Artifact 0f61dcd44e7926e09ccfef301b2f94d912480e752bd973f59a18a047a04158b0:
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r37/packages/susy2/susy2.red
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2011-09-02 18:13:33
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— Some historical releases purely for archival purposes
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module susy2; %version 1.2 %changes and bugs compare to version 1.0 %8.12.1996 %vericication of cp,axp; %12.05.1997 %lyst in order to consider n(0)*d(1)^3+m(0)*d(1)^3; %10.09.1997 %verification of jacob(wx,wx1,wx2) i fjacob %1.10.1997 %introduction chiral1 and %b_chiral,f_chiral,b_antychiral,f_antychiral %introduction lyst2 %introduction of matrix(expression,boson or fermion, full or boson sector) %3.10.1997 %verification of coordinates %6.10.1997 %verification of wcomb,fcomb; %19.04.1999 %changes and bugs compare to version 1.1 %new command s_int algebraic; operator !@f_f,!@g_g,newton,delta,b_part,bf_part,pg,chan,s_part,prykr,prykl; operator bos,fer,der,del,d,axp,axx,zan,zen,fun,tet,gras,ber,fir,berz,firr,dr, stp,byk,r_r,!&a,p_p,s_s,waga; noncom bos,fer,der,del,d,axp,axx,zan,zen,fun,tet,gras,ber,fir,berz,firr,dr, stp,byk,r_r; factor !&a,byk; factor fer,bos,fun,gras; %fer,bos,axp superfunctions %der,del,d,dr operations %zan,zen needs to divergency %fun,tet,gras,axx classical part %stp to adjoint %ber,fir,berz,firr corresponds to Gato %byk,r_r,!&a,p_p,s_s,waga for super integration %*******************************************% %*** declaration of chirality and **********% %*** cutoff and drr ************************% %*******************************************% chiral:={ abra_kadabra => 1 , der(~n)**2 => 0, del(~n)**2 => 0,del(2)*del(1) => - d(1) - del(1)*del(2), der(2)*der(1) => -d(1) - der(1)*der(2),der(~n)*del(~n) => 0, del(~n)*der(~n) => 0,der(1)*del(2)=> -d(1) - del(2)*der(1), der(2)*del(1)=> -d(1) - del(1)*der(2), %b_chiral,f_chiral,b_antychiral,f_antychiral as lists b_chiral => {}, f_chiral => {}, b_antychiral => {}, f_antychiral => {}, fer(~f,1,~m) => 0 when not freeof(b_chiral,f), bos(~f,3,~m) => -bos(f,0,m+1) when not freeof(b_chiral,f), bos(~f,3,~m,~k) => (-1)**k*bos(f,0,m+1,k) when not freeof(b_chiral,f), bos(~f,1,~m) => 0 when not freeof(f_chiral,f), bos(~f,1,~m,~k) => 0 when not freeof(f_chiral,f), fer(~f,3,~m) => -fer(f,0,m+1) when not freeof(f_chiral,f), fer(~f,2,~m) => 0 when not freeof(b_antychiral,f), bos(~f,3,~m) => 0 when not freeof(b_antychiral,f), bos(~f,3,~m,~k) => 0 when not freeof(b_antychiral,f), bos(~f,2,~m) => 0 when not freeof(f_antychiral,f), bos(~f,2,~m,~k) => 0 when not freeof(f_antychiral,f), fer(~f,3,~m) => 0 when not freeof(f_antychiral,f), der(1)*fer(~f,1,~m) => - fer(f,1,m)*der(1), der(1)*fer(~f,2,~m) => bos(f,3,m) - fer(f,2,m)*der(1), der(1)*fer(~f,3,~m) => - fer(f,3,m)*der(1), fer(~f,1,~m)*del(1) => - del(1)*fer(f,1,m), fer(~f,2,~m)*del(1) => bos(f,3,m) - del(1)*fer(f,2,m), fer(~f,3,~m)*del(1) => - del(1)*fer(f,3,m), der(2)*fer(~f,1,~m) => -bos(f,0,m+1) - bos(f,3,m) - fer(f,1,m)*der(2), der(2)*fer(~f,2,~m) => - fer(f,2,m)*der(2), der(2)*fer(~f,3,~m) => -bos(f,2,m+1)-fer(f,3,m)*der(2), fer(~f,1,~m)*del(2) => -bos(f,0,m+1) - bos(f,3,m) - del(2)*fer(f,1,m), fer(~f,2,~m)*del(2) => - del(2)*fer(f,2,m), fer(~f,3,~m)*del(2) => -bos(f,2,m+1)-del(2)*fer(f,3,m), der(1)*bos(~f,1,~m) => bos(f,1,m)*der(1), der(1)*bos(~f,2,~m) => fer(f,3,m) + bos(f,2,m)*der(1), der(1)*bos(~f,3,~m) => bos(f,3,m)*der(1), bos(~f,1,~m)*del(1) => del(1)*bos(f,1,m), bos(~f,2,~m)*del(1) => -fer(f,3,m)+del(1)*bos(f,2,m), bos(~f,3,~m)*del(1) => del(1)*bos(f,3,m), der(2)*bos(~f,1,~m) => -fer(f,0,m+1) - fer(f,3,m) + bos(f,1,m)*der(2), der(2)*bos(~f,2,~m) => bos(f,2,m)*der(2), der(2)*bos(~f,3,~m) => -fer(f,2,m+1) + bos(f,3,m)*der(2), bos(~f,1,~m)*del(2) => fer(f,0,m+1) + fer(f,3,m) + del(2)*bos(f,1,m), bos(~f,2,~m)*del(2) => del(2)*bos(f,2,m), bos(~f,3,~m)*del(2) => fer(f,2,m+1) + del(2)*bos(f,3,m), der(1)*bos(~f,1,~m,~l) => bos(f,1,m,l)*der(1), der(1)*bos(~f,2,~m,~l) => l*fer(f,3,m)*bos(f,2,m,l-1)+bos(f,2,m,l)*der(1), der(1)*bos(~f,3,~m,~l) => bos(f,3,m,l)*der(1), bos(~f,1,~m,~l)*del(1) => del(1)*bos(f,1,m,l), bos(~f,2,~m,~l)*del(1) => -l*fer(f,3,m)*bos(f,2,m,l-1)+del(1)*bos(f,2,m,l), bos(~f,3,~m,~l)*del(1) => del(1)*bos(f,3,m,l), der(2)*bos(~f,1,~m,~l) => -l*(fer(f,0,m+1)+fer(f,3,m))*bos(f,1,m,l-1) + bos(f,1,m,l)*der(2), der(2)*bos(~f,2,~m,~l) => bos(f,2,m,l)*der(2), der(2)*bos(~f,3,~m,~l) => -l*fer(f,2,m+1)*bos(f,3,m,l-1) + bos(f,3,m,l)*der(2), bos(~f,1,~m,~l)*del(2) => l*(fer(f,0,m+1)+fer(f,3,m))*bos(f,1,m,l-1) + del(2)*bos(f,1,m,l), bos(~f,2,~m,~l)*del(2) => del(2)*bos(f,2,m,l), bos(~f,3,~m,~l)*del(2) => l*fer(f,2,m+1)*bos(f,3,m,l-1) + del(2)*bos(f,3,m,l)}$ chiral1:={abra_kadabra => 3 , %der(3) as commutator,bos(f,3,n) or fer(f,3,4) as commutator der(~n)**2 => 0 when n neq 3, del(~n)**2 => 0 when n neq 3, der(3)^2 => d(1)^2, del(3)^2 => d(1)^2, der(2)*der(1) => -(d(1)+der(3))/2,der(1)*der(2) => (-d(1)+der(3))/2, del(2)*del(1) => -(d(1)+del(3))/2,del(1)*del(2) => (-d(1)+del(3))/2, der(1)*der(3) => d(1)*der(1), der(2)*der(3) => -d(1)*der(2), der(3)*der(1) => -d(1)*der(1), der(3)*der(2) => d(1)*der(2), del(1)*del(3) => d(1)*del(1), del(2)*del(3) => -d(1)*del(2), del(3)*del(1) => -d(1)*del(1), del(3)*del(2) => d(1)*del(2), der(~n)*del(~n) => if n neq 3 then 0 else if n eq 3 then d(1)^2, del(~n)*der(~n) => if n neq 3 then 0 else if n eq 3 then d(1)^2, der(1)*del(2) => -d(1) -del(2)*der(1),der(2)*del(1) => -d(1) -del(1)*der(2), der(1)*del(3) => d(1)*del(1), der(2)*del(3) => -d(1)*del(2), b_chiral => {}, f_chiral => {}, b_antychiral => {}, f_antychiral => {}, fer(~f,1,~m) => 0 when not freeof(b_chiral,f), bos(~f,3,~m) => -bos(f,0,m+1) when not freeof(b_chiral,f), bos(~f,3,~m,~k) => (-1)**k*bos(f,0,m+1,k) when not freeof(b_chiral,f), bos(~f,1,~m) => 0 when not freeof(f_chiral,f), bos(~f,1,~m,~k) => 0 when not freeof(f_chiral,f), fer(~f,3,~m) => -fer(f,0,m+1) when not freeof(f_chiral,f), fer(~f,2,~m) => 0 when not freeof(b_antychiral,f), bos(~f,3,~m) => bos(f,0,m+1) when not freeof(b_antychiral,f), bos(~f,3,~m,~k) => bos(f,0,m+1,k) when not freeof(b_antychiral,f), bos(~f,2,~m) => 0 when not freeof(f_antychiral,f), bos(~f,2,~m,~k) => 0 when not freeof(f_antychiral,f), fer(~f,3,~m) => fer(f,0,m+1) when not freeof(f_antychiral,f), der(1)*fer(~f,1,~m) => - fer(f,1,m)*der(1), der(1)*fer(~f,2,~m) => -bos(f,0,m+1)/2+bos(f,3,m)/2 - fer(f,2,m)*der(1), der(1)*fer(~f,3,~m) => bos(f,1,m+1)-fer(f,3,m)*der(1), fer(~f,1,~m)*del(1) => - del(1)*fer(f,1,m), fer(~f,2,~m)*del(1) => -bos(f,0,m+1)/2 + bos(f,3,m)/2 - del(1)*fer(f,2,m), fer(~f,3,~m)*del(1) => bos(f,1,m+1) - del(1)*fer(f,3,m), der(2)*fer(~f,1,~m) => -bos(f,0,m+1)/2 - bos(f,3,m)/2 - fer(f,1,m)*der(2), der(2)*fer(~f,2,~m) => - fer(f,2,m)*der(2), der(2)*fer(~f,3,~m) => -bos(f,2,m+1)-fer(f,3,m)*der(2), fer(~f,1,~m)*del(2) => -bos(f,0,m+1)/2 - bos(f,3,m)/2 - del(2)*fer(f,1,m), fer(~f,2,~m)*del(2) => - del(2)*fer(f,2,m), fer(~f,3,~m)*del(2) => -bos(f,2,m+1)-del(2)*fer(f,3,m), der(3)*fer(~f,0,~m) => fer(f,3,m) + fer(f,0,m)*der(3) - 2*bos(f,1,m)*der(2) + 2*bos(f,2,m)*der(1), der(3)*fer(~f,1,~m) => -fer(f,1,m+1) - bos(f,0,m+1)*der(1) -bos(f,3,m)*der(1) +fer(f,1,m)*der(3), der(3)*fer(~f,2,~m) => bos(f,0,m+1)*der(2) - bos(f,3,m)*der(2) + fer(f,2,m+1) +fer(f,2,m)*der(3), der(3)*fer(~f,3,~m) => fer(f,0,m+2) + fer(f,3,m)*der(3) -2*bos(f,1,m+1)*der(2)-2*bos(f,2,m+1)*der(1), fer(~f,0,~m)*del(3) => fer(f,3,m)+ del(3)*fer(f,0,m) + 2*del(2)*bos(f,1,m) - 2*del(1)*bos(f,2,m), fer(~f,1,~m)*del(3) => -fer(f,1,m+1)+ del(1)*bos(f,0,m+1) + del(1)*bos(f,3,m) + del(3)*fer(f,1,m), fer(~f,2,~m)*del(3) => - del(2)*bos(f,0,m+1) +del(2)*bos(f,3,m) +fer(f,2,m+1) +del(3)*fer(f,2,m), fer(~f,3,~m)*del(3) => fer(f,0,m+2) + del(3)*fer(f,3,m) + 2*del(2)*bos(f,1,m+1) + 2*del(1)*bos(f,2,m+1), der(1)*bos(~f,1,~m) => bos(f,1,m)*der(1), der(1)*bos(~f,2,~m) => -fer(f,0,m+1)/2 + fer(f,3,m)/2 + bos(f,2,m)*der(1), der(1)*bos(~f,3,~m) => fer(f,1,m+1) + bos(f,3,m)*der(1), bos(~f,1,~m)*del(1) => del(1)*bos(f,1,m), bos(~f,2,~m)*del(1) => fer(f,0,m+1)/2 - fer(f,3,m)/2 +del(1)*bos(f,2,m), bos(~f,3,~m)*del(1) => - fer(f,1,m+1) + del(1)*bos(f,3,m), der(2)*bos(~f,1,~m) => -fer(f,0,m+1)/2 - fer(f,3,m)/2 + bos(f,1,m)*der(2), der(2)*bos(~f,2,~m) => bos(f,2,m)*der(2), der(2)*bos(~f,3,~m) => -fer(f,2,m+1) + bos(f,3,m)*der(2), bos(~f,1,~m)*del(2) => fer(f,0,m+1)/2 + fer(f,3,m)/2 + del(2)*bos(f,1,m), bos(~f,2,~m)*del(2) => del(2)*bos(f,2,m), bos(~f,3,~m)*del(2) => fer(f,2,m+1) + del(2)*bos(f,3,m), der(3)*bos(~f,0,~m) => bos(f,3,m) + bos(f,0,m)*der(3) + 2*fer(f,1,m)*der(2) -2*fer(f,2,m)*der(1), der(3)*bos(~f,1,~m) => -bos(f,1,m+1) + fer(f,0,m+1)*der(1) + fer(f,3,m)*der(1) + bos(f,1,m)*der(3), der(3)*bos(~f,2,~m) => - fer(f,0,m+1)*der(2) + fer(f,3,m)*der(2) + bos(f,2,m)*der(3) +bos(f,2,m+1), der(3)*bos(~f,3,~m) => bos(f,0,m+2) + 2*fer(f,2,m+1)*der(1) + 2*fer(f,1,m+1)*der(2) + bos(f,3,m)*der(3), bos(~f,0,~m)*del(3) => bos(f,3,m) + del(3)*bos(f,0,m) + 2*del(2)*fer(f,1,m) - 2*del(1)*fer(f,2,m), bos(~f,1,~m)*del(3) => del(1)*fer(f,0,m+1) + del(1)*fer(f,3,m) -bos(f,1,m+1) + del(3)*bos(f,1,m), bos(~f,2,~m)*del(3) => -del(2)*fer(f,0,m+1) + del(2)*fer(f,3,m) + del(3)*bos(f,2,m) +bos(f,2,m+1) , bos(~f,3,~m)*del(3) => bos(f,0,m+2) + 2*del(1)*fer(f,2,m+1) + 2*del(2)*fer(f,1,m+1) + del(3)*bos(f,3,m), der(1)*bos(~f,1,~m,~l) => bos(f,1,m,l)*der(1), der(1)*bos(~f,2,~m,~l) => l*bos(f,2,m,l-1)*(-fer(f,0,m+1)/2 + fer(f,3,m)/2) + bos(f,2,m,l)*der(1), der(1)*bos(~f,3,~m,~l) => l*bos(f,3,m,l-1)*fer(f,1,m+1) + bos(f,3,m,l)*der(1), bos(~f,1,~m,~l)*del(1) => del(1)*bos(f,1,m,l), bos(~f,2,~m,~l)*del(1) => - l*bos(f,2,m,l-1)*(-fer(f,0,m+1)/2 + fer(f,3,m)/2) +del(1)*bos(f,2,m,l), bos(~f,3,~m,~l)*del(1) => - l*bos(f,3,m,l-1)*fer(f,1,m+1) + del(1)*bos(f,3,m,l), der(2)*bos(~f,1,~m,~l) => - l*bos(f,1,m,l-1)*(fer(f,0,m+1)/2+ fer(f,3,m)/2) + bos(f,1,m,l)*der(2), der(2)*bos(~f,2,~m,~l) => bos(f,2,m,l)*der(2), der(2)*bos(~f,3,~m,~l) => -l*fer(f,2,m+1)*bos(f,3,m,l-1) + bos(f,3,m,l)*der(2), bos(~f,1,~m,~l)*del(2) => l*(fer(f,0,m+1)/2+fer(f,3,m)/2)*bos(f,1,m,l-1) + del(2)*bos(f,1,m,l), bos(~f,2,~m,~l)*del(2) => del(2)*bos(f,2,m,l), bos(~f,3,~m,~l)*del(2) => l*fer(f,2,m+1)*bos(f,3,m,l-1) + del(2)*bos(f,3,m,l), der(3)*bos(~f,~k,~m,~l) => der(1)*prykr(bos(f,k,m,l),2)-der(2)*prykr(bos(f,k,m,l),1), bos(~f,~k,~m,~l)*del(3) => -prykl(bos(f,k,m,l),2)*del(1)+prykl(bos(f,k,m,l),1)*del(2) }$ trad:={abra_kadabra => 2 , der(~n)**2 => d(1),del(~n)**2 => d(1),del(2)*del(1) => -del(1)*del(2), der(2)*der(1) => -der(1)*der(2), der(~n)*del(~n) => d(1), del(~n)*der(~n) => d(1), der(1)*del(2) => -del(2)*der(1), der(2)*del(1) => -del(1)*der(2), der(1)*fer(~f,1,~m) => bos(f,0,m+1) - fer(f,1,m)*der(1), der(1)*fer(~f,2,~m) => bos(f,3,m) - fer(f,2,m)*der(1), der(1)*fer(~f,3,~m) => bos(f,2,m+1) - fer(f,3,m)*der(1), fer(~f,1,~m)*del(1) => bos(f,0,m+1) - del(1)*fer(f,1,m), fer(~f,2,~m)*del(1) => bos(f,3,m) - del(1)*fer(f,2,m), fer(~f,3,~m)*del(1) => bos(f,2,m+1) - del(1)*fer(f,3,m), der(2)*fer(~f,1,~m) => -bos(f,3,m) - fer(f,1,m)*der(2), der(2)*fer(~f,2,~m) => bos(f,0,m+1) - fer(f,2,m)*der(2), der(2)*fer(~f,3,~m) => -bos(f,1,m+1)-fer(f,3,m)*der(2), fer(~f,1,~m)*del(2) => -bos(f,3,m) - del(2)*fer(f,1,m), fer(~f,2,~m)*del(2) => bos(f,0,m+1) - del(2)*fer(f,2,m), fer(~f,3,~m)*del(2) => -bos(f,1,m+1)-del(2)*fer(f,3,m), der(1)*bos(~f,1,~m) => fer(f,0,m+1) + bos(f,1,m)*der(1), der(1)*bos(~f,2,~m) => fer(f,3,m)+bos(f,2,m)*der(1), der(1)*bos(~f,3,~m) => fer(f,2,m+1) + bos(f,3,m)*der(1), bos(~f,1,~m)*del(1) => -fer(f,0,m+1) + del(1)*bos(f,1,m), bos(~f,2,~m)*del(1) => -fer(f,3,m)+del(1)*bos(f,2,m), bos(~f,3,~m)*del(1) => -fer(f,2,m+1) + del(1)*bos(f,3,m), der(2)*bos(~f,1,~m) => -fer(f,3,m) + bos(f,1,m)*der(2), der(2)*bos(~f,2,~m) => fer(f,0,m+1) + bos(f,2,m)*der(2), der(2)*bos(~f,3,~m) => -fer(f,1,m+1) + bos(f,3,m)*der(2), bos(~f,1,~m)*del(2) => fer(f,3,m) + del(2)*bos(f,1,m), bos(~f,2,~m)*del(2) => -fer(f,0,m+1) + del(2)*bos(f,2,m), bos(~f,3,~m)*del(2) => fer(f,1,m+1) + del(2)*bos(f,3,m), der(1)*bos(~f,1,~m,~l) => l*fer(f,0,m+1)*bos(f,1,m,l-1) + bos(f,1,m,l)*der(1), der(1)*bos(~f,2,~m,~l) => l*fer(f,3,m)*bos(f,2,m,l-1)+ bos(f,2,m,l)*der(1), der(1)*bos(~f,3,~m,~l) => l*fer(f,2,m+1)*bos(f,3,m,l-1) + bos(f,3,m,l)*der(1), bos(~f,1,~m,~l)*del(1) => -l*fer(f,0,m+1)*bos(f,1,m,l-1) + del(1)*bos(f,1,m,l), bos(~f,2,~m,~l)*del(1) => -l*fer(f,3,m)*bos(f,2,m,l-1)+ del(1)*bos(f,2,m,l), bos(~f,3,~m,~l)*del(1) => -l*fer(f,2,m+1)*bos(f,3,m,l-1) + del(1)*bos(f,3,m,l), der(2)*bos(~f,1,~m,~l) => -l*fer(f,3,m)*bos(f,1,m,l-1) + bos(f,1,m,l)*der(2), der(2)*bos(~f,2,~m,~l) => l*fer(f,0,m+1)*bos(f,2,m,l-1) + bos(f,2,m,l)*der(2), der(2)*bos(~f,3,~m,~l) => -l*fer(f,1,m+1)*bos(f,3,m,l-1) + bos(f,3,m,l)*der(2), bos(~f,1,~m,~l)*del(2) => l*fer(f,3,m)*bos(f,1,m,l-1) + del(2)*bos(f,1,m,l), bos(~f,2,~m,~l)*del(2) => -l*fer(f,0,m+1)*bos(f,2,m,l-1) + del(2)*bos(f,2,m,l), bos(~f,3,~m,~l)*del(2) => l*fer(f,1,m+1)*bos(f,3,m,l-1) + del(2)*bos(f,3,m,l)}$ drr:= { d(-1)**(~n) => dr(-n) when n neq 1,d(-1) => dr(-1) }$ nodrr:={ dr(-~n) => d(-1)**n when n neq 1, dr(-1) => d(-1) }$ cutoff:= { dr(~n) => 0 when n < - cut }$ inverse:={bos(~f,~n,~m) => bos(f,n,m,1) , fun(~f,~n) => fun(f,n,1) }$ %*******************************************% %*** module ordering **********************% %*******************************************% %ordering of bos with 4 and 3 indices and (fer,axp,ber,fir,zen,zan) let { bos(~f,~n,~m,~k)*bos(~g,~x,~z,~v) => bos(g,x,z,v)*bos(f,n,m,k) when ordp(f,g) and f neq g or f equal g and n<x or f equal g and n equal x and m<z, bos(~f,~n,~m)*bos(~g,~x,~z) => bos(g,x,z)*bos(f,n,m) when ordp(f,g) and f neq g or f equal g and n<x or f equal g and n equal x and m<z, bos(~f,~n,~m,~k)*bos(~g,~x,~z) => bos(g,x,z)*bos(f,n,m,k) when ordp(f,g) and f neq g or f equal g and n<x or f equal g and n equal x and m<z, bos(~g,~x,~z)*bos(~f,~n,~m,~k) =>bos(f,n,m,k)*bos(g,x,z) when ordp(g,f) and f neq g or f equal g and n>x or f equal g and n equal x and m>z, bos(~f,~n,~m,~k)*bos(~f,~n,~m,~l) => bos(f,n,m,k+l), bos(~f,~n,~m,~k)**2 => bos(f,n,m,2k), bos(~f,~n,~m,0) => 1, bos(0,~f,~n,~m) => 0, bos(0,~f,~n) => 0, bos(~f,~n,~m,~k)*bos(~f,~n,~m) => bos(f,n,m,k+1), bos(~f,~n,~m)*bos(~f,~n,~m,~k) => bos(f,n,m,k+1), ber(~f,~n,~m)*bos(~g,~k,~x,~l) => bos(g,k,x,l)*ber(f,n,m), fir(~f,~n,~m)*bos(~g,~k,~x,~l) => bos(g,k,x,l)*fir(f,n,m), ber(~f,~n,~m)*bos(~g,~k,~l) => bos(g,k,l)*ber(f,n,m), ber(~f,~n,~m)*fer(~g,~k,~l) => fer(g,k,l)*ber(f,n,m), fir(~f,~n,~m)*bos(~g,~k,~l) => bos(g,k,l)*fir(f,n,m), fir(~f,~n,~m)*fer(~g,~k,~l) => -fer(g,k,l)*fir(f,n,m), %ordering of fer, fer(0,~n,~m) => 0, bos(~f,~n,~m,~y)*fer(~g,~x,~h) => fer(g,x,h)*bos(f,n,m,y), bos(~f,~n,~m)*fer(~g,~x,~h) => fer(g,x,h)*bos(f,n,m), fer(~f,~n,~m)**2 => 0, fer(~f,~n,~m)*fer(~g,~k,~l) => - fer(g,k,l)*fer(f,n,m) when ordp(f,g) and f neq g or f equal g and n<k or f equal g and n equal k and m<l, %ordering classical, fun(~f,~n,~m)*fun(~g,~k,~l) => fun(g,k,l)*fun(f,n,m) when ordp(f,g) and f neq g or f equal g and n<k or f equal g and n equal k and m<l, fun(~f,~n,~m)*fun(~g,~x) => fun(g,x)*fun(f,n,m) when ordp(f,g) and f neq g or f equal g and n<x , fun(~g,~x)*fun(~f,~n,~m) => fun(f,n,m)*fun(g,x) when ordp(g,f) and f neq g or f equal g and n>x, fun(~f,~n)*fun(~g,~m) => fun(g,m)*fun(f,n) when ordp(f,g) and f neq g or f equal g and n<m, fun(~f,~n,~m,~k,~l)*fun(~s,~x) => fun(s,x)*fun(f,n,m,k,l), fun(~f,~n,~m,~k,~l)*fun(~s,~x,~z) => fun(s,x,z)*fun(f,n,m,k,l), fun(~f,~n,~m,~k,~l)*gras(~s,~x) => gras(s,x)*fun(f,n,m,k,l), fun(~f,~n,~m,~k,~l)*tet(~s) => tet(s)*fun(f,n,m,k,l), fun(~f,~n,~m,~k,~l)*fun(~s,~x,~z) => fun(s,x,z)*fun(f,n,m,k,l), fun(~f,~n,~m)*gras(~g,~x) => gras(g,x)*fun(f,n,m), fun(~f,~n)*gras(~g,~x) => gras(g,x)*fun(f,n), gras(~f,~n)*gras(~g,~m) =>-gras(g,m)*gras(f,n) when ordp(f,g) and f neq g or f equal g and n<m, ber(~f,~n)*fun(~g,~m) => fun(g,m)*fun(f,n), ber(~f,~n)*gras(~g,~m) => gras(g,m)*fun(f,n), fir(~f,~n)*fun(~g,~m) => fun(g,m)*fir(f,n), fir(~f,~n)*gras(~g,~m) => - fir(g,m)*gras(f,n), gras(~f,~n)^2 => 0, fun(~f,~n,0) => 1,fun(0,~n,~m) => 0, fun(0,~n) => 0, gras(0,~n) => 0, fun(~f,~n,~m)*fun(~f,~n,~k) => fun(f,n,m+k), fun(~f,~n,~m)**2 => fun(f,n,2m), fun(~f,~n,~m)*tet(~k) => tet(k)*fun(f,n,m), fun(~f,~n)*tet(~k) => tet(k)*fun(f,n), gras(~f,~n)*tet(~k) => - tet(k)*gras(f,n), axx(~f)*fun(~g,~n) => fun(g,n)*axx(f), axx(~f)*gras(~g,~n) => gras(g,n)*axx(f), axx(~f)*fun(~g,~n,~m) => fun(g,n,m)*axx(f), axx(~f)*fun(~g,~n,~m,~k,~l) => fun(g,n,m,~k,~l)*axx(f), fun(~f,~n,~g,~m,~k) => (for s:=0:k sum (-1)**s*newton(k,s)*fun(f,n,k-s)*fun(g,m,s)/(2**s)) when numberp(k) and k >=0, %ordering other, bos(~g,~x,~h)*zan(~f,~n,~m) => zan(f,n,m)*bos(g,x,h), bos(~g,~x,~h,~l)*zan(~f,~n,~m) => zan(f,n,m)*bos(g,x,h,l), fer(~g,~x,~h)*zan(~f,~n,~m) => zan(f,n,m)*fer(g,x,h), bos(~g,~x,~h)*zen(~f,~n,~m) => zen(f,n,m)*bos(g,x,h), bos(~g,~x,~h,~l)*zen(~f,~n,~m) => zen(f,n,m)*bos(g,x,h,l), fer(~g,~x,~h)*zen(~f,~n,~m) => - zen(f,n,m)*fer(g,x,h), axp(~g)*zan(~f,~n,~m) => zan(f,n,m)*axp(g), axp(~g)*zen(~f,~n,~m) => zen(f,n,m)*axp(g), axp(~g)*bos(~f,~n,~m) => bos(f,n,m)*axp(g), axp(~g)*bos(~f,~n,~m,~l) => bos(f,n,m,l)*axp(g), axp(~g)*fer(~f,~n,~m) => fer(f,n,m)*axp(g), axp(~f)*axp(~g) => axp(f+g), axp(~f)**(~n) => axp(n*f)}$ %other; let { dr(~n)*dr(~m) => dr(n+m),tet(~n)^2 => 0, tet(~n)*tet(~m) => -tet(m)*tet(n) when n<m, !@g_g(~f,~m,~n)**2 => 0 when m=0 or m=3, dr(~n)*d(1) => dr(n+1), d(1)*dr(~n) => dr(n+1),dr(~n)*d(2) => dr(n+1), d(2)*dr(~n) => dr(n+1),der(~m)*dr(~n) => dr(n)*der(m), dr(~n)*del(~m) => del(m)*dr(n), dr(~n)**2 => dr(2n),axp(0)=> 1, axx(0)=> 1,dr(0)=> 1, der(0) => 1, der(~n)*d(~m) => d(m)*der(n) when m neq t, d(1)*del(~n) => del(n)*d(1),del(~n)*d(2) => d(2)*del(n), d(-1)*del(~n) => del(n)*d(-1),del(~n)*d(-2) => d(-2)*del(n), d(-3)*del(~n) => del(n)*d(-3),del(~n)*d(-4) => d(-4)*del(n), d(1)*d(-1)=> 1,d(-1)*d(1)=>1,d(1)*d(-2)=> 1,d(-2)*d(1)=> 1, d(1)*d(-3)=> 1,d(-3)*d(1)=>1,d(1)*d(-4)=> 1,d(-4)*d(1)=> 1, d(3)*d(-1)=> 1,d(-1)*d(3)=>1,d(3)*d(-2)=> 1,d(-2)*d(3)=> 1, d(3)*d(-4)=> 1,d(-4)*d(3)=>1,d(2)*d(-1)=> 1,d(-1)*d(2)=> 1, d(2)*d(-2)=> 1,d(-2)*d(2)=>1,d(2)*d(-3)=> 1,d(-3)*d(2)=> 1, d(2)*d(-4)=> 1,d(-4)*d(2)=>1,d(3)*d(-3)=>1, d(1)*d(3)=> d(3)*d(1), d(-3)*d(3)=>1, d(t)*d(1)=> d(1)*d(t),d(t)*d(2)=>d(2)*d(t), d(t)*der(~n)=>der(n)*d(t),d(t)*del(~n)=>del(n)*d(t), d(t)*d(-1)=>d(-1)*d(t),d(t)*d(-2)=> d(-2)*d(t),!@x_y^2 =>1, d(t)*d(-3)=>d(-3)*d(t),d(t)*d(-4)=> d(-4)*d(t),abs(!#ll) =>1, delta(~f,~g) => if f equal g then 1 else 0, bf_part(~wx,~n) => part(fpart(wx),n+1), b_part(~wx,~n) => part(bpart(wx),n+1), pg(~n,~x) => sub(d(1)=0,d(1)**n*x), chan(~x) => sub(d(2)=d(1),sub(d(1)=d(2),x)), s_part(~x,~n) => coeffn(sub(der(1)=!@k,der(2)=(!@k)^2,der(3)=(!@k)^3, del(1)=!@k,del(2)=(!@k)^2,der(3)=(!@k)^3,x),!@k,n), newton(~n,~m) => factorial(n)/(factorial(m)*factorial(n-m)), prykr(~f,~g) => if g eq 1 then der(1)*f else if g eq 2 then der(2)*f, prykl(~f,~g) => if g eq 1 then f*del(1) else if g eq 2 then f*del(2)}$ %adjoint let { bos(~f,~n,~m)*stp(~x) => stp(x)*bos(f,n,m), bos(~f,~n,~m,~l)*stp(~x) => stp(x)*bos(f,n,m,l), axp(~f)*stp(~x) => stp(x)*axp(f), d(~n)*stp(~x) => -stp(x)*d(n) , der(~k)*stp(1) => stp(2)*der(k) when k neq 3, der(~k)*stp(2) => - stp(1)*der(k) when k neq 3, del(~k)*stp(1) => - del(k) when k neq 3, del(~k)*stp(2) => - stp(1)*del(k) when k neq 3, der(3)*stp(~x) => stp(x)*der(3),del(3)*stp(~x) => stp(x)*der(3), del(~x)*stp(10) => stp(20)*del(x) when x neq 3, del(~x)*stp(20) => -stp(10)*del(x) when x neq 3, fer(~f,~n,~m)*stp(1) => stp(10)*fer(f,n,m), fer(~f,~n,~m)*stp(10) => -stp(20)*fer(f,n,m), fer(~f,~n,~m)*stp(20) => stp(10)*fer(f,n,m), fer(~f,~n,~m)*stp(2) => stp(10)*fer(f,n,m)}$ %***********************************% %*** Local action ******************% %***********************************% tryk:={ d(~f)*fer(~g,~n,~m) => delta(f,g)*zen(g,n,m) + fer(g,n,m)*d(f), d(~f)*bos(~g,~n,~m) => delta(f,g)*zan(g,n,m) + bos(g,n,m)*d(f), d(~f)*bos(~g,~n,~m,~l) => l*delta(f,g)*zan(g,n,m)*bos(g,n,m,l-1)+ bos(g,n,m,l)*d(f), d(~f)*axp(~g) => axp(g)*(d(f)*g-g*d(f))+axp(g)*d(f) }$ tryk1:={zan(~f,0,~m) => (-1)**m*d(1)**m, zan(~f,3,~m) => if abra_kadabra eq 2 then (-1)**m*der(1)*der(2)*d(1)**m else if abra_kadabra eq 1 then (-1)**(m+1)*der(2)*der(1)*d(1)**m else if abra_kadabra eq 3 then (-1)**m*der(3)*d(1)**m, zan(~f,1,~m) => (-1)**m*der(1)*d(1)**m, zan(~f,2,~m) => (-1)**m*der(2)*d(1)**m, zen(~f,1,~m) => (-1)**(m+1)*der(1)*d(1)**m, zen(~f,2,~m) => (-1)**(m+1)*der(2)*d(1)**m, zen(~f,0,~m) => (-1)**m*d(1)**m, zen(~f,3,~m) => if abra_kadabra eq 2 then (-1)**m*der(1)*der(2)*d(1)**m else if abra_kadabra eq 1 then (-1)**(m+1)*der(2)*der(1)*d(1)**m else if abra_kadabra eq 3 then (-1)**m*der(3)*d(1)**m}$ tryk2:={zan(~f,0,~m) => bos(f,0,0)*(-1)**m*d(1)**m, zan(~f,3,~m) => bos(f,0,0)*( if abra_kadabra eq 2 then (-1)**m*der(1)*der(2)*d(1)**m else if abra_kadabra eq 1 then (-1)**(m+1)*der(2)*der(1)*d(1)**m else if abra_kadabra eq 3 then (-1)**m*der(3)*d(1)**m), zan(~f,1,~m) => fer(f,0,0)*(-1)**m*der(1)*d(1)**m, zan(~f,2,~m) => fer(f,0,0)*(-1)**m*der(2)*d(1)**m, zen(~f,1,~m) => bos(f,0,0)*(-1)**(m+1)*der(1)*d(1)**m, zen(~f,2,~m) => bos(f,0,0)*(-1)**(m+1)*der(2)*d(1)**m, zen(~f,0,~m) => fer(f,0,0)*(-1)**m*d(1)**m, zen(~f,3,~m) => fer(f,0,0)*( if abra_kadabra eq 2 then (-1)**m*der(1)*der(2)*d(1)**m else if abra_kadabra eq 1 then (-1)**(m+1)*der(2)*der(1)*d(1)**m else if abra_kadabra eq 3 then (-1)**m*der(3)*d(1)**m)}$ tryk3:={fer(~f,~n,~m) =>1, bos(~f,~n,~m) =>1, bos(~f,~n,~m,~l) =>1, axp(~f) => 1 }$ tryk4:={fer(~f,~n,~m) => 1, bos(~f,~n,~m) => 1, axp(~f) => 1, bos(~f,~n,~m,~l) => 1, der(~n) => 1, d(~n) => 1, del(~n) => 1}$ %only for trad tryk5:={bos(~f,~m,~n) => if m=0 then fun(mkid(f,0),n)+tet(1)*gras(mkid(f,mkid(f,1)),n)+ tet(2)*gras(mkid(f,mkid(f,2)),n)+tet(2)*tet(1)*fun(mkid(f,1),n) else if m=1 then fun(mkid(f,0),n) - tet(2)*gras(mkid(f,mkid(f,2)),n) + tet(1)*gras(mkid(f,mkid(f,1)),n+1) - tet(2)*tet(1)*fun(mkid(f,1),n+1) else if m=2 then fun(mkid(f,1),n) + tet(1)*gras(mkid(f,mkid(f,2)),n) + tet(2)*gras(mkid(f,mkid(f,1)),n+1) + tet(2)*tet(1)*fun(mkid(f,0),n+1) else if m=3 then tet(1)*gras(mkid(f,mkid(f,2)),n+1) + fun(mkid(f,1),n) - tet(2)*tet(1)*fun(mkid(f,0),n+2) - tet(2)*gras(mkid(f,mkid(f,1)),n+1) else rederr " wrong values of arguments", fer(~f,~m,~n) => if m=0 then gras(mkid(f,mkid(f,1)),n)+tet(1)*fun(mkid(f,0),n)+ tet(2)*fun(mkid(f,1),n)+tet(2)*tet(1)*gras(mkid(f,mkid(f,2)),n) else if m=1 then gras(mkid(f,mkid(f,1)),n)-tet(2)*fun(mkid(f,1),n)+ tet(1)*fun(mkid(f,0),n+1)-tet(2)*tet(1)*gras(mkid(f,mkid(f,2)),n+1) else if m=2 then gras(mkid(f,mkid(f,2)),n)+tet(1)*fun(mkid(f,1),n)+ tet(2)*fun(mkid(f,0),n+1)+tet(2)*tet(1)*gras(mkid(f,mkid(f,1)),n+1) else if m=3 then tet(1)*fun(mkid(f,1),n+1) + gras(mkid(f,mkid(f,2)),n) - tet(2)*tet(1)*gras(mkid(f,mkid(f,1)),n+2) - tet(2)*fun(mkid(f,0),n+1) else rederr "wrong values of arguments" , bos(~f,~m,~n,~l) => if m equal 0 then fun(mkid(f,0),n,l) + l*fun(mkid(f,0),n,l-1)*(tet(1)*gras(mkid(f,1),n)+ tet(2)*gras(mkid(g,2),n) +tet(2)*tet(1)*(fun(mkid(f,1),n,1)+ (l-1)*fun(mkid(f,0),n,-1)*gras(mkid(g,1),n)*gras(mkid(g,2),n))) else if m=1 then fun(mkid(f,0),n,l)+l*fun(mkid(f,0),n,l-1)*(tet(1)*gras(mkid(g,1),n+1)- tet(2)*gras(mkid(g,2),n)+tet(2)*tet(1)*(fun(mkid(f,1),n+1,1)- (l-1)*fun(mkid(f,0),n,-1)*gras(mkid(g,1),n+1)*gras(mkid(g,2),n))) else if m=2 then fun(mkid(f,1),n,l)+l*fun(mkid(f,1),n,l-1)*(tet(1)*gras(mkid(g,2),n)+ tet(2)*gras(mkid(g,1),n+1)+tet(2)*tet(1)*(fun(mkid(f,0),n+1,1) - (l-1)*fun(mkid(f,1),n,-1)*gras(mkid(g,1),n+1)*gras(mkid(g,2),n))) else if m=3 then fun(mkid(f,1),n,l)+l*fun(mkid(f,1),n,l-1)*(tet(1)*gras(mkid(g,2),n+1)- tet(2)*gras(mkid(g,1),n+1) + tet(2)*tet(1)*(-fun(mkid(f,0),n+1,1)+ (l-1)*fun(mkid(f,1),n,-1)*gras(mkid(g,1),n+1)*gras(mkid(g,2),n+1) ) ) else rederr "wrong values of arguments" , axp(~f) => axx(bf_part(f,0))*(1+ tet(1)*bf_part(f,1)+ tet(2)*bf_part(f,2) + tet(2)*tet(1)*(bf_part(f,3)+ 2*bf_part(f,1)*bf_part(f,2))) }$ tryk6:={ gras(~f,~n) =>0 }$ tryk7:={ !