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-1 217 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 218 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 219 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 220 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 221 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 222 "T imes" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 223 17 "LIST OF FUNCTIONS" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 223 41 "The code consists of two main programs - " }{TEXT 207 4 "symm" }{TEXT 223 8 " and " }{TEXT 208 9 "matrices " }{TEXT 223 33 " and two auxiliary functions - " }{TEXT 209 6 "simp le" }{TEXT 223 7 " and " }{TEXT 210 6 " l_f.\n" }{TEXT 223 13 "The p rogram " }{TEXT 211 4 "symm" }{TEXT 223 26 " is the main function. \+ \n" }{TEXT 223 52 "The input consists of a complex-valued polynomial \+ " }{TEXT 212 4 "f(p)" }{TEXT 223 20 " considered as the\n" }{TEXT 223 49 "projective form of homogeneous binary polynomial " }{TEXT 213 6 "F(x,y)" }{TEXT 223 6 ", and\n" }{TEXT 223 11 "the degree " }{TEXT 214 9 " n=deg(F)" }{TEXT 223 40 ". The program computes the invarian ts " }{TEXT 215 4 " J " }{TEXT 223 5 "and " }{TEXT 216 1 "K" }{TEXT 223 124 " in reduced form, determines the dimension of the symmetry g roup, and, in the case of a finite symmetry group, applies the\n" } {TEXT 223 16 " Maple command " }{TEXT 217 5 "solve" }{TEXT 223 64 " \+ to solve the two polynomial symmetry equations (3,4) to find\n" } {TEXT 223 29 "explicit form of symmetries. " }}{PARA 0 "" 0 "" {TEXT 223 15 "The output of " }{TEXT 218 4 "symm" }{TEXT 223 81 " consists of the projective index of the form and the explicit formulae for its \n" }{TEXT 223 77 "discrete projective symmetries. The program also notifies the user if the\n" }{TEXT 223 79 "symmetry group is not disc rete, or is in the maximal discrete symmetry class. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "symm:=proc(form,n)\n" }{MPLTEXT 1 0 15 "global tr,err;\n" }{MPLTEXT 1 0 63 "local Q,Qp,Qpp,Qppp,Qpppp,H, T,V,U,J,K,j,k, Eq1,Eq2,i,eqtr,ans;\n" }{MPLTEXT 1 0 11 " tr:='tr':\n" }{MPLTEXT 1 0 13 " Q:=form(p);\n" }{MPLTEXT 1 0 16 " Qp:=diff(Q,p);\n" }{MPLTEXT 1 0 18 " Qpp:=diff(Qp,p);\n" }{MPLTEXT 1 0 20 " Qppp:=diff( Qpp,p);\n" }{MPLTEXT 1 0 22 " Qpppp:=diff(Qppp,p);\n" }{MPLTEXT 1 0 34 " H:=n*(n-1)*(Q*Qpp-(n-1)/n*Qp^2);\n" }{MPLTEXT 1 0 13 " if H=0 the n\n" }{MPLTEXT 1 0 57 " ans:=`Hessian is zero: two-dimensional symmet ry group`\n" }{MPLTEXT 1 0 6 " else\n" }{MPLTEXT 1 0 76 " T:=-n^2*(n- 1)*(Q^2*Qppp-3*(n-2)/n*Q*Qp*Qpp+2*(n-1)*(n-2)/n^2 *Qp^3);\n" } {MPLTEXT 1 0 97 " V:=Q^3*Qpppp-4*(n-3)/n*Q^2*Qp*Qppp+6*(n-2)*(n-3)/n^2 *Q*Qp^2 *Qpp-3*(n-1)*(n-2)*(n-3)/n^3*Qp^4;\n" }{MPLTEXT 1 0 35 " U:=n ^3*(n-1)*V-3*(n-2)/(n-1)*H^2;\n" }{MPLTEXT 1 0 38 " J:=simple(T^2/H^3) ;K:=simple(U/H^2);\n" }{MPLTEXT 1 0 32 " j:=subs(p=P,J);k:=subs(p=P,K) ;\n" }{MPLTEXT 1 0 53 " Eq1:=simplify(numer(J)*denom(j)-numer(j)*denom (J));\n" }{MPLTEXT 1 0 53 " Eq2:=simplify(numer(K)*denom(k)-numer(k)*d enom(K));\n" }{MPLTEXT 1 0 15 " if Eq1=0 then\n" }{MPLTEXT 1 0 53 " \+ ans:=`Form has a one-dimensional symmetry group`;\n" }{MPLTEXT 1 0 6 " else\n" }{MPLTEXT 1 0 18 " if Eq2=0 then\n" }{MPLTEXT 1 0 83 " \+ print (` Form has the maximal possible discrete symmetry \+ group`);\n" }{MPLTEXT 1 0 24 " eqtr:= [solve(Eq1,P)];\n" }{MPLTEXT 1 0 46 " tr:=map(radsimp,map(allvalues,eqtr));\n" }{MPLTEXT 1 0 10 " else\n" }{MPLTEXT 1 0 36 " eqtr:=[solve(\{Eq1,Eq2\},P) ];\n" }{MPLTEXT 