| version 1.4, 2000/06/09 08:04:54 |
version 1.12, 2000/08/09 03:45:27 |
|
|
| $OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.3 2000/06/08 08:37:53 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 544 In(10)=reductionTable : |
|
| Line 544 In(10)=reductionTable : |
|
| In(11)=bases: |
In(11)=bases: |
| [ %[null] , [ 0 , y^2 , -x ] , [ -y , x , 1 ] ] |
[ %[null] , [ 0 , y^2 , -x ] , [ -y , x , 1 ] ] |
| In(12)= $B$3$l$O(B, [3, 2, 1] $B$N85$N$&$A(B, 2,1 $B$,$b$H$^$C$F$$$k(B. |
In(12)= $B$3$l$O(B, [3, 2, 1] $B$N85$N$&$A(B, 2,1 $B$,$b$H$^$C$F$$$k(B. |
| [ 2 ] $B$N7W;;$K(B 0 $BHVL\$,I,MW$G$3$l$,$^$@$J$$(B. |
$B:G8e$N(B [ 2 ] $B$N7W;;$K(B 0 $BHVL\$,I,MW$G$3$l$,$^$@$J$$(B. |
| |
$BMW$9$k$K(B 1 $BHVL\$H(B 3 $BHVL\$r>C$9(B operator [1, 0, -y^2] |
| |
[ y^3 , 0 , -x^2 ] |
| |
[ 0 , y^2 , -x ] |
| |
[ y , -x , 0 ] |
| |
$B$N(B reduction $B$,I,MW(B. |
| |
|
| |
----------------------------------------- |
| |
June 11, 2000 (Tue), 20:05 |
| |
V-strict $B$+$I$&$+$r%A%'%C%/$9$k4X?t$r=q$-$?$$(B. |
| |
$B0BA4$K(B ring (schreyer order) $B$rDj5A$9$k4X?t$,M_$7$$(B. |
| |
$B0BA4$K(B parse $B$9$k4X?t$bM_$7$$(B. |
| |
$B%Y%/%H%k$H(B es $BI=8=$NJQ494X?t$b$$$k(B. |
| |
|
| |
AvoidTheSameRing == 1 $B$J$i(B, schreyer $B$N(B gbList $B$bJQ99$G$-$J$$$h$&$K(B |
| |
$B$9$Y$-$+!)(B |
| |
$B4XO"JQ?t(B: |
| |
needWarningForAvoidTheSameRing |
| |
isTheSameRing() : ring $B$,F1$8$+(B check. pointer $B$G$J$/Cf?H$^$G$_$k(B. |
| |
see poly4.c. $B$3$3$N%3%a%s%H$O;29M$K$J$k(B. |
| |
3.If Schreyer = 1, then the system always generates a new ring. |
| |
|
| |
define_ring $B$K(B gbList $B$bEO$;$k$N(B? |
| |
==> set_up_ring@ $B$r8+$k(B. grep set_up_ring ==> |
| |
primitive.c KsetUpRing() grep KsetUpRing ==> |
| |
keyword gbListTower $B$,;H$($k$,(B, list $B$GM?$($J$$$H$$$1$J$$(B. |
| |
list $B$KJQ49$9$k$N$O(B, (list) dc. |
| |
|
| |
tparse $B$NI,MW$J$o$1(B? |
| |
?? $B$*$b$$$@$;$J$$(B. |
| |
|
| |
ring_def $B$G(B ring (schreyer order) $B$rDj5A$9$k$H(B, $B7W;;$N$H$-$N(B |
| |
order $B$b(B tower $B$G$d$C$F$/$l$k$N(B? |
| |
$BB?J,(B NO. |
| |
grep ppAdd *.c ==> |
| |
poly2.c |
| |
checkRing(f,g); |
| |
|
| |
while (f != POLYNULL && g != POLYNULL) { |
| |
/*printf("%s + %s\n",POLYToString(f,'*',1),POLYToString(g,'*',1));*/ |
| |
checkRing2(f,g); /* for debug */ |
| |
gt = (*mmLarger)(f,g); |
| |
|
| |
mmLarger $B$OJQ$($F$J$$$h$&$K8+$($k(B. checkRing $B$O%^%/%m(B. |
| |
|
| |
mmLarger_tower $B$O(B |
| |
if (!(f->m->ringp->schreyer) || !(g->m->ringp->schreyer)) |
| |
return(mmLarger_matrix(f,g)); |
| |
$B$H$J$C$F$k$N$G(B mmLarger_tower $B$r(B default $B$K$7$F$*$1$P?4G[$J$$$h$&$K8+$($k(B. |
| |
|
| |
ring_def $B$O@5$7$/F0$/(B? |
| |
|
| |
TODO: |
| |
$B4X?t$N;EMM(B: ( new.sm1 $B$^$?$O(B complex.sm1 $B$K$*$$$H$/(B ) |
| |
mmLarger $B$O(B tower $B$KJQ$($F$7$^$&(B. |
| |
$BJQ?tL>(B, weight vector, $B%7%U%H%Y%/%H%k(B m $B$rM?$($k$H(B ring (with schreyer order) |
| |
$B$r:n$k(B. ==> weyl<m>, weyl |
| |
parser $B$O$H$/$K:n$kI,MW$,$J$$$h$&$K8+$($k$,(B...(tparse) ==> name |
| |
$B%Y%/%H%k(B <---> es $BI=8=(B cf. toVectors, [(toe_) f] gbext ==> name |
| |
$BE,@Z$J(B homogenization $B4X?t(B ==> homogenize<m> |
| |
ord_w $B$N(B schreyer $BHG(B ==> ord_w<m> |
| |
init $B$N(B schreyer $BHG(B ==> init<m> |
| |
gb_h, syz_h $B$NBP1~HG(B ==> [ ii vv ww m] syz_h |
| |
resolution $B$+$i(B shift vector $B$r7W;;$9$k4X?t(B. |
| |
|
| |
$B7k2L$N(B check $B$r$9$k(B assert $B4X?t$bI,MW(B. |
| |
|
| |
$B>e$N(B $B%7%U%H%Y%/%H%kBP1~HG$N4X?t$OEvJ,(B new.sm1 $B$X(B. $B$=$N$"$H(B complex.sm1 $B$X(B. |
| |
|
| |
cohom.sm1 $B$N(B interface $B4X?t$O(B cohom.k $B$X(B. |
| |
Help key word $B$O(B (Cohom.deRham) $B$_$?$$$K(B, . $B$G$/$.$C$F=q$/(B. |
| |
|
| |
---------------------- |
| |
$B%(%i!<$N860x$,$h$&$d$/$o$+$k(B: June 14, 19:00 |
| |
Schreyer frame $B$NCJ3,$G(B syz $B$K(B 1 $B$,$"$k$H(B strategy $B$,(B |
| |
$B$O$?$i$+$J$$(B. |
| |
|
| |
test13() GKZ $B$N(B minimal free resolution. 2 $BEY<B9T$9$k$HJQ(B. |
| |
grade $B$,JQ99$5$l$k$H(B, $BJQ$J$3$H$,$*$-$k$N$G(B, |
| |
ScheckIfSchreyer() $B4X?t$G(B, $B$3$l$r(B scheck $B$9$k$3$H$K$7$?(B. |
| |
sm1(" (report) (mmLarger) switch_function /ss set "); |
| |
$B$O$^$@$d$a$H$/(B. matrix $B$K$J$C$F$k$N$G(B. |
| |
|
| |
------------------------------------------ |
| |
June 15, 2000 |
| |
TODO: |
| |
1.if (IdenfityIntegerAndUniversalNumber) $B$N$H$-(B --- default |
| |
lt, gt, eq $B$G(B integer $B$H(B universalNumber $B$NHf3S$,$G$-$k$h$&$K$9$k(B. |
| |
rational $B$H$NHf3S$b2DG=$K$9$k(B. |
| |
|
| |
2. sm1_push_int0 $B$KBP1~$9$k$3$H$r(B, sm1 $B$NB&$G$d$k(B. |
| |
$B%^%/%mL>(B obj to_int --> Done. |
| |
weight_vector $B$N(B universalNumber ==> $B$^$@(B. $B%(%i!<$r$@$5$J$$$N$,$3$o$$(B. |
| |
s_weight_vector |
| |
weightv |
| |
ord_w |
| |
toVectors |
| |
define_ring |
| |
init |
| |
gkz |
| |
|
| |
------------- |
| |
Schreyer skelton $B$,$I$&$7$F(B 1 $B$rMWAG$K$b$D$+$7$i$Y$k(B. |
| |
|
| |
June 24 (Sat), 22:30 at Posthouse (Heathrow) www.posthouse-hotels.com |
| |
Sevilla $BBZ:_(B, Mega $B$b$h$&$d$/$*$o$j(B minimal resolution $B$N(B check $B$KLa$k(B. |
| |
resol1.c $B$K<!$N(B line $B$r2C$($?(B. |
| |
/* If isConstant(sv.a) is added, (x^3 - y^2 z^2) deRham stops |
| |
with an error. I've not yet understood the reason. |
| |
At Posthouse at Heathrow. June 24, 2000 */ |
| |
if (isConstant(sv.b)) { |
| |
s->deleted = 1; |
| |
} |
| |
===> $B$*$+$7$$$N$G:o=|(B. |
| |
|
| |
isConstant(sv.a) $B$,$J$$$H(B, $B$3$s$I$O(B, |
| |
Sminimal([x^2+y^2,x*y]); $B$,%(%i!<$G$H$^$k(B. |
| |
(x,y $B$N(B weight $B$O(B -1). |
| |
LaScala-Stillman $B$NO@J8$r$b$&0lEY$J$,$a$h$&(B. |
| |
|
| |
commit $B$9$Y$-(B: misc/mega2000 (cvs-misc add) Done. |
| |
OpenXM/src/kan96xx Done. |
| |
OpenXM/src/k097/lib/minimal Done. |