@f_f(~f,0,~n) => bos(f,0,n), !@f_f(~f,1,~n) => if abra_kadabra eq 2 then bos(f,1,n) else if not freeof(f_chiral,f) then 0 else bos(f,1,n), !@f_f(~f,2,~n) => if abra_kadabra eq 2 then bos(f,2,n) else if not freeof(f_antychiral,f) then 0 else bos(f,2,n), !@f_f(~f,3,~n) => if abra_kadabra eq 2 then bos(f,3,n) else if not freeof(b_chiral,f) then - bos(f,0,n+1) else if abra_kadabra eq 1 and not freeof(b_antychiral,f) then 0 else if abra_kadabra eq 3 and not freeof(b_antychiral,f) then bos(f,0,n+1) else bos(f,3,n), !@g_g(~f,0,~n) => fer(f,0,n), !@g_g(~f,1,~n) => if abra_kadabra eq 2 then fer(f,1,n) else if not freeof(b_chiral,f) then 0 else fer(f,1,n), !@g_g(~f,2,~n) => if abra_kadabra eq 2 then fer(f,2,n) else if not freeof(b_antychiral,f) then 0 else fer(f,2,n), !@g_g(~f,3,~n) => if abra_kadabra eq 2 then fer(f,3,n) else if not freeof(f_chiral,f) then -fer(f,0,n+1) else if abra_kadabra eq 1 and not freeof(f_antychiral,f) then 0 else if abra_kadabra eq 3 and not freeof(f_antychiral,f) then fer(f,0,n+1) else fer(f,3,n)}$ tryk8:={ bos(~f,~n,~m) => berz(f,n,m)+eps*ber(f,n,m), fer(~f,~n,~m) => firr(f,n,m)+eps*fir(f,n,m), bos(~f,~n,~m,~l) => berz(f,n,m,l)+l*eps*berz(f,n,m,l-1)*ber(f,n,m)}$ tryk9:={ berz(~f,~n,~m) => bos(f,n,m), firr(~f,~n,~m) => fer(f,n,m), berz(~f,~n,~m,~l) => bos(f,n,m,l)}$ tryk10:= { fir(~f,~n,~m) => pg(m,pr(n,bos(f))), ber(~f,~n,~m) => pg(m,pr(n,bos(f)))}$ tryk11:= { !#a(~n) => !#aa(n), !#b(~n) => !#bb(n), !#c(~n) => !#cc(n) }$ tryk12:= { !#aa(~n) => !#b(n), !#bb(~n) => !#c(n), !#cc(~n) => !#a(n) }$ tryk13:= { !#aa(~n) => !#c(n), !#bb(~n) => !#a(n), !#cc(~n) => !#b(n) }$ tryk14:={ bos(~f,~n,~m,t,t) => pg(m,pr(n,bos(f,t))), fer(~f,~n,~m,t) => pg(m,pr(n,bos(f,t))) }$ tryk15:={ bos(~f,~n,~m,~l) => if n equal 0 or n equal 3 then berz(f,n,m,l) else if n equal 1 then (-1)**l*berz(f,2,m,l) else if n equal 2 then berz(f,1,m,l), bos(~f,~n,~m) => if n equal 0 or n equal 3 then berz(f,n,m) else if n equal 1 then -berz(f,2,m) else if n equal 2 then berz(f,1,m), fer(~f,~n,~m) => if n equal 0 or n equal 3 then firr(f,n,m) else if n equal 1 then -firr(f,2,m) else if n equal 2 then firr(f,1,m) }$ %only for chiral tryk16:={ bos(~f,0,~n) => if not freeof(b_chiral,f) then fun(mkid(f,0),n)+tet(2)*gras(mkid(f,mkid(f,2)),n)- tet(2)*tet(1)*fun(mkid(f,0),n+1)/2 else if not freeof(b_antychiral,f) then fun(mkid(f,0),n)+tet(1)*gras(mkid(f,mkid(f,1)),n) + tet(2)*tet(1)*fun(mkid(f,0),n+1)/2 else fun(mkid(f,0),n)+tet(1)*gras(mkid(f,mkid(f,1)),n)+ tet(2)*gras(mkid(f,mkid(f,2)),n)+tet(2)*tet(1)*fun(mkid(f,1),n), bos(~f,1,~n) => if not freeof(f_chiral,f) then 0 else if not freeof(f_antychiral,f) then fun(mkid(f,0),n) - tet(2)*gras(mkid(f,mkid(f,1)),n+1) - tet(2)*tet(1)*fun(mkid(f,0),n+1)/2 else fun(mkid(f,0),n) - tet(2)*gras(mkid(f,mkid(f,2)),n) - tet(2)*gras(mkid(f,mkid(f,1)),n+1)/2 - tet(2)*tet(1)*fun(mkid(f,0),n+1)/2, bos(~f,2,~n) => if not freeof(f_chiral,f) then fun(mkid(f,0),n) - tet(2)*gras(mkid(f,mkid(f,1)),n+1) - tet(2)*tet(1)*fun(mkid(f,0),n+1)/2 else if not freeof(f_antychiral,f) then 0 else fun(mkid(f,1),n) + tet(1)*gras(mkid(f,mkid(f,2)),n) - tet(1)*gras(mkid(f,mkid(f,1)),n+1) + tet(2)*tet(1)*fun(mkid(f,1),n+1)/2, bos(~f,3,~n) => if abra_kadabra eq 1 then if not freeof(b_chiral,f) then - bos(f,0,n+1) else if not freeof(b_antychiral,f) then 0 else fun(mkid(f,1),n) - fun(mkid(f,0),n+1)/2 - tet(2)*gras(mkid(f,mkid(f,2)),n+1) - tet(2)*tet(1)*fun(mkid(f,1),n+1)/2 + tet(2)*tet(1)*fun(mkid(f,0),n+2)/4 else if abra_kadabra eq 3 then if not freeof(b_chiral,f) then - bos(f,0,n+1) else if not freeof(b_antychiral,f) then bos(f,0,n+1) else 2*fun(mkid(f,1),n) - tet(2)*gras(mkid(f,mkid(f,2)),n+1) + tet(1)*gras(mkid(f,mkid(f,1)),n+1)+tet(2)*tet(1)*fun(mkid(f,0),n+2)/2, bos(~f,0,~n,~k) => if not freeof(b_chiral,f) then fun(mkid(f,0),n,k)+k*tet(2)*fun(mkid(f,0),n,k-1)* (gras(mkid(f,mkid(f,2)),n) -tet(1)*fun(mkid(f,0),n+1,1)/2) else if not freeof(b_antychiral,f) then fun(mkid(f,0),n,k)+k*tet(1)*fun(mkid(f,0),n,k-1)* (gras(mkid(f,mkid(f,1)),n) - tet(2)*fun(mkid(f,0),n+1,1)/2) else fun(mkid(f,0),n,k)+ k*tet(1)*gras(mkid(f,mkid(f,1)),n)*fun(mkid(f,0),n,k-1)+ k*tet(2)*gras(mkid(f,mkid(f,2)),n)*fun(mkid(f,0),n,k-1)+ +tet(2)*tet(1)*(k*fun(mkid(f,1),n,1)*fun(mkid(f,0),n,k-1)+ k*(k-1)*gras(mkid(f,mkid(f,1)),n)*gras(mkid(f,mkid(f,2)),n)* fun(mkid(f,0),n,k-2)), bos(~f,1,~n,~k) => if not freeof(f_chiral,f) then 0 else if not freeof(f_antychiral,f) then fun(mkid(f,0),n,k) - k*fun(mkid(f,0),n,k-1)*tet(2)*( gras(mkid(f,mkid(f,1)),n+1) + tet(1)*fun(mkid(f,0),n+1,1)/2) else fun(mkid(f,0),n,k) -k*fun(mkid(f,0),n,k-1)*tet(2)* (gras(mkid(f,mkid(f,2)),n) + gras(mkid(f,mkid(f,1)),n+1)/2 + tet(1)*fun(mkid(f,0),n+1,1)/2), bos(~f,2,~n,~k) => if not freeof(f_chiral,f) then fun(mkid(f,0),n,k) - k*tet(2)*fun(mkid(f,0),n,k-1)* (gras(mkid(f,mkid(f,1)),n+1) + tet(1)*fun(mkid(f,0),n+1)/2) else if not freeof(f_antychiral,f) then 0 else fun(mkid(f,1),n,k) + k*tet(1)*fun(mkid(f,1),n,k-1)* (gras(mkid(f,mkid(f,2)),n) - gras(mkid(f,mkid(f,1)),n+1) - tet(2)*fun(mkid(f,1),n+1,1)/2), bos(~f,3,~n,~k) => if abra_kadabra eq 1 then if not freeof(b_chiral,f) then (-1)**k*bos(f,0,n+1,k) else if not freeof(b_antychiral,f) then 0 else fun(mkid(f,1),n,mkid(f,0),n+1,k) - k*fun(mkid(f,1),n,mkid(f,0),n+1,k-1)* (tet(2)*gras(mkid(f,mkid(f,2)),n+1) + tet(2)*tet(1)*fun(mkid(f,1),n+1,1)/2 - tet(2)*tet(1)*fun(mkid(f,0),n+2,1)/4) else if abra_kadabra eq 3 then if not freeof(b_chiral,f) then (-1)**k*bos(f,0,n+1,k) else if not freeof(b_antychiral,f) then bos(f,0,n+1,k) else 2**k*fun(mkid(f,1),n,k) + k*2**(k-1)*fun(mkid(f,1),n,k-1)* (- tet(2)*gras(mkid(f,mkid(f,2)),n+1) + tet(1)*gras(mkid(f,mkid(f,1)),n+1)+tet(2)*tet(1)*fun(mkid(f,0),n+2,1)/2) -k*(k-1)*2**(k-2)*tet(2)*tet(1)*fun(mkid(f,1),n,k-2)* gras(mkid(f,mkid(f,1)),n+1)*gras(mkid(f,mkid(f,2)),n+1), fer(~f,0,~n) => if not freeof(f_chiral,f) then gras(mkid(f,mkid(f,1)),n)+tet(2)*fun(mkid(f,1),n) - tet(2)*tet(1)*gras(mkid(f,mkid(f,1)),n+1)/2 else if not freeof(f_antychiral,f) then gras(mkid(f,mkid(f,1)),n)+tet(1)*fun(mkid(f,0),n)+ tet(2)*tet(1)*gras(mkid(f,mkid(f,1)),n+1)/2 else gras(mkid(f,mkid(f,1)),n)+tet(1)*fun(mkid(f,0),n)+ tet(2)*fun(mkid(f,1),n)+tet(2)*tet(1)*gras(mkid(f,mkid(f,2)),n), fer(~f,1,~n) => if not freeof(b_chiral,f) then 0 else if not freeof(b_antychiral,f) then gras(mkid(f,mkid(f,1)),n) - tet(2)*fun(mkid(f,0),n+1) - tet(2)*tet(1)*gras(mkid(f,mkid(f,1)),n+1)/2 else gras(mkid(f,mkid(f,1)),n) - tet(2)*fun(mkid(f,1),n)- tet(2)*fun(mkid(f,0),n+1)/2 - tet(2)*tet(1)*gras(mkid(f,mkid(f,1)),n+1)/2, fer(~f,2,~n) => if not freeof(b_chiral,f) then gras(mkid(f,mkid(f,2)),n) -tet(1)*fun(mkid(f,0),n+1)+ tet(2)*tet(1)*gras(mkid(f,mkid(f,2)),n+1)/2 else if not freeof(b_antychiral,f) then 0 else gras(mkid(f,mkid(f,2)),n)+tet(1)*fun(mkid(f,1),n) - tet(1)*fun(mkid(f,0),n+1)/2 + tet(2)*tet(1)*gras(mkid(f,mkid(f,2)),n+1)/2, fer(~f,3,~n) => if abra_kadabra eq 1 then if not freeof(f_chiral,f) then - fer(f,0,n+1) else if not freeof(f_antychiral,f) then 0 else gras(mkid(f,mkid(f,2)),n) - gras(mkid(f,mkid(f,1)),n+1)/2 - tet(2)*fun(mkid(f,1),n+1) - tet(2)*tet(1)*gras(mkid(f,mkid(f,1)),n+2)/4 - tet(2)*tet(1)*gras(mkid(f,mkid(f,2)),n+1)/2 else if abra_kadabar eq 3 then if not freeof(f_chiral,f) then - fer(f,0,n+1) else if not freeof(f_antychiral,f) then fer(f,0,n+1) else 2*gras(mkif(f,mkid(f,2)),n) - tet(2)*fun(mkid(f,1),n+1) + tet(1)*fun(mkid(f,0),n+1) +tet(2)*tet(1)*gras(mkid(f,mkid(f,1)),n+2)/2, axp(~f) => axx(bf_part(f,0))*(1+ tet(1)*bf_part(f,1)+ tet(2)*bf_part(f,2) + tet(2)*tet(1)*(bf_part(f,3)+ 2*bf_part(f,1)*bf_part(f,2))) }$ %***********************************% %*** module - operators ***********% %***********************************% %differentations let { d(1)*fer(~f,~n,~m) => fer(f,n,m+1)+fer(f,n,m)*d(1), d(1)*bos(~f,~n,~m) => bos(f,n,m+1)+bos(f,n,m)*d(1), fer(~f,~n,~m)*d(2) => -fer(f,n,m+1)+d(2)*fer(f,n,m), bos(~f,~n,~m)*d(2) => -bos(f,n,m+1)+d(2)*bos(f,n,m), d(1)*bos(~f,~n,~m,~l) => l*bos(f,n,m+1,1)*bos(f,n,m,l-1)+bos(f,n,m,l)*d(1), bos(~f,~n,~m,~l)*d(2) => -l*bos(f,n,m+1,1)*bos(f,n,m,l-1)+d(2)*bos(f,n,m,l), der(~k)*fer(~f,0,~m) => bos(f,k,m)-fer(f,0,m)*der(k) when