1 0 17 " tr:= [];\n" }{MPLTEXT 1 0 38 " \+ for i from 1 to nops(eqtr) do\n" }{MPLTEXT 1 0 63 " tr:=map(ra dsimp,[op(tr),allvalues(rhs(eqtr[i][1]))]);\n" }{MPLTEXT 1 0 11 " \+ od\n" }{MPLTEXT 1 0 10 " fi;\n" }{MPLTEXT 1 0 63 " print(`T he number of the projective symmetries`=nops(tr));\n" }{MPLTEXT 1 0 22 " ans:=map(l_f,tr);\n" }{MPLTEXT 1 0 18 " if err=1 then\n" } {MPLTEXT 1 0 55 " print(`ERROR: Some of the transformations are no t\n" }{MPLTEXT 1 0 25 " linear-fractional`)\n" }{MPLTEXT 1 0 9 " \+ else\n" }{MPLTEXT 1 0 19 " if err=2 then\n" }{MPLTEXT 1 0 110 " \+ print(`WARNING: Some of the transformations are not written in the form polynomial over polynomial`)\n" }{MPLTEXT 1 0 9 " fi;\n" }{MPLTEXT 1 0 8 " fi;\n" }{MPLTEXT 1 0 7 " fi;\n" }{MPLTEXT 1 0 5 " fi;\n" }{MPLTEXT 1 0 5 " ans\n" }{MPLTEXT 1 0 5 "end:\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 219 0 "" }{TEXT 223 13 "The program " } {TEXT 220 8 "matrices" }{TEXT 223 24 " determines the matrix\n" } {TEXT 223 104 "symmetry corresponding to a given (list of) projective \+ symmetries. As discussed in the text, this only\n" }{TEXT 223 99 "req uires determining an overall scalar multiple, which can be found by su bstituting the projective\n" }{TEXT 223 100 "symmetry into the form. \+ The output consists of each projective symmetry, the scalar factor $ \\mu $,\n" }{TEXT 223 34 "and the resulting matrix symmetry." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "matrices:=proc(form,n,L::lis t)\n" }{MPLTEXT 1 0 24 " local Q,ks,ksi,i,Sf,M;\n" }{MPLTEXT 1 0 13 " \+ ksi:='ksi';\n" }{MPLTEXT 1 0 28 " for i from 1 to nops(L) do\n" } {MPLTEXT 1 0 42 " Sf:=simplify(denom(L[i])^n*form(L[i]));\n" } {MPLTEXT 1 0 25 " ks:=quo(Sf,form(p),p);\n" }{MPLTEXT 1 0 44 " ksi:= simplify(ks^(1/n),radical,symbolic);\n" }{MPLTEXT 1 0 46 " M[i]:=matr ix(2,2,[coeff(numer(L[i]),p)/ksi,\n" }{MPLTEXT 1 0 55 " coeff(numer(L [i]),p,0)/ksi,coeff(denom(L[i]),p)/ksi,\n" }{MPLTEXT 1 0 32 " coeff(d enom(L[i]),p,0)/ksi]);\n" }{MPLTEXT 1 0 49 " print(L[i], mu=ksi, \+ map(simplify,M[i])); \n" }{MPLTEXT 1 0 5 " od;\n" }{MPLTEXT 1 0 5 "en d:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 223 24 "The auxiliary function " }{TEXT 221 5 "l_f " }{TEXT 223 29 "uses polynomial division to\n" }{TEXT 223 71 "reduce rational expressions to linear fractional form ( when possible). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "l_f:=pr oc(x)\n" }{MPLTEXT 1 0 26 " local A,B,C,S,de,nu,r,R;\n" }{MPLTEXT 1 0 24 " global err;err:='err';\n" }{MPLTEXT 1 0 17 " nu:=numer(x); \n" } {MPLTEXT 1 0 15 " de:=denom(x);\n" }{MPLTEXT 1 0 70 " if type(nu,polyn om(anything,p)) and type(de,polynom(anything,p))\n" }{MPLTEXT 1 0 6 " then\n" }{MPLTEXT 1 0 38 " if degree(nu,p)+1=degree(de,p) then\n" }{MPLTEXT 1 0 24 " A:=quo(de,nu,p,'B');\n" }{MPLTEXT 1 0 15 " S:= 1/A;R:=0\n" }{MPLTEXT 1 0 8 " else \n" }{MPLTEXT 1 0 24 " A:=quo(nu ,de,p,'B');\n" }{MPLTEXT 1 0 21 " if B=0 then S:=A;\n" }{MPLTEXT 1 0 14 " R:=0; \n" }{MPLTEXT 1 0 8 " else\n" }{MPLTEXT 1 0 38 " \+ C:=quo(de,B,p,'r');R:=simple(r);\n" }{MPLTEXT 1 0 25 " S:=simp lify(A+1/C) \n" }{MPLTEXT 1 0 7 " fi;\n" }{MPLTEXT 1 0 6 " fi;\n" } {MPLTEXT 1 0 17 " if R=0 then \n" }{MPLTEXT 1 0 17 " collect(S,p )\n" }{MPLTEXT 1 0 7 " else\n" }{MPLTEXT 1 0 13 " err:=1;x\n" } {MPLTEXT 1 0 6 " fi;\n" }{MPLTEXT 1 0 6 " else\n" }{MPLTEXT 1 0 65 " \+ err:=2;x fi;\n" } {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 223 23 "The auxi liary function " }{TEXT 222 7 "simple " }{TEXT 223 28 " helps to simpl ify rational\n" }{TEXT 223 71 " expressions by manipulating the numera tor and denominator separately.