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"plot(T(x/2)-1/2*T(x),x=0..1,numpoints=1024);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "R:=proc(n) global a0,a1,g0,g1; local i,L1,L2, x,y;\n" }{MPLTEXT 1 0 65 " if n=0 then RETURN([[0,g0(0)/(1-a0)],[1,g 1(1)/(1-a1)]]); fi;\n" }{MPLTEXT 1 0 26 " L1:=R(n-1); L2:=NULL;\n" } {MPLTEXT 1 0 31 " for i from 1 to 2^(n-1) do\n" }{MPLTEXT 1 0 33 " \+ x:=L1[i][1]; y:=L1[i][2];\n" }{MPLTEXT 1 0 32 " L2:=L2,[x/2,a 0*y+g0(x)];\n" }{MPLTEXT 1 0 8 " od;\n" }{MPLTEXT 1 0 33 " for i f rom 1 to 2^(n-1)+1 do\n" }{MPLTEXT 1 0 33 " x:=L1[i][1]; y:=L1[i] [2];\n" }{MPLTEXT 1 0 36 " L2:=L2,[(x+1)/2,a1*y+g1(x)];\n" } {MPLTEXT 1 0 8 " od;\n" }{MPLTEXT 1 0 10 " [L2];\n" }{MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "a0:=1/2; a1:=1/2 ; g0:=x->sin(Pi*x); g1:=x->-sin(Pi*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "R(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "fo r i from 0 to 10 do Plots[i]:=plot(R(i),0..1,-1.5..1.5,numpoints=1024, title=`n = `||i): od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "di splay([seq(Plots[i],i=0..10)],insequence=true);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 111 "for i from 1 to 9 do Plots[i]:=plot([R(i),R(i +1)],0..1,-1.5..1.5,numpoints=1024,color=red,title=`n = `||i): od:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "display([seq(Plots[i],i=1.. 9)],insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Rd iff:=proc(n) local R1,R2,i,L,x,y;\n" }{MPLTEXT 1 0 35 " R1:=R(n-1); \+ R2:=R(n); L:=NULL;\n" }{MPLTEXT 1 0 29 " for i from 1 to 2^n+1 do\n" }{MPLTEXT 1 0 20 " x:=R2[i][1];\n" }{MPLTEXT 1 0 27 " if ty pe(i,odd) then\n" }{MPLTEXT 1 0 38 " y:=R2[i][2]-R1[(i+1)/2][2 ];\n" }{MPLTEXT 1 0 12 " else\n" }{MPLTEXT 1 0 57 " y:=R2 [i][2]-1/2*(R1[(i)/2][2]+R1[(i+2)/2][2]);\n" }{MPLTEXT 1 0 11 " f i;\n" }{MPLTEXT 1 0 16 " L:=L,[x,y];\n" }{MPLTEXT 1 0 8 " od;\n" } {MPLTEXT 1 0 9 " [L];\n" }{MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Rdiff(1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "for i from 1 to 10 do 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amma;\n" }{MPLTEXT 1 0 8 " od;\n" }{MPLTEXT 1 0 33 " for i from 0 \+ to 2^(n-2)-1 do\n" }{MPLTEXT 1 0 73 " gamma:=g1((2*i+1)/2^(n-1))- 1/2*(g1(i/2^(n-2))+g1((i+1)/2^(n-2)));\n" }{MPLTEXT 1 0 32 " L1:= L1,a1*L0[i+1]+gamma;\n" }{MPLTEXT 1 0 8 " od;\n" }{MPLTEXT 1 0 21 " \+ map(evalf,[L1]);\n" }{MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "schauder(3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "for i from 1 to +5 do schauder(i)*2^i; od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for i from 1 to 12 do min(op(map(abs,schau der(i)*2^i))); od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a0:=0 .6; a1:=0.6;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for i from \+ 1 to 12 do min(op(map(abs,schauder(i)*2^i))); od;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 61 "for i from 1 to 12 do min(op(map(abs,schaude