File: [local] / OpenXM / src / k097 / lib / restriction / restriction.k (download)
Revision 1.2, Thu Jan 24 00:27:54 2013 UTC (11 years, 7 months ago) by takayama
Branch: MAIN
Changes since 1.1: +46 -1
lines
Added the function Sintegration(F,v,intv).
It uses the minimal filtered resolution to get integrals of F along intv.
This function is faster than integration in sm1, because it uses the minimal resolution.
Example: load["lib/all.k"];;
RingD("x,t");
Sintegration([Dt-(3*t^2-x),Dx+t],[t,x],[t]):
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/* $OpenXM: OpenXM/src/k097/lib/restriction/restriction.k,v 1.2 2013/01/24 00:27:54 takayama Exp $ */
load["lib/minimal/minimal-test.k"];;
Load_sm1(["Srestall_s.sm1","lib/restriction/Srestall_s.sm1"],"Srestall_s.sm1.loaded");
def Srestall(gg,ttxx,tt,k1) {
local cohom, gg, ttxx, tt, k1, cohom0, cohomd,ans;
sm1("gg dehomogenize /gg set");
gg = ToString_array(gg);
sm1(" [(x) ring_of_differential_operators [[(x) 1]] weight_vector 0] define_ring ]");
sm1("[(Homogenize_vec) 1] system_variable");
Println([Tag(gg),Tag(ttxx),Tag(tt)]);
sm1("gg ttxx tt k1 (integer) dc Srestall1 /cohom0 set");
sm1(" cohom0 {deRham.simp} map /cohomd set ");
ans = [cohomd,cohom0];
return(ans);
}
HelpAdd(["Srestall",
[ "Srestall(gg,v,rv,k1) evaluates the dimensions of all restictions of",
"gg along the list of variables rv. Here, v is a list of variables and",
" k1 is the maximal integral root of the b-function of gg.",
"Srestall uses the function Sminimal to get a (-w,w)-minimal free resolution.",
"cf. Bfroots1(ii,vv)"
]]);
def DeRham2(f) {
local s;
s = ToString(f);
II = Sannfs(f,"x,y");
Println("Step 1: Annhilating ideal (II)"); Println(II);
sm1(" II 0 get { [(x) (y) (Dx) (Dy) ] laplace0 } map /II set ");
Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
pp = Map(II,"Spoly");
Res = Sminimal(pp);
Res0 = Res[0];
Println("Step2: (-1,1)-minimal resolution (Res0) "); sm1_pmat(Res0);
R = BfRoots1(Res0[0],"x,y");
Println("Step3: computing the cohomology of the truncated complex.");
Print("Roots and b-function are "); Println(R);
R0 = R[0];
Ans=Srestall(Res0, ["x", "y"], ["x", "y"],R0[Length(R0)-1]);
Print("Answer is "); Println(Ans[0]);
return(Ans);
}
def DeRham3(f) {
local s;
s = ToString(f);
II = Sannfs(f,"x,y,z");
Print("Step 1: Annhilating ideal (II)"); Println(II);
sm1(" II 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /II set ");
Sweyl("x,y,z",[["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]]);
pp = Map(II,"Spoly");
Res = Sminimal(pp);
Res0 = Res[0];
Print("Step2: (-1,1)-minimal resolution (Res0) "); sm1_pmat(Res0);
R = BfRoots1(Res0[0],"x,y,z");
Println("Step3: computing the cohomology of the truncated complex.");
Print("Roots and b-function are "); Println(R);
R0 = R[0];
Ans=Srestall(Res0, ["x", "y", "z"], ["x", "y", "z"],R0[Length(R0)-1] );
Print("Answer is ");Println(Ans[0]);
return(Ans);
}
def DeRham1(f) {
local s;
s = ToString(f);
II = Sannfs(f,"x");
