> <\body> \S> with flux only is actually a case where I can check whether my guessed oxidation gives a different physical result from GSKPR. \ I have now saved ttOnlyH2H5.cdb as ttOnlyH3H4GSKPR.cdb. \ Adapting the notation in this, > run from 1 to 7, > run over the >, > run over the >, > run from 1 to 2 and are the latitude coords on the >, while 3 is the longitude coord on the >. \ I have now clicked Run all for the first version of 55OnlyH3H4GSKPR.cdb at 1.48 pm on Sat 19 Feb '11. \ This version used > for 3, and defined these to be Integer(3..3). \ So I don't know whether Cadabra will treat a range like that correctly. \ It's now 2.47 pm and the run has finished. \ A lot of > etc. have not been substituted out. \ I have now saved it as ttOnlyH3H4GSKPRv2.cdb. \ In fact three substitution rules in the original version were wrong. \ I have clicked Run all for v2 at 3.02 pm. \ I should be able to work out the flux bilinears for the new coordinate ranges from the old ones. \ Now from about a third of the way down BulkVacuumEnergy32.tm: Thus we find: H=8 |cb>+ 12 |cb>GG-GG>, H=2 |cb>+ 12 |cb>+ 6 |cb>GG>, and 3 similar to that, and H=6 |cb>+ 12 |cb> \ + 2 |b>GG -GG>. In the notation used there, indices > run over the >, and indices > run over the >. \ is radius of > and is radius of >. \ While here, > run from 1 to 7, > run over the >, > run over the >, > run from 1 to 2 and are the latitude coords on the >, while 3 is the longitude coord on the >. \ So in the present notation, the above become: H=8 |cb>+ 12 |cb>GG-GG>, H=2 |cb>+ 12 |cb>+ 6 |cb>GG>, and 3 similar to that, and H=6 |cb>+ 12 |cb> \ + 2 |b>GG -GG>. H=8 |cb>+ 12 |cb>GG-GG>, H=8 |cb>+ 12 |cb>G>, H=2 |cb>+ 12 |cb>+ 6 |cb>GG>, and 3 similar to that, and H=2 |cb>+ 12 |cb>+ 6 |cb>G>, and 3 similar to that. It is now 3.29 pm, and the ttOnlyH3H4GSKPRv2.cdb run has finished. \ The result is: <\equation*> 576 s s s s s s s s + 1536 u u u u u u u u + 3072 s s s s w w w w + 1536 s s s s x x x x + 12288 u u u u w w w w + 3072 u u u u x x x x + 2304 s s s s v v v v + 4608 u u u u v v v v + 24576 w w w w w w w w + 30720 w w w w x x x x + 4608 x x x x x x x x + 55296 v v v v w w w w + 27648 v v v v x x x x + 13824 v v v v v v v v + 768 s s s s u u u u + 12288 s s u u w w x x + 18432 s s v v w w w w + 36864 u u v v w w x x <\equation*> 576*s*s*s*s*s*s*s*s + 1536*u*u*u*u*u*u*u*u + 3072*s*s*s*s*w*w*w*w + 1536*s*s*s*s*x*x*x*x + 12288*u*u*u*u*w*w*w*w + 3072*u*u*u*u*x*x*x*x + 2304*s*s*s*s*v*v*v*v + 4608*u*u*u*u*v*v*v*v + 24576*w*w*w*w*w*w*w*w + 30720*w*w*w*w*x*x*x*x + 4608*x*x*x*x*x*x*x*x + 55296*v*v*v*v*w*w*w*w + 27648*v*v*v*v*x*x*x*x + 13824*v*v*v*v*v*v*v*v + 768*s*s*s*s*u*u*u*u + 12288*s*s*u*u*w*w*x*x + 18432*s*s*v*v*w*w*w*w + 36864*u*u*v*v*w*w*x*x The substitutions used to derive this are: X_{Q R S T} -\ v v y_{Q T} y_{R S} - v v y_{Q S} y_{R T} X_{a Q b R} -\ - w w q_{a b} y_{Q R} and 3 similar X_{3 Q 3 R} -\ - x x y_{Q R} and 3 similar X_{a 3 b 3} -\ - u u q_{a b} and 3 similar X_{a b c d} -\ s s q_{a d} q_{b c} - s s q_{a c} q_{b d} It's the ones with a 3 index that will differ between my oxidation, and straight GSKPR. I have now corrected 3 substitution rules in ttOnlyH3H4GSKPR.cdb, only two of the corrections were essential, and clicked Run all for it at 3.40 pm. \ It is now 4.05 pm, and the run has finished. \ The result is: <\equation*> 576 s s s s s s s s + 1536 u u u u u u u u + 3072 s s s s w w w w + 1536 s s s s x x x x + 12288 u u u u w w w w + 3072 u u u u x x x x + 2304 s s s s v v v v + 4608 u u u u v v v v + 24576 w w w w w w w w + 30720 w w w w x x x x + 4608 x x x x x x x x + 55296 v v v v w w w w + 27648 v v v v x x x x + 13824 v v v v v v v v + 768 s s s s u u u u + 12288 s s u u w w x x + 18432 s s v v w w w w + 36864 u u v v w w x x <\equation*> 576*s*s*s*s*s*s*s*s + 1536*u*u*u*u*u*u*u*u + 3072*s*s*s*s*w*w*w*w + 1536*s*s*s*s*x*x*x*x + 12288*u*u*u*u*w*w*w*w + 3072*u*u*u*u*x*x*x*x + 2304*s*s*s*s*v*v*v*v + 4608*u*u*u*u*v*v*v*v + 24576*w*w*w*w*w*w*w*w + 30720*w*w*w*w*x*x*x*x + 4608*x*x*x*x*x*x*x*x + 55296*v*v*v*v*w*w*w*w + 27648*v*v*v*v*x*x*x*x + 13824*v*v*v*v*v*v*v*v + 768*s*s*s*s*u*u*u*u + 12288*s*s*u*u*w*w*x*x + 18432*s*s*v*v*w*w*w*w + 36864*u*u*v*v*w*w*x*x <\input|maxima] > 576*s*s*s*s*s*s*s*s + 1536*u*u*u*u*u*u*u*u + 3072*s*s*s*s*w*w*w*w + 1536*s*s*s*s*x*x*x*x + 12288*u*u*u*u*w*w*w*w + 3072*u*u*u*u*x*x*x*x + 2304*s*s*s*s*v*v*v*v + 4608*u*u*u*u*v*v*v*v + 24576*w*w*w*w*w*w*w*w + 30720*w*w*w*w*x*x*x*x + 4608*x*x*x*x*x*x*x*x + 55296*v*v*v*v*w*w*w*w + 27648*v*v*v*v*x*x*x*x + 13824*v*v*v*v*v*v*v*v + 768*s*s*s*s*u*u*u*u + 12288*s*s*u*u*w*w*x*x + 18432*s*s*v*v*w*w*w*w + 36864*u*u*v*v*w*w*x*x; <\output> ) >4608*x+30720*w*x+27648*v*x+3072*u*x+1536*s*x+36864*u*v*w*x+12288*s*u*w*x+24576*w+55296*v*w+18432*s*v*w+12288*u*w+3072*s*w+13824*v+4608*u*v+2304*s*v+1536*u+768*s*u+576*s> <\input|181) >> 576*s*s*s*s*s*s*s*s + 1536*u*u*u*u*u*u*u*u + 3072*s*s*s*s*w*w*w*w + 1536*s*s*s*s*x*x*x*x + 12288*u*u*u*u*w*w*w*w + 3072*u*u*u*u*x*x*x*x + 2304*s*s*s*s*v*v*v*v + 4608*u*u*u*u*v*v*v*v + 24576*w*w*w*w*w*w*w*w + 30720*w*w*w*w*x*x*x*x + 4608*x*x*x*x*x*x*x*x + 55296*v*v*v*v*w*w*w*w + 27648*v*v*v*v*x*x*x*x + 13824*v*v*v*v*v*v*v*v + 768*s*s*s*s*u*u*u*u + 12288*s*s*u*u*w*w*x*x + 18432*s*s*v*v*w*w*w*w + 36864*u*u*v*v*w*w*x*x; <\output> ) >4608*x+30720*w*x+27648*v*x+3072*u*x+1536*s*x+36864*u*v*w*x+12288*s*u*w*x+24576*w+55296*v*w+18432*s*v*w+12288*u*w+3072*s*w+13824*v+4608*u*v+2304*s*v+1536*u+768*s*u+576*s> <\input|182) >> radcan(%o180 - %o181); <\output> ) >0> <\input|183) >> \; > So the two versions of ttOnlyH3H4GSKPR.cdb give the same result. \ It should be possible to adapt the first version to, e.g., \H\H>. Now from above, in the present notation: H=6 |cb>+ 12 |cb> \ + 2 |b>GG -GG>. H=8 |cb>+ 12 |cb>GG-GG>, H=8 |cb>+ 12 |cb>G>, H=2 |cb>+ 12 |cb>+ 6 |cb>GG>, and 3 similar to that, and H=2 |cb>+ 12 |cb>+ 6 |cb>G>, and 3 similar to that. <\equation*> X=RG-HH+HH For the 3-sphere, =>GG-GG>. \ Hence =>GG-GG>, and =>G>. Cases: <\equation*> X=RG-HH+HH H=8 |cb>+ 12 |cb>GG-GG> H=8 |cb>+ 12 |cb>GG-GG> H=8 |cb>+ 12 |cb>GG-GG> <\equation*> X=RG-8 |cb>+ 12 |cb>GG-GG+8 |cb>+ 12 |cb>GG-GG <\equation*> X=RG+8 |cb>+ 12 |cb>GG-GG <\equation*> X=->+ |c*b>+|2*c*b>GG-GG <\equation*> X=RG-HH+HH <\equation*> X=RG+HH H=8 |cb>+ 12 |cb>G>, H=-8 |cb>+ 12 |cb>G>, <\equation*> X=RG- |cb>+ |2*cb>G <\equation*> X=-->+ |cb>+ |2*cb>G <\equation*> X=RG-HH+HH H=6 |cb>+ 12 |cb> \ + 2 |b>GG -GG>. H=6 |cb>+ 12 |cb> \ + 2 |b>GG -GG>. H=6 |cb>+ 12 |cb> \ + 2 |b>GG -GG>. <\equation*> X=RG-6 |cb>+ 12 |cb> \ + 2 |b>GG -GG+6 |cb>+ 12 |cb> \ + 2 |b>GG -GG <\equation*> X=RG+6 |cb>+ 12 |cb> \ + 2 |b> GG -GG <\equation*> X=>+ |4*c*b>+ |2*c*b> \ + |4*b> GG -GG <\equation*> X=RG-HH+HH <\equation*> X=RG+HH H=2 |cb>+ 12 |cb>+ 6 |cb>GG>, and 3 similar to that, and H=-2 |cb>+ 12 |cb>+ 6 |cb>GG>, and 3 similar to that, and <\equation*> X=- |4*c*b>+ \ |2*c*b>+ |4*cb>GG <\equation*> X=RG-HH+HH <\equation*> X=RG+HH H=2 |cb>+ 12 |cb>+ 6 |cb>G>, and 3 similar to that. H=-2 |cb>+ 12 |cb>+ 6 |cb>G>, and 3 similar to that. <\equation*> X=- |4*c*b>+ \ |2*cb>+ |4*cb>G So in summary: <\equation*> X=->+ |c*b>+|2*c*b>GG-GG <\equation*> X=-->+ |cb>+ |2*cb>G <\equation*> X=>+ |4*c*b>+ |2*c*b> \ + |4*b> GG -GG <\equation*> X=- |4*c*b>+ \ |2*c*b>+ |4*cb>GG <\equation*> X=- |4*c*b>+ \ |2*cb>+ |4*cb>G The substitutions used in the Cadabra scripts, and the corresponding results from above, are: X_{Q R S T} -\ v v y_{Q T} y_{R S} - v v y_{Q S} y_{R T} <\equation*> X=>+ |4*c*b>+ |2*c*b> \ + |4*b> GG -GG Hence >+ |4*c*b>+ |2*c*b> \ + |4*b>>. X_{a Q b R} -\ - w w q_{a b} y_{Q R} and 3 similar <\equation*> X=- |4*c*b>+ \ |2*c*b>+ |4*cb>GG Hence |4*c*b>+ \ |2*c*b>+ |4*cb>>. X_{3 Q 3 R} -\ - x x y_{Q R} and 3 similar <\equation*> X=- |4*c*b>+ \ |2*cb>+ |4*cb>G Hence |4*c*b>+ \ |2*cb>+ |4*cb>>. X_{a 3 b 3} -\ - u u q_{a b} and 3 similar <\equation*> X=-->+ |cb>+ |2*cb>G Hence ->+ |cb>+ |2*cb>>. X_{a b c d} -\ s s q_{a d} q_{b c} - s s q_{a c} q_{b d} <\equation*> X=->+ |c*b>+|2*c*b>GG-GG Hence ->+ |c*b>+|2*c*b>>. It's the ones with a 3 index that will differ between my oxidation, and straight GSKPR. \ The values of etc. deduced above are for my oxidation. \ In summary, they are: >+ |4*c*b>+ |2*c*b> \ + |4*b>>. \ |4*c*b>+ \ |2*c*b>+ |4*cb>>. |4*c*b>+ \ |2*cb>+ |4*cb>>. \ ->+ |cb>+ |2*cb>>. ->+ |c*b>+|2*c*b>>. <\input|183) >> subst(sqrt(V),v,subst(sqrt(W),w,subst(sqrt(X),x,subst(sqrt(U), u,subst(sqrt(S),s,%o180))))); <\output> ) >4608*X+30720*W*X+27648*V*X+3072*U*X+1536*S*X+36864*U*V*W*X+12288*S*U*W*X+24576*W+55296*V*W+18432*S*V*W+12288*U*W+3072*S*W+13824*V+4608*U*V+2304*S*V+1536*U+768*S*U+576*S> <\input|184) >> factor(radcan(subst(((1/(b^2))+ ((3*f^2)/(4*c^4*b^4))+ ((3*g^2)/(2*c^2*b^6)) \ + ((h^2)/(4*b^8))),V, subst(( ((e^2)/(4*c^6*b^2))+ \ ((3*f^2)/(2*c^4*b^4))+ ((3*g^2)/(4*c^2*b^6))),W,subst(( ((e^2)/(4*c^6*b^2))+ \ ((3*f^2)/(2*c^4*b^4))+ ((3*g^2)/(4*c^2*b^6))),X, subst((-(1/(c^2))+ ((e^2)/(c^6*b^2))+ ((3*f^2)/(2*c^4*b^4))),U, subst((-(1/(c^2))+ ((e^2)/(c^6*b^2))+((3*f^2)/(2*c^4*b^4))), S,%o183))))))); <\output> ) >3*c*h+72*b*c*g*h+36*b*c*f*h+48*b*c*h+810*b*c*g*h+1296*b*c*f*g*h+108*b*c*e*g*h+864*b*c*g*h+864*b*c*f*h+288*b*c*e*f*h+432*b*c*f*h-72*b*c*f*h+42*b*c*e*h-48*b*c*e*h+288*b*c*h+24*b*c*h+4536*b*c*g*h+13284*b*c*f*g*h+1728*b*c*e*g*h+6480*b*c*g*h-432*b*c*g*h+16848*b*c*f*g*h+6264*b*c*e*f*g*h+10368*b*c*f*g*h-2592*b*c*f*g*h+792*b*c*e*g*h+864*b*c*e*g*h-864*b*c*e*g*h+3456*b*c*g*h+288*b*c*g*h+7128*b*c*f*h+4320*b*c*e*f*h+6912*b*c*f*h-2160*b*c*f*h+900*b*c*e*f*h+2304*b*c*e*f*h-864*b*c*e*f*h+1728*b*c*f*h-576*b*c*f*h+144*b*c*f*h+48*b*c*e*h+336*b*c*e*h-48*b*c*e*h-384*b*c*e*h+768*b*c*h+192*b*c*h+10773*b*c*g+49248*b*c*f*g+7884*b*c*e*g+18144*b*c*g-2592*b*c*g+100926*b*c*f*g+39960*b*c*e*f*g+53136*b*c*f*g-17280*b*c*f*g+4950*b*c*e*g+6912*b*c*e*g-5472*b*c*e*g+12960*b*c*g-1728*b*c*g+1872*b*c*g+99144*b*c*f*g+63828*b*c*e*f*g+67392*b*c*f*g-30240*b*c*f*g+15120*b*c*e*f*g+25056*b*c*e*f*g-16128*b*c*e*f*g+20736*b*c*f*g-10368*b*c*f*g+4896*b*c*f*g+1116*b*c*e*g+3168*b*c*e*g-1632*b*c*e*g+1728*b*c*e*g-3456*b*c*e*g+672*b*c*e*g+4608*b*c*g+1152*b*c*g+41067*b*c*f+38880*b*c*e*f+28512*b*c*f-20088*b*c*f+15606*b*c*e*f+17280*b*c*e*f-18576*b*c*e*f+13824*b*c*f-8640*b*c*f+6408*b*c*f+3096*b*c*e*f+3600*b*c*e*f-6048*b*c*e*f+4608*b*c*e*f-3456*b*c*e*f+4224*b*c*e*f+2304*b*c*f-1152*b*c*f+576*b*c*f-960*b*c*f+285*b*e+192*b*c*e-864*b*c*e+672*b*c*e-192*b*c*e+1072*b*c*e-768*b*c*e-640*b*c*e+768*b*c+384*b*c+160*b*c|b*c>> <\input|185) >> radcan(%o184 - subst(-1,eta,%o3)); <\output> ) >0> <\input|186) >> \; > So the old result for my oxidation has been recovered in the new coordinate system. \ Now consider what the substitutions would have been, instead, for GSKPR. X_{Q R S T} -\ v v y_{Q T} y_{R S} - v v y_{Q S} y_{R T} <\equation*> X=>+ |4*c*b>+ |2*c*b> \ + |4*b> GG -GG Hence >+ |4*c*b>+ |2*c*b> \ + |4*b>>. X_{a Q b R} -\ - w w q_{a b} y_{Q R} and 3 similar <\equation*> X=- |4*c*b>+ \ |2*c*b>+ |4*cb>GG Hence |4*c*b>+ \ |2*c*b>+ |4*cb>>. X_{3 Q 3 R} -\ - x x y_{Q R} and 3 similar <\equation*> X=0 Hence . X_{a 3 b 3} -\ - u u q_{a b} and 3 similar <\equation*> X=-->G Hence ->>. X_{a b c d} -\ s s q_{a d} q_{b c} - s s q_{a c} q_{b d} <\equation*> X=->+ |c*b>+|2*c*b>GG-GG Hence ->+ |c*b>+|2*c*b>>. It's the ones with a 3 index that will differ between my oxidation, and straight GSKPR. \ The differences are that now, , and ->>. <\input|186) >> factor(radcan(subst(((1/(b^2))+ ((3*f^2)/(4*c^4*b^4))+ ((3*g^2)/(2*c^2*b^6)) \ + ((h^2)/(4*b^8))),V, subst(( ((e^2)/(4*c^6*b^2))+ \ ((3*f^2)/(2*c^4*b^4))+ ((3*g^2)/(4*c^2*b^6))),W,subst(0,X, subst((-(1/(c^2))),U, subst((-(1/(c^2))+ ((e^2)/(c^6*b^2))+((3*f^2)/(2*c^4*b^4))), S,%o183))))))); <\output> ) >9*c*h+216*b*c*g*h+108*b*c*f*h+144*b*c*h+2268*b*c*g*h+3240*b*c*f*g*h+216*b*c*e*g*h+2592*b*c*g*h+1836*b*c*f*h+504*b*c*e*f*h+1296*b*c*f*h-72*b*c*f*h+60*b*c*e*h-48*b*c*e*h+864*b*c*h+72*b*c*h+11664*b*c*g*h+29808*b*c*f*g*h+3024*b*c*e*g*h+18144*b*c*g*h-432*b*c*g*h+32400*b*c*f*g*h+9504*b*c*e*f*g*h+25920*b*c*f*g*h-2592*b*c*f*g*h+1008*b*c*e*g*h+1728*b*c*e*g*h-864*b*c*e*g*h+10368*b*c*g*h+864*b*c*g*h+11664*b*c*f*h+5616*b*c*e*f*h+14688*b*c*f*h-2160*b*c*f*h+1008*b*c*e*f*h+4032*b*c*e*f*h-864*b*c*e*f*h+5184*b*c*f*h-576*b*c*f*h+432*b*c*f*h+48*b*c*e*h+480*b*c*e*h-48*b*c*e*h-384*b*c*e*h+2304*b*c*h+576*b*c*h+24624*b*c*g+95904*b*c*f*g+12096*b*c*e*g+46656*b*c*g-2592*b*c*g+164916*b*c*f*g+51408*b*c*e*f*g+119232*b*c*f*g-15120*b*c*f*g+5040*b*c*e*g+12096*b*c*e*g-4032*b*c*e*g+36288*b*c*g-1728*b*c*g+4032*b*c*g+133488*b*c*f*g+66744*b*c*e*f*g+129600*b*c*f*g-21600*b*c*f*g+12096*b*c*e*f*g+38016*b*c*e*f*g-8928*b*c*e*f*g+51840*b*c*f*g-10368*b*c*f*g+8352*b*c*f*g+672*b*c*e*g+4032*b*c*e*g-672*b*c*e*g+3456*b*c*e*g-3456*b*c*e*g+960*b*c*e*g+13824*b*c*g+3456*b*c*g+44469*b*c*f+31752*b*c*e*f+46656*b*c*f-10584*b*c*f+9612*b*c*e*f+22464*b*c*e*f-8208*b*c*e*f+29376*b*c*f-8640*b*c*f+7992*b*c*f+1584*b*c*e*f+4032*b*c*e*f-2736*b*c*e*f+8064*b*c*e*f-3456*b*c*e*f+4032*b*c*e*f+6912*b*c*f-1152*b*c*f+1728*b*c*f-960*b*c*f+144*b*e+192*b*c*e-448*b*c*e+960*b*c*e-192*b*c*e+864*b*c*e-768*b*c*e-640*b*c*e+2304*b*c+1152*b*c+480*b*c|b*c>> <\input|187) >> \; > But actually, for GSKPR, we have to set , , and to . <\input|187) >> radcan(subst(0,f,subst(0,g,subst(0,h,%o184 - %o186)))); <\output> ) >+2304*c-12864*b*c*e+6336*c-2304*b*c+14112*b*c*e+-9216*b*c-7680*b*c*e|b*c>> <\input|188) >> \; > So there are differences between my oxidation and GSKPR even in that case, but do they change whether a solution exists or not? \ The classical action for the \S> case is: <\equation*> >->-*96 *|cb> + 432 *|cb> + 288 *|cb> + 24 *|b> \; <\input|188) >> a^4*b^4*c^3*((6/(c^2))-(12/(b^2))-(1/48)*(96 *((e^2)/(c^6*b^2)) + 432 *((f^2)/(c^4*b^4)) + 288 *((g^2)/(c^2*b^6)) + 24 *((h^2)/(b^8))) + %o186/18); <\output> ) >a*b*c**h+216*b*c*g*h+108*b*c*f*h+144*b*c*h+2268*b*c*g*h+3240*b*c*f*g*h+216*b*c*e*g*h+2592*b*c*g*h+1836*b*c*f*h+504*b*c*e*f*h+1296*b*c*f*h-72*b*c*f*h+60*b*c*e*h-48*b*c*e*h+864*b*c*h+72*b*c*h+11664*b*c*g*h+29808*b*c*f*g*h+3024*b*c*e*g*h+18144*b*c*g*h-432*b*c*g*h+32400*b*c*f*g*h+9504*b*c*e*f*g*h+25920*b*c*f*g*h-2592*b*c*f*g*h+1008*b*c*e*g*h+1728*b*c*e*g*h-864*b*c*e*g*h+10368*b*c*g*h+864*b*c*g*h+11664*b*c*f*h+5616*b*c*e*f*h+14688*b*c*f*h-2160*b*c*f*h+1008*b*c*e*f*h+4032*b*c*e*f*h-864*b*c*e*f*h+5184*b*c*f*h-576*b*c*f*h+432*b*c*f*h+48*b*c*e*h+480*b*c*e*h-48*b*c*e*h-384*b*c*e*h+2304*b*c*h+576*b*c*h+24624*b*c*g+95904*b*c*f*g+12096*b*c*e*g+46656*b*c*g-2592*b*c*g+164916*b*c*f*g+51408*b*c*e*f*g+119232*b*c*f*g-15120*b*c*f*g+5040*b*c*e*g+12096*b*c*e*g-4032*b*c*e*g+36288*b*c*g-1728*b*c*g+4032*b*c*g+133488*b*c*f*g+66744*b*c*e*f*g+129600*b*c*f*g-21600*b*c*f*g+12096*b*c*e*f*g+38016*b*c*e*f*g-8928*b*c*e*f*g+51840*b*c*f*g-10368*b*c*f*g+8352*b*c*f*g+672*b*c*e*g+4032*b*c*e*g-672*b*c*e*g+3456*b*c*e*g-3456*b*c*e*g+960*b*c*e*g+13824*b*c*g+3456*b*c*g+44469*b*c*f+31752*b*c*e*f+46656*b*c*f-10584*b*c*f+9612*b*c*e*f+22464*b*c*e*f-8208*b*c*e*f+29376*b*c*f-8640*b*c*f+7992*b*c*f+1584*b*c*e*f+4032*b*c*e*f-2736*b*c*e*f+8064*b*c*e*f-3456*b*c*e*f+4032*b*c*e*f+6912*b*c*f-1152*b*c*f+1728*b*c*f-960*b*c*f+144*b*e+192*b*c*e-448*b*c*e+960*b*c*e-192*b*c*e+864*b*c*e-768*b*c*e-640*b*c*e+2304*b*c+1152*b*c+480*b*c|3*b*c>-|b>+|b*c>+|b*c>+|b*c>|48>+>->> <\input|189) >> subst(0,f,subst(0,g,subst(0,h,%o188))); <\output> ) >a*b*c**e+192*b*c*e-448*b*c*e+960*b*c*e-192*b*c*e+864*b*c*e-768*b*c*e-640*b*c*e+2304*b*c+1152*b*c+480*b*c|3*b*c>-|b*c>+>->> <\input|190) >> factor(radcan(diff(%o189,a))); <\output> ) >*72*e+96*c*e-224*b*c*e+480*c*e-96*b*c*e+432*b*c*e-3*b*c*e-384*b*c*e-320*b*c*e-18*b*c+1152*c+9*b*c+576*b*c+240*b*c|3*b*c>> <\input|191) >> factor(radcan(diff(%o189,b))); <\output> ) >-*144*e+192*c*e-224*b*c*e+960*c*e-96*b*c*e+3*b*c*e-384*b*c*e+320*b*c*e+18*b*c+2304*c-18*b*c-480*b*c|3*b*c>> <\input|192) >> factor(radcan(diff(%o189,c))); <\output> ) >-*1512*e+1440*c*e-3808*b*c*e+4320*c*e-1056*b*c*e+5616*b*c*e-9*b*c*e-1920*b*c*e-2880*b*c*e+54*b*c-3456*c-9*b*c+576*b*c+1200*b*c|3*b*c>> <\input|193) >> factor(radcan(%o190*3*b^4*c^21/(8*a^3))); <\output> ) >72*e+96*c*e-224*b*c*e+480*c*e-96*b*c*e+432*b*c*e-3*b*c*e-384*b*c*e-320*b*c*e-18*b*c+1152*c+9*b*c+576*b*c+240*b*c> <\input|194) >> factor(radcan(%o191*3*b^5*c^21/(-4*a^4))); <\output> ) >144*e+192*c*e-224*b*c*e+960*c*e-96*b*c*e+3*b*c*e-384*b*c*e+320*b*c*e+18*b*c+2304*c-18*b*c-480*b*c> <\input|195) >> factor(radcan(%o192*3*b^4*c^22/(-2*a^4))); <\output> ) >1512*e+1440*c*e-3808*b*c*e+4320*c*e-1056*b*c*e+5616*b*c*e-9*b*c*e-1920*b*c*e-2880*b*c*e+54*b*c-3456*c-9*b*c+576*b*c+1200*b*c> <\input|196) >> factor(radcan(subst(sqrt(B),b,subst(sqrt(C),c,subst(sqrt(E), e,%o193))))); <\output> ) >72*E+96*C*E-224*B*C*E+480*C*E-96*B*C*E+432*B*C*E-3*B*C*E-384*B*C*E-320*B*C*E-18*B*C+1152*C+9*B*C+576*B*C+240*B*C> <\input|197) >> factor(radcan(subst(sqrt(B),b,subst(sqrt(C),c,subst(sqrt(E), e,%o194))))); <\output> ) >144*E+192*C*E-224*B*C*E+960*C*E-96*B*C*E+3*B*C*E-384*B*C*E+320*B*C*E+18*B*C+2304*C-18*B*C-480*B*C> <\input|198) >> factor(radcan(subst(sqrt(B),b,subst(sqrt(C),c,subst(sqrt(E), e,%o195))))); <\output> ) >1512*E+1440*C*E-3808*B*C*E+4320*C*E-1056*B*C*E+5616*B*C*E-9*B*C*E-1920*B*C*E-2880*B*C*E+54*B*C-3456*C-9*B*C+576*B*C+1200*B*C> <\input|199) >> factor(radcan(subst(B*X,C,subst(B^3*Y,E,%o196)))); <\output> ) >B*72*Y+96*X*Y-224*X*Y+480*X*Y-96*X*Y+432*X*Y-3*B*X*Y-384*X*Y-320*X*Y-18*B*X+1152*X+9*B*X+576*X+240*X> <\input|200) >> factor(radcan(subst(B*X,C,subst(B^3*Y,E,%o197)))); <\output> ) >B*144*Y+192*X*Y-224*X*Y+960*X*Y-96*X*Y+3*B*X*Y-384*X*Y+320*X*Y+18*B*X+2304*X-18*B*X-480*X> <\input|201) >> factor(radcan(subst(B*X,C,subst(B^3*Y,E,%o198)))); <\output> ) >B*1512*Y+1440*X*Y-3808*X*Y+4320*X*Y-1056*X*Y+5616*X*Y-9*B*X*Y-1920*X*Y-2880*X*Y+54*B*X-3456*X-9*B*X+576*X+1200*X> <\input|202) >> taylor(%o199/B^12,B,0,5); <\output> ) >240*X+576*X+1152*X+-384*X-320*X*Y+480*X-96*X+432*X*Y+96*X-224*X*Y+72*Y+-3*X*Y-18*X+9*X*B+\> <\input|203) >> taylor(%o200/B^12,B,0,5); <\output> ) >-480*X+2304*X+-384*X+320*X*Y+960*X-96*X*Y+192*X-224*X*Y+144*Y+3*X*Y+18*X-18*X*B+\> <\input|204) >> taylor(%o201/B^12,B,0,5); <\output> ) >1200*X+576*X-3456*X+-1920*X-2880*X*Y+4320*X-1056*X+5616*X*Y+1440*X-3808*X*Y+1512*Y+-9*X*Y+54*X-9*X*B+\> <\input|205) >> qu1: 240*X^8+576*X^10+1152*X^12+(-384*X^8-320*X^6)*Y+(480*X^6-96*X^5+432*X^4)*Y^2+(96*X^3-224*X^2)*Y^3+72*Y^4; <\output> ) >72*Y+96*X-224*X*Y+480*X-96*X+432*X*Y+-384*X-320*X*Y+1152*X+576*X+240*X> <\input|206) >> cl1: -(-3*X^9*Y-18*X^12+9*X^11); <\output> ) >3*X*Y+18*X-9*X> <\input|207) >> radcan(qu1 - B^3*cl1 - %o199/B^12); <\output> ) >0> <\input|208) >> qu2: -480*X^8+2304*X^12+(-384*X^8+320*X^6)*Y+(960*X^6-96*X^5)*Y^2+(192*X^3-224*X^2)*Y^3+144*Y^4; <\output> ) >144*Y+192*X-224*X*Y+960*X-96*X*Y+320*X-384*X*Y+2304*X-480*X> <\input|209) >> cl2: -(3*X^9*Y+18*X^12-18*X^11); <\output> ) >-3*X*Y-18*X+18*X> <\input|210) >> radcan(qu2 - B^3*cl2 - %o200/B^12); <\output> ) >0> <\input|211) >> qu3: 1200*X^8+576*X^10-3456*X^12+(-1920*X^8-2880*X^6)*Y+(4320*X^6-1056*X^5+5616*X^4)*Y^2+(1440*X^3-3808*X^2)*Y^3+1512*Y^4; <\output> ) >1512*Y+1440*X-3808*X*Y+4320*X-1056*X+5616*X*Y+-1920*X-2880*X*Y-3456*X+576*X+1200*X> <\input|212) >> cl3: -(-9*X^9*Y+54*X^12-9*X^11); <\output> ) >9*X*Y-54*X+9*X> <\input|213) >> radcan(qu3 - B^3*cl3 - %o201/B^12); <\output> ) >0> <\input|214) >> factor(radcan(cl1 + cl2)); <\output> ) >9*X> <\input|215) >> cla: factor(radcan(cl1+cl2)); <\output> ) >9*X> <\input|216) >> qua: factor(radcan(qu1+qu2)); <\output> ) >8*27*Y+36*X*Y-56*X*Y+180*X*Y-24*X*Y+54*X*Y-96*X*Y+432*X+72*X-30*X> <\input|217) >> radcan(qua - B^3*cla - %o199/B^12 - %o200/B^12); <\output> ) >0> <\input|218) >> \; > In the same way as about a third of the way down BulkVacuumEnergy39.tm, choose the indep eqn lh sides to be qua - B^3*cla, qu2 - B^3*cl2, and qu3 - B^3*cl3, which means that by choosing cla to occur in both homogeneous eqns, we can avoid spurious solutions. <\input|218) >> factor(radcan(qua*cl2 - qu2*cla)); <\output> ) >-24*X*27*Y+198*X*Y-164*X*Y+396*X*Y-504*X*Y+306*X*Y+1080*X*Y-960*X*Y+432*X*Y-324*X*Y+432*X*Y-576*X*Y+504*X*Y+90*X*Y+2592*X-1728*X+432*X-432*X-180*X> <\input|219) >> factor(radcan(qua*cl3 - qu3*cla)); <\output> ) >72*X*27*Y-126*X*Y-218*X*Y-36*X*Y+168*X*Y+474*X*Y-1080*X*Y-312*X*Y-216*X*Y-648*X*Y+432*X*Y+576*X*Y+216*X*Y+330*X*Y-2592*X+864*X-432*X+180*X-180*X> <\input|220) >> factor(radcan(%o2189/(-24*X^9))); <\output> ) >-|24*X>> <\input|221) >> factor(radcan(%o218/(-24*X^9))); <\output> ) >27*Y+198*X*Y-164*X*Y+396*X*Y-504*X*Y+306*X*Y+1080*X*Y-960*X*Y+432*X*Y-324*X*Y+432*X*Y-576*X*Y+504*X*Y+90*X*Y+2592*X-1728*X+432*X-432*X-180*X> <\input|222) >> factor(radcan(%o219/(72*X^9))); <\output> ) >27*Y-126*X*Y-218*X*Y-36*X*Y+168*X*Y+474*X*Y-1080*X*Y-312*X*Y-216*X*Y-648*X*Y+432*X*Y+576*X*Y+216*X*Y+330*X*Y-2592*X+864*X-432*X+180*X-180*X> <\input|223) >> heq1: %o221; <\output> ) >27*Y+198*X*Y-164*X*Y+396*X*Y-504*X*Y+306*X*Y+1080*X*Y-960*X*Y+432*X*Y-324*X*Y+432*X*Y-576*X*Y+504*X*Y+90*X*Y+2592*X-1728*X+432*X-432*X-180*X> <\input|224) >> heq2: %o222; <\output> ) >27*Y-126*X*Y-218*X*Y-36*X*Y+168*X*Y+474*X*Y-1080*X*Y-312*X*Y-216*X*Y-648*X*Y+432*X*Y+576*X*Y+216*X*Y+330*X*Y-2592*X+864*X-432*X+180*X-180*X> <\input|225) >> plot3d(heq1/(0.000000000000000000001 +abs(heq1)), [X,0,5],[Y,0,5],[grid,20,20], [gnuplot_preamble,"set terminal postscript; set output \\"qoo1.eps\\";"]); <\output> <\errput> \; ) > <\input|226) >> plot3d(heq2/(0.000000000000000000001 +abs(heq2)), [X,0,5],[Y,0,5],[grid,20,20], [gnuplot_preamble,"set terminal postscript; set output \\"qoo2.eps\\";"]); <\output> <\errput> \; ) > <\input|227) >> \; > The interesting region from the 1st plot is 1>, 0.5>. <\input|227) >> plot3d(heq1/(0.000000000000000000001 +abs(heq1)), [X,0,1],[Y,0,0.5],[grid,20,20], [gnuplot_preamble,"set terminal postscript; set output \\"qoo3.eps\\";"]); <\output> <\errput> \; ) > <\input|228) >> \; > That last plot for some reason caused quite a bit of paging, so I am going to close this TeXmacs window to try to regain some memory. \ Well, try another plot first. \; <\input|228) >> plot3d(heq2/(0.000000000000000000001 +abs(heq2)), [X,0,1],[Y,0,0.5],[grid,20,20], [gnuplot_preamble,"set terminal postscript; set output \\"qoo4.eps\\";"]); <\output> <\errput> \; ) > <\input|229) >> \; > Perhaps there was a big rebuffering, because the above plot only took a second or two. <\input|229) >> plot3d(heq1/(0.000000000000000000001 +abs(heq1)), [X,0,0.2],[Y,0,0.2],[grid,20,20], [gnuplot_preamble,"set terminal postscript; set output \\"qoo5.eps\\";"]); <\output> <\errput> \; ) > <\input|230) >> plot3d(heq2/(0.000000000000000000001 +abs(heq2)), [X,0,0.2],[Y,0,0.2],[grid,20,20], [gnuplot_preamble,"set terminal postscript; set output \\"qoo6.eps\\";"]); <\output> <\errput> \; ) > <\input|231) >> \; > There seems to be no chance of a solution at all, even with negative , this time. \ So the two approaches are in agreement that there is no solution, but the plots of the zeros of the equations are much simpler this time. \ Which casts some doubt on the validity of using my guessed oxidation. <\input|231) >> plot3d(heq1/(0.000000000000000000001 +abs(heq1)), [X,0,0.1],[Y,0,0.1],[grid,20,20], [gnuplot_preamble,"set terminal postscript; set output \\"qoo7.eps\\";"]); <\output> <\errput> \; ) > <\input|232) >> plot3d(heq2/(0.000000000000000000001 +abs(heq2)), [X,0,0.1],[Y,0,0.1],[grid,20,20], [gnuplot_preamble,"set terminal postscript; set output \\"qoo8.eps\\";"]); <\output> <\errput> \; ) > <\input|233) >> \; > It seems clear there is going to be no solution at all this time: there is only one curve for each eqn., which near the origin is like a parabola for both, although for heq1 it curves to the right and back down to the axis again, while for heq2 it heads upwards, tending to become parallel to the axis. \ And the heq2 curve is above the heq1 curve everywhere away from the origin. \ So at least the two approaches are in agreement about there being no solution in this case. However this suggests the two approaches might disagree about whether there is a solution in the \S> case, which is another case where I can compare the two approaches. Move straight on to the \H\H> cases, for now, using my guessed oxidation, for now. \ I have now saved ttOnlyH3H4GSKPR.cdb as ttOnlyH3H2H2.cdb. \ Indices > now run over the >, > over one >, and > over the other >. \ Let now be the > curvature radius, and and be the curvature radii of the first and second > respectively. \ I have now clicked Run all for ttOnlyH3H2H2.cdb at 9.03 pm on Sat 19 Feb '11. \ It is now 9.21 pm, and the run has completed. Now consider the possible flux bilinears. \ The possible fluxes are >, >, >, >, >, >, >, >. \ Let their bilinears be: <\equation*> HH=6|b*c>\GGGG |b*c>>. <\equation*> HH=6|b*d>\GGGG |b*d>>. <\equation*> HH=|b*c>GG-GGGG-GG |b*c>>. <\equation*> HH=|b*c*d>GG-GGGG |b*c*d>>. <\equation*> HH=|b*d>GG-GGGG-GG |b*d>>. <\equation*> HH=|b*c*d>GGG-GGG |b*c*d>>. <\equation*> HH=|b*c*d>GGGG-GG |b*c*d>>. <\equation*> HH=|c*d>GG-GGGG-GG |c*d>>. I have now done cp FluxBilinears3.cdb FluxBilinears5.cdb. \ I have now clicked Run all for FLuxBilinears5.cdb at 11.00 pm. \ It is now 11.03 pm, and the run has completed. \ The result is fairly long. \ However only the first part of the result is needed, due to the @sumsort. \ The result has 78 important terms. \ I have now put in some substitutions to set the vanishing > to 0. \ I have now clicked Run all at 11.18 pm. \ It is now 11.20 pm and the run has completed. \ The result still has 78 important terms. \ The 78 important terms in the result are: <\equation*> 4 EE T - 4 EE T + 2 EE T - 2 EE T - 2 EE T + 2 EE T + 4 FF T - 4 FF T + 2 FF T - 2 FF T - 2 FF T + 2 FF T + 2 GG T - 2 GG T + 4 GG T - 4 GG T - 4 GG T + 4 GG T + 6 GG T - 6 GG T + 8 HH T - 8 HH T + 8 HH T - 8 HH T + 8 HH T - 8 HH T - 8 HH T + 8 HH T + 6 HH T - 6 HH T - 8 HH T + 8 HH T - 6 HH T + 6 HH T + 2 KK T - 2 KK T + 4 KK T - 4 KK T - 4 KK T + 4 KK T + 6 KK T - 6 KK T + 4 LL T - 4 LL T + 2 LL T - 2 LL T - 4 LL T + 4 LL T + 12 LL T - 12 LL T + 6 LL T - 6 LL T - 2 LL T + 2 LL T - 6 LL T + 6 LL T + 2 MM T - 2 MM T + 4 MM T - 4 MM T - 2 MM T + 2 MM T + 6 MM T - 6 MM T - 4 MM T + 4 MM T - 6 MM T + 6 MM T + 12 MM T - 12 MM T + 2 NN T - 2 NN T + 2 NN T - 2 NN T - 2 NN T + 2 NN T + 2 NN T - 2 NN T= <\equation*> =4 EE T + 4 FF T+ 2 GG T+ 8 HH T+ 2 KK T- 4 EE T \ - 4 FF T- 2 GG T- 8 HH T- 2 KK T + 2 EE T + 4 GG T+ 8 HH T+ 4 LL T+ 2 MM T - 2 EE T- 4 GG T- 8 HH T - 4 LL T- 2 MM T - 2 EE T - 4 GG T - 8 HH T- 4 LL T- 2 MM T+ 2 EE T \ + 4 GG T+ 8 HH T + 4 LL T + 2 MM T + 2 FF T + 8 HH T + 4 KK T+ 2 LL T+ 4 MM T- 2 FF T - 8 HH T - 4 KK T- 2 LL T- 4 MM T- 2 FF T - 8 HH T- 4 KK T - 2 LL T- 4 MM T+ 2 FF T \ + 8 HH T + 4 KK T + 2 LL T+ 4 MM T+ 6 GG T \ \ \ + 12 LL T + 2 NN T- 6 GG T \ \ \ \ - 12 LL T \ \ \ - 2 NN T \ \ + 6 HH T + 6 LL T+ 6 MM T + 2 NN T - 6 HH T - 6 LL T - 6 MM T - 2 NN T- 6 HH T \ - 6 LL T - 6 MM T \ - 2 NN T + 6 HH T \ \ \ \ + 6 LL T + 6 MM T + 2 NN T + 6 KK T+ 12 MM T + 2 NN T- 6 KK T \ \ \ \ - 12 MM T \ \ \ \ \ - 2 NN T= <\equation*> =4 EE \ + 4 FF+ 2 GG + 8 HH+ 2 KK T- 4 EE T \ + 4 FF T+ 2 GG T+ 8 HH T+ 2 KK T + 2 EE \ + 4 GG + 8 HH + 4 LL+ 2 MM T - 2 EE+ 4 GG + 8 HH \ + 4 LL+ 2 MM T - 2 EE + 4 GG+ 8 HH + 4 LL + 2 MM T+ 2 EE \ \ + 4 GG+ 8 HH \ + 4 LL + 2 MM T + 2 FF \ + 8 HH \ + 4 KK+ 2 LL+ 4 MM T- 2 FF + 8 HH \ + 4 KK + 2 LL + 4 MM T- 2 FF + 8 HH+ 4 KK + 2 LL + 4 MM T+ 2 FF \ \ + 8 HH \ + 4 KK \ + 2 LL + 4 MM T+ 6 GG \ \ \ \ + 12 LL + 2 NN T- 6 GG \ + 12 LL \ \ \ + 2 NN T \ \ + 6 HH \ + 6 LL + 6 MM + 2 NN T - 6 HH + 6 LL \ + 6 MM \ + 2 NN T- 6 HH \ + 6 LL + 6 MM \ \ + 2 NN T + 6 HH \ \ \ \ + 6 LL + 6 MM \ + 2 NN T + 6 KK+ 12 MM + 2 NN T- 6 KK \ \ \ + 12 MM + 2 NN T= <\equation*> =4 EE \ + 4 FF+ 2 GG + 8 HH+ 2 KK T- \ T + 2 EE \ + 4 GG + 8 HH + 4 LL+ 2 MM T - T - T+T + 2 FF \ + 8 HH \ + 4 KK+ 2 LL+ 4 MM T-T- T+T+ 6 GG \ \ \ \ + 12 LL + 2 NN T- \ T \ + 6 HH \ + 6 LL + 6 MM + 2 NN T - T- T + \ T + 6 KK+ 12 MM + 2 NN T- \ T=T H H= <\equation*> =4 EE \ + 4 FF+ 2 GG + 8 HH+ 2 KK GG- \ GG T+ 2 EE \ + 4 GG + 8 HH + 4 LL+ 2 MM T - T - T+T + 2 FF \ + 8 HH \ + 4 KK+ 2 LL+ 4 MM T-T- T+T+ 6 GG \ \ \ \ + 12 LL + 2 NN T- \ T \ + 6 HH \ + 6 LL + 6 MM + 2 NN T - T- T + \ T + 6 KK+ 12 MM + 2 NN T- \ T=T H H= <\equation*> T H H=4 EE \ + 4 FF+ 2 GG + 8 HH+ 2 KK GG- \ GG T+ 2 EE \ + 4 GG + 8 HH + 4 LL+ 2 MM T - T - T+T + 2 FF \ + 8 HH \ + 4 KK+ 2 LL+ 4 MM T-T- T+T+ 6 GG \ \ \ \ + 12 LL + 2 NN T- \ T \ + 6 HH \ + 6 LL + 6 MM + 2 NN T - T- T + \ T + 6 KK+ 12 MM + 2 NN T- \ T |b*c>>. \ |b*d>>. \ |b*c>>. \ |b*c*d>>. \ |b*d>>. \ |b*c*d>>. \ |b*c*d>>. \ |c*d>>. <\equation*> T H H=4|b*c> \ \ + 4 |b*d>+ 2 |b*c> + 8 |b*c*d>+ 2 |b*d> GG- \ GG T+ 2 |b*c> \ + 4 |b*c> + 8|b*c*d> \ + 4 |b*c*d>+ 2 |b*c*d> T - T - T+T + 2 |b*d> \ + 8 |b*c*d> \ + 4|b*d> + 2 |b*c*d>+ 4|b*c*d> \ T-T- T+T+ 6 |b*c> \ \ \ \ + 12|b*c*d> \ + 2 |c*d> T- \ T \ + 6 |b*c*d> \ + 6 |b*c*d>+ 6 |b*c*d> + 2|c*d> \ T - T- T + \ T + 6 |b*d>+ 12|b*c*d> \ + 2 |c*d> T- \ T Hence: H=4|b*c> \ \ + 4 |b*d>+ 2 |b*c> + 8 |b*c*d>+ 2 |b*d> GG- \ GG > H= 2 |b*c> \ + 4 |b*c> + 8|b*c*d> \ + 4 |b*c*d>+ 2 |b*c*d>GG> and 3 similar H= 2 |b*d> \ + 8 |b*c*d> \ + 4|b*d> + 2 |b*c*d>+ 4|b*c*d> \ GG> and 3 similar H= 6 |b*c> \ \ \ \ + 12|b*c*d> \ + 2 |c*d> GG- GG \ > H= 6 |b*c*d> \ + 6 |b*c*d>+ 6 |b*c*d> + 2|c*d> \ GG> and 3 similar H= 6 |b*d>+ 12|b*c*d> \ + 2 |c*d> GG- \ GG> Now the result of ttOnlyH3H2H2.cdb is: <\equation*> 576 s s s s s s s s + 384 s s s s t t t t + 6144 u u u u u u u u + 576 t t t t t t t t + 2304 s s s s w w w w + 2304 s s s s x x x x + 18432 u u u u w w w w + 18432 u u u u x x x x + 2304 t t t t w w w w + 2304 t t t t x x x x + 1152 s s s s v v v v + 4608 u u u u v v v v + 1152 t t t t v v v v + 13824 w w w w w w w w + 32256 w w w w x x x x + 13824 x x x x x x x x + 16128 v v v v w w w w + 16128 v v v v x x x x + 2880 v v v v v v v v + 1536 s s s s u u u u + 3072 s s t t u u u u + 1536 t t t t u u u u + 18432 s s u u w w x x + 18432 t t u u w w x x + 9216 s s v v w w w w + 36864 u u v v w w x x + 9216 t t v v x x x x <\equation*> 576*s*s*s*s*s*s*s*s + 384*s*s*s*s*t*t*t*t + 6144*u*u*u*u*u*u*u*u + 576*t*t*t*t*t*t*t*t + 2304*s*s*s*s*w*w*w*w + 2304*s*s*s*s*x*x*x*x + 18432*u*u*u*u*w*w*w*w + 18432*u*u*u*u*x*x*x*x + 2304*t*t*t*t*w*w*w*w + 2304*t*t*t*t*x*x*x*x + 1152*s*s*s*s*v*v*v*v + 4608*u*u*u*u*v*v*v*v + 1152*t*t*t*t*v*v*v*v + 13824*w*w*w*w*w*w*w*w + 32256*w*w*w*w*x*x*x*x + 13824*x*x*x*x*x*x*x*x + 16128*v*v*v*v*w*w*w*w + 16128*v*v*v*v*x*x*x*x + 2880*v*v*v*v*v*v*v*v + 1536*s*s*s*s*u*u*u*u + 3072*s*s*t*t*u*u*u*u + 1536*t*t*t*t*u*u*u*u + 18432*s*s*u*u*w*w*x*x + 18432*t*t*u*u*w*w*x*x + 9216*s*s*v*v*w*w*w*w + 36864*u*u*v*v*w*w*x*x + 9216*t*t*v*v*x*x*x*x <\input|233) >> 576*s*s*s*s*s*s*s*s + 384*s*s*s*s*t*t*t*t + 6144*u*u*u*u*u*u*u*u + 576*t*t*t*t*t*t*t*t + 2304*s*s*s*s*w*w*w*w + 2304*s*s*s*s*x*x*x*x + 18432*u*u*u*u*w*w*w*w + 18432*u*u*u*u*x*x*x*x + 2304*t*t*t*t*w*w*w*w + 2304*t*t*t*t*x*x*x*x + 1152*s*s*s*s*v*v*v*v + 4608*u*u*u*u*v*v*v*v + 1152*t*t*t*t*v*v*v*v + 13824*w*w*w*w*w*w*w*w + 32256*w*w*w*w*x*x*x*x + 13824*x*x*x*x*x*x*x*x + 16128*v*v*v*v*w*w*w*w + 16128*v*v*v*v*x*x*x*x + 2880*v*v*v*v*v*v*v*v + 1536*s*s*s*s*u*u*u*u + 3072*s*s*t*t*u*u*u*u + 1536*t*t*t*t*u*u*u*u + 18432*s*s*u*u*w*w*x*x + 18432*t*t*u*u*w*w*x*x + 9216*s*s*v*v*w*w*w*w + 36864*u*u*v*v*w*w*x*x + 9216*t*t*v*v*x*x*x*x; <\output> ) >13824*x+32256*w*x+16128*v*x+9216*t*v*x+18432*u*x+2304*t*x+2304*s*x+36864*u*v*w*x+18432*t*u*w*x+18432*s*u*w*x+13824*w+16128*v*w+9216*s*v*w+18432*u*w+2304*t*w+2304*s*w+2880*v+4608*u*v+1152*t*v+1152*s*v+6144*u+1536*t*u+3072*s*t*u+1536*s*u+576*t+384*s*t+576*s> <\input|234) >> subst(sqrt(U),u,subst(sqrt(V),v,subst(sqrt(W),w,subst(sqrt(X), x,subst(sqrt(S),s,subst(sqrt(T),t,%o233)))))); <\output> ) >13824*X+32256*W*X+16128*V*X+9216*T*V*X+18432*U*X+2304*T*X+2304*S*X+36864*U*V*W*X+18432*T*U*W*X+18432*S*U*W*X+13824*W+16128*V*W+9216*S*V*W+18432*U*W+2304*T*W+2304*S*W+2880*V+4608*U*V+1152*T*V+1152*S*V+6144*U+1536*T*U+3072*S*T*U+1536*S*U+576*T+384*S*T+576*S> <\input|235) >> \; > The substitutions used to derive the above are: X_{Q R S T} -\ v v y_{Q T} y_{R S} - v v y_{Q S} y_{R T} <\equation*> X=RG-HH+HH For the 3-sphere, =>GG-GG>. \ Cases: <\equation*> X=RG-HH+HH H=4|b*c> \ \ + 4 |b*d>+ 2 |b*c> + 8 |b*c*d>+ 2 |b*d> GG- \ GG > H=4|b*c> \ \ + 4 |b*d>+ 2 |b*c> + 8 |b*c*d>+ 2 |b*d> GG- \ GG > H=4|b*c> \ \ + 4 |b*d>+ 2 |b*c> + 8 |b*c*d>+ 2 |b*d> GG- \ GG > <\equation*> X=RG-4|b*c> \ \ + 4 |b*d>+ 2 |b*c> + 8 |b*c*d>+ 2 |b*d> GG- \ GG +4|b*c> \ \ + 4 |b*d>+ 2 |b*c> + 8 |b*c*d>+ 2 |b*d> GG- \ GG <\equation*> X=RG+4|b*c> \ \ + 4 |b*d>+ 2 |b*c> + 8 |b*c*d>+ 2 |b*d> \ GG - \ GG <\equation*> X=>+|2*b*c> \ \ + |2*b*d>+ \ |4*b*c> +|b*c*d>+ |4*b*d> \ GG - \ GG The is for >. X_{Q R S T} -\ v v y_{Q T} y_{R S} - v v y_{Q S} y_{R T} >+|2*b*c> \ \ + |2*b*d>+ \ |4*b*c> +|b*c*d>+ |4*b*d>>. X_{a Q b R} -\ - w w q_{a b} y_{Q R} and 3 similar. <\equation*> X=RG-HH+HH <\equation*> X=HH H= 2 |b*c> \ + 4 |b*c> + 8|b*c*d> \ + 4 |b*c*d>+ 2 |b*c*d>GG> H= -2 |b*c> \ + 4 |b*c> + 8|b*c*d> \ + 4 |b*c*d>+ 2 |b*c*d>GG> <\equation*> X=- |4*b*c> \ + \ |2*b*c> +|b*c*d> \ + |2*b*c*d>+ |4*b*c*d>GG X_{a Q b R} -\ - w w q_{a b} y_{Q R} and 3 similar. |4*b*c> \ + \ |2*b*c> +|b*c*d> \ + |2*b*c*d>+ |4*b*c*d>>. X_{i Q j R} -\ - x x r_{i j} y_{Q R} <\equation*> X=RG-HH+HH <\equation*> X=HH H= 2 |b*d> \ + 8 |b*c*d> \ + 4|b*d> + 2 |b*c*d>+ 4|b*c*d> \ GG> H= -2 |b*d> \ + 8 |b*c*d> \ + 4|b*d> + 2 |b*c*d>+ 4|b*c*d> \ GG> <\equation*> X=- |4*b*d> \ + \ |b*c*d> \ +|2*b*d> + |4*b*c*d>+ |2*b*c*d> \ GG X_{i Q j R} -\ - x x r_{i j} y_{Q R} |4*b*d> \ + \ |b*c*d> \ +|2*b*d> + |4*b*c*d>+ |2*b*c*d> >. X_{a i b j} -\ - u u q_{a b} r_{i j} <\equation*> X=RG-HH+HH <\equation*> X=HH H= 6 |b*c*d> \ + 6 |b*c*d>+ 6 |b*c*d> + 2|c*d> GG> H=- 6 |b*c*d> \ + 6 |b*c*d>+ 6 |b*c*d> + 2|c*d> GG> <\equation*> X=- |4*b*c*d> \ + |4*b*c*d>+ \ |4*b*c*d> + |4*c*d> GG X_{a i b j} -\ - u u q_{a b} r_{i j} |4*b*c*d> \ + |4*b*c*d>+ \ |4*b*c*d> + |4*c*d> >. X_{a b c d} -\ s s q_{a d} q_{b c} - s s q_{a c} q_{b d} <\equation*> X=RG-HH+HH H= 6 |b*c> \ \ \ \ + 12|b*c*d> \ + 2 |c*d> GG- GG \ > H= 6 |b*c> \ \ \ \ + 12|b*c*d> \ + 2 |c*d> GG- GG \ > H= 6 |b*c> \ \ \ \ + 12|b*c*d> \ + 2 |c*d> GG- GG \ > <\equation*> X=RG-6 |b*c> \ \ \ \ + 12|b*c*d> \ + 2 |c*d> GG- GG \ +6 |b*c> \ \ \ \ + 12|b*c*d> \ + 2 |c*d> GG- GG \ <\equation*> X=RG+6 |b*c> \ \ \ \ + 12|b*c*d> \ + 2 |c*d> GG \ - GG <\equation*> X=|c>+ |4*b*c> \ \ \ \ +|2*b*c*d> \ + |4*c*d> GG \ - GG > is 1 for the hyperbolic case. X_{a b c d} -\ s s q_{a d} q_{b c} - s s q_{a c} q_{b d} |c>+ |4*b*c> \ \ \ \ +|2*b*c*d> \ + |4*c*d> >. X_{i j k l} -\ t t r_{i l} r_{j k} - t t r_{i k} r_{j l} <\equation*> X=RG-HH+HH H= 6 |b*d>+ 12|b*c*d> \ + 2 |c*d> GG- \ GG> H= 6 |b*d>+ 12|b*c*d> \ + 2 |c*d> GG- \ GG> H= 6 |b*d>+ 12|b*c*d> \ + 2 |c*d> GG- \ GG> <\equation*> X=RG- 6 |b*d>+ 12|b*c*d> \ + 2 |c*d> GG- \ GG+ 6 |b*d>+ 12|b*c*d> \ + 2 |c*d> GG- \ GG <\equation*> X=RG+ 6 |b*d>+ 12|b*c*d> \ + 2 |c*d> \ GG- \ GG <\equation*> X=|d>+ |4*b*d>+|2*b*c*d> \ + |4*c*d> \ GG- \ GG > is 1 for the hyperbolic case. X_{i j k l} -\ t t r_{i l} r_{j k} - t t r_{i k} r_{j l} |d>+ |4*b*d>+|2*b*c*d> \ + |4*c*d> >. So in summary: >+|2*b*c> \ \ + |2*b*d>+ \ |4*b*c> +|b*c*d>+ |4*b*d>>. |4*b*c> \ + \ |2*b*c> +|b*c*d> \ + |2*b*c*d>+ |4*b*c*d>>. |4*b*d> \ + \ |b*c*d> \ +|2*b*d> + |4*b*c*d>+ |2*b*c*d> >. |4*b*c*d> \ + |4*b*c*d>+ \ |4*b*c*d> + |4*c*d> >. |c>+ |4*b*c> \ \ \ \ +|2*b*c*d> \ + |4*c*d> >. \ |d>+ |4*b*d>+|2*b*c*d> \ + |4*c*d> >. <\input|235) >> factor(radcan(subst(((1/(b^2))+((e^2)/(2*b^6*c^2)) \ \ + ((f^2)/(2*b^6*d^2))+ \ ((g^2)/(4*b^4*c^4)) +((h^2)/(b^4*c^2*d^2))+ ((k^2)/(4*b^4*d^4))),V, subst(( ((e^2)/(4*b^6*c^2)) \ + \ ((g^2)/(2*b^4*c^4)) +((h^2)/(b^4*c^2*d^2)) \ + ((l^2)/(2*b^2*c^4*d^2))+ ((m^2)/(4*b^2*c^2*d^4))),W, subst(( ((f^2)/(4*b^6*d^2)) \ + \ ((h^2)/(b^4*c^2*d^2)) \ +((k^2)/(2*b^4*d^4)) + ((l^2)/(4*b^2*c^4*d^2))+ ((m^2)/(2*b^2*c^2*d^4)) ),X, subst(( ((3*h^2)/(4*b^4*c^2*d^2)) \ + ((3*l^2)/(4*b^2*c^4*d^2))+ \ ((3*m^2)/(4*b^2*c^2*d^4)) + ((n^2)/(4*c^4*d^4)) ),U, subst(((eta/(c^2))+ ((3*g^2)/(4*b^4*c^4)) \ \ \ \ +((3*l^2)/(2*b^2*c^4*d^2)) \ + ((n^2)/(4*c^4*d^4)) ),S, subst(((sigma/(d^2))+ ((3*k^2)/(4*b^4*d^4))+((3*m^2)/(2*b^2*c^2*d^4)) \ + ((n^2)/(4*c^4*d^4)) ),T,%o234)))))))); <\output> ) >96*b*c*d*\+96*b*c*d*n*\+576*b*c*d*m*\+288*b*c*d*k*\+56*b*c*d*n*\+528*b*c*d*m*n*\+144*b*c*d*l*n*\+216*b*c*d*k*n*\+96*b*c*d*h*n*\+24*b*c*d*g*n*\+32*b*c*d*\*n*\+1560*b*c*d*m*\+480*b*c*d*l*m*\+1488*b*c*d*k*m*\+864*b*c*d*h*m*\+96*b*c*d*g*m*\+96*b*c*d*f*m*\+48*b*c*d*e*m*\+408*b*c*d*l*\+96*b*c*d*k*l*\+864*b*c*d*h*l*\+336*b*c*d*g*l*\+48*b*c*d*f*l*\+192*b*c*d*\*l*\+96*b*c*d*e*l*\+432*b*c*d*k*\+480*b*c*d*h*k*\+24*b*c*d*g*k*\+144*b*c*d*f*k*\+48*b*c*d*e*k*\+96*b*c*d*k*\+1104*b*c*d*h*\+480*b*c*d*g*h*\+384*b*c*d*f*h*\+384*b*c*d*e*h*\+384*b*c*d*h*\+144*b*c*d*g*\+48*b*c*d*f*g*\+96*b*c*d*\*g*\+144*b*c*d*e*g*\+96*b*c*d*g*\+72*b*c*d*f*\+96*b*c*d*e*f*\+192*b*c*d*f*\+64*b*c*d*\*\+72*b*c*d*e*\+192*b*c*d*e*\+192*b*c*d*\+24*b*c*d*n*\+264*b*c*d*m*n*\+168*b*c*d*l*n*\+84*b*c*d*k*n*\+96*b*c*d*h*n*\+36*b*c*d*g*n*\+48*b*c*d*\*n*\+1236*b*c*d*m*n*\+1344*b*c*d*l*m*n*\+984*b*c*d*k*m*n*\+1440*b*c*d*h*m*n*\+456*b*c*d*g*m*n*\+96*b*c*d*f*m*n*\+288*b*c*d*\*m*n*\+120*b*c*d*e*m*n*\+660*b*c*d*l*n*\+456*b*c*d*k*l*n*\+1440*b*c*d*h*l*n*\+408*b*c*d*g*l*n*\+120*b*c*d*f*l*n*\+288*b*c*d*\*l*n*\+96*b*c*d*e*l*n*\+216*b*c*d*k*n*\+768*b*c*d*h*k*n*\+240*b*c*d*g*k*n*\+72*b*c*d*f*k*n*\+48*b*c*d*\*k*n*\+120*b*c*d*e*k*n*\+48*b*c*d*k*n*\+1392*b*c*d*h*n*\+768*b*c*d*g*h*n*\+384*b*c*d*f*h*n*\+192*b*c*d*\*h*n*\+384*b*c*d*e*h*n*\+192*b*c*d*h*n*\+72*b*c*d*g*n*\+120*b*c*d*f*g*n*\+48*b*c*d*\*g*n*\+72*b*c*d*e*g*n*\+48*b*c*d*g*n*\+36*b*c*d*f*n*\+96*b*c*d*e*f*n*\+96*b*c*d*f*n*\+32*b*c*d*\*n*\+36*b*c*d*e*n*\+96*b*c*d*e*n*\+96*b*c*d*n*\+2376*b*c*d*m*\+2880*b*c*d*l*m*\+3300*b*c*d*k*m*\+4992*b*c*d*h*m*\+1176*b*c*d*g*m*\+624*b*c*d*f*m*\+288*b*c*d*\*m*\+624*b*c*d*e*m*\+384*b*c*d*m*\+3096*b*c*d*l*m*\+1968*b*c*d*k*l*m*\+8016*b*c*d*h*l*m*\+2400*b*c*d*g*l*m*\+768*b*c*d*f*l*m*\+1152*b*c*d*\*l*m*\+912*b*c*d*e*l*m*\+384*b*c*d*l*m*\+1776*b*c*d*k*m*\+5328*b*c*d*h*k*m*\+984*b*c*d*g*k*m*\+1056*b*c*d*f*k*m*\+888*b*c*d*e*k*m*\+1056*b*c*d*k*m*\+8880*b*c*d*h*m*\+3984*b*c*d*g*h*m*\+3024*b*c*d*f*h*m*\+576*b*c*d*\*h*m*\+2784*b*c*d*e*h*m*\+2688*b*c*d*h*m*\+432*b*c*d*g*m*\+528*b*c*d*f*g*m*\+288*b*c*d*\*g*m*\+432*b*c*d*e*g*m*\+288*b*c*d*g*m*\+408*b*c*d*f*m*\+624*b*c*d*e*f*m*\+960*b*c*d*f*m*\+192*b*c*d*\*m*\+216*b*c*d*e*m*\+576*b*c*d*e*m*\+576*b*c*d*m*\+720*b*c*d*l*\+1212*b*c*d*k*l*\+2976*b*c*d*h*l*\+528*b*c*d*g*l*\+336*b*c*d*f*l*\+288*b*c*d*\*l*\+192*b*c*d*e*l*\+96*b*c*d*l*\+240*b*c*d*k*l*\+3600*b*c*d*h*k*l*\+1176*b*c*d*g*k*l*\+312*b*c*d*f*k*l*\+288*b*c*d*\*k*l*\+624*b*c*d*e*k*l*\+384*b*c*d*k*l*\+5232*b*c*d*h*l*\+2064*b*c*d*g*h*l*\+1440*b*c*d*f*h*l*\+576*b*c*d*\*h*l*\+1104*b*c*d*e*h*l*\+768*b*c*d*h*l*\+336*b*c*d*f*g*l*\+96*b*c*d*f*l*\+240*b*c*d*e*f*l*\+192*b*c*d*f*l*\+420*b*c*d*k*\+1488*b*c*d*h*k*\+132*b*c*d*g*k*\+504*b*c*d*f*k*\+264*b*c*d*e*k*\+528*b*c*d*k*\+4728*b*c*d*h*k*\+1680*b*c*d*g*h*k*\+1920*b*c*d*f*h*k*\+1632*b*c*d*e*h*k*\+2112*b*c*d*h*k*\+216*b*c*d*g*k*\+168*b*c*d*f*g*k*\+144*b*c*d*\*g*k*\+216*b*c*d*e*g*k*\+144*b*c*d*g*k*\+324*b*c*d*f*k*\+336*b*c*d*e*f*k*\+672*b*c*d*f*k*\+96*b*c*d*\*k*\+108*b*c*d*e*k*\+288*b*c*d*e*k*\+288*b*c*d*k*\+3840*b*c*d*h*\+1752*b*c*d*g*h*\+2112*b*c*d*f*h*\+288*b*c*d*\*h*\+1344*b*c*d*e*h*\+1536*b*c*d*h*\+480*b*c*d*f*g*h*\+480*b*c*d*f*h*\+528*b*c*d*e*f*h*\+768*b*c*d*f*h*\+24*b*c*d*f*g*\+48*b*c*d*f*\+48*b*c*d*e*f*\+96*b*c*d*f*\+9*b*n+108*b*c*m*n+108*b*d*l*n+18*b*c*k*n+72*b*c*d*h*n+18*b*d*g*n+24*b*c*d*\*n+645*b*c*m*n+1140*b*c*d*l*m*n+366*b*c*k*m*n+1296*b*c*d*h*m*n+282*b*c*d*g*m*n+84*b*c*d*f*m*n+168*b*c*d*\*m*n+78*b*c*d*e*m*n+645*b*d*l*n+282*b*c*d*k*l*n+1296*b*c*d*h*l*n+366*b*d*g*l*n+78*b*c*d*f*l*n+264*b*c*d*\*l*n+84*b*c*d*e*l*n+96*b*c*k*n+540*b*c*d*h*k*n+132*b*c*d*g*k*n+78*b*c*d*f*k*n+36*b*c*d*\*k*n+66*b*c*d*e*k*n+36*b*c*d*k*n+1188*b*c*d*h*n+540*b*c*d*g*h*n+288*b*c*d*f*h*n+96*b*c*d*\*h*n+288*b*c*d*e*h*n+144*b*c*d*h*n+96*b*d*g*n+66*b*c*d*f*g*n+84*b*c*d*\*g*n+78*b*c*d*e*g*n+36*b*c*d*g*n+33*b*c*d*f*n+60*b*c*d*e*f*n+72*b*c*d*f*n+56*b*c*d*\*n+33*b*c*d*e*n+72*b*c*d*e*n+72*b*c*d*n+1926*b*c*m*n+4878*b*c*d*l*m*n+2013*b*c*k*m*n+6648*b*c*d*h*m*n+1509*b*c*d*g*m*n+660*b*c*d*f*m*n+660*b*c*d*\*m*n+624*b*c*d*e*m*n+312*b*c*d*m*n+4878*b*c*d*l*m*n+3120*b*c*d*k*l*m*n+13272*b*c*d*h*l*m*n+3120*b*c*d*g*l*m*n+1308*b*c*d*f*l*m*n+1344*b*c*d*\*l*m*n+1308*b*c*d*e*l*m*n+672*b*c*d*l*m*n+924*b*c*k*m*n+5424*b*c*d*h*k*m*n+1332*b*c*d*g*k*m*n+780*b*c*d*f*k*m*n+408*b*c*d*\*k*m*n+738*b*c*d*e*k*m*n+600*b*c*d*k*m*n+11448*b*c*d*h*m*n+5376*b*c*d*g*h*m*n+3144*b*c*d*f*h*m*n+1440*b*c*d*\*h*m*n+3120*b*c*d*e*h*m*n+2592*b*c*d*h*m*n+768*b*c*d*g*m*n+756*b*c*d*f*g*m*n+456*b*c*d*\*g*m*n+858*b*c*d*e*g*m*n+696*b*c*d*g*m*n+294*b*c*d*f*m*n+96*b*c*d*\*f*m*n+576*b*c*d*e*f*m*n+624*b*c*d*f*m*n+144*b*c*d*\*m*n+120*b*c*d*e*\*m*n+318*b*c*d*e*m*n+672*b*c*d*e*m*n+432*b*c*d*m*n+1926*b*d*l*n+1509*b*c*d*k*l*n+6648*b*c*d*h*l*n+2013*b*d*g*l*n+624*b*c*d*f*l*n+1236*b*c*d*\*l*n+660*b*c*d*e*l*n+312*b*c*d*l*n+768*b*c*d*k*l*n+5376*b*c*d*h*k*l*n+1332*b*c*d*g*k*l*n+858*b*c*d*f*k*l*n+456*b*c*d*\*k*l*n+756*b*c*d*e*k*l*n+696*b*c*d*k*l*n+11448*b*c*d*h*l*n+5424*b*c*d*g*h*l*n+3120*b*c*d*f*h*l*n+1440*b*c*d*\*h*l*n+3144*b*c*d*e*h*l*n+2592*b*c*d*h*l*n+924*b*d*g*l*n+738*b*c*d*f*g*l*n+984*b*c*d*\*g*l*n+780*b*c*d*e*g*l*n+600*b*c*d*g*l*n+318*b*c*d*f*l*n+120*b*c*d*\*f*l*n+576*b*c*d*e*f*l*n+672*b*c*d*f*l*n+528*b*c*d*\*l*n+96*b*c*d*e*\*l*n+294*b*c*d*e*l*n+624*b*c*d*e*l*n+432*b*c*d*l*n+105*b*c*k*n+1212*b*c*d*h*k*n+327*b*c*d*g*k*n+126*b*c*d*f*k*n+72*b*c*d*\*k*n+186*b*c*d*e*k*n+132*b*c*d*k*n+4644*b*c*d*h*k*n+2328*b*c*d*g*h*k*n+1464*b*c*d*f*h*k*n+768*b*c*d*\*h*k*n+1416*b*c*d*e*h*k*n+1440*b*c*d*h*k*n+327*b*c*d*g*k*n+408*b*c*d*f*g*k*n+240*b*c*d*\*g*k*n+408*b*c*d*e*g*k*n+456*b*c*d*g*k*n+81*b*c*d*f*k*n+72*b*c*d*\*f*k*n+240*b*c*d*e*f*k*n+168*b*c*d*f*k*n+24*b*c*d*\*k*n+120*b*c*d*e*\*k*n+48*b*c*d*\*k*n+129*b*c*d*e*k*n+264*b*c*d*e*k*n+72*b*c*d*k*n+6480*b*c*d*h*n+4644*b*c*d*g*h*n+2880*b*c*d*f*h*n+1392*b*c*d*\*h*n+2880*b*c*d*e*h*n+2880*b*c*d*h*n+1212*b*c*d*g*h*n+1416*b*c*d*f*g*h*n+768*b*c*d*\*g*h*n+1464*b*c*d*e*g*h*n+1440*b*c*d*g*h*n+456*b*c*d*f*h*n+384*b*c*d*\*f*h*n+888*b*c*d*e*f*h*n+864*b*c*d*f*h*n+96*b*c*d*\*h*n+384*b*c*d*e*\*h*n+192*b*c*d*\*h*n+456*b*c*d*e*h*n+864*b*c*d*e*h*n+288*b*c*d*h*n+105*b*d*g*n+186*b*c*d*f*g*n+216*b*c*d*\*g*n+126*b*c*d*e*g*n+132*b*c*d*g*n+129*b*c*d*f*g*n+120*b*c*d*\*f*g*n+240*b*c*d*e*f*g*n+264*b*c*d*f*g*n+216*b*c*d*\*g*n+72*b*c*d*e*\*g*n+48*b*c*d*\*g*n+81*b*c*d*e*g*n+168*b*c*d*e*g*n+72*b*c*d*g*n+12*b*c*d*f*n+36*b*c*d*\*f*n+60*b*c*d*e*f*n+24*b*c*d*f*n+96*b*c*d*e*\*f*n+96*b*c*d*\*f*n+60*b*c*d*e*f*n+96*b*c*d*e*f*n+96*b*c*d*\*n+36*b*c*d*e*\*n+96*b*c*d*e*\*n+96*b*c*d*\*n+12*b*c*d*e*n+24*b*c*d*e*n+2613*b*c*m+7692*b*c*d*l*m+4542*b*c*k*m+13392*b*c*d*h*m+2748*b*c*d*g*m+1812*b*c*d*f*m+720*b*c*d*\*m+1536*b*c*d*e*m+1152*b*c*d*m+11709*b*c*d*l*m+8568*b*c*d*k*l*m+35856*b*c*d*h*l*m+8766*b*c*d*g*l*m+4230*b*c*d*f*l*m+3096*b*c*d*\*l*m+4248*b*c*d*e*l*m+2736*b*c*d*l*m+3732*b*c*k*m+19524*b*c*d*h*k*m+3585*b*c*d*g*k*m+3570*b*c*d*f*k*m+528*b*c*d*\*k*m+2670*b*c*d*e*k*m+2868*b*c*d*k*m+35952*b*c*d*h*m+15420*b*c*d*g*h*m+11544*b*c*d*f*h*m+2976*b*c*d*\*h*m+10248*b*c*d*e*h*m+9744*b*c*d*h*m+2274*b*c*d*g*m+