r(i)/a0^i))); od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT 205 22 "Bernoulli convolutions" }} {EXCHG {PARA 0 "> " 0 "" 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(1-q))*(1-q)/2 else 1 fi;\n" }{MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f1:=x->Bq(f0,x):\n" }{MPLTEXT 1 0 18 "f2:=x->Bq(f1,x):\n" }{MPLTEXT 1 0 18 "f3:=x->Bq(f2,x):\n" } {MPLTEXT 1 0 18 "f4:=x->Bq(f3,x):\n" }{MPLTEXT 1 0 18 "f5:=x->Bq(f4,x) :\n" }{MPLTEXT 1 0 18 "f6:=x->Bq(f5,x):\n" }{MPLTEXT 1 0 18 "f7:=x->Bq (f6,x):\n" }{MPLTEXT 1 0 18 "f8:=x->Bq(f7,x):\n" }{MPLTEXT 1 0 18 "f9: =x->Bq(f8,x):\n" }{MPLTEXT 1 0 19 "f10:=x->Bq(f9,x):\n" }{MPLTEXT 1 0 20 "f11:=x->Bq(f10,x):\n" }{MPLTEXT 1 0 20 "f12:=x->Bq(f11,x):\n" } {MPLTEXT 1 0 20 "f13:=x->Bq(f12,x):\n" }{MPLTEXT 1 0 20 "f14:=x->Bq(f1 3,x):\n" }{MPLTEXT 1 0 20 "f15:=x->Bq(f14,x):\n" }{MPLTEXT 1 0 20 "f16 :=x->Bq(f15,x):\n" }{MPLTEXT 1 0 20 "f17:=x->Bq(f16,x):\n" }{MPLTEXT 1 0 20 "f18:=x->Bq(f17,x):\n" }{MPLTEXT 1 0 20 "f19:=x->Bq(f18,x):\n" }{MPLTEXT 1 0 20 "f20:=x->Bq(f19,x):\n" }{MPLTEXT 1 0 20 "f21:=x->Bq(f 20,x):\n" }{MPLTEXT 1 0 20 "f22:=x->Bq(f21,x):\n" }{MPLTEXT 1 0 20 "f2 3:=x->Bq(f22,x):\n" }{MPLTEXT 1 0 20 "f24:=x->Bq(f23,x):\n" }{MPLTEXT 1 0 20 "f25:=x->Bq(f24,x):\n" }{MPLTEXT 1 0 18 "f26:=x->Bq(f25,x):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "q:=evalf(1/3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(f15,-1.6..1.6,numpoints=1024); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "q:=evalf(1/2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(f15,-2.5..2.5,numpoints =1024);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "q:=evalf(2/3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(f15,-3.5..3! .5,nump oints=1024);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q:=evalf((s qrt(5)-1)/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(f15,- 3..3,numpoints=1024);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Bq :=proc(f,x)\n" }{MPLTEXT 1 0 46 " if x<-1/(1-q) then 0 elif x>1/(1-q) then 0\n" }{MPLTEXT 1 0 41 " else (f((x+1)/q)+f((x-1)/q))/(2*q);\n " }{MPLTEXT 1 0 7 " fi;\n" }{MPLTEXT 1 0 5 "end; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f0:=proc(x)\n" }{MPLTEXT 1 0 64 " if x<-1 /(1-q) then 0 elif x<1/(1-q) then (1-q)/2 else 0 fi;\n" }{MPLTEXT 1 0 4 "end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "q:=0.5;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(f15,-2.5..2.5,numpoints =1024);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "q:=sqrt(0.5);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(f15,-3.5..3.5,numpoin ts=1024);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "q:=evalf(2/3); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(f15,-3.5..3.5,nump oints=1024);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q:=evalf((s qrt(5)-1)/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(f15,- 3..3,numpoints=1024);" }}}{EXCHG {PARA 0 "> " 0 "" {TEXT 204 0 "" }}}} } {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }