=================================================================== RCS file: /home/cvs/OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v retrieving revision 1.8 retrieving revision 1.12 diff -u -p -r1.8 -r1.12 --- OpenXM/src/k097/lib/minimal/minimal-note-ja.txt 2000/07/26 02:21:31 1.8 +++ OpenXM/src/k097/lib/minimal/minimal-note-ja.txt 2000/08/09 03:45:27 1.12 @@ -1,4 +1,4 @@ -$OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.7 2000/06/26 11:14:00 takayama Exp $ +$OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.11 2000/08/02 05:14:30 takayama Exp $ SpairAndReduction() : 与えられた pair を reduction する. @@ -692,3 +692,487 @@ July 26. test11(); /* a = test_ann3("x^3-y^2*z^2"); */ test14(); /* gkz (1,2,3) */ +July 30. Removed unnecessary code. + 例: + 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 のテストコードをいれた. + IsExact_h で 変数 c の値がかわる. 原因不明. + 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. */ + このあとなぜか, c が b の値にかわってしまう. + なお def IsExact(c,...) と定義されており, この c を別の変数名に + 変えればこの問題はおきない. + Println("Why is the initial c rewritten by b? (buggy) ");sm1_pmat(c[0]); + + ===> complex.sm1 の isExact_h (isExact) で popVariables を忘れてただけ. + + betti 数は, 行列の消去をやるまでわからないの? + SbettiTable(). + + Sminimal は [(Homogenize_vec) 0] system_variable にするようで, + これが cohomology の計算には邪魔. + +August 2, 2000. + +Sminimal は [(Homogenize_vec) 0] system_variable にするようで, +これが cohomology の計算には邪魔. +( cf. 大阿久氏のスクリプト. 現在神戸に滞在中. ) + +/restoreEnvAfterResolution { + [(AvoidTheSameRing)] pushEnv + [ [(AvoidTheSameRing) 0] system_variable + [(gbListTower) [[ ]] (list) dc] system_variable + ] pop popEnv + setupEnvForResolution.opts restoreOptions <=== 変更. opts はいろんなところで使ってる. +} def + +このマクロをよべばいいのか？ + sm1(" restoreEnvAfterResolution "); +を Sminimal のおわりに呼ぶように変えた. +test17b(), test18() は正常動作. + + +August 7, Mon 13:00JST ( 5:00 St.Andrews, Scotland, 4039 号室) +example-ja.tex を書くための出力. + +% 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)= \ No newline at end of file