numberp k and k < 3, der(~k)*bos(~f,0,~m) => fer(f,k,m)+bos(f,0,m)*der(k) when numberp k and k < 3, fer(~f,0,~m)*del(~k) => bos(f,k,m)-del(k)*fer(f,0,m) when numberp k and k < 3, bos(~f,0,~m)*del(~k) => -fer(f,k,m)+del(k)*bos(f,0,m) when numberp k and k < 3, der(~k)*bos(~f,0,~m,~l) => l*fer(f,k,m)*bos(f,0,m,l-1)+bos(f,0,m,l)*der(k) when numberp k and k < 3, bos(~f,0,~m,~l)*del(~k) => -l*fer(f,k,m)*bos(f,0,m,l-1)+del(k)*bos(f,0,m,l) when numberp k and k < 3, d(1)*axp(~g) => pg(1,g)*axp(g)+axp(g)*d(1), der(1)*axp(~g) => pr(1,g)*axp(g)+axp(g)*der(1), der(2)*axp(~g) => pr(2,g)*axp(g)+axp(g)*der(2), axp(~g)*d(2) => -pg(1,g)*axp(g)+d(2)*axp(g), axp(~g)*del(1) => -pr(1,g)*axp(g)+del(1)*axp(g), axp(~g)*del(2) => -pr(2,g)*axp(g)+del(2)*axp(g), d(1)*fun(~f,~m) => fun(f,m+1)+fun(f,m)*d(1), fun(~f,~m)*d(2) => -fun(f,m+1)+d(2)*fun(f,m), d(1)*fun(~f,~n,~m) => m*fun(f,n+1,1)*fun(f,n,m-1)+fun(f,n,m)*d(1), fun(~f,~n,~m)*d(2) => -m*fun(f,n+1,1)*fun(f,n,m-1)+d(2)*fun(f,n,m), gras(~f,~m)*d(2) => -gras(f,m+1)+d(2)*gras(f,m), d(1)*gras(~f,~m) => gras(f,m+1)+gras(f,m)*d(1), d(1)*axx(~f) => pg(1,f)*axx(f)+axx(f)*d(1), axx(~f)*d(2) => -pg(1,f)*axx(f)+d(2)*axx(f)}$ %integrations; let { d(-1)*fer(~f,~n,~m) => if numberp(ww) then for k:=0:ww-1 sum (-1)**k*fer(f,n,m+k)*d(-1)**(k+1) else rederr "introduce the precision e.g. give the value of ww > 0", fer(~f,~n,~m)*d(-2) => if numberp(ww) then for k:=0:ww-1 sum d(-2)**(k+1)*fer(f,n,m+k) else rederr "introduce the precision e.g. give the value of ww > 0", d(-1)*bos(~f,~n,~m) => if numberp(ww) then for k:=0:ww-1 sum (-1)**k*bos(f,n,m+k)*d(-1)**(k+1) else rederr "introduce the precision e.g. give the value of ww > 0", bos(~f,~n,~m)*d(-2) => if numberp(ww) then for k:=0:ww-1 sum d(-2)**(k+1)*bos(f,n,m+k) else rederr "introduce the precision e.g. give the value of ww > 0", d(-1)*bos(~f,~n,~m,~l) => if numberp(ww) then for k:=0:ww-1 sum (-1)**k*pg(k,bos(f,n,m,l))*d(-1)**(k+1) else rederr "introduce the precision e.g. give the value of ww > 0", bos(~f,~n,~m,~l)*d(-2) => if numberp(ww) then for k:=0:ww-1 sum d(-2)**(k+1)*pg(k,bos(f,n,m,l)) else rederr "introduce the precision e.g. give the value of ww > 0", d(-1)*axp(~f) => if numberp(ww) then for k:=0:ww-1 sum (-1)**k*pg(k,axp(f))*d(-1)**(k+1) else rederr "introduce the precision e.g. give the value of ww > 0", axp(~f)*d(-2) => if numberp(ww) then for k:=0:ww-1 sum d(-2)**(k+1)*pg(k,axp(f)) else rederr "introduce the precision e.g. give the value of ww > 0", %acceleration; dr(~x)*bos(~f,~n,~m) => if numberp(ww) then for s:=0:ww sum (-1)**s*newton(-x+s-1,-x-1)*bos(f,n,m+s)*dr(x-s) else rederr "introduce the precision e.g. give the value of ww > 0", dr(~x)*fer(~f,~n,~m) => if numberp(ww) then for s:=0:ww sum (-1)**s*newton(-x+s-1,-x-1)*fer(f,n,m+s)*dr(x-s) else rederr "introduce the precision e.g. give the value of ww > 0", dr(~x)*bos(~f,~n,~m,~l) => if numberp(ww) then for s:=0:ww sum (-1)**s*newton(-x+s-1,-x-1)*pg(s,bos(f,n,m,l))*dr(x-s) else rederr "introduce the precision e.g. give the value of ww > 0", dr(~x)*fun(~f,~n) => if numberp(ww) then for s:=0:ww sum (-1)**s*newton(-x+s-1,-x-1)*fun(f,n+s)*dr(x-s) else rederr "introduce the precision e.g. give the value of ww > 0", dr(~x)*gras(~f,~n) => if numberp(ww) then for s:=0:ww sum (-1)**s*newton(-x+s-1,-x-1)*gras(f,n+s)*dr(x-s) else rederr "introduce the precision e.g. give the value of ww > 0", dr(~x)*fun(~f,~n,~l) => if numberp(ww) then for s:=0:ww sum (-1)**s*newton(-x+s-1,-x-1)*pg(s,fun(f,n,l))*dr(x-s) else rederr "introduce the precision e.g. give the value of ww > 0", %classical d(-1)*fun(~f,~n,~m) => if numberp(ww) then for k:=0:ww-1 sum (-1)**k*pg(k,fun(f,n,m))*d(-1)**(k+1) else rederr "introduce the precision e.g. give the value of ww > 0", fun(~f,~n,~m)*d(-2) => if numberp(ww) then for k:=0:ww-1 sum d(-2)**(k+1)*pg(k,fun(f,n,m)) else rederr "introduce the precision e.g. give the value of ww > 0", d(-1)*fun(~f,~n) => if numberp(ww) then for k:=0:ww-1 sum (-1)**k*fun(f,n+k)*d(-1)**(k+1) else rederr "introduce the precision e.g. give the value of ww > 0", fun(~f,~n)*d(-2) => if numberp(ww) then for k:=0:ww-1 sum d(-2)**(k+1)*fun(f,n+k) else rederr "introduce the precision e.g. give the value of ww > 0", d(-1)*gras(~f,~n) => if numberp(ww) then for k:=0:ww-1 sum (-1)**k*gras(f,n+k)*d(-1)**(k+1) else rederr "introduce the precision e.g. give the value of ww > 0", gras(~f,~n)*d(-2) => if numberp(ww) then for k:=0:ww-1 sum d(-2)**(k+1)*gras(f,n+k) else rederr "introduce the precision e.g. give the value of ww > 0", d(-1)*axx(~f) => if numberp(ww) then for k:=0:ww-1 sum (-1)**k*pg(k,axx(f))*d(-1)**(k+1) else rederr "introduce the precision e.g. give the value of ww > 0", axx(~f)*d(-2) => if numberp(ww) then for k:=0:ww-1 sum d(-2)**(k+1)*pg(k,axx(f)) else rederr "introduce the precision e.g. give the value of ww > 0" }$ %other time let { d(t)*axp(~f) => axp(f)*d(t)*f, d(t)*bos(~f,~n,~m) => bos(f,n,m,t,t) + bos(f,n,m)*d(t), d(t)*fer(~f,~n,~m) => fer(f,n,m,t) +fer(f,n,m)*d(t), d(t)*bos(~f,~n,~m,~l) => l*bos(f,n,m,l-1)*bos(f,n,m,t,t)+bos(f,n,m,l)*d(t) }$ %******************************************% %*** module - actions ********************% %******************************************% procedure rzut(rr,n); begin scalar ola,ewa; ola:=chan(rr); ewa:=sub(d(-1)=0,d(-2)=0,d(-3)=0,d(-4)=0,ola); if n = 0 then return ewa; if n = 1 then ewa:=ewa-sub(der(1)=0,der(2)=0,der(3)=0,d(1)=0,ewa); if n = 2 then ewa:=ewa-sd_part(ewa,0,0)-sd_part(ewa,1,0)*der(1)- sd_part(ewa,2,0)*der(2)-sd_part(ewa,0,1)*d(1); return ewa end$ procedure sd_part(wr,n,m); begin scalar ewa,ola; ewa:=sub(d(1)=!@kk,d(2)=!@kk,d(-2)=!@ss,d(-1)=!@ss, d(-3)=!@ss*d(-33),d(-4)=!@ss*d(-44),wr); ola:=if m greaterp 0 then coeffn(ewa,!@kk,m) else if m equal 0 then sub(!@ss=0,!@kk=0,ewa) else coeffn(ewa,!@ss,-m); return s_part(sub(!@ss=1,!@kk=1,d(-33)=d(-3),d(-44)=d(-4),ola),n) end$ procedure d_part(ww,n); begin scalar ewa,ola; ewa:=sub(d(1)=!@kk,d(2)=!@kk,d(-1)=!@ss,d(-2)=!@ss, d(-3)=!@ss*d(-33),d(-4)=d(-44)*!@ss,ww); ola:=if n greaterp 0 then coeffn(ewa,!@kk,n) else if n=0 then sub(!@kk=0,!@ss=0,ewa) else coeffn(ewa,!@ss,-n); return sub(d(-33)=d(-3),d(-44)=d(-4),ola) end$ procedure pr(n,ww); begin scalar ewa; if n=0 then ewa:=ww; if n=1 then ewa:=der(1)*ww; if n=2 then ewa:=der(2)*ww; if n=3 then if abra_kadabra eq 3 then ewa:=der(3)*ww else ewa:=der(1)*pr(2,ww); return sub(der(1)=0,der(2)=0,der(3)=0,ewa) end$ %*********************************% %*** module adjoint **********% %*********************************% % stp(1),stp(10),stp(20) if does not appeare der or apeapare der(1)*der(2); % stp(2) if appeare der; !@rak:={ stp(1)=1, stp(10)=1, stp(20)=1, stp(2)= - 1 }$ procedure cp(xwx); begin scalar kap,kap1,ess,k,l; if xwx equal 0 then return 0;kap:=length(xwx); if numberp(kap) then return cp1(xwx); kap1:=first kap; matrix !@z_z(kap1,kap1);matrix !@s_s(kap1,kap1); for k:=1:kap1 do for l:=1:kap1 do << ess:=sub(!@krr=1,!@krr*xwx(k,l));!@z_z(k,l):=cp1(ess); clear ess; >>; clear !@krr;!@s_s:=tp(!@z_z);clear !@z_z; return !@s_s end$ procedure cp1(yyz); begin scalar ewa,ola,xx,yyy; if yyz equal 0 then return 0; yyy:=if length(yyz) equal 1 and arglength(yyz) equal -1 then !@*yyz else yyz; factor d,der,del; ewa:=lyst(yyy); ola:=for each xx in ewa collect begin scalar mew,wem,em1,em2,em,em3,licz,mian; licz:=num(xx);mian:=den(xx); if numberp(licz) then return xx; mew:=licz*stp(1);wem:=sub(!@rak,mew);em:=if part(wem,0) equal minus then -1 else 1; em1:= cp2(em*wem); em2:=part(reverse(em1),0):=*; return em2*em/mian end; remfac d,der,del; return sub(!@=1,part(ola,0):=+) end$ procedure cp2(zz); begin scalar ewa,ola,ela,el1; if arglength(@*zz) equal 2 then return {zz}; ewa:=(zz where tryk4); ola:=zz/ewa; ela:=if arglength(!