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lĪ:#Sʙݳ6Gvq!1\&sQQs=[LX]l2M,Hw%ԦyS,h]eDcswc{a <-nrUHwRf9Qw%,I,;I3[DMؗ<FqXثQZivK+cd˩h[>vF2F6>~]<;K,^Iff9ۙa|{-3DSETT"lll l   #l #Physics 433: Computational PhysicsDSETT&&&lll p   'l'Sridhara Dasu, University of WisconsinDSET2Hl *PPPPPPPPPPPPPDSET2Hl$  $T Bp*DSETT  l0l(l,  l4 Slide TitleDSETTnnnl@l8l<X}}( $+<!U)(~" .#8$9<:U+<n> olHoBlack Header Red level-1 sub-header Blue level-2 sub-header Green level-3 sub-header Purple level-4 sub-headerDSET2H lL  $T  3eR *DSETT"" "lXlPlT\  #l\#Physics 433: Computational PhysicsDSETT  lhl`ldl H 41E1Y1my $-45mx%y- llLecture 4 Quadrature Numerical methods to integrate Trapezoidal rule Romberg integration Gaussian quadrature Application Cornu spiral Calculation of heat capcity of a solidDSET2H {lp  $T  Om`*DSETT l|ltlx  lNumerical IntegrationDSETTtu u llllX^^0C9U;CJt2E$89W:^;EJ s8tVH9H] hE=ZtH @hE=Z vlvTrapezoidal rule Two points where function is calculated  Simpsons rule Three points where function is calculated DSET2H@lPh l@6*P V VE @@(dxpr) CWEE)EGrphbj (, Symbol `.( "( currentpoint ",Times(PAreas:) )hf 8+J1 `(P,) hf 8+R2 `(Average of two measures:+df)/()(x)/))$dx 8( $a(R$b (  `(=,Times)Lh(51*2"17(f 8+1 `(+("1*2"7(vf 8+"2 `(x  * (  (x * ( (+)HO)C()$h 8(3 `+0+f)/')'))/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5664 div 2208 3 -1 roll exch div scale currentpoint translate 64 32 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2180 1648 rndmvto 224 0 rlineto stroke 3131 1648 rndmvto 224 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (Areas:) 0 288 sh ( ) 1021 288 sh (Average of two measures:) 0 864 sh 384 /Times-Italic f1 (hf) 1117 288 sh (hf) 1650 288 sh (f) 400 1744 sh (x) 748 1744 sh (dx) 1080 1744 sh (h) 1848 1744 sh (f) 2532 1744 sh (f) 3483 1744 sh (O) 4224 1744 sh (h) 4636 1744 sh (f) 5052 1744 sh 224 ns (a) 80 2144 sh (b) 80 1320 sh 224 /Times-Roman f1 (1) 1412 384 sh (2) 1977 384 sh (1) 2635 1840 sh (2) 3618 1840 sh (3) 4860 1572 sh 384 ns (1) 2196 1506 sh (2) 2196 2040 sh (1) 3147 1506 sh (2) 3147 2040 sh 384 /Times-Roman f1 (,) 1522 288 sh (\() 590 1744 sh (\)) 938 1744 sh (\() 4494 1744 sh (\251) 5242 1744 sh (\251) 5338 1744 sh (\)) 5434 1744 sh 576 /Symbol f1 (\362) 64 1808 sh 384 ns (=) 1544 1744 sh (+) 2827 1744 sh (\351) 2038 1474 sh (\353) 2038 2030 sh (\352) 2038 1858 sh (\371) 3744 1474 sh (\373) 3744 2030 sh (\372) 3744 1858 sh (+) 3936 1744 sh end MTsave restore dEMATH9>W Areas: hf 1 ,hf 2 Average of two measures:f(x)dx ab  =h12f 1 +12f 2 []+O(h 3 f) mDSET2H@lPh@l#6*P E @@(dxpr) CWEE)EGrphbj (, Symbol `.( "( currentpoint ",Times(PAreas:) 2)Khf 8+J1 `(P,)2)3hf 8+R2 `(P,)2)4hf 8+R3 `( Weighted sum:+hf)/()(x)/))$dx 8( $a(R$b (  `(=,Times)Lh(51+3"17(f 8+1 `(+(&4+3"?(~f 8+"2 `(+($1+3" 7(xf 8+"3 `(x  * (  (x * ( (+)HO)D()#h 8(5 `+0+f 8(0()4)#) `++)/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6816 div 2208 3 -1 roll exch div scale currentpoint translate 64 32 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2180 1648 rndmvto 224 0 rlineto stroke 3131 1648 rndmvto 256 0 rlineto stroke 4162 1648 rndmvto 224 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (Areas:) 0 288 sh ( 2) 1021 288 sh (Weighted sum:) -16 864 sh 384 /Times-Italic f1 (hf) 1323 288 sh (hf) 2046 288 sh (hf) 2817 288 sh (f) 400 1744 sh (x) 748 1744 sh (dx) 1080 1744 sh (h) 1848 1744 sh (f) 2532 1744 