2130*b*c*d*f*g*m+1212*b*c*d*\*g*m+2598*b*c*d*e*g*m+2076*b*c*d*g*m+1257*b*c*d*f*m+192*b*c*d*\*f*m+1956*b*c*d*e*f*m+2568*b*c*d*f*m+408*b*c*d*\*m+336*b*c*d*e*\*m+96*b*c*d*\*m+1014*b*c*d*e*m+2280*b*c*d*e*m+1704*b*c*d*m+7692*b*c*d*l*m+8766*b*c*d*k*l*m+35856*b*c*d*h*l*m+8568*b*c*d*g*l*m+4248*b*c*d*f*l*m+2880*b*c*d*\*l*m+4230*b*c*d*e*l*m+2736*b*c*d*l*m+3984*b*c*d*k*l*m+31044*b*c*d*h*k*l*m+7788*b*c*d*g*k*l*m+4632*b*c*d*f*k*l*m+2400*b*c*d*\*k*l*m+4380*b*c*d*e*k*l*m+3696*b*c*d*k*l*m+63960*b*c*d*h*l*m+31044*b*c*d*g*h*l*m+18168*b*c*d*f*h*l*m+8016*b*c*d*\*h*l*m+18168*b*c*d*e*h*l*m+15072*b*c*d*h*l*m+3984*b*c*d*g*l*m+4380*b*c*d*f*g*l*m+1968*b*c*d*\*g*l*m+4632*b*c*d*e*g*l*m+3696*b*c*d*g*l*m+1644*b*c*d*f*l*m+912*b*c*d*\*f*l*m+3108*b*c*d*e*f*l*m+3360*b*c*d*f*l*m+480*b*c*d*\*l*m+768*b*c*d*e*\*l*m+384*b*c*d*\*l*m+1644*b*c*d*e*l*m+3360*b*c*d*e*l*m+2208*b*c*d*l*m+1542*b*c*k*m+11352*b*c*d*h*k*m+1662*b*c*d*g*k*m+2412*b*c*d*f*k*m+1596*b*c*d*e*k*m+2040*b*c*d*k*m+36564*b*c*d*h*k*m+14040*b*c*d*g*h*k*m+13980*b*c*d*f*h*k*m+2064*b*c*d*\*h*k*m+10872*b*c*d*e*h*k*m+12336*b*c*d*h*k*m+2064*b*c*d*g*k*m+2184*b*c*d*f*g*k*m+1176*b*c*d*\*g*k*m+2514*b*c*d*e*g*k*m+2376*b*c*d*g*k*m+1566*b*c*d*f*k*m+2208*b*c*d*e*f*k*m+2976*b*c*d*f*k*m+336*b*c*d*\*k*m+336*b*c*d*e*\*k*m+1038*b*c*d*e*k*m+2520*b*c*d*e*k*m+1776*b*c*d*k*m+44496*b*c*d*h*m+28884*b*c*d*g*h*m+22512*b*c*d*f*h*m+5232*b*c*d*\*h*m+20040*b*c*d*e*h*m+20160*b*c*d*h*m+7656*b*c*d*g*h*m+8460*b*c*d*f*g*h*m+3600*b*c*d*\*g*h*m+9240*b*c*d*e*g*h*m+7824*b*c*d*g*h*m+4584*b*c*d*f*h*m+1104*b*c*d*\*f*h*m+7128*b*c*d*e*f*h*m+8832*b*c*d*f*h*m+864*b*c*d*\*h*m+1440*b*c*d*e*\*h*m+768*b*c*d*\*h*m+3408*b*c*d*e*h*m+7296*b*c*d*e*h*m+4896*b*c*d*h*m+456*b*c*d*g*m+1104*b*c*d*f*g*m+240*b*c*d*\*g*m+828*b*c*d*e*g*m+624*b*c*d*g*m+744*b*c*d*f*g*m+624*b*c*d*\*f*g*m+1464*b*c*d*e*f*g*m+1584*b*c*d*f*g*m+96*b*c*d*\*g*m+312*b*c*d*e*\*g*m+384*b*c*d*\*g*m+540*b*c*d*e*g*m+984*b*c*d*e*g*m+672*b*c*d*g*m+312*b*c*d*f*m+678*b*c*d*e*f*m+816*b*c*d*f*m+96*b*c*d*\*f*m+240*b*c*d*e*\*f*m+564*b*c*d*e*f*m+1296*b*c*d*e*f*m+672*b*c*d*f*m+48*b*c*d*e*\*m+96*b*c*d*e*\*m+192*b*c*d*e*\*m+120*b*c*d*e*m+336*b*c*d*e*m+336*b*c*d*e*m+2613*b*d*l+2748*b*c*d*k*l+13392*b*c*d*h*l+4542*b*d*g*l+1536*b*c*d*f*l+2376*b*c*d*\*l+1812*b*c*d*e*l+1152*b*c*d*l+2274*b*c*d*k*l+15420*b*c*d*h*k*l+3585*b*c*d*g*k*l+2598*b*c*d*f*k*l+1176*b*c*d*\*k*l+2130*b*c*d*e*k*l+2076*b*c*d*k*l+35952*b*c*d*h*l+19524*b*c*d*g*h*l+10248*b*c*d*f*h*l+4992*b*c*d*\*h*l+11544*b*c*d*e*h*l+9744*b*c*d*h*l+3732*b*d*g*l+2670*b*c*d*f*g*l+3300*b*c*d*\*g*l+3570*b*c*d*e*g*l+2868*b*c*d*g*l+1014*b*c*d*f*l+624*b*c*d*\*f*l+1956*b*c*d*e*f*l+2280*b*c*d*f*l+1560*b*c*d*\*l+624*b*c*d*e*\*l+384*b*c*d*\*l+1257*b*c*d*e*l+2568*b*c*d*e*l+1704*b*c*d*l+456*b*c*d*k*l+7656*b*c*d*h*k*l+2064*b*c*d*g*k*l+828*b*c*d*f*k*l+432*b*c*d*\*k*l+1104*b*c*d*e*k*l+624*b*c*d*k*l+28884*b*c*d*h*k*l+14040*b*c*d*g*h*k*l+9240*b*c*d*f*h*k*l+3984*b*c*d*\*h*k*l+8460*b*c*d*e*h*k*l+7824*b*c*d*h*k*l+1662*b*c*d*g*k*l+2514*b*c*d*f*g*k*l+984*b*c*d*\*g*k*l+2184*b*c*d*e*g*k*l+2376*b*c*d*g*k*l+540*b*c*d*f*k*l+432*b*c*d*\*f*k*l+1464*b*c*d*e*f*k*l+984*b*c*d*f*k*l+96*b*c*d*\*k*l+528*b*c*d*e*\*k*l+288*b*c*d*\*k*l+744*b*c*d*e*k*l+1584*b*c*d*e*k*l+672*b*c*d*k*l+44496*b*c*d*h*l+36564*b*c*d*g*h*l+20040*b*c*d*f*h*l+8880*b*c*d*\*h*l+22512*b*c*d*e*h*l+20160*b*c*d*h*l+11352*b*c*d*g*h*l+10872*b*c*d*f*g*h*l+5328*b*c*d*\*g*h*l+13980*b*c*d*e*g*h*l+12336*b*c*d*g*h*l+3408*b*c*d*f*h*l+2784*b*c*d*\*f*h*l+7128*b*c*d*e*f*h*l+7296*b*c*d*f*h*l+864*b*c*d*\*h*l+3024*b*c*d*e*\*h*l+2688*b*c*d*\*h*l+4584*b*c*d*e*h*l+8832*b*c*d*e*h*l+4896*b*c*d*h*l+1542*b*d*g*l+1596*b*c*d*f*g*l+1776*b*c*d*\*g*l+2412*b*c*d*e*g*l+2040*b*c*d*g*l+1038*b*c*d*f*g*l+888*b*c*d*\*f*g*l+2208*b*c*d*e*f*g*l+2520*b*c*d*f*g*l+1488*b*c*d*\*g*l+1056*b*c*d*e*\*g*l+1056*b*c*d*\*g*l+1566*b*c*d*e*g*l+2976*b*c*d*e*g*l+1776*b*c*d*g*l+120*b*c*d*f*l+216*b*c*d*\*f*l+564*b*c*d*e*f*l+336*b*c*d*f*l+48*b*c*d*\*f*l+624*b*c*d*e*\*f*l+576*b*c*d*\*f*l+678*b*c*d*e*f*l+1296*b*c*d*e*f*l+336*b*c*d*f*l+576*b*c*d*\*l+96*b*c*d*e*\*l+408*b*c*d*e*\*l+960*b*c*d*e*\*l+576*b*c*d*\*l+312*b*c*d*e*l+816*b*c*d*e*l+672*b*c*d*e*l+351*b*c*k+2532*b*c*d*h*k+177*b*c*d*g*k+810*b*c*d*f*k+354*b*c*d*e*k+708*b*c*d*k+11292*b*c*d*h*k+3552*b*c*d*g*h*k+4896*b*c*d*f*h*k+3432*b*c*d*e*h*k+4416*b*c*d*h*k+573*b*c*d*g*k+396*b*c*d*f*g*k+216*b*c*d*\*g*k+672*b*c*d*e*g*k+480*b*c*d*g*k+810*b*c*d*f*k+792*b*c*d*e*f*k+1584*b*c*d*f*k+144*b*c*d*\*k+348*b*c*d*e*k+960*b*c*d*e*k+960*b*c*d*k+24120*b*c*d*h*k+13920*b*c*d*g*h*k+14460*b*c*d*f*h*k+1752*b*c*d*\*h*k+11244*b*c*d*e*h*k+13176*b*c*d*h*k+3552*b*c*d*g*h*k+4632*b*c*d*f*g*h*k+1680*b*c*d*\*g*h*k+4632*b*c*d*e*g*h*k+4560*b*c*d*g*h*k+3348*b*c*d*f*h*k+4836*b*c*d*e*f*h*k+6720*b*c*d*f*h*k+480*b*c*d*\*h*k+480*b*c*d*e*\*h*k+2076*b*c*d*e*h*k+5376*b*c*d*e*h*k+4128*b*c*d*h*k+177*b*c*d*g*k+672*b*c*d*f*g*k+132*b*c*d*\*g*k+396*b*c*d*e*g*k+480*b*c*d*g*k+297*b*c*d*f*g*k+216*b*c*d*\*f*g*k+852*b*c*d*e*f*g*k+696*b*c*d*f*g*k+24*b*c*d*\*g*k+168*b*c*d*e*\*g*k+144*b*c*d*\*g*k+297*b*c*d*e*g*k+696*b*c*d*e*g*k+360*b*c*d*g*k+378*b*c*d*f*k+594*b*c*d*e*f*k+1188*b*c*d*f*k+144*b*c*d*\*f*k+474*b*c*d*e*f*k+1392*b*c*d*e*f*k+1392*b*c*d*f*k+48*b*c*d*e*\*k+96*b*c*d*\*k+24*b*c*d*e*\*k+102*b*c*d*e*k+444*b*c*d*e*k+720*b*c*d*e*k+480*b*c*d*k+24660*b*c*d*h+24120*b*c*d*g*h+17424*b*c*d*f*h+3840*b*c*d*\*h+17424*b*c*d*e*h+18144*b*c*d*h+11292*b*c*d*g*h+11244*b*c*d*f*g*h+4728*b*c*d*\*g*h+14460*b*c*d*e*g*h+13176*b*c*d*g*h+5568*b*c*d*f*h+1344*b*c*d*\*f*h+9816*b*c*d*e*f*h+12528*b*c*d*f*h+1104*b*c*d*\*h+2112*b*c*d*e*\*h+1536*b*c*d*\*h+5568*b*c*d*e*h+12528*b*c*d*e*h+8688*b*c*d*h+2532*b*c*d*g*h+3432*b*c*d*f*g*h+1488*b*c*d*\*g*h+4896*b*c*d*e*g*h+4416*b*c*d*g*h+2076*b*c*d*f*g*h+1632*b*c*d*\*f*g*h+4836*b*c*d*e*f*g*h+5376*b*c*d*f*g*h+480*b*c*d*\*g*h+1920*b*c*d*e*\*g*h+2112*b*c*d*\*g*h+3348*b*c*d*e*g*h+6720*b*c*d*e*g*h+4128*b*c*d*g*h+888*b*c*d*f*h+2208*b*c*d*e*f*h+3120*b*c*d*f*h+384*b*c*d*\*f*h+528*b*c*d*e*\*f*h+2208*b*c*d*e*f*h+5856*b*c*d*e*f*h+4224*b*c*d*f*h+384*b*c*d*e*\*h+384*b*c*d*\*h+480*b*c*d*e*\*h+768*b*c*d*e*\*h+888*b*c*d*e*h+3120*b*c*d*e*h+4224*b*c*d*e*h+1920*b*c*d*h+351*b*d*g+354*b*c*d*f*g+420*b*c*d*\*g+810*b*c*d*e*g+708*b*c*d*g+348*b*c*d*f*g+264*b*c*d*\*f*g+792*b*c*d*e*f*g+960*b*c*d*f*g+432*b*c*d*\*g+504*b*c*d*e*\*g+528*b*c*d*\*g+810*b*c*d*e*g+1584*b*c*d*e*g+960*b*c*d*g+102*b*c*d*f*g+108*b*c*d*\*f*g+474*b*c*d*e*f*g+444*b*c*d*f*g+48*b*c*d*\*f*g+336*b*c*d*e*\*f*g+288*b*c*d*\*f*g+594*b*c*d*e*f*g+1392*b*c*d*e*f*g+720*b*c*d*f*g+288*b*c*d*\*g+144*b*c*d*e*\*g+96*b*c*d*\*g+324*b*c*d*e*\*g+672*b*c*d*e*\*g+288*b*c*d*\*g+378*b*c*d*e*g+1188*b*c*d*e*g+1392*b*c*d*e*g+480*b*c*d*g+81*c*d*f+204*c*d*e*f+408*b*c*d*f+72*b*c*d*\*f+285*c*d*e*f+888*b*c*d*e*f+888*b*c*d*f+96*b*c*d*e*\*f+192*b*c*d*\*f+48*b*c*d*e*\*f+204*c*d*e*f+888*b*c*d*e*f+1440*b*c*d*e*f+960*b*c*d*f+96*b*c*d*\+72*b*c*d*e*\+192*b*c*d*e*\+192*b*c*d*\+48*b*c*d*e*\+96*b*c*d*e*\+81*c*d*e+408*b*c*d*e+888*b*c*d*e+960*b*c*d*e+480*b*c*d|b*c*d>> <\input|236) >> \; > Now we need the classical action. \ From FluxBilinears6.cdb: <\equation*> H H=48 EE + 48 FF + 72 GG + 288 HH + 72 KK + 144 LL + 144 MM + 24 NN |b*c>>. \ |b*d>>. \ |b*c>>. \ |b*c*d>>. \ |b*d>>. \ |b*c*d>>. \ |b*c*d>>. \ |c*d>>. <\equation*> H H=48* |b*c> + 48* |b*d> + 72* |b*c> + 288* |b*c*d> + 72* |b*d> + 144* |b*c*d>+ 144* |b*c*d> + 24*|c*d>\ Hence the classical action for the \ \H\H> case is: <\equation*> ->-|c>-|d>-48* |b*c> + 48* |b*d> + 72* |b*c> + 288* |b*c*d> + 72* |b*d> + 144* |b*c*d>+ 144* |b*c*d> + 24*|c*d> <\input|236) >> -(6/(b^2))-((2*eta)/(c^2))-((2*sigma)/(d^2))-(1/48)*(48* ((e^2)/(b^6*c^2)) + 48* ((f^2)/(b^6*d^2)) + 72* ((g^2)/(b^4*c^4)) + 288* ((h^2)/(b^4*c^2*d^2)) + 72* ((k^2)/(b^4*d^4)) + 144* ((l^2)/(b^2*c^4*d^2))+ 144* ((m^2)/(b^2*c^2*d^4)) + 24*((n^2)/(c^4*d^4))); <\output> ) >-|d>-|c*d>+|b*c*d>+|b*c*d>+|b*d>+|b*c*d>+|b*c>+|b*d>+|b*c>|48>-|c>->> <\input|237) >> \; > I might as well reduce the coefficients as far as possible, by removing the factor of 6 from %o235. <\input|237) >> factor(radcan(a^4*b^3*c^2*d^2*(%o236 + (%o235/6)))); <\output> ) >*192*b*c*d*\+192*b*c*d*n*\+1152*b*c*d*m*\+576*b*c*d*k*\+112*b*c*d*n*\+1056*b*c*d*m*n*\+288*b*c*d*l*n*\+432*b*c*d*k*n*\+192*b*c*d*h*n*\+48*b*c*d*g*n*\+64*b*c*d*\*n*\+3120*b*c*d*m*\+960*b*c*d*l*m*\+2976*b*c*d*k*m*\+1728*b*c*d*h*m*\+192*b*c*d*g*m*\+192*b*c*d*f*m*\+96*b*c*d*e*m*\+816*b*c*d*l*\+192*b*c*d*k*l*\+1728*b*c*d*h*l*\+672*b*c*d*g*l*\+96*b*c*d*f*l*\+384*b*c*d*\*l*\+192*b*c*d*e*l*\+864*b*c*d*k*\+960*b*c*d*h*k*\+48*b*c*d*g*k*\+288*b*c*d*f*k*\+96*b*c*d*e*k*\+192*b*c*d*k*\+2208*b*c*d*h*\+960*b*c*d*g*h*\+768*b*c*d*f*h*\+768*b*c*d*e*h*\+768*b*c*d*h*\+288*b*c*d*g*\+96*b*c*d*f*g*\+192*b*c*d*\*g*\+288*b*c*d*e*g*\+192*b*c*d*g*\+144*b*c*d*f*\+192*b*c*d*e*f*\+384*b*c*d*f*\+128*b*c*d*\*\+144*b*c*d*e*\+384*b*c*d*e*\+384*b*c*d*\+48*b*c*d*n*\+528*b*c*d*m*n*\+336*b*c*d*l*n*\+168*b*c*d*k*n*\+192*b*c*d*h*n*\+72*b*c*d*g*n*\+96*b*c*d*\*n*\+2472*b*c*d*m*n*\+2688*b*c*d*l*m*n*\+1968*b*c*d*k*m*n*\+2880*b*c*d*h*m*n*\+912*b*c*d*g*m*n*\+192*b*c*d*f*m*n*\+576*b*c*d*\*m*n*\+240*b*c*d*e*m*n*\+1320*b*c*d*l*n*\+912*b*c*d*k*l*n*\+2880*b*c*d*h*l*n*\+816*b*c*d*g*l*n*\+240*b*c*d*f*l*n*\+576*b*c*d*\*l*n*\+192*b*c*d*e*l*n*\+432*b*c*d*k*n*\+1536*b*c*d*h*k*n*\+480*b*c*d*g*k*n*\+144*b*c*d*f*k*n*\+96*b*c*d*\*k*n*\+240*b*c*d*e*k*n*\+96*b*c*d*k*n*\+2784*b*c*d*h*n*\+1536*b*c*d*g*h*n*\+768*b*c*d*f*h*n*\+384*b*c*d*\*h*n*\+768*b*c*d*e*h*n*\+384*b*c*d*h*n*\+144*b*c*d*g*n*\+240*b*c*d*f*g*n*\+96*b*c*d*\*g*n*\+144*b*c*d*e*g*n*\+96*b*c*d*g*n*\+72*b*c*d*f*n*\+192*b*c*d*e*f*n*\+192*b*c*d*f*n*\+64*b*c*d*\*n*\+72*b*c*d*e*n*\+192*b*c*d*e*n*\+192*b*c*d*n*\+4752*b*c*d*m*\+5760*b*c*d*l*m*\+6600*b*c*d*k*m*\+9984*b*c*d*h*m*\+2352*b*c*d*g*m*\+1248*b*c*d*f*m*\+576*b*c*d*\*m*\+1248*b*c*d*e*m*\+768*b*c*d*m*\+6192*b*c*d*l*m*\+3936*b*c*d*k*l*m*\+16032*b*c*d*h*l*m*\+4800*b*c*d*g*l*m*\+1536*b*c*d*f*l*m*\+2304*b*c*d*\*l*m*\+1824*b*c*d*e*l*m*\+768*b*c*d*l*m*\+3552*b*c*d*k*m*\+10656*b*c*d*h*k*m*\+1968*b*c*d*g*k*m*\+2112*b*c*d*f*k*m*\+1776*b*c*d*e*k*m*\+2112*b*c*d*k*m*\+17760*b*c*d*h*m*\+7968*b*c*d*g*h*m*\+6048*b*c*d*f*h*m*\+1152*b*c*d*\*h*m*\+5568*b*c*d*e*h*m*\+5376*b*c*d*h*m*\+864*b*c*d*g*m*\+1056*b*c*d*f*g*m*\+576*b*c*d*\*g*m*\+864*b*c*d*e*g*m*\+576*b*c*d*g*m*\+816*b*c*d*f*m*\+1248*b*c*d*e*f*m*\+1920*b*c*d*f*m*\+384*b*c*d*\*m*\+432*b*c*d*e*m*\+1152*b*c*d*e*m*\+1152*b*c*d*m*\+1440*b*c*d*l*\+2424*b*c*d*k*l*\+5952*b*c*d*h*l*\+1056*b*c*d*g*l*\+672*b*c*d*f*l*\+576*b*c*d*\*l*\+384*b*c*d*e*l*\+192*b*c*d*l*\+480*b*c*d*k*l*\+7200*b*c*d*h*k*l*\+2352*b*c*d*g*k*l*\+624*b*c*d*f*k*l*\+576*b*c*d*\*k*l*\+1248*b*c*d*e*k*l*\+768*b*c*d*k*l*\+10464*b*c*d*h*l*\+4128*b*c*d*g*h*l*\+2880*b*c*d*f*h*l*\+1152*b*c*d*\*h*l*\+2208*b*c*d*e*h*l*\+1536*b*c*d*h*l*\+672*b*c*d*f*g*l*\+192*b*c*d*f*l*\+480*b*c*d*e*f*l*\+384*b*c*d*f*l*\+840*b*c*d*k*\+2976*b*c*d*h*k*\+264*b*c*d*g*k*\+1008*b*c*d*f*k*\+528*b*c*d*e*k*\+1056*b*c*d*k*\+9456*b*c*d*h*k*\+3360*b*c*d*g*h*k*\+3840*b*c*d*f*h*k*\+3264*