@*ola) equal 2 then {ola} else part(ola,0):=list; el1:=append({ewa},ela); return el1 end$ %************************************% %*** module O(2) invariance ********% %************************************% procedure odwa(wx); begin scalar ewa,ola; let tryk15; ewa:=sub(der(1)=-der(20),der(2)=der(10), del(1)=-del(20),del(2)=del(10),wx); clearrules tryk15; let tryk9; ola:=ewa; clearrules tryk9; return sub(der(10)=der(1),der(20)=der(2),del(10)=del(1),del(20)=del(2),ola) end$ %************************************% %*** module - coefficients **********% %************************************% procedure lyst(wx); begin scalar ewa,ola,kap,kap1,adam; if wx=0 then return {0}; factor d,der,del;kap:=length(wx);kap1:=arglength(wx); if kap equal 1 and kap1 equal -1 then return {wx}; if kap1>kap then return {wx}; on div; ewa:=wx; if part(ewa,0) = plus then adam:=part(ewa,0):=list else adam:={ewa};off div; remfac d,der,del;return adam; end$ procedure lyst1(wy); begin scalar ewa,ola; ewa:=lyst(wy); ola:=(ewa where tryk3);return ola end$ procedure lyst2(wy); begin scalar ewa,ola; ewa:=lyst(wy); ola:=(ewa where tryk4);return ola end$ %************************************% %*** module - gradients *************% %************************************% procedure war(wa,f); begin scalar ewa,ola,adam,mew; let tryk; ewa:=d(f)*wa-wa*d(f); clearrules tryk; ola:=(ewa where tryk1); ewa:=sub(d(1)=0,der(1)=0,der(2)=0,der(3)=0,ola); if ewa=0 then return 0; adam:=lyst(ewa); mew:=(adam where tryk3); return if mew equal 0 then {} else mew end$ procedure dyw(wa,f); begin scalar ewa,ola; ewa:=(d(f)*wa-wa*d(f) where tryk); ola:=(ewa where tryk2); ewa:=sub(d(1)=0,der(1)=0,der(2)=0,der(3)=0,ola); if ewa=0 then return 0; return lyst(ewa) end$ procedure gra(wa,f); begin scalar ewa,ola; ewa:=(d(f)*wa-wa*d(f) where tryk); ola:=(ewa where tryk1); return sub(d(1)=0,der(1)=0,der(2)=0,der(3)=0,ola) end$ %***************************************% %*** module - coordinates **************% %***************************************% procedure fpart(wx); begin scalar ewa,ola,adam; ewa:=if abra_kadabra eq 2 then (wx where tryk5) else (wx where tryk16); ola:=sub(tet(1)=!#qw,tet(2)=!#qq,ewa); adam:= {coeffn(coeffn(ola,!#qw,0),!#qq,0), coeffn(coeffn(ola,!#qw,1),!#qq,0), coeffn(coeffn(ola,!#qw,0),!#qq,1), coeffn(coeffn(ola,!#qw,1),!#qq,1)}; return adam end$ procedure bpart(wx); begin scalar ewa,ola,adam; ewa:=if abra_kadabra eq 2 then (wx where tryk5) else (wx where tryk16); let tryk6; ola:=sub(tet(1)=!#qw,tet(2)=!#qq,ewa);clearrules tryk6; adam:= {coeffn(coeffn(ola,!#qw,0),!#qq,0), coeffn(coeffn(ola,!#qw,1),!#qq,0), coeffn(coeffn(ola,!#qw,0),!#qq,1), coeffn(coeffn(ola,!#qw,1),!#qq,1)}; return adam end$ %******************************************% %*** module combinations ******************% %******************************************% procedure koza(wx,wi,wn); begin scalar ew1,ew2,am; ew3:=part(wx,3);ew1:=part(wx,1); am:= if ew3 eq f and wi eq 0 or ew3 eq f and wi eq 3 then !@x_y*!@g_g(ew1,wi,wn) else if ew3 eq f and wi eq 1 or ew3 eq f and wi eq 2 then !@x_y**2*!@f_f(ew1,wi,wn) else if ew3 eq b and wi eq 0 or ew3 eq b and wi eq 3 then !@x_y**2*!@f_f(ew1,wi,wn) else if ew3 eq b and wi eq 1 or ew3 eq b and wi eq 2 then !@x_y*!@g_g(ew1,wi,wn); return am end$ procedure w_comb(as,m,a,bb); begin scalar kap,ewa,ola,wic,wid,wod,wod1,wx,k,s,!*precise; kap:=length(as); if m = 0 then return 0;if m = 0.5 then return w_comb1(as,a,bb); (!#l)^(m+1):=0;(!#l)^(m+1/2):=0; ewa:=for s:=0:floor(m) sum for k:=1:kap sum (!#l)^(part(as,k,2)+s)*(koza(part(as,k),0,s)*2+ (!#l)*koza(part(as,k),3,s)+ (!#l)^(1/2)*koza(part(as,k),1,s)+ (!#l)^(1/2)*koza(part(as,k),2,s)); ola:=ewa;wic:=ewa; for k:=0:floor(m) do << ola:=ewa*ola; ewa:=for s:=0:m-k+1 sum for r:=1:kap sum (!#l)^(part(as,r,2)+s)*(koza(part(as,r),0,s)*2+ (!#l)*koza(part(as,r),3,s)+ (!#l)^(1/2)*koza(part(as,r),1,s)+ (!#l)^(1/2)*koza(part(as,r),2,s)); wic:=wic+ola;>>; wid:=sub((!#l)=(!#ll)^2,wic); wod:=coeffn(wid,(!#ll),2m); wod:=if bb eq b then sub(!@x_y=0,wod) else if bb eq f then coeffn(wod,!@x_y,1); wod1:=lyst((wod where tryk7)); clear (!#l)^(m+1),(!#l)^(m+1/2); kap:=length(wod1);ewa:=0; for k:=1:kap do <<adam:= if wod1 eq 0 then 0 else part(wod1,k); ola:=if adam eq 0 then 0 else if part(adam,0)=minus then -adam else adam; wx:=if ola eq 0 then 1 else if part(ola,0)=times then if fixp(part(ola,1)) then part(ola,1) else 1 else 1; ewa:=ewa+mkid(a,k)*ola/wx;>>; return ewa end$ procedure w_comb1(as,a,bb); begin scalar ew,kap1,ew1,kap; kap:=length(as); ew:=for n:=1:kap join if part(as,n,2) neq 1/2 then {} else if part(as,n,3) eq f and bb eq f then {fer(part(as,n,1),0,0)} else if part(as,n,3) eq f and bb eq b then {} else if part(as,n,3) eq b and bb eq f then {} else if part(as,n,3) eq b and bb eq b then {bos(part(as,n,1),0,0)}; kap1:=length(ew); ew1:=if kap1 eq 0 then 0 else for n:=1:kap1 sum mkid(a,n)*part(ew,n); return ew1 end$ procedure fcomb(as,n,b,bb); begin scalar ewa,ola,ala,k,kap,wx,wy,kap1,wod,wod1,ema,wz,wz1; operator b;ewa:=w_comb(as,n,a,bb);kap:=length(as);ala:={}; wz:=ewa;wz1:=ewa;ema:={}; for k:=1:kap do << wz:=sub(part(as,k,1)=0,wz); wx:=wz1-wz; wz1:=wz;ema:=append(ema,{{wx}});>>; for k:=1:kap do << wx:=dyw(part(ema,k,1),part(as,k,1)); wy:=if wx=0 then {} else wx; ala:=append(ala,wy);>>; kap1:=length(ala); ewa:=0; for k:=1:kap1 do << wod:=part(ala,k); wod1:=(wod where tryk3);ewa:=ewa+b(k)*wod/wod1;>>; return ewa end$ procedure pse_ele(n,ww,ss); begin scalar ewa,ola,kap,k,maj,maj1,ela; ewa:=0; operator ss; for k:=1:n do << ewa:=ewa+ (w_comb(ww,k,mkid(mkid(a,k),a),b)+ w_comb(ww,k-1/2,mkid(mkid(a,k),b),f)*der(1) + w_comb(ww,k-1/2,mkid(mkid(a,k),c),f)*der(2)+ w_comb(ww,k-1,mkid(mkid(a,k),d),b)*(if abra_kadabra eq 3 then der(3) else der(1)*der(2)))*d(1)**(n-k);>>; remfac fer,bos; kap:=length(ewa); ola:=0; for k:=1:kap do << maj:=if ewa eq 0 then 0 else if kap equal 1 then ewa else part(ewa,k); maj1:=if maj eq 0 then 1 else (maj where tryk4); ola:=ola+ss(k+1)*maj/maj1;>>; ela:=ss(0)*d(1)**n+ss(1)*(if abra_kadabra eq 3 then der(3) else der(1)*der(2))*d(1)**(n-1)+ola;factor fer,bos; if abra_kadabra eq 3 then ela:=(ela where {der(1)*der(2) => der(3)}); return ela end$ %********************************************% %*** module jacobi *************************% %********************************************% %wim as {bos(f)=>expression}; procedure n_gat(pp,wim); begin scalar kap,niech,zyje; kap:=length(wim); niech:=gato(pp); let wim;zyje:=(niech where tryk10); clearrules wim; return zyje end$ procedure gato(p); begin scalar as,ess,mess; if numberp(length(p)) then return gat1(p); as:=first length(p); matrix !#zz(as,as); for k:=1:as do for l:=1:as do << ess:=(!#zab*p(k,l) where tryk8); mess:=(ess/!#zab where tryk9);!#zz(k,l):=sub(eps=0,df(mess,eps));>>; return !#zz end$ procedure gat1(p); begin scalar ess,zz,mess; ess:=(p where tryk8); mess:=(ess where tryk9); zz:=sub(eps=0,df(mess,eps)); return zz end$ % p is a hamiltonian operator in d,der; % w is a list of functions with the ordering such way that %first corresponds to the (1,1) element of p corresponds to {f,f}; % m is a list of components of the test vecor functions; procedure fjacob(p,w); begin scalar as,as1,es1,wod0,wod1,wod2,wod3,wodx; if numberp(length(p)) then return jacob1(p,w); as:=first length(p); matrix !#ala(as,as),!#ela(as,as); !#ala:=gato(p); operator !#a,!#b,!#c; as1:=for k:=1:as collect for l:=1:as sum sub(d(1)=0,der(1)=0,der(2)=0,p(k,l)*bos(!#b(l),0,0)); for k:=1:as do << bos(part(w,k)):=part(as1,k);>>; !#ela:=(!#ala where tryk10); wod:=for k:=1:as sum for l:=1:as sum bos(!#c(k),0,0)*!#ela(k,l)*bos(!#a(l),0,0); wod1:=sub(d(1)=0,der(1)=0,der(2)=0,wod); %permutation; wodx:=(wod1 where tryk11); wod2:=(wodx where tryk12); wod3:=(wodx where tryk13); return wod1+wod2+wod3 end$ procedure jacob(p,w,m); begin scalar woda; woda:=for wx:=1:3 sum begin scalar trys,as1,as,wod; operator !#a,!#b,!#c,!@a,!@b,!@c; as:=first length(p); trys:=for k:=1:as join { !@a(k)=if k equal first(m) then !#a(k) else 0, !