x , 1 , 0 ]  ]  ]
   [    level= , 2 ]
   [    tower2= , [    [    -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ]  ]  ]
   [    Dx , -3*es^3*y*Dy ]
   [gi, gj] = [    9*y^2*Dy+2*es^2*x+es^3+3*y*h^2 , 3*y*Dx-2*es*x+es^2 ]
   1
   Reduce the element 6*es*x*y*Dy+2*es^2*x*Dx-3*es^2*y*Dy+es^3*Dx-6*y*Dx*h^2+2*es^2*h^2
   by  [    [    9*y^2*Dy+3*y*h^2 , 0 , 2*x , 1 ]  , %[null] , [    2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]  , [    3*y*Dx , -2*x , 1 , 0 ]  ]
   result is [    -3*es^2*y*Dy+es^3*Dx+4*es^2*h^2-4*x^2*Dy*h , 1 , [    0 , 0 , -2*x , 2*h^2 ]  ]
   vdegree of the original = 0
   vdegree of the remainder = 0
   [    -3*es^2*y*Dy+es^3*Dx+4*es^2*h^2-4*x^2*Dy*h , [    Dx , 0 , -2*x , -3*y*Dy+2*h^2 ]  , 0 , 1 , 0 , 0 ]
   [    0 , 3 ]
   Processing [level,i]= [    0 , 3 ]    Strategy = 4
   [    1 , 1 ]
   Processing [level,i]= [    1 , 1 ]    Strategy = 4
   Betti numbers are ------
     [    2 , 1 , 0 ]
   [seq,level,q]=[    3 , 1 , 1 ]
   [    level, q = , 1 , 1 ]
   bases=
    [
      [    -Dx , 1 , 2*x , 3*y*Dy-2*h^2 ]
    ]
   dr=
     [    Dx , -1 , -2*x , -3*y*Dy+2*h^2 ]
   newbases=
    [
      [    0 , 0 , 0 , 0 ]
    ]
   [seq,level,q]=[    2 , 0 , 3 ]
   [    level, q = , 0 , 3 ]
   bases=
    [
      [    9*y^2*Dy+3*y*h^2 , 0 , 2*x , 1 ]
      [    -4*x^2*Dy*h , 0 , -3*y*Dy+4*h^2 , Dx ]
      [    2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]
      [    3*y*Dx , -2*x , 1 , 0 ]
    ]
   dr=
     [    -9*y^2*Dy-3*y*h^2 , 0 , -2*x , -1 ]
   newbases=
    [
      [    0 , 0 , 0 , 0 ]
      [    -9*y^2*Dx*Dy-3*y*Dx*h^2-4*x^2*Dy*h , 0 , -2*x*Dx-3*y*Dy+2*h^2 , 0 ]
      [    2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]
      [    3*y*Dx , -2*x , 1 , 0 ]
    ]
   [seq,level,q]=[    1 , 0 , 2 ]
   [    level, q = , 0 , 2 ]
   bases=
    [
      [    0 , 0 , 0 , 0 ]
      [    -9*y^2*Dx*Dy-3*y*Dx*h^2-4*x^2*Dy*h , 0 , -2*x*Dx-3*y*Dy+2*h^2 , 0 ]
      [    2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]
      [    3*y*Dx , -2*x , 1 , 0 ]
    ]
   dr=
     [    -3*y*Dx , 2*x , -1 , 0 ]
   newbases=
    [
      [    0 , 0 , 0 , 0 ]
      [    6*x*y*Dx^2-4*x^2*Dy*h , -4*x^2*Dx-6*x*y*Dy , 0 , 0 ]
      [    -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy , 0 , 0 ]
      [    0 , 0 , 0 , 0 ]
    ]
   [    level= , 0 ]
    [
      [    -2*x*Dx-3*y*Dy+h^2 ]
      [    -3*y*Dx^2+2*x*Dy*h ]
    ]
    [
      [    -2*x*Dx-3*y*Dy+h^2 ]
      [    -3*y*Dx^2+2*x*Dy*h ]
    ]
   [    level= , 1 ]
    [
      [    0 , 0 , 0 , 0 ]
      [    -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy , 0 , 0 ]
      [    0 , 0 , 0 , 0 ]
    ]
    [
      [    0 , 0 ]
      [    -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy ]
      [    0 , 0 ]
    ]
   [    level= , 2 ]
    [
      [    0 , 0 , 0 , 0 ]
    ]
    [
      [    0 , 0 , 0 ]
    ]
   ------------ Note -----------------------------
   To get shift vectors, use Reparse and SgetShifts(resmat,w)
   To get initial of the complex, use Reparse and Sinit_w(resmat,w)
   0: minimal resolution, 3: Schreyer resolution
   ------------ Resolution Summary  --------------
   Betti numbers : [    2 , 1 ]
   Betti numbers of the Schreyer frame: [    4 , 4 , 1 ]