b*c*d*e*h*k*\+4224*b*c*d*h*k*\+432*b*c*d*g*k*\+336*b*c*d*f*g*k*\+288*b*c*d*\*g*k*\+432*b*c*d*e*g*k*\+288*b*c*d*g*k*\+648*b*c*d*f*k*\+672*b*c*d*e*f*k*\+1344*b*c*d*f*k*\+192*b*c*d*\*k*\+216*b*c*d*e*k*\+576*b*c*d*e*k*\+576*b*c*d*k*\+7680*b*c*d*h*\+3504*b*c*d*g*h*\+4224*b*c*d*f*h*\+576*b*c*d*\*h*\+2688*b*c*d*e*h*\+3072*b*c*d*h*\+960*b*c*d*f*g*h*\+960*b*c*d*f*h*\+1056*b*c*d*e*f*h*\+1536*b*c*d*f*h*\+48*b*c*d*f*g*\+96*b*c*d*f*\+96*b*c*d*e*f*\+192*b*c*d*f*\-4*b*c*d*\+18*b*n+216*b*c*m*n+216*b*d*l*n+36*b*c*k*n+144*b*c*d*h*n+36*b*d*g*n+48*b*c*d*\*n+1290*b*c*m*n+2280*b*c*d*l*m*n+732*b*c*k*m*n+2592*b*c*d*h*m*n+564*b*c*d*g*m*n+168*b*c*d*f*m*n+336*b*c*d*\*m*n+156*b*c*d*e*m*n+1290*b*d*l*n+564*b*c*d*k*l*n+2592*b*c*d*h*l*n+732*b*d*g*l*n+156*b*c*d*f*l*n+528*b*c*d*\*l*n+168*b*c*d*e*l*n+192*b*c*k*n+1080*b*c*d*h*k*n+264*b*c*d*g*k*n+156*b*c*d*f*k*n+72*b*c*d*\*k*n+132*b*c*d*e*k*n+72*b*c*d*k*n+2376*b*c*d*h*n+1080*b*c*d*g*h*n+576*b*c*d*f*h*n+192*b*c*d*\*h*n+576*b*c*d*e*h*n+288*b*c*d*h*n+192*b*d*g*n+132*b*c*d*f*g*n+168*b*c*d*\*g*n+156*b*c*d*e*g*n+72*b*c*d*g*n+66*b*c*d*f*n+120*b*c*d*e*f*n+144*b*c*d*f*n+112*b*c*d*\*n+66*b*c*d*e*n+144*b*c*d*e*n+144*b*c*d*n+3852*b*c*m*n+9756*b*c*d*l*m*n+4026*b*c*k*m*n+13296*b*c*d*h*m*n+3018*b*c*d*g*m*n+1320*b*c*d*f*m*n+1320*b*c*d*\*m*n+1248*b*c*d*e*m*n+624*b*c*d*m*n+9756*b*c*d*l*m*n+6240*b*c*d*k*l*m*n+26544*b*c*d*h*l*m*n+6240*b*c*d*g*l*m*n+2616*b*c*d*f*l*m*n+2688*b*c*d*\*l*m*n+2616*b*c*d*e*l*m*n+1344*b*c*d*l*m*n+1848*b*c*k*m*n+10848*b*c*d*h*k*m*n+2664*b*c*d*g*k*m*n+1560*b*c*d*f*k*m*n+816*b*c*d*\*k*m*n+1476*b*c*d*e*k*m*n+1200*b*c*d*k*m*n+22896*b*c*d*h*m*n+10752*b*c*d*g*h*m*n+6288*b*c*d*f*h*m*n+2880*b*c*d*\*h*m*n+6240*b*c*d*e*h*m*n+5184*b*c*d*h*m*n+1536*b*c*d*g*m*n+1512*b*c*d*f*g*m*n+912*b*c*d*\*g*m*n+1716*b*c*d*e*g*m*n+1392*b*c*d*g*m*n+588*b*c*d*f*m*n+192*b*c*d*\*f*m*n+1152*b*c*d*e*f*m*n+1248*b*c*d*f*m*n+288*b*c*d*\*m*n+240*b*c*d*e*\*m*n+636*b*c*d*e*m*n+1344*b*c*d*e*m*n+864*b*c*d*m*n+3852*b*d*l*n+3018*b*c*d*k*l*n+13296*b*c*d*h*l*n+4026*b*d*g*l*n+1248*b*c*d*f*l*n+2472*b*c*d*\*l*n+1320*b*c*d*e*l*n+624*b*c*d*l*n+1536*b*c*d*k*l*n+10752*b*c*d*h*k*l*n+2664*b*c*d*g*k*l*n+1716*b*c*d*f*k*l*n+912*b*c*d*\*k*l*n+1512*b*c*d*e*k*l*n+1392*b*c*d*k*l*n+22896*b*c*d*h*l*n+10848*b*c*d*g*h*l*n+6240*b*c*d*f*h*l*n+2880*b*c*d*\*h*l*n+6288*b*c*d*e*h*l*n+5184*b*c*d*h*l*n+1848*b*d*g*l*n+1476*b*c*d*f*g*l*n+1968*b*c*d*\*g*l*n+1560*b*c*d*e*g*l*n+1200*b*c*d*g*l*n+636*b*c*d*f*l*n+240*b*c*d*\*f*l*n+1152*b*c*d*e*f*l*n+1344*b*c*d*f*l*n+1056*b*c*d*\*l*n+192*b*c*d*e*\*l*n+588*b*c*d*e*l*n+1248*b*c*d*e*l*n+864*b*c*d*l*n+210*b*c*k*n+2424*b*c*d*h*k*n+654*b*c*d*g*k*n+252*b*c*d*f*k*n+144*b*c*d*\*k*n+372*b*c*d*e*k*n+264*b*c*d*k*n+9288*b*c*d*h*k*n+4656*b*c*d*g*h*k*n+2928*b*c*d*f*h*k*n+1536*b*c*d*\*h*k*n+2832*b*c*d*e*h*k*n+2880*b*c*d*h*k*n+654*b*c*d*g*k*n+816*b*c*d*f*g*k*n+480*b*c*d*\*g*k*n+816*b*c*d*e*g*k*n+912*b*c*d*g*k*n+162*b*c*d*f*k*n+144*b*c*d*\*f*k*n+480*b*c*d*e*f*k*n+336*b*c*d*f*k*n+48*b*c*d*\*k*n+240*b*c*d*e*\*k*n+96*b*c*d*\*k*n+258*b*c*d*e*k*n+528*b*c*d*e*k*n+144*b*c*d*k*n+12960*b*c*d*h*n+9288*b*c*d*g*h*n+5760*b*c*d*f*h*n+2784*b*c*d*\*h*n+5760*b*c*d*e*h*n+5760*b*c*d*h*n+2424*b*c*d*g*h*n+2832*b*c*d*f*g*h*n+1536*b*c*d*\*g*h*n+2928*b*c*d*e*g*h*n+2880*b*c*d*g*h*n+912*b*c*d*f*h*n+768*b*c*d*\*f*h*n+1776*b*c*d*e*f*h*n+1728*b*c*d*f*h*n+192*b*c*d*\*h*n+768*b*c*d*e*\*h*n+384*b*c*d*\*h*n+912*b*c*d*e*h*n+1728*b*c*d*e*h*n+576*b*c*d*h*n+210*b*d*g*n+372*b*c*d*f*g*n+432*b*c*d*\*g*n+252*b*c*d*e*g*n+264*b*c*d*g*n+258*b*c*d*f*g*n+240*b*c*d*\*f*g*n+480*b*c*d*e*f*g*n+528*b*c*d*f*g*n+432*b*c*d*\*g*n+144*b*c*d*e*\*g*n+96*b*c*d*\*g*n+162*b*c*d*e*g*n+336*b*c*d*e*g*n+144*b*c*d*g*n+24*b*c*d*f*n+72*b*c*d*\*f*n+120*b*c*d*e*f*n+48*b*c*d*f*n+192*b*c*d*e*\*f*n+192*b*c*d*\*f*n+120*b*c*d*e*f*n+192*b*c*d*e*f*n+192*b*c*d*\*n+72*b*c*d*e*\*n+192*b*c*d*e*\*n+192*b*c*d*\*n+24*b*c*d*e*n+48*b*c*d*e*n-b*c*d*n+5226*b*c*m+15384*b*c*d*l*m+9084*b*c*k*m+26784*b*c*d*h*m+5496*b*c*d*g*m+3624*b*c*d*f*m+1440*b*c*d*\*m+3072*b*c*d*e*m+2304*b*c*d*m+23418*b*c*d*l*m+17136*b*c*d*k*l*m+71712*b*c*d*h*l*m+17532*b*c*d*g*l*m+8460*b*c*d*f*l*m+6192*b*c*d*\*l*m+8496*b*c*d*e*l*m+5472*b*c*d*l*m+7464*b*c*k*m+39048*b*c*d*h*k*m+7170*b*c*d*g*k*m+7140*b*c*d*f*k*m+1056*b*c*d*\*k*m+5340*b*c*d*e*k*m+5736*b*c*d*k*m+71904*b*c*d*h*m+30840*b*c*d*g*h*m+23088*b*c*d*f*h*m+5952*b*c*d*\*h*m+20496*b*c*d*e*h*m+19488*b*c*d*h*m+4548*b*c*d*g*m+4260*b*c*d*f*g*m+2424*b*c*d*\*g*m+5196*b*c*d*e*g*m+4152*b*c*d*g*m+2514*b*c*d*f*m+384*b*c*d*\*f*m+3912*b*c*d*e*f*m+5136*b*c*d*f*m+816*b*c*d*\*m+672*b*c*d*e*\*m+192*b*c*d*\*m+2028*b*c*d*e*m+4560*b*c*d*e*m+3408*b*c*d*m+15384*b*c*d*l*m+17532*b*c*d*k*l*m+71712*b*c*d*h*l*m+17136*b*c*d*g*l*m+8496*b*c*d*f*l*m+5760*b*c*d*\*l*m+8460*b*c*d*e*l*m+5472*b*c*d*l*m+7968*b*c*d*k*l*m+62088*b*c*d*h*k*l*m+15576*b*c*d*g*k*l*m+9264*b*c*d*f*k*l*m+4800*b*c*d*\*k*l*m+8760*b*c*d*e*k*l*m+7392*b*c*d*k*l*m+127920*b*c*d*h*l*m+62088*b*c*d*g*h*l*m+36336*b*c*d*f*h*l*m+16032*b*c*d*\*h*l*m+36336*b*c*d*e*h*l*m+30144*b*c*d*h*l*m+7968*b*c*d*g*l*m+8760*b*c*d*f*g*l*m+3936*b*c*d*\*g*l*m+9264*b*c*d*e*g*l*m+7392*b*c*d*g*l*m+3288*b*c*d*f*l*m+1824*b*c*d*\*f*l*m+6216*b*c*d*e*f*l*m+6720*b*c*d*f*l*m+960*b*c*d*\*l*m+1536*b*c*d*e*\*l*m+768*b*c*d*\*l*m+3288*b*c*d*e*l*m+6720*b*c*d*e*l*m+4416*b*c*d*l*m+3084*b*c*k*m+22704*b*c*d*h*k*m+3324*b*c*d*g*k*m+4824*b*c*d*f*k*m+3192*b*c*d*e*k*m+4080*b*c*d*k*m+73128*b*c*d*h*k*m+28080*b*c*d*g*h*k*m+27960*b*c*d*f*h*k*m+4128*b*c*d*\*h*k*m+21744*b*c*d*e*h*k*m+24672*b*c*d*h*k*m+4128*b*c*d*g*k*m+4368*b*c*d*f*g*k*m+2352*b*c*d*\*g*k*m+5028*b*c*d*e*g*k*m+4752*b*c*d*g*k*m+3132*b*c*d*f*k*m+4416*b*c*d*e*f*k*m+5952*b*c*d*f*k*m+672*b*c*d*\*k*m+672*b*c*d*e*\*k*m+2076*b*c*d*e*k*m+5040*b*c*d*e*k*m+3552*b*c*d*k*m+88992*b*c*d*h*m+57768*b*c*d*g*h*m+45024*b*c*d*f*h*m+10464*b*c*d*\*h*m+40080*b*c*d*e*h*m+40320*b*c*d*h*m+15312*b*c*d*g*h*m+16920*b*c*d*f*g*h*m+7200*b*c*d*\*g*h*m+18480*b*c*d*e*g*h*m+15648*b*c*d*g*h*m+9168*b*c*d*f*h*m+2208*b*c*d*\*f*h*m+14256*b*c*d*e*f*h*m+17664*b*c*d*f*h*m+1728*b*c*d*\*h*m+2880*b*c*d*e*\*h*m+1536*b*c*d*\*h*m+6816*b*c*d*e*h*m+14592*b*c*d*e*h*m+9792*b*c*d*h*m+912*b*c*d*g*m+2208*b*c*d*f*g*m+480*b*c*d*\*g*m+1656*b*c*d*e*g*m+1248*b*c*d*g*m+1488*b*c*d*f*g*m+1248*b*c*d*\*f*g*m+2928*b*c*d*e*f*g*m+3168*b*c*d*f*g*m+192*b*c*d*\*g*m+624*b*c*d*e*\*g*m+768*b*c*d*\*g*m+1080*b*c*d*e*g*m+1968*b*c*d*e*g*m+1344*b*c*d*g*m+624*b*c*d*f*m+1356*b*c*d*e*f*m+1632*b*c*d*f*m+192*b*c*d*\*f*m+480*b*c*d*e*\*f*m+1128*b*c*d*e*f*m+2592*b*c*d*e*f*m+1344*b*c*d*f*m+96*b*c*d*e*\*m+192*b*c*d*e*\*m+384*b*c*d*e*\*m+240*b*c*d*e*m+672*b*c*d*e*m+672*b*c*d*e*m-6*b*c*d*m+5226*b*d*l+5496*b*c*d*k*l+26784*b*c*d*h*l+9084*b*d*g*l+3072*b*c*d*f*l+4752*b*c*d*\*l+3624*b*c*d*e*l+2304*b*c*d*l+4548*b*c*d*k*l+30840*b*c*d*h*k*l+7170*b*c*d*g*k*l+5196*b*c*d*f*k*l+2352*b*c*d*\*k*l+4260*b*c*d*e*k*l+4152*b*c*d*k*l+71904*b*c*d*h*l+39048*b*c*d*g*h*l+20496*b*c*d*f*h*l+9984*b*c*d*\*h*l+23088*b*c*d*e*h*l+19488*b*c*d*h*l+7464*b*d*g*l+5340*b*c*d*f*g*l+6600*b*c*d*\*g*l+7140*b*c*d*e*g*l+5736*b*c*d*g*l+2028*b*c*d*f*l+1248*b*c*d*\*f*l+3912*b*c*d*e*f*l+4560*b*c*d*f*l+3120*b*c*d*\*l+1248*b*c*d*e*\*l+768*b*c*d*\*l+2514*b*c*d*e*l+5136*b*c*d*e*l+3408*b*c*d*l+912*b*c*d*k*l+15312*b*c*d*h*k*l+4128*b*c*d*g*k*l+1656*b*c*d*f*k*l+864*b*c*d*\*k*l+2208*b*c*d*e*k*l+1248*b*c*d*k*l+57768*b*c*d*h*k*l+28080*b*c*d*g*h*k*l+18480*b*c*d*f*h*k*l+7968*b*c*d*\*h*k*l+16920*b*c*d*e*h*k*l+15648*b*c*d*h*k*l+3324*b*c*d*g*k*l+5028*b*c*d*f*g*k*l+1968*b*c*d*\*g*k*l+4368*b*c*d*e*g*k*l+4752*b*c*d*g*k*l+1080*b*c*d*f*k*l+864*b*c*d*\*f*k*l+2928*b*c*d*e*f*k*l+1968*b*c*d*f*k*l+192*b*c*d*\*k*l+1056*b*c*d*e*\*k*l+576*b*c*d*\*k*l+1488*b*c*d*e*k*l+3168*b*c*d*e*k*l+1344*b*c*d*k*l+88992*b*c*d*h*l+73128*b*c*d*g*h*l+40080*b*c*d*f*h*l+17760*b*c*d*\*h*l+45024*b*c*d*e*h*l+40320*b*c*d*h*l+22704*b*c*d*g*h*l+21744*b*c*d*f*g*h*l+10656*b*c*d*\*g*h*l+27960*b*c*d*e*g*h*l+24672*b*c*d*g*h*l+6816*b*c*d*f*h*l+5568*b*c*d*\*f*h*l+14256*b*c*d*e*f*h*l+14592*b*c*d*f*h*l+1728*b*c*d*\*h*l+6048*b*c*d*e*\*h*l+5376*b*c*d*\*h*l+9168*b*c*d*e*h*l+17664*b*c*d*e*h*l+9792*b*c*d*h*l+3084*b*d*g*l+3192*b*c*d*f*g*l+3552*b*c*d*\*g*l+4824*b*c*d*e*g*l+4080*b*c*d*g*l+2076*b*c*d*f*g*l+1776*b*c*d*\*f*g*l+4416*b*c*d*e*f*g*l+5040*b*c*d*f*g*l+2976*b*c*d*\*g*l+2112*b*c*d*e*\*g*l+2112*b*c*d*\*g*l+3132*b*c*d*e*g*l+5952*b*c*d*e*g*l+3552*b*c*d*g*l+240*b*c*d*f*l+432*b*c*d*\*f*l+1128*b*c*d*e*f*l+672*b*c*d*f*l+96*b*c*d*\*f*l+1248*b*c*d*e*\*f*l+1152*b*c*d*\*f*l+1356*b*c*d*e*f*l+2592*b*c*d*e*f*l+672*b*c*d*f*l+1152*b*c*d*\*l+192*b*c*d*e*\*l+816*b*c*d*e*\*l+1920*b*c*d*e*\*l+1152*b*c*d*\*l+624*b*c*d*e*l+1632*b*c*d*e*l+1344*b*c*d*e*l-6*b*c*d*l+702*b*c*k+5064*b*c*d*h*k+354*b*c*d*g*k+1620*b*c*d*f*k+708*b*c*d*e*k+1416*b*c*d*k+22584*b*c*d*h*k+7104*b*c*d*g*h*k+9792*b*c*d*f*h*k+6864*b*c*d*e*h*k+8832*b*c*d*h*k+1146*b*c*d*g*k+792*b*c*d*f*g*k+432*b*c*d*\*g*k+1344*b*c*d*e*g*k+960*b*c*d*g*k+1620*b*c*d*f*k+1584*b*c*d*e*f*k+3168*b*c*d*f*k+288*b*c*d*\*k+696*b*c*d*e*k+1920*b*c*d*e*k+1920*b*c*d*k+48240*b*c*d*h*k+27840*b*c*d*g*h*k+28920*b*c*d*f*h*k+3504*b*c*d*\*h*k+22488*b*c*d*e*h*k+26352*b*c*d*h*k+7104*b*c*d*g*h*k+9264*b*c*d*f*g*h*k+3360*b*c*d*\*g*h*k+9264*b*c*d*e*g*h*k+9120*b*c*d*g*h*k+6696*b*c*d*f*h*k+9672*b*c*d*e*f*h*k+13440*b*c*d*f*h*k+960*b*c*d*\*h*k+960*b*c*d*e*\*h*k+4152*b*c*d*e*h*k+10752*b*c*d*e*h*k+8256*b*c*d*h*k+354*b*c*d*g*k+1344*b*c*d*f*g*k+264*b*c*d*\*g*k+792*b*c*d*e*g*k+960*b*c*d*g*k+594*b*c*d*f*g*k+432*b*c*d*\*f*g*k+1704*b*c*d*e*f*g*k+1392*b*c*d*f*g*k+48*b*c*d*\*g*k+336*b*c*d*e*\*g*k+288*b*c*d*\*g*k+594*b*c*d*e*g*k+1392*b*c*d*e*g*k+720*b*c*d*g*k+756*b*c*d*f*k+1188*b*c*d*e*f*k+2376*b*c*d*f*k+288*b*c*d*\*f*k+948*b*c*d*e*f*k+2784*b*c*d*e*f*k+2784*b*c*d*f*k+96*b*c*d*e*\*k+192*b*c*d*\*k+48*b*c*d*e*\*k+204*b*c*d*e*k+888*b*c*d*e*k+1440*b*c*d*e*k-3*b*c*d*k+960*b*c*d*k+49320*b*c*d*h+48240*b*c*d*g*h+34848*b*c*d*f*h+7680*b*c*d*\*h+34848*b*c*d*e*h+36288*b*c*d*h+22584*b*c*d*g*h+22488*b*c*d*f*g*h+9456*b*c*d*\*g*h+28920*b*c*d*e*g*h+26352*b*c*d*g*h+11136*b*c*d*f*h+2688*b*c*d*\*f*h+19632*b*c*d*e*f*h+25056*b*c*d*f*h+2208*b*c*d*\*h+4224*b*c*d*e*\*h+3072*b*c*d*\*h+11136*b*c*d*e*h+25056*b*c*d*e*h+17376*b*c*d*h+5064*b*c*d*g*h+6864*b*c*d*f*g*h+2976*b*c*d*\*g*h+9792*b*c*d*e*g*h+8832*b*c*d*g*h+4152*b*c*d*f*g*h+3264*b*c*d*\*f*g*h+9672*b*c*d*e*f*g*h+10752*b*c*d*f*g*h+960*b*c*d*\*g*h+3840*b*c*d*e*\*g*h+4224*b*c*d*\*g*h+6696*b*c*d*e*g*h+13440*b*c*d*e*g*h+8256*b*c*d*g*h+1776*b*c*d*f*h+4416*b*c*d*e*f*h+6240*b*c*d*f*h+768*b*c*d*\*f*h+1056*b*c*d*e*\*f*h+4416*b*c*d*e*f*h+11712*b*c*d*e*f*h+8448*b*c*d*f*h+768*b*c*d*e*\*h+768*b*c*d*\*h+960*b*c*d*e*\*h+1536*b*c*d*e*\*h+1776*b*c*d*e*h+6240*b*c*d*e*h+8448*b*c*d*e*h-12*b*c*d*h+3840*b*c*d*h+702*b*d*g+708*b*c*d*f*g+840*b*c*d*\*g+1620*b*c*d*e*g+1416*b*c*d*g+696*b*c*d*f*g+528*b*c*d*\*f*g+1584*b*c*d*e*f*g+1920*b*c*d*f*g+864*b*c*d*\*g+1008*b*c*d*e*\*g+1056*b*c*d*\*g+1620*b*c*d*e*g+3168*b*c*d*e*g+1920*b*c*d*g+204*b*c*d*f*g+216*b*c*d*\*f*g+948*b*c*d*e*f*g+888*b*c*d*f*g+96*b*c*d*\*f*g+672*b*c*d*e*\*f*g+576*b*c*d*\*f*g+1188*b*c*d*e*f*g+2784*b*c*d*e*f*g+1440*b*c*d*f*g+576*b*c*d*\*g+288*b*c*d*e*\*g+192*b*c*d*\*g+648*b*c*d*e*\*g+1344*b*c*d*e*\*g+576*b*c*d*\*g+756*b*c*d*e*g+2376*b*c*d*e*g+2784*b*c*d*e*g-3*b*c*d*g+960*b*c*d*g+162*c*d*f+408*c*d*e*f+816*b*c*d*f+144*b*c*d*\*f+570*c*d*e*f+1776*b*c*d*e*f+1776*b*c*d*f+192*b*c*d*e*\*f+384*b*c*d*\*f+96*b*c*d*e*\*f+408*c*d*e*f+1776*b*c*d*e*f+2880*b*c*d*e*f-2*b*c*d*f+1920*b*c*d*f+192*b*c*d*\+144*b*c*d*e*\+384*b*c*d*e*\+384*b*c*d*\+96*b*c*d*e*\+192*b*c*d*e*\-4*b*c*d*\+162*c*d*e+816*b*c*d*e+1776*b*c*d*e-2*b*c*d*e+1920*b*c*d*e-12*b*c*d+960*b*c*d|2*b*c*d>> <\input|238) >> \; > %o237 