{VERSION 2 3 "APPLE_PPC_MAC" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 42 "Chapter 1 Section 5. Alternate K is Small" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 229 " In this section we give the second of two concepts for \+ \"K is small.\" The iteration of the previous section continues to be applicable with this second concept of small. We introduce the notion of resolvent in this section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 12 "Exercise 1.5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Exercise 1." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "K:=(x,y)->piecewise(x <=y,f(x)*g(y),h(x)*j(y));" }}} {PARA 0 "" 0 "" {TEXT -1 42 "This problem might better be done by hand ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Exer cise 2. There are two K's from the previous section." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "K:=(x,t)->Heaviside(x-t)*(x-t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "assume(x>0): additionally(x<1):\nint(int(K(x, t)^2,t=0..1),x=0..1);\nx:='x'; t:='t';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "K:=(x,t)->Heaviside(x-t)*(1-x)+Heaviside(t-x)*(1-t); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "assume(x>0): additional ly(x<1):\nint(int(K(x,t)^2,t=0..1),x=0..1);\nx:='x'; t:='t';" }}} {PARA 0 "" 0 "" {TEXT -1 10 "Exercise 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "k:=(x,y)->piecewise(t<1/4,a,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "int(k(x,t),t=0..1);\nint(int(k(x,t)^2,t=0..1),x=0..1) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "phi[0]:=x->x;\nint(k(x ,t)*phi[0](t),t=0..1)+x;\nphi[1]:=unapply(\",x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "int(k(x,t)*phi[1](t),t=0..1)+x;\nphi[2]:=unap ply(\",x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "int(k(x,t)*ph i[2](t),t=0..1)+x;\nphi[3]:=unapply(\",x);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 53 "int(k(x,t)*phi[3](t),t=0..1)+x;\nphi[4]:=unapply(\" ,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "2048/32; 512/32; 12 8/32; 32/32;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "guess:=x+a* 1/32*1/(1-a/4);\nsubs(a=2,\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "y:=x->x+1/8;\nint(k(x,t)*y(t),t=0..1)+x;" }}}{PARA 0 "" 0 "" {TEXT -1 11 "Exercise 4." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " restart:\nassume(x>0): additionally(x<1):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "k:=(x,y)->piecewise(t " 0 "" {MPLTEXT 1 0 53 "int(k(x,t),t=0..1);\nint(int(k(x,t)^2,t=0..1) ,x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "phi[0]:=x->x; \nint(k(x,t)*phi[0](t),t=0..1)+x;\nphi[1]:=unapply(\",x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "int(k(x,t)*phi[1](t),t=0..1)+x;\nph i[2]:=unapply(\",x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "int (k(x,t)*phi[2](t),t=0..1)+x;\nphi[3]:=unapply(\",x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "guess:=x->exp(x)-1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "int(k(x,t)*guess(t),t=0..1)+x;" }}}{PARA 0 " " 0 "" {TEXT -1 87 "Exercise 5. In this problem, it is well to realize that if R(x,t) is the resolvent then" }}{PARA 0 "" 0 "" {TEXT -1 31 " R(x,t) = k(x,t) + " }{XPPEDIT 18 0 "int(k(x,s)*R(s,t),s=0 ..1)" "-%$intG6$*&-%\"kG6$%\"xG%\"sG\"\"\"-%\"RG6$F*%\"tGF+/F*;\"\"!F+ " }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "k:=(x,t)->x*t;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k(x,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "int(k(x,s)*k(s,t),s=0..1);\nk2:=unapply(\",(x ,t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "int(k(x,s)*k2(s,t) ,s=0..1);\nk3:=unapply(\",(x,t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "guess:=(x,t)->x*t*1/(1-1/3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "k(x,t)+int(k(x,s)*guess(s,t),s=0..1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "4 0" 12 } {VIEWOPTS 1 1 0 1 1 1803 }