@b(k)=if k equal second(m) then !#b(k) else 0, !@c(k)=if k equal third(m) then !#c(k) else 0}; let trys; matrix !@ala(as,as),!@ela(as,as); !@ala:=gato(p); as1:=for k:=1:as collect for l:=1:as sum sub(d(1)=0,der(1)=0,der(2)=0,p(k,l)*bos( if wx eq 1 then !@b(l) else if wx eq 2 then !@c(l) else if wx eq 3 then !@a(l),0,0)); for k:=1:as do << bos(part(w,k)):=part(as1,k);>>; !@ela:=(!@ala where tryk10); wod:=for k:=1:as sum for l:=1:as sum bos(if wx eq 1 then !@c(k) else if wx eq 2 then !@a(k) else if wx eq 3 then !@b(k),0,0)*!@ela(k,l)* bos(if wx eq 1 then !@a(l) else if wx eq 2 then !@b(l) else if wx eq 3 then !@c(l),0,0); for k:=1:as do clear !@a(k),!@b(k),!@c(k),bos(part(w,k)); return sub(d(1)=0,der(1)=0,der(2)=0,wod) end; return woda end$ procedure jacob1(p,w); begin scalar ala,ela,wod,wod1,wod2,wodx,ewa; ala:=gat1(p);bos(w):=sub(d(1)=0,der(1)=0,der(2)=0,p*bos(!#b,0,0)); let tryk10;ela:=ala;clearrules tryk10;wod:=bos(!#c,0,0)*ela*bos(!#a,0,0); wod1:=sub(d(1)=0,der(1)=0,der(2)=0,wod); %permutation; wodx:=sub(!#a=!#aa,!#b=!#bb,!#c=!#cc,wod1); wod2:=sub(!#aa=!#b,!#bb=!#c,!#cc=!#a,wodx); wod3:=sub(!#aa=!#c,!#bb=!#a,!#cc=!#b,wodx); clear bos(w); return wod1+wod2+wod3 end$ %************************************************ %********* module macierz *********************** %************************************************ %wx pse_ele %xx f (fermion) or b (boson) %yy bosonic part or fermionic part procedure macierz(wx,xx,yy); begin scalar ewa,ola,ew1,ew2; matrix !@z_z_x(4,4); ewa:=if xx eq f then sub(der(1)=0,der(2)=0,der(3)=0,d(1)=0,wx*fer(!#z_z,0,0)) else if xx eq b then sub(der(1)=0,der(2)=0,der(3)=0,d(1)=0,wx*bos(!#z_z,0,0)) else rederr "wrong value of second argument which should be b or f"; ola:=if yy eq b then bpart(ewa) else if yy eq f then fpart(ewa) else rederr "wrong value of third argument which should be b or f"; ew1:=(ola where {fun(!#z_z0,~n) => ber(1,n),fun(!#z_z1,~n) => ber(2,n), gras(!#z_z!#z_z1,~n) => fir(1,n),gras(!#z_z!#z_z2,~n) => fir(2,n)}); ew2:=(ew1 where { ber(1,~n) => !#s_s*d(1)^n, ber(2,~n) => !#s_s^4*d(1)^n, fir(1,~n) => !#s_s^2*d(1)^n,fir(2,~n) => !#s_s^3*d(1)^n}); for k:=1:4 do for l:=1:4 do !@z_z_x(k,l):=chan(coeffn(part(ew2,k),!#s_s,l)); return !@z_z_x end$ %********************************************% %*** module dot_ham *************************% %********************************************% procedure dot_ham(ww,mm); begin scalar ewa,ola,ala,as; as:=length(ww); ewa:=d(t)*mm-mm*d(t); for k:=1:as do bos(part(part(ww,k),1),t):=part(part(ww,k),2); ola:=(ewa where tryk14); for k:=1:as do clear bos(part(part(ww,k),1),t); return ola end$ %module supersymmetric integration %############################################################################ %######### F U N C T I O N A L S U S Y I N T E G R A T I O N ###### %######### O N L Y F O R T R A D ###### %############################################################################ % s_int(number,expression,variable) % numbers corespond to integration over: 0 => d(1), % 1=> der(1), 2=> der(2), 3 => der(1)*der(2) % variable {f,g,...} the names of the superfunctions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% G L O B A L A C T I O N %%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% let { waga(~k,~s,~m) => 2*m+delta(k,s)+2*delta(3,s)+delta(3-k,s)*(if m>0 then 1 else 0), s_s(~f,~n) =>1, s_s(1,~f,~n) => 1, der(1)*del(-1)=>1, der(2)*del(-2) => 1, der(3)*del(-3) =>1, del(0)=>d(-3), der(1)*del(-3) => 1, der(2)*del(-3) =>del(-3), der(1)*der(2)*del(-3)=>1 }$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&&&&&&&&&&&&&&&&& %%%% L O C A L A C T I O N %%%%%%%%%%%%%%%%&&&&&&&&&&&&&&&&& %############################################################## %%%%%%%%%%%%%%%%%%%%%%% n_dyw %%%%%%%%%%%%%%%%%%%%%% %############################################################## %scaling dryk:= {d(~f,~n)*fer(~g,~k,~m) => fer(g,k,m)*d(f,n+delta(f,g)), d(~f,~n)*bos(~g,~k,~m) => bos(g,k,m)*d(f,n+delta(f,g)), d(~f,~n)*zan(~g,~k,~m) => zan(g,k,m)*d(f,n+1), d(~f,~n)*zen(~g,~k,~m) => zen(g,k,m)*d(f,n+1) }$ %wariation wariat_0:={ zen(~f,~k,~n) => (-1)^n*fer(f,k,0)*d(1)^n, zan(~f,~k,~n) => (-1)^n*bos(f,k,0)*d(1)^n }$ wariat_1:={ zen(~f,~k,~n) => if k = 3 or k = 1 then (-1)^(n+1)*bos(f,k-1,0)*der(1)*d(1)^n else (-1)^n*fer(f,k,0)*d(1)^n , zan(~f,~k,~n) => if k = 3 or k = 1 then (-1)^n*fer(f,k-1,0)*der(1)*d(1)^n else (-1)^n*bos(f,k,0)*d(1)^n }$ wariat_2:={ zen(~f,~k,~n) => if k = 3 or k = 2 then (-1)^(k-1+n)*bos(f,k-2,0)*der(2)*d(1)^n else (-1)^n*fer(f,k,0)*d(1)^n , zan(~f,~k,~n) => if k = 3 or k = 2 then (-1)^(k+n)*fer(f,k-2,0)*der(2)*d(1)^n else (-1)^n*bos(f,k,0)*d(1)^n }$ wariat_3:={ zen(~f,0,~n) => if n > 1 then (-1)^(n-1)*(-fer(f,0,0)*d(1)^n + n*d(1)*fer(f,0,0)*d(1)^(n-1)) else fer(f,0,n) , zen(~f,1,~n) => (-1)^n*(fer(f,1,0)*d(1)^n + n*d(1)*bos(f,0,0)*der(1)*d(1)^(n-1)), zen(~f,2,~n) => (-1)^n*(fer(f,2,0)*d(1)^n + n*d(1)*bos(f,0,0)*der(2)*d(1)^(n-1)), zen(~f,3,~n) => -zen(f,0,n)*der(1)*der(2) + zan(f,1,n)*der(2) - zan(f,2,n)*der(1), zan(~f,0,~n) => if n>1 then (-1)^(n-1)*(-bos(f,0,0)*d(1)^n + n*d(1)*bos(f,0,0)*d(1)^(n-1)) else bos(f,0,n), zan(~f,1,~n) => (-1)^n*(bos(f,1,0)*d(1)^n - n*d(1)*fer(f,0,0)*der(1)*d(1)^(n-1)), zan(~f,2,~n) => (-1)^n*(bos(f,2,0)*d(1)^n - n*d(1)*fer(f,0,0)*der(2)*d(1)^(n-1)), zan(~f,3,~n) => -zan(f,0,n)*der(1)*der(2) - zen(f,1,n)*der(2)+zen(f,2,n)*der(1) }$ %########################################################### %%%%%%%%%%%%%%%%%%%%%% maxi %%%%%%%%%%%%%%%%%%% %########################################################### szukaj0:={ byk(~n,~g)*fer(~f,~k,~m) => if n<=2*m then fer(f,k,m)*byk(2*m,f) else fer(f,k,m)*byk(n,g), byk(~n,~g)*bos(~f,~k,~m) => if n<=2*m then bos(f,k,m)*byk(2*m,f) else bos(f,k,m)*byk(n,g) }$ szukaj1:={ byk(~k,~g)*fer(~f,~n,~m) => if k <= waga(1,n,m) then fer(f,n,m)*byk(waga(1,n,m),f) else fer(f,n,m)*byk(k,g), byk(~k,~g)*bos(~f,~n,~m) =>if k <= waga(1,n,m) then bos(f,n,m)*byk(waga(1,n,m),f) else bos(f,n,m)*byk(k,g) }$ szukaj2:={ byk(~k,~g)*fer(~f,~n,~m) => if k <= waga(2,n,m) then fer(f,n,m)*byk(waga(2,n,m),f) else fer(f,n,m)*byk(k,g), byk(~k,~g)*bos(~f,~n,~m) => if k<= waga(2,n,m) then bos(f,n,m)*byk(waga(2,n,m),f) else bos(f,n,m)*byk(k,g) }$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% szukaj3:={ byk(~k,~g)*fer(~f,0,~m) => if m < 2 then fer(f,0,m)*byk(k,g) else << if k <= 2*m then fer(f,0,m)*byk(2*m,f) else fer(f,0,m)*byk(k,g) >>, byk(~k,~g)*fer(~f,1,~m) => if m < 1 then fer(f,1,m)*byk(k,g) else << if k <= 2*m+1 then fer(f,1,m)*byk(2*m+1,f) else fer(f,1,m)*byk(k,g) >>, byk(~k,~g)*fer(~f,2,~m) => if m < 1 then fer(f,2,m)*byk(k,g) else << if k <= 2*m+1 then fer(f,2,m)*byk(2*m+1,f) else fer(f,2,m)*byk(k,g) >> , byk(~k,~g)*fer(~f,3,~m) => if k <= 2*m+2 then fer(f,3,m)*byk(2*m+2,f) else fer(f,3,m)*byk(k,g), byk(~k,~g)*bos(~f,0,~m) => if m < 2 then bos(f,0,m)*byk(k,g) else << if k <= 2*m then bos(f,0,m)*byk(2*m,f) else bos(f,0,m)*byk(k,g) >> , byk(~k,~g)*bos(~f,1,~m) => if m < 1 then bos(f,1,m)*byk(k,g) else << if k <= 2*m+1 then bos(f,1,m)*byk(2*m+1,f) else bos(f,1,m)*byk(k,g) >>, byk(~k,~g)*bos(~f,2,~m) => if m < 1 then bos(f,2,m)*byk(k,g) else << if k <= 2*m+1 then bos(f,2,m)*byk(2*m+1,f) else bos(f,2,m)*byk(k,g) >>, byk(~k,~g)*bos(~f,3,~m) => if k<=2*m+2 then bos(f,3,m)*byk(2*m+2,f) else bos(f,3,m)*byk(k,g) }$ %########################################################################### poszukaj0:={ fer(~f,~s,~m)*r_r(~k,~g) => if k = 2*m and g equal f then r_r(2*m,f)*zen(f,s,m) else r_r(k,g)*fer(f,s,m), bos(~f,~s,~m)*r_r(~k,~g) => if k = 2*m and g equal f then r_r(2*m,f)*zan(f,s,m) else r_r(k,g)*bos(f,s,m) }$ poszukaj1:={ fer(~f,~s,~m)*r_r(~k,~g) => if k = waga(1,s,m) and g equal f then r_r(k,f)*zen(f,s,m) else r_r(k,g)*fer(f,s,m) , bos(~f,~s,~m)*r_r(~k,~g) => if k = waga(1,s,m) and g equal f then r_r(k,f)*zan(f,s,m) else r_r(k,g)*bos(f,s,m) }$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% poszukaj2:={ fer(~f,~s,~m)*r_r(~k,~g) => if k = waga(2,s,m) and g equal f then