is the action that is to be varied to get the field eqns. \ First, find out if there is a solution in the \S\S> case, with flux only, that would be analogous to the \S> solution. <\input|238) >> factor(radcan(subst(0,e,subst(0,f,subst(0,g,subst(0,h, subst(0,k,subst(0,l,subst(0,m,%o237))))))))); <\output> ) >*192*b*c*d*\+192*b*c*d*n*\+112*b*c*d*n*\+64*b*c*d*\*n*\+128*b*c*d*\*\+384*b*c*d*\+48*b*c*d*n*\+96*b*c*d*\*n*\+64*b*c*d*\*n*\+192*b*c*d*n*\-4*b*c*d*\+18*b*n+48*b*c*d*\*n+112*b*c*d*\*n+144*b*c*d*n+192*b*c*d*\*n+192*b*c*d*\*n-b*c*d*n+192*b*c*d*\+384*b*c*d*\-4*b*c*d*\-12*b*c*d+960*c*d|2*b*c*d>> <\input|239) >> factor(radcan(subst(-1,eta,subst(-1,sigma,%o238)))); <\output> ) >*18*b*n-48*b*c*d*n-48*b*c*d*n+144*b*c*d*n+112*b*c*d*n+96*b*c*d*n+112*b*c*d*n-b*c*d*n-192*b*c*d*n-192*b*c*d*n-192*b*c*d*n-64*b*c*d*n-64*b*c*d*n-192*b*c*d*n-12*b*c*d+960*c*d+4*b*c*d+384*b*c*d+192*b*c*d+4*b*c*d+384*b*c*d+128*b*c*d+192*b*c*d|2*b*c*d>> <\input|240) >> factor(radcan(diff(%o239,a))); <\output> ) >*18*b*n-48*b*c*d*n-48*b*c*d*n+144*b*c*d*n+112*b*c*d*n+96*b*c*d*n+112*b*c*d*n-b*c*d*n-192*b*c*d*n-192*b*c*d*n-192*b*c*d*n-64*b*c*d*n-64*b*c*d*n-192*b*c*d*n-12*b*c*d+960*c*d+4*b*c*d+384*b*c*d+192*b*c*d+4*b*c*d+384*b*c*d+128*b*c*d+192*b*c*d|b*c*d>> <\input|241) >> factor(radcan(diff(%o239,b))); <\output> ) >*18*b*n-48*b*c*d*n-48*b*c*d*n-48*b*c*d*n+112*b*c*d*n+96*b*c*d*n+112*b*c*d*n-b*c*d*n+64*b*c*d*n-192*b*c*d*n+64*b*c*d*n-64*b*c*d*n-64*b*c*d*n-192*b*c*d*n-4*b*c*d-1600*c*d+4*b*c*d-128*b*c*d+192*b*c*d+4*b*c*d-128*b*c*d+128*b*c*d+192*b*c*d|2*b*c*d>> <\input|242) >> factor(radcan(diff(%o239,c))); <\output> ) >-*126*b*n-288*b*c*d*n-240*b*c*d*n+432*b*c*d*n+560*b*c*d*n+384*b*c*d*n+336*b*c*d*n-b*c*d*n-384*b*c*d*n-768*b*c*d*n-192*b*c*d*n-192*b*c*d*n-128*b*c*d*n-192*b*c*d*n+12*b*c*d-960*c*d+384*b*c*d+576*b*c*d-4*b*c*d-384*b*c*d+128*b*c*d-192*b*c*d|b*c*d>> <\input|243) >> factor(radcan(diff(%o239,d))); <\output> ) >-*126*b*n-240*b*c*d*n-288*b*c*d*n+432*b*c*d*n+336*b*c*d*n+384*b*c*d*n+560*b*c*d*n-b*c*d*n-192*b*c*d*n-192*b*c*d*n-384*b*c*d*n-128*b*c*d*n-192*b*c*d*n-768*b*c*d*n+12*b*c*d-960*c*d-4*b*c*d-384*b*c*d-192*b*c*d+384*b*c*d+128*b*c*d+576*b*c*d|b*c*d>> <\input|244) >> radcan(subst(c,d,(%o242 - %o243))); <\output> ) >0> <\input|245) >> \; > So if I seem a solution of the field eqns with and only flux, I only need to solve 3 field eqns. <\input|245) >> factor(radcan(subst(c,d,%o240*b^5*c^14*d^14/(2*a^3)))); <\output> ) >18*b*n-96*b*c*n+144*b*c*n+320*b*c*n-b*c*n-384*b*c*n-512*b*c*n-12*b*c+960*c+8*b*c+768*b*c+512*b*c> <\input|246) >> factor(radcan(subst(c,d,%o241*2*b^6*c^14*d^14/(3*a^4)))); <\output> ) >18*b*n-96*b*c*n-48*b*c*n+320*b*c*n-b*c*n+128*b*c*n-512*b*c*n-4*b*c-1600*c+8*b*c-256*b*c+512*b*c> <\input|247) >> factor(radcan(subst(c,d,%o242*b^5*c^15*d^14/(-a^4)))); <\output> ) >126*b*n-528*b*c*n+432*b*c*n+1280*b*c*n-b*c*n-576*b*c*n-1280*b*c*n+12*b*c-960*c-4*b*c+512*b*c> <\input|248) >> factor(radcan(%o245 - %o246)); <\output> ) >8*c*24*b*n-64*b*c*n-b*c+320*c+128*b*c> <\input|249) >> subst(sqrt(B),b,subst(sqrt(C),c,subst(sqrt(N),n,%o248))); <\output> ) >8*C*24*B*N-64*B*C*N-B*C+320*C+128*B*C> <\input|250) >> solve(%o249,N); <\output> ) >N=-*C*3*B-960*C-256*B>-16*B*C|12*B>,N=*C*3*B-960*C-256*B>+16*B*C|12*B>> <\input|251) >> subst(sqrt(B),b,subst(sqrt(C),c,subst(sqrt(N),n,%o245))); <\output> ) >18*B*N-96*B*C*N+144*B*C*N+320*B*C*N-B*C*N-384*B*C*N-512*B*C*N-12*B*C+960*C+8*B*C+768*B*C+512*B*C> <\input|252) >> subst(sqrt(B),b,subst(sqrt(C),c,subst(sqrt(N),n,%o246))); <\output> ) >18*B*N-96*B*C*N-48*B*C*N+320*B*C*N-B*C*N+128*B*C*N-512*B*C*N-4*B*C-1600*C+8*B*C-256*B*C+512*B*C> <\input|253) >> subst(sqrt(B),b,subst(sqrt(C),c,subst(sqrt(N),n,%o247))); <\output> ) >126*B*N-528*B*C*N+432*B*C*N+1280*B*C*N-B*C*N-576*B*C*N-1280*B*C*N+12*B*C-960*C-4*B*C+512*B*C> <\input|254) >> factor(radcan(subst(((sqrt(2)*C^3*sqrt((3*B^3-960)*C^2-256*B^2)+16*B*C^3)/(12*B)),N,%o251))); <\output> ) >-*8**B*3*B-960*C-256*B>-3*B*C+2496*B*C-215040*C-640*B|96>> <\input|255) >> factor(radcan(subst(((sqrt(2)*C^3*sqrt((3*B^3-960)*C^2-256*B^2)+16*B*C^3)/(12*B)),N,%o252))); <\output> ) >-*8**B*3*B-960*C-256*B>-3*B*C+2496*B*C-215040*C-640*B|96>> <\input|256) >> factor(radcan(subst(((sqrt(2)*C^3*sqrt((3*B^3-960)*C^2-256*B^2)+16*B*C^3)/(12*B)),N,%o253))); <\output> ) >-*8**B*C*3*B-960*C-256*B>-48**B*3*B-960*C-256*B>+10752**B*3*B-960*C-256*B>-21*B*C+10560*B*C-1505280*C+512*B*C+1536*B-344064*B|96>> <\input|257) >> factor(radcan(%o254*96/(-C^15))); <\output> ) >8**B*3*B-960*C-256*B>-3*B*C+2496*B*C-215040*C-640*B> <\input|258) >> factor(radcan(%o255*96/(-C^15))); <\output> ) >8**B*3*B-960*C-256*B>-3*B*C+2496*B*C-215040*C-640*B> <\input|259) >> factor(radcan(%o256*96/(-C^15))); <\output> ) >*B*C*3*B-960*C-256*B>-48**B*3*B-960*C-256*B>+10752**B*3*B-960*C-256*B>-21*B*C+10560*B*C-1505280*C+512*B*C+1536*B-344064*B|C>> <\input|260) >> factor(radcan(%o256*96/(-C^14))); <\output> ) >8**B*C*3*B-960*C-256*B>-48**B*3*B-960*C-256*B>+10752**B*3*B-960*C-256*B>-21*B*C+10560*B*C-1505280*C+512*B*C+1536*B-344064*B> <\input|261) >> \; > <\equation*> 8**B*3*B-960*C-256*B>-3*B*C+2496*B*C-215040*C-640*B=0 <\equation*> 8**B*3*B-960*C-256*B>=3*B*C-2496*B*C+215040*C+640*B <\equation*> 3*B-960*C-256*B>=*C-2496*B*C+215040*C+640*B|8**B> <\input|261) >> factor(radcan(subst(((3*B^6*C-2496*B^3*C+215040*C+640*B^4)/(8*sqrt(2)*B^3)), sqrt((3*B^3-960)*C^2-256*B^2),%o260))); <\output> ) >-B*C-448*B*C+71680*B*C+B*C-1120*B*C+258048*B*C-16056320*C+128*B-28672*B|B>> <\input|262) >> \; > It's not clear why I have a quadratic equation for , here, while for the analogous \S> problem, I got a linear equation for what was then . <\input|262) >> solve(%o261,C); <\output> ) >C=--2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+B-1120*B+258048*B-16056320|2*B-896*B+143360*B>,C=-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>-B+1120*B-258048*B+16056320|2*B-896*B+143360*B>> <\input|263) >> factor(radcan(subst(((sqrt(B^18-2752*B^15+2114560*B^12-698220544*B^9+110775762944*B^6-8286602526720*B^3+257805411942400)-B^9+1120*B^6-258048*B^3+16056320)/(2*B^8-896*B^5+143360*B^2)),C,%o257))); <\output> ) >*3*B-4320*B+1849344*B-295895040*B+15414067200*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+3*B-8960*B+8682496*B-4003528704*B+980892516352*B-129889206272000*B+8728554661478400*B-247493195464704000>-3*B*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+2496*B*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>-215040*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+3*B-7136*B+4358144*B-1024851968*B+95567216640*B-3452751052800|2*B*B-448*B+71680>> <\input|264) >> factor(radcan(%o263*2*B^2*(B^6-448*B^3+71680))); <\output> ) >16*B*3*B-4320*B+1849344*B-295895040*B+15414067200*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+3*B-8960*B+8682496*B-4003528704*B+980892516352*B-129889206272000*B+8728554661478400*B-247493195464704000>-3*B*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+2496*B*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>-215040*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+3*B-7136*B+4358144*B-1024851968*B+95567216640*B-3452751052800> <\input|265) >> plot2d(%o264, [B,0,5], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\errput> Warning: empty y range [-6.9055e+12:-6.9055e+12], adjusting to [-6.83645e+12:-6.97456e+12] <\input|266) >> plot2d(%o264, [B,5,10], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; <\errput> \; gnuplot\ plot [5.:10.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: warning: Skipping data file with no valid points \; gnuplot\ plot [5.:10.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: all points y value undefined! \; ) > <\input|267) >> plot2d(%o264, [B,10,20], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\errput> \; gnuplot\ plot [10.:20.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: warning: Skipping data file with no valid points \; gnuplot\ plot [10.:20.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: all points y value undefined! \; <\input|268) >> plot2d(%o264, [B,20,40], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; gnuplot\ plot [20.:40.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: warning: Skipping data file with no valid points \; gnuplot\ plot [20.:40.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: all points y value undefined! \; ) > <\input|269) >> plot2d(%o264, [B,40,80], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\errput> \; gnuplot\ plot [40.:80.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: warning: Skipping data file with no valid points \; gnuplot\ plot [40.:80.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: all points y value undefined! \; <\input|270) >> plot2d(%o264, [B,80,160], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> ) > <\errput> \; gnuplot\ plot [80.:160.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: warning: Skipping data file with no valid points \; gnuplot\ plot [80.:160.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: all points y value undefined! \; <\input|271) >> plot2d(%o264, [B,160,320], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> ) > <\errput> \; gnuplot\ plot [160.:320.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: warning: Skipping data file with no valid points \; gnuplot\ plot [160.:320.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: all points y value undefined! \; <\input|272) >> plot2d(%o264, [B,320,640], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\errput> \; gnuplot\ plot [320.:640.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: warning: Skipping data file with no valid points \; gnuplot\ plot [320.:640.]'