sh (f) 3515 1744 sh (f) 4514 1744 sh (O) 5255 1744 sh (h) 5667 1744 sh (f) 6083 1744 sh 224 ns (a) 80 2144 sh (b) 80 1320 sh 224 /Times-Roman f1 (1) 1618 384 sh (2) 2373 384 sh (3) 3144 384 sh (1) 2635 1840 sh (2) 3650 1840 sh (3) 4649 1840 sh (5) 5891 1572 sh (4) 6369 1572 sh 384 ns (2) 1840 288 sh (2) 2611 288 sh (1) 2196 1506 sh (3) 2212 2040 sh (4) 3163 1506 sh (3) 3179 2040 sh (1) 4178 1506 sh (3) 4194 2040 sh 384 /Times-Roman f1 (,) 1728 288 sh (,) 2499 288 sh (\() 590 1744 sh (\)) 938 1744 sh (\() 5525 1744 sh (\)) 6605 1744 sh 224 ns (\() 6275 1572 sh (\)) 6511 1572 sh 576 /Symbol f1 (\362) 64 1808 sh 384 ns (=) 1544 1744 sh (+) 2827 1744 sh (+) 3858 1744 sh (\351) 2038 1474 sh (\353) 2038 2030 sh (\352) 2038 1858 sh (\371) 4775 1474 sh (\373) 4775 2030 sh (\372) 4775 1858 sh (+) 4967 1744 sh end MTsave restore drMATHf> Areas: 2hf 1 ,2hf 2 ,2hf 3 Weighted sum:f(x)dx ab  =h13f 1 +43f 2 +13f 3 []+O(h 5 f (4) )m:DSET2HHl  $T 8 e`*DSETTlll  lNumerical IntegrationDSETT  llllX H,'.+_2a+4+2" ,(-0./_3a/=/79793,$Ii#F 4_H0 8HQx6$;HhQy6$; lExtended Trapezoidal rule Definite integral  Divide the interval [a,b] into N-1 sub-intervals  The integral is given by the sum of areas of trapezoids  For equally spaced points, xi+1-xi=h, DSET2H lP#il#6*PF @@0dxpr CWEEFGrphbj 0, Symbol `.(8 "0 currentpoint ",Times(xI)8=,Times)f)/()(x)/))$dx 8(a(3b (/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2240 div 736 3 -1 roll exch div scale currentpoint translate 64 38 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (I) 0 442 sh (f) 1072 442 sh (x) 1420 442 sh (dx) 1752 442 sh 224 ns (a) 775 636 sh (b) 832 168 sh 384 /Symbol f1 (=) 224 442 sh 576 ns (\362) 592 506 sh 384 /Times-Roman f1 (\() 1262 442 sh (\)) 1610 442 sh end MTsave restore dCMATH7  I=f(x)dx ab 9DSET2HlP{0l#6*P    @@dxpr CWEE Grphbj , Symbol `.( " currentpoint ",Times(Px 8+0i `(PW:)  )x 8+(1 `(P=,Times)La),,)$x 8+0N `(P=)Hb)+,)$x 8+(1 `(P<)Px 8+02 `(P<)H...)T<)Px 8+0N),-)1 `(PB<)Px 8+0N `( Evaluate function at ('f 8+"i `(u=)\f)/()(x 8+0i `(o)/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6112 div 1024 3 -1 roll exch div scale currentpoint translate 64 32 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (x) 16 288 sh (x) 510 288 sh (a) 1182 288 sh (x) 1500 288 sh (b) 2268 288 sh (x) 2586 288 sh (x) 3274 288 sh (x) 4634 288 sh (x) 5642 288 sh (f) 3165 864 sh (f) 3844 864 sh (x) 4192 864 sh 224 ns (i) 208 384 sh (N) 1692 384 sh (N) 4826 384 sh (N) 5834 384 sh (i) 3300 960 sh (i) 4384 960 sh 384 /Times-Roman f1 (:) 286 288 sh (,) 1356 288 sh (,) 2442 288 sh (...) 3978 288 sh (\() 4034 864 sh (\)) 4478 864 sh 384 /Times-Roman f1 ( ) 414 288 sh (Evaluate function at ) -16 864 sh 224 /Times-Roman f1 (1) 670 384 sh (1) 2746 384 sh (2) 3466 384 sh (1) 5114 384 sh 384 /Symbol f1 (=) 878 288 sh (=) 1980 288 sh (<) 2954 288 sh (<) 3690 288 sh (<) 4314 288 sh (<) 5322 288 sh (=) 3476 864 sh 224 ns (-) 5002 384 sh end MTsave restore d)MATH5 x i : x 1 =a,x N =b,x 1 <x 2 <...<x N-1 <x N Evaluate function at f i =f(x i )DSET2HlP{Ql#6*P 6x @@dxpr CWEE6xGrphbj , Symbol `.(0 " currentpoint ",Times(Z T 8+.i `(Zf=,Times 8(81+>2"@' `(Z()'x 8+0i)+)1 `(Z-)Lx 8+0i `(Z1)()Sf 8+"i)+)1 `(Z+)\f 8+!i `(Z)(<I)<)LI 8+T `(<=)T 8+.i(Ri)=) 1(FN),-)1 (JB0/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3840 div 1728 3 -1 roll exch div scale currentpoint translate 64 56