| |
|
| |
July 26. |
| |
resol.c $B$N(B schreyerSkelton0 $B$G(B, skelton $B$,(B minimal $B$K$J$k$h$&$K(B |
| |
$B%3!<%I$rA^F~(B. |
| |
$B%F%9%H$O(B |
| |
cd src/k097/lib/minimal |
| |
k0 |
| |
load["minimal.k"];; |
| |
Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]); |
| |
Sminimal([x^2+y^2,x*y]); |
| |
$B$G(B. |
| |
|
| |
LaScala-Stillman $B$NO@J85U$G(B i<j $B$J$i(B e_i > e_j $B$H$J$k(B. |
| |
(order.c mmLarger_tower()) |
| |
|
| |
$B%F%9%H(B 2. |
| |
cd src/k097/lib/minimal |
| |
k0 |
| |
load["minimal-test.k"];; |
| |
v: |
| |
Sminimal(v); |
| |
|
| |
test11(); /* a = test_ann3("x^3-y^2*z^2"); */ |
| |
test14(); /* gkz (1,2,3) */ |
| |
|
| |
July 30. Removed unnecessary code. |
| |
$BNc(B: |
| |
Sminimal("x^3-y^2"); |
| |
test12() ( x^3-y^2 z^2) |
| |
test15() GKZ 1,2,3 with a check. |
| |
test15b() toric |
| |
test15c() (u,v) = (-1,1) |
| |
|
| |
August 1. |
| |
(u,v)-minimal $B$N%F%9%H%3!<%I$r$$$l$?(B. |
| |
IsExact_h $B$G(B $BJQ?t(B c $B$NCM$,$+$o$k(B. $B860xITL@(B. |
| |
c=Sinit_w(b,w); |
| |
Println("Resolution (b)----"); |
| |
sm1_pmat(b); |
| |
Println("Initial (c)----"); |
| |
sm1_pmat(c); cc=c; |
| |
Println("Exactness of the resolution ---"); |
| |
Println(IsExact_h(b,v)); /* IsExact_h breaks the variable c. |
| |
THIS BUG SHOULD BE FIXED. */ |
| |
$B$3$N$"$H$J$<$+(B, c $B$,(B b $B$NCM$K$+$o$C$F$7$^$&(B. |
| |
$B$J$*(B def IsExact(c,...) $B$HDj5A$5$l$F$*$j(B, $B$3$N(B c $B$rJL$NJQ?tL>$K(B |
| |
$BJQ$($l$P$3$NLdBj$O$*$-$J$$(B. |
| |
Println("Why is the initial c rewritten by b? (buggy) ");sm1_pmat(c[0]); |
| |
|
| |
===> complex.sm1 $B$N(B isExact_h (isExact) $B$G(B popVariables $B$rK:$l$F$?$@$1(B. |
| |
|
| |
betti $B?t$O(B, $B9TNs$N>C5n$r$d$k$^$G$o$+$i$J$$$N(B? |
| |
SbettiTable(). |
| |
|
| |
Sminimal $B$O(B [(Homogenize_vec) 0] system_variable $B$K$9$k$h$&$G(B, |
| |
$B$3$l$,(B cohomology $B$N7W;;$K$O<YKb(B. |
| |
|
| |
August 2, 2000. |
| |
|
| |
Sminimal $B$O(B [(Homogenize_vec) 0] system_variable $B$K$9$k$h$&$G(B, |
| |
$B$3$l$,(B cohomology $B$N7W;;$K$O<YKb(B. |
| |
( cf. $BBg0$5W;a$N%9%/%j%W%H(B. $B8=:_?@8M$KBZ:_Cf(B. ) |
| |
|
| |
/restoreEnvAfterResolution { |
| |
[(AvoidTheSameRing)] pushEnv |
| |
[ [(AvoidTheSameRing) 0] system_variable |
| |
[(gbListTower) [[ ]] (list) dc] system_variable |
| |
] pop popEnv |
| |
setupEnvForResolution.opts restoreOptions <=== $BJQ99(B. opts $B$O$$$m$s$J$H$3$m$G;H$C$F$k(B. |
| |
} def |
| |
|
| |
$B$3$N%^%/%m$r$h$Y$P$$$$$N$+!)(B |
| |
sm1(" restoreEnvAfterResolution "); |
| |
$B$r(B Sminimal $B$N$*$o$j$K8F$V$h$&$KJQ$($?(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)= |
| |
|
![[BACK]](/icons/cvsweb/back.gif){kind=icon}
Return to minimal-note-ja.txt CVS log ![[TXT]](/icons/cvsweb/text.gif){kind=icon}
![[DIR]](/icons/cvsweb/dir.gif){kind=icon}
Up to [local] / OpenXM / src / k097 / lib / minimal