   -----------------------------------------------
   In(4)=sm1_pmat(a);
    [
     [
      [
        [    -2*x*Dx-3*y*Dy+h^2 ]
        [    -3*y*Dx^2+2*x*Dy*h ]
      ]
      [
        [    -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy ]
      ]
     ]
     [
      [
        [    -2*x*Dx-3*y*Dy+h^2 ]
        [    -3*y*Dx^2+2*x*Dy*h ]
      ]
      [
        [    0 , 0 ]
        [    -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy ]
        [    0 , 0 ]
      ]
      [
        [    0 , 0 , 0 ]
      ]
     ]
     [
      [
       [
         [    -2*x*Dx-3*y*Dy+h^2 ]
         [    -3*y*Dx^2+2*x*Dy*h ]
         [    9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
         [    27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]
       ]
       [
         [    0 , 0 , 0 , 0 ]
         [    6*x*y*Dx^2-4*x^2*Dy*h , -4*x^2*Dx-6*x*y*Dy , 0 , 0 ]
         [    -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy , 0 , 0 ]
         [    0 , 0 , 0 , 0 ]
       ]
       [
         [    0 , 0 , 0 , 0 ]
       ]
      ]
      [
        [    0 , 0 , 1 , 2 ]
        [    0 , 3 , 0 , 0 ]
        [    0 ]
      ]
      [
        [    %[null] , %[null] , [    -3*y*Dx , 2*x , -1 , 0 ]  , [    -9*y^2*Dy-3*y*h^2 , 0 , -2*x , -1 ]  ]
        [    %[null] , [    Dx , -1 , -2*x , -3*y*Dy+2*h^2 ]  , %[null] , %[null] ]
        [    %[null] ]
      ]
       [    1 , 4 , 4 , 1 ]
      [
        [    0 , 0 , 1 , 2 ]
        [    0 , 3 , %[null] , 0 ]
        [    0 ]
      ]
     ]
     [
      [
        [    -2*x*Dx-3*y*Dy+h^2 ]
        [    -3*y*Dx^2+2*x*Dy*h ]
        [    9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
        [    27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]
      ]
      [
        [    9*y^2*Dy+3*y*h^2 , 0 , 2*x , 1 ]
        [    -4*x^2*Dy*h , 0 , -3*y*Dy+4*h^2 , Dx ]
        [    2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]
        [    3*y*Dx , -2*x , 1 , 0 ]
      ]
      [
        [    -Dx , 1 , 2*x , 3*y*Dy-2*h^2 ]
      ]
     ]
     [
      [
       [
         [    -9*y^2*Dy , 0 , 2*x , 0 ]
         [    0 , 0 , -3*y*Dy , Dx ]
         [    0 , -3*y*Dy , Dx , 0 ]
         [    -3*y*Dx , 2*x , 0 , 0 ]
       ]
       [
         [    -Dx , 0 , 0 , 3*y*Dy ]
       ]
        [   ]
      ]
      [
        [    -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ]
        [    -9*y^2*Dy , -3*es^2*y*Dy , -3*es*y*Dy , -3*y*Dx ]
        [    -Dx ]
      ]
      [
        [   ]
       [
        [
          [    0 , 2 ]
          [    -9*y^2*Dy , 2*x ]
        ]
        [
          [    2 , 3 ]
          [    -3*y*Dy , Dx ]
        ]
        [
          [    1 , 2 ]
          [    -3*y*Dy , Dx ]
        ]
        [
          [    0 , 1 ]
          [    -3*y*Dx , 2*x ]
        ]
       ]
       [
        [
          [    0 , 3 ]
          [    -Dx , 3*y*Dy ]
        ]
       ]
        [   ]
      ]
      [
        [    -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , -9*y^2*Dx*Dy-3*y*Dx*h^2-4*x^2*Dy*h , -27*y^3*Dy^2-27*y^2*Dy*h^2+3*y*h^4+8*x^3*Dy*h ]
      ]
     ]
    ]
   In(5)=
   
Legend:
Removed from v.1.11  
changed lines
  Added in v.1.12
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