/home/chris/maxout.gnuplot_pipes' index 0 \ notitle with lines 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ \ line 0: all points y value undefined! \; <\input|273) >> plot2d(%o264, [B,640,1000000], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|274) >> B^18-2752*B^15+2114560*B^12-698220544*B^9+110775762944*B^6-8286602526720*B^3+257805411942400; <\output> ) >B-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400> <\input|275) >> \; > %o274 has to be 0> for to be real. <\input|275) >> plot2d(%o274/(0.00000000000001 + abs(%o274)), [B,0,1], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|276) >> plot2d(%o274/(0.00000000000001 + abs(%o274)), [B,1,10], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> ) > <\errput> \; <\input|277) >> plot2d(%o274/(0.00000000000001 + abs(%o274)), [B,0.5,1], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|278) >> \; > There is a 0 at about , where it goes from -ve to +ve. \ Now it's positive for , so there must be another 0 at a smaller positive value of . \ Which must be at 1>. \ But the graphs are nonsense when the upper limit is . <\input|278) >> plot2d(%o274, [B,0.5,1], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> ) > <\errput> \; <\input|279) >> \; > The function is positive and decreassing in the %o278 range. <\input|279) >> plot2d(%o274, [B,0,0.5], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|280) >> \; > It's positive and decreasing in the %o279 range. <\input|280) >> plot2d(%o274, [B,1,2], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> ) > <\errput> \; <\input|281) >> plot2d(%o274, [B,2,10], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|282) >> \; > It's positive and decreasing in the %o280 range. \ %o281 shows it being about 0 up to about , after which it goes . \ The opposite of what the regularized sign function plot showed. <\input|282) >> plot2d(%o274, [B,2,7], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|283) >> \; > It is positive but with possibly a shallow minumum, and a deeper minimum above that, throughout the %o282 range. <\input|283) >> plot2d(%o274, [B,7,8], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> ) > <\errput> \; <\input|284) >> \; > It is decreasing throughout the above range, and passes through 0 at about . <\input|284) >> sqrt(-(3*B^12-4320*B^9+1849344*B^6-295895040*B^3+15414067200)*sqrt(B^18-2752*B^15+2114560*B^12-698220544*B^9+110775762944*B^6-8286602526720*B^3+257805411942400)+3*B^21-8960*B^18+8682496*B^15-4003528704*B^12+980892516352*B^9-129889206272000*B^6+8728554661478400*B^3-247493195464704000)-3*B^6*sqrt(B^18-2752*B^15+2114560*B^12-698220544*B^9+110775762944*B^6-8286602526720*B^3+257805411942400); <\output> ) >-3*B+4320*B-1849344*B+295895040*B-15414067200*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+3*B-8960*B+8682496*B-4003528704*B+980892516352*B-129889206272000*B+8728554661478400*B-247493195464704000>-3*B*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>> <\input|285) >> sqrt(-(3*B^12-4320*B^9+1849344*B^6-295895040*B^3+15414067200)*sqrt(B^18-2752*B^15+2114560*B^12-698220544*B^9+110775762944*B^6-8286602526720*B^3+257805411942400)+3*B^21-8960*B^18+8682496*B^15-4003528704*B^12+980892516352*B^9-129889206272000*B^6+8728554661478400*B^3-247493195464704000); <\output> ) >-3*B+4320*B-1849344*B+295895040*B-15414067200*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+3*B-8960*B+8682496*B-4003528704*B+980892516352*B-129889206272000*B+8728554661478400*B-247493195464704000>> <\input|286) >> %o285^2; <\output> ) >-3*B+4320*B-1849344*B+295895040*B-15414067200*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+3*B-8960*B+8682496*B-4003528704*B+980892516352*B-129889206272000*B+8728554661478400*B-247493195464704000> <\input|287) >> plot2d(%o286, [B,0,7.3], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|288) >> 15414067200.0^(1/12); <\output> ) >7.063063390029239> <\input|303) >> 247493195464704000.0^(1/21); <\output> ) >6.733872936375738> <\input|304) >> 257805411942400.0^(1/18); <\output> ) >6.318694160346535> <\input|305) >> \; > %o287 is negative throughout the range. <\input|288) >> plot2d(%o286, [B,7.3,7.43], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|289) >> \; > %o288 is negative and decreasing until the curve disappears at about . <\input|289) >> plot2d(%o274, [B,7.4,20], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|290) >> \; > %o289 is 0 until about and then becomes positive. <\input|290) >> plot2d(%o274, [B,7.4,15], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|291) >> \; > %o290 is 0 until about and then becomes positive. <\input|291) >> plot2d(%o274, [B,7.4,11], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|292) >> \; > %o291 is 0 then negative throughout the range. <\input|292) >> plot2d(%o274, [B,11,12], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|293) >> \; > %o292 is negative throughout the range, with a minimum near . <\input|293) >> plot2d(%o274, [B,12,12.5], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> ) > <\errput> \; <\input|294) >> \; > %o293 shows the curve increasing through a 0 at about . <\input|294) >> plot2d(%o274, [B,7.4,7.5], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|295) >> \; > %o294 shows the curve decreasing in almost a straight line through 0 at about . <\input|295) >> plot2d(%o274, [B,7.5,7.7], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> ) > <\errput> \; <\input|296) >> \; > negative and decreasing throughout the range. <\input|296) >> plot2d(%o286, [B,12.03,7.43], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> Bad range: [B,12.03,7.43]. Range must be of the form [variable,min,max] \ -- an error. \ To debug this try debugmode(true); <\input|297) >> plot2d(%o286, [B,12.03,12.06], [gnuplot_preamble,"set terminal postscript; set output \\"qoo9.eps\\";"]); <\output> <\errput> \; ) > <\input|298) >> \; > The curve first appears at about 12.058, where it is and rapidly decreasing. \ So it ought to be possible to plot the original graph at least in the range 12.058 to 12.06. <\input|298) >> plot2d(%o286, [B,12.05,12.2], [gnuplot_preamble,"set terminal postscript; set output \\"qoo10.eps\\";"]); <\output> <\errput> \; ) > <\input|299) >> \; > rapidly decreasing throughout the range, essentially vertically when the curve first appears, and goes negative through 0 at about . <\input|299) >> plot2d(%o264, [B,12.05,12.1], [gnuplot_preamble,"set terminal postscript; set output \\"qoo11.eps\\";"]); <\output> <\errput> \; ) > <\input|300) >> \; > Positive and decreasing throughout the range, decreasing essentially vertically when the curve first appears, and also decreasing essentially vertically when it disappears again. \ It is about 10> when the curve first appears, and decreases to about 10> where the curve disappears again. \ See if there is any other region where %o264 will be real. <\input|300) >> plot2d(%o286, [B,12.1,15], [gnuplot_preamble,"set terminal postscript; set output \\"qoo12.eps\\";"]); <\output> <\errput> \; ) > <\input|301) >> \; > 0>, and decreasing ever more rapidly througout the range. <\input|301) >> plot2d(%o286, [B,15,25], [gnuplot_preamble,"set terminal postscript; set output \\"qoo12.eps\\";"]); <\output> <\errput> \; ) > <\input|302) >> \; > negative and decreasing ever more rapidly throughout the range. \ Now %o302 to %o304, just after %o286 above, show that no further 0's are expected in %o286. \ So there appear to be no 0's on the square root branches chosen above, which are the most likely to give positive and . \ Note, however, that would do no harm: it would mean \H>. <\input|305) >> factor(radcan(subst(-((sqrt(B^18-2752*B^15+2114560*B^12-698220544*B^9+110775762944*B^6-8286602526720*B^3+257805411942400)+B^9-1120*B^6+258048*B^3-16056320)/(2*B^8-896*B^5+143360*B^2)),C,%o257))); <\output> ) >*3*B-4320*B+1849344*B-295895040*B+15414067200*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+3*B-8960*B+8682496*B-4003528704*B+980892516352*B-129889206272000*B+8728554661478400*B-247493195464704000>+3*B*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>-2496*B*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+215040*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+3*B-7136*B+4358144*B-1024851968*B+95567216640*B-3452751052800|2*B*B-448*B+71680>> <\input|306) >> factor(radcan(%o305*2*B^2*(B^6-448*B^3+71680))); <\output> ) >16*B*3*B-4320*B+1849344*B-295895040*B+15414067200*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+3*B-8960*B+8682496*B-4003528704*B+980892516352*B-129889206272000*B+8728554661478400*B-247493195464704000>+3*B*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>-2496*B*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+215040*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+3*B-7136*B+4358144*B-1024851968*B+95567216640*B-3452751052800> <\input|307) >> plot2d(%o306, [B,0,7.4], [gnuplot_preamble,"set terminal postscript; set output \\"qoo13.eps\\";"]); <\output> <\errput> \; ) > <\input|308) >> \; > %o307 has the vertical axis running from to , but there is no sign of the curve, except right up against the vertical axis. <\input|308) >> (3*B^12-4320*B^9+1849344*B^6-295895040*B^3+15414067200)*sqrt(B^18-2752*B^15+2114560*B^12-698220544*B^9+110775762944*B^6-8286602526720*B^3+257805411942400)+3*B^21-8960*B^18+8682496*B^15-4003528704*B^12+980892516352*B^9-129889206272000*B^6+8728554661478400*B^3-247493195464704000; <\output> ) >3*B-4320*B+1849344*B-295895040*B+15414067200*-2752*B+2114560*B-698220544*B+110775762944*B-8286602526720*B+257805411942400>+3*B-8960*B+8682496*B-4003528704*B+980892516352*B-129889206272000*B+8728554661478400*B-247493195464704000> <\input|309) >> \; > %o308 has to be positive for the first square root in %o306 to be real. <\input|309) >> plot2d(%o308, [B,0,7.4], [gnuplot_preamble,"set terminal postscript; set output \\"qoo13.eps\\";"]); <\output> <\errput> \; ) > <\input|310) >> \; > %o309 has the axes the other way round, it is approximately 0, but possibly a tiny bit positive, until about , when it goes negative. <\input|310) >> plot2d(%o308, [B,0,3], [gnuplot_preamble,"set terminal postscript; set output \\"qoo14.eps\\";"]); <\output> <\errput> \; \; ) > <\input|311) >> \; > 0 or negative throughout the range. <\input|311) >> plot2d(%o308, [B,0,1], [gnuplot_preamble,"set terminal postscript; set output \\"qoo14.eps\\";"]); <\output> <\errput> \; ) > <\input|312) >> \; > 0 or negative throughout the range. <\input|312) >> plot2d(%o306, [B,12,14], [gnuplot_preamble,"set terminal postscript; set output \\"qoo14.eps\\";"]); <\output> <\errput> \; ) > <\input|313) >> \; > Just above 0 from about 12.06 where the curve starts, positive and increasing throughout the range. <\input|313) >> plot2d(%o306, [B,12,12.1], [gnuplot_preamble,"set terminal postscript; set output \\"qoo14.eps\\";"]); <\output> <\errput> \; ) > <\input|314) >> \; > Positive and increasing from just below 12.06 where the curve starts. <\input|314) >> plot2d(%o306, [B,14,24], [gnuplot_preamble,"set terminal postscript; set output \\"qoo15.eps\\";"]); <\output> ) > <\input|315) >> \; > 0> and increasing throughout the range. \ That was the branch that seemed to have the best chance of having a solution. \ Continue in a new file, BulkVacuumEnergy41.tm. \; <\initial> <\collection>