r_r(k,f)*zen(f,s,m) else r_r(k,g)*fer(f,s,m), bos(~f,~s,~m)*r_r(~k,~g) => if k = waga(2,s,m) and g equal f then r_r(k,f)*zan(f,s,m) else r_r(k,g)*bos(f,s,m) }$ poszukaj3:={ fer(~f,0,~m)*r_r(~k,~g) => if m < 2 then r_r(k,g)*fer(f,0,m) else << if k = 2*m and f equal g then r_r(1,2*m,f)*zen(f,0,m) else r_r(k,g)*fer(f,0,m) >>, fer(~f,1,~m)*r_r(~k,~g) => if m < 1 then r_r(k,g)*fer(f,1,m) else <<if k = 2*m+1 and f equal g then r_r(1,2*m+1,f)*zen(f,1,m) else r_r(k,g)*fer(f,1,m)>>, fer(~f,2,~m)*r_r(~k,~g) => if m < 1 then r_r(k,g)*fer(f,2,m) else << if k = 2*m+1 and f e qual g then r_r(1,2*m+1,f)*zen(f,2,m) else r_r(k,g)*fer(f,2,m) >> , fer(~f,3,~m)*r_r(~k,~g) => if k = 2*m+2 and f equal g then r_r(1,2*m+2,f)*zen(f,3,m) else r_r(k,g)*fer(f,3,m), bos(~f,0,~m)*r_r(~k,~g) => if m < 2 then r_r(k,g)*bos(f,0,m) else << if k = 2*m and f equal g then r_r(1,2*m,f)*zan(f,0,m) else r_r(k,g)*bos(f,0,m) >>, bos(~f,1,~m)*r_r(~k,~g) => if m < 1 then r_r(k,g)*bos(f,1,m) else << if k = 2*m+1 and f equal g then r_r(1,2*m+1,f)*zan(f,1,m) else r_r(k,g)*bos(f,1,m) >>, bos(~f,2,~m)*r_r(~k,~g) => if m < 1 then r_r(k,g)*bos(f,2,m) else << if k = 2*m+1 and f e qual g then r_r(1,2*m+1,f)*zan(f,2,m) else r_r(k,g)*bos(f,2,m) >> , bos(~f,3,~m)*r_r(~k,~g) => if k = 2*m+2 and f equal g then r_r(1,2*m+2,f)*zan(f,3,m) else r_r(k,g)*bos(f,3,m) }$ %####################################################################### %%%%%%%%%%%%%%%%%%%%% I N T E G R A T I O N %%%%%%%%%%%%%%%%%%%%%%%%%%% %####################################################################### calkuj0:={ zen(~f,~n,~m) => fer(f,n,m-1),zan(~f,~n,~m) => bos(f,n,m-1) }$ pocalkuj0:={ zen(~f,~n,~m) => -fer(f,n,m-1)*d(1), zan(~f,~n,~m) => -bos(f,n,m-1)*d(1) }$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% calkuj1:={ zen(~f,~k,~n) => if k = 3 or k = 1 then bos(f,k-1,n) else if k = 0 and n > 0 or k = 2 and n > 0 then bos(f,k+1,n-1), zan(~f,~k,~n) => if k = 3 or k = 1 then fer(f,k-1,n) else if k = 0 and n > 0 or k = 2 and n > 0 then fer(f,k+1,n-1) }$ pocalkuj1:={ zen(~f,~k,~n) => if k = 3 or k = 1 then -bos(f,k-1,n)*der(1) else if k = 0 and n > 0 or k = 2 and n > 0 then -bos(f,k+1,n-1)*der(1), zan(~f,~k,~n) => if k = 3 or k = 1 then fer(f,k-1,n)*der(1) else if k = 0 and n > 0 or k = 2 and n > 0 then fer(f,k+1,n-1)*der(1) }$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% calkuj2:={ zen(~f,~k,~n) => if k = 3 or k = 2 then (-1)^k*bos(f,k-2,n) else if k = 0 and n > 0 or k = 1 and n > 0 then (-1)^k*bos(f,k+2,n-1) , zan(~f,~k,~n) => if k = 3 or k = 2 then (-1)^k*fer(f,k-2,n) else if k = 0 and n > 0 or k = 1 then (-1)^k*fer(f,k+2,n-1) }$ pocalkuj2:={ zen(~f,~k,~n) => if k = 3 or k = 2 then -(-1)^k*bos(f,k-2,n)*der(2) else if k = 0 and n > 0 or k = 1 and n > 0 then -(-1)^k*bos(f,k+2,n-1)*der(2) , zan(~f,~k,~n) => if k = 3 or k = 2 then (-1)^k*fer(f,k-2,n)*der(2) else if k = 0 and n > 0 or k = 1 and n > 0 then (-1)^k*fer(f,k+2,n-1)*der(2) }$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% calkuj3:={ zen(~f,0,~n) => if n < 2 then fer(f,0,n) else -fer(f,3,n-2), zen(~f,1,~n) => if n < 1 then fer(f,1,0) else fer(f,2,n-1), zen(~f,2,~n) => if n < 1 then fer(f,2,0) else -fer(f,1,n-1), zen(~f,3,~n) => fer(f,0,n), zan(~f,0,~n) => if n < 2 then fer(f,0,n) else -bos(f,3,n-2), zan(~f,1,~n) => if n < 1 then fer(f,1,0) else bos(f,2,n-1), zan(~f,2,~n) => if n < 1 then fer(f,2,0) else -bos(f,1,n-1), zan(~f,3,~n) => bos(f,0,n) }$ pocalkuj3:={ zen(~f,0,~n) => -bos(f,2,n-1)*der(2)-bos(f,1,n-1)*der(1)+ fer(f,3,n-2)*der(1)*der(2) , zen(~f,1,~n) => bos(f,3,n-1)*der(2)-bos(f,0,n)*der(1)- fer(f,2,n-1)*der(1)*der(2) , zen(~f,2,~n) => -bos(f,0,n)*der(2)-bos(f,3,n-1)*der(1)+ fer(f,1,n-1)*der(1)*der(2), zen(~f,3,~n) => -fer(f,0,n)*der(1)*der(2)+bos(f,1,n)*der(2)-bos(f,2,n)*der(1), zan(~f,0,~n) => bos(f,3,n-2)*der(1)*der(2)+fer(f,2,n-1)*der(2)+ fer(f,1,n-1)*der(1), zan(~f,1,~n) => -fer(f,3,n-1)*der(2)+fer(f,0,n)*der(1)- bos(f,2,n-1)*der(1)*der(2) , zan(~f,2,~n) => fer(f,3,n-1)*der(1)+fer(f,0,n)*der(2)+ bos(f,1,n-1)*der(1)*der(2) , zan(~f,3,~n) => -bos(f,0,n)*der(1)*der(2)-fer(f,1,n)*der(2)+fer(f,2,n)*der(1) }$ %$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ %$$$$$ $$$$$$ %$$$$$ P R O C E D U R E S $$$$$$ %$$$$$ $$$$$$ %$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ %%%%%%%%%%%%%%%%% one maximum %%%%%%%%%%%%%%%%%%%%% procedure maxi(wrt,wx); begin scalar kr,kr1,ew1,ew2,ew3,ew4,ew5; if wx equal 0 then return {0,0}; kr:=num wx; kr1:=den wx; ew1:=(byk(0,0)*kr where (help!* := mkid(szukaj,wrt))); ew2:=(ew1 where {byk(~n,~f) => (!l_a_!@m)^n*p_p(n,f)}); ew2:=sub(p_p=r_r,lcof(ew2,!l_a_!@m)); ew2:=(ew2 where (help!* := mkid(poszukaj,wrt))); ew3:=sub(r_r=s_s,ew2); ew1:=if part(ew3,0) equal minus then -1 else 1; ew4:=sub(x_x=0,if length(ew1*ew3) < arglength(ew1*ew3) then ew1*ew3 else part(ew1*ew3+x_x,1)); ew5:=kr-sub(zen=fer,zan=bos,ew1*ew4); return {ew1*ew4/kr1,ew5/kr1} end$ %################################################################# %%%%%%%%%%%%%%%%% dywergent terms %%%%%%%%%%%%%%%%%%%%%%%%%%% %################################################################# procedure n_dyw(wrt,wx,wz); begin scalar eks0,eks,eks1,eks2,osa1,osa2,osa3,osa4; kap:=length wz; eks:=num wx; eks0:=den wx; eks1:=if part(eks,0) equal minus then -1 else 1; eks2:=eks1*eks;osa4:=0; for k:=1:kap do << osa1:=eks2-sub(part(wz,k)=0,eks2); eks2:=eks2-osa1; osa2:=sub(d(part(wz,k))=0,(d(part(wz,k))*osa1 where tryk)); %scaling osa3:=(d(part(wz,k),0)*osa2 where dryk); osa3:=(osa3 where {d(~f,~n) => 1/n when n>0}); %end scaling osa3:=sub(der(1)=0,d(1)=0,der(3)=0,der(2)=0, (osa3 where (help!* := mkid(wariat_,wrt)))); osa4:=osa4+osa3; >>; return {wx-eks1*osa4/eks0,eks1*osa4/eks0} end$ %&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& %*************** ******************************* %*************** MAIN PROCEDURE ******************************* %*************** ******************************* %&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& procedure s_int(wrt,wx,wz); begin scalar kak,kak1,kak2,pak,pak1,pak2,kap,pak3,pak4; if wx equal 0 then return 0; if abra_kadabra equal 1 and wrt > 0 or abra_kadabra equal 3 and wrt > 0 then rederr " **** I T is Impossible to define in a proper manner this integral => use trad representation for computation only"; kak:=n_dyw(wrt,wx,wz); kak1:=first kak; kak2:=second kak; %if kak2 neq 0 then return del(-wrt)*wx; pak:=hom(kak1); pak1:=first first pak; pak2:=second pak; kap:=length pak2; pak3:=if pak2 equal 0 then 0 else for k:=1:kap sum cal(wrt,part(pak2,k)); pak4:= pak1*pak3+del(-wrt)*kak2 ; return pak4 end$ procedure cal(wrt,wx); begin scalar wem,wem1,wem2,wem3,wem4,wem5,wem6,z_z_z; if wx equal 0 then return 0; wem:=maxi(wrt,wx); wem1:=first(wem); wem2:=second(wem); z_z_z:=0; while wem1 neq 0 do << wem3:=sub(zen=fer,zan=bos,wem1); wem4:=(wem1 where (help!* := mkid(calkuj,wrt))); wem5:=sub(der(3)=0,der(1)=0,d(1)=0,der(2)=0,(-wem1 where (help!* := mkid(pocalkuj,wrt)))); if wem4 = 0 then z_z_z:=z_z_z+del(-wrt)*wem3 else << xxx:=(-!l_a_!@m*wem4+wem3+wem5 where {wem3=>koz}); wem6:=(rhs first solve(xxx,koz));clear xxx; z_z_z:=z_z_z+coeffn(wem6,!l_a_!@m,1); wem6:=sub(!l_a_!@m=0,wem6); wem2:=wem2+wem6 >>; wem6:=maxi(wrt,wem2); wem1:=first wem6; wem2:=second wem6; >>; return z_z_z end$ procedure hom(wx); begin scalar zet1,zet2,iks,iks1,iks2; if wx equal 0 then return {{0},0}; iks:=num wx;iks2:=den wx; iks1:=if part(iks,0) equal minus then -1 else 1; iks:=iks1*iks; zet1:=(iks where { fer(~f,~k,~n) => !&a(f)*!&a(!@)^(2n+(if k = 1 or k = 2 then 1 else if k = 3 then 2 else 0))*zen(f,k,n), bos(~f,~k,~n) => !&a(f)*!&a(!@)^(2n+(if k = 1 or k = 2 then 1 else if k = 3 then 2 else 0))*zan(f,k,n)}); zet2:=part(zet1+x_x,0):=list; zet1:=reverse rest reverse zet2; return {{iks1/iks2},sub(zen=fer,zan=bos,(zet1 where {!&a(~f) =>1,!&a(!@) => 1}))} end$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% D E C L A R A T I O N %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% let trad; endmodule; end;