Println("Step 1: Annhilating ideal (II)"); Println(II);
sm1(" II 0 get { [(x) (Dx) ] laplace0 } map /II set ");
Sweyl("x",[["x",-1,"Dx",1]]);
pp = Map(II,"Spoly");
Res = Sminimal(pp);
Res0 = Res[0];
Println("Step2: (-1,1)-minimal resolution (Res0) "); sm1_pmat(Res0);
R = BfRoots1(Res0[0],"x");
Println("Step3: computing the cohomology of the truncated complex.");
Print("Roots and b-function are "); Println(R);
R0 = R[0];
Ans=Srestall(Res0, ["x"], ["x"],R0[Length(R0)-1]);
Print("Answer is "); Println(Ans[0]);
return(Ans);
}
/* Demo for non-quasi */
def nonquasi(p,q) {
local s,ans,f;
f = x^p+y^q+x*y^(q-1);
Print("f=");Println(f);
s = ToString(f);
sm1(" Onverbose ");
sm1(" s [(s) (x) (y)] genericAnn /ans set ");
sm1(" ans 0 get (ring) dc ring_def ");
sm1("[ ans { [[(s). (-1).]] replace } map ] /II set ");
Println("Step 1: Annhilating ideal (II)"); Println(II);
sm1(" II 0 get { [(x) (y) (Dx) (Dy) ] laplace0 } map /II set ");
Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
pp = Map(II,"Spoly");
Res = Sminimal(pp);
Res0 = Res[0];
Println("Step2: (-1,1)-minimal resolution (Res0) "); sm1_pmat(Res0);
R = BfRoots1(Res0[0],"x,y");
Println("Step3: computing the cohomology of the truncated complex.");
Print("Roots and b-function are "); Println(R);
R0 = R[0];
Ans=Srestall(Res0, ["x", "y"], ["x", "y"], R0[Length(R0)-1]);
Print("Answer is "); Println(Ans[0]);
return(Ans);
}
def t_addD(v) {
return(AddString(["D",ToString(v)]));
}
def Sintegration(gg,v,intv) {
local i,vstr,n,dv,wv,vall;
II=Map(gg,"ToString");
v = Map(v,"ToString");
intv = Map(intv,"ToString");
vstr=" "; n=Length(v);
for (i=0; i<n-1; i++) {
vstr=AddString([vstr,v[i],","]);
}
vstr=AddString([vstr,v[n-1]]);
dv=Map(v,"t_addD");
wv=[ ];
for (i=0; i<n; i++) { wv = Join(wv,[v[i],-1]); }
for (i=0; i<n; i++) { wv = Join(wv,[dv[i],1]); }
/* Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]); */
Sweyl(vstr,[wv]);
II=Map(II,"Poly");
/* sm1(" II 0 get { [(x) (y) (Dx) (Dy) ] laplace0 } map /II set "); */
vall = Join(v,dv);
sm1(" II { vall laplace0 } map /II set ");
pp = Map(II,"Spoly");
Println("Step1: Laplace transform "); Println(pp);
Res = Sminimal(pp);
Res0 = Res[0];
Println("Step2: (-1,1)-minimal resolution (Res0) "); sm1_pmat(Res0);
/* R = BfRoots1(Res0[0],"x,y"); */
R = BfRoots1(Res0[0],vstr);
Println("Step3: computing the cohomology of the truncated complex.");
Print("Roots and b-function are "); Println(R);
R0 = R[0];
/* Ans=Srestall(Res0, ["x", "y"], ["x", "y"],R0[Length(R0)-1]); */
Ans=Srestall(Res0, v, intv,R0[Length(R0)-1]);
Print("Answer is "); Println(Ans[0]);
return(Ans);
}
HelpAdd(["Sintegration",
[ "Sintegration(gg,v,intv) evaluates the dimensions of all integrations of",
"gg along the list of variables intv. Here, v is a list of variables.",
"Interation variables intv must be in the first part of v.",
"Sintegration uses the function Srestall.",
"Example: RingD(\"x,t\"); Sintegration([Dt-(3*t^2-x),Dx+t],[t,x],[t]):"
]]);