\n" }{TEXT 223 48 "The simplified ratio nal expression is returned.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simple:=proc(x)\n" }{MPLTEXT 1 0 21 "local nu,de,num,den;\n" } {MPLTEXT 1 0 14 "nu:=numer(x);\n" }{MPLTEXT 1 0 14 "de:=denom(x);\n" } {MPLTEXT 1 0 40 "num:=(simplify((nu,radical,symbolic)));\n" }{MPLTEXT 1 0 40 "den:=(simplify((de,radical,symbolic)));\n" }{MPLTEXT 1 0 19 "s implify(num/den);\n" }{MPLTEXT 1 0 5 "end:\n" }{MPLTEXT 1 0 1 "\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 223 8 "Example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=p->p^3+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6 #I\"pG6\"F%6$I)operatorGF%I&arrowGF%F%,&*$9$\"\"$\"\"\"F-F-F%F%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "symm(f,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I^o~Form~has~the~maximal~possible~discrete~symmetry~~~ ~~~~~~~~~~~groupG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "/IHThe~number~of ~the~projective~symmetriesG6\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7 (I\"pG6\"*$F#!\"\"*&,&#F&\"\"#\"\"\"*&^##F+F*F+\"\"$F.F+F+F#F&*&,&F)F+ *&^#F)F+F/F.F+F+F#F&*&F(F+F#F+*&F1F+F#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "matrices(f,3,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%I \"pG6\"/I#muGF$\"\"\"=F$6$;F'\"\"#F*E\\[l%6$F+F'\"\"!6$F'F+F.6$F+F+F'6 $F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%*$I\"pG6\"!\"\"/I#muGF%\"\" \"=F%6$;F)\"\"#F,E\\[l%6$F-F)F)6$F)F-F)6$F-F-\"\"!6$F)F)F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%*&,&#!\"\"\"\"#\"\"\"*&^##F(F'F(\"\"$F+F(F(I\" pG6\"F&/I#muGF.F'=F.6$;F(F'F3E\\[l%6$F'F(F(6$F(F'F$6$F'F'\"\"!6$F(F(F8 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%*&,&#!\"\"\"\"#\"\"\"*&^#F%F(\"\"$ #F(F'F(F(I\"pG6\"F&/I#muGF.F'=F.6$;F(F'F3E\\[l%6$F'F(F(6$F(F'F$6$F'F' \"\"!6$F(F(F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%*&,&#!\"\"\"\"#\"\"\" *&^##F(F'F(\"\"$F+F(F(I\"pG6\"F(/I#muGF.F'=F.6$;F(F'F3E\\[l%6$F'F(\"\" !6$F(F'F66$F'F'F(6$F(F(F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%*&,&#!\" \"\"\"#\"\"\"*&^#F%F(\"\"$#F(F'F(F(I\"pG6\"F(/I#muGF.F'=F.6$;F(F'F3E \\[l%6$F'F(\"\"!6$F(F'F66$F'F'F(6$F(F(F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=p->p^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"pG 6\"F%6$I)operatorGF%I&arrowGF%F%*$9$\"\"#F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "symm(f,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "IJ Form~has~a~one-dimensional~symmetry~groupG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:=p->p^4+2*I*sqrt(3)*p^2+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"pG6\"F%6$I)operatorGF%I&arrowGF%F%,(*$9$\"\"%\"\" \"*(^#\"\"#F--I%sqrtG6$%*protectedGI(_syslibGF%6#\"\"$F-F+F0F-F-F-F%F% F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "symm(f,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I^o~Form~has~the~maximal~possible~discrete~sym metry~~~~~~~~~~~~~~groupG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "/IHThe~n umber~of~the~projective~symmetriesG6\"\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "7.,$*$I\"pG6\"!\"\"F'F%,$F%F'F$,$*&,&\"\"\"F,*&^#F,F,F%F, F,F,,&F'F,F-F,F'F',$*&F/F,F+F'F'F1F**(F.F,,&F%F,F,F,F,,&F%F,F'F,F'*(^# F'F,F3F,F4F'*(F.F,F4F,F3F'*(F6F,F4F,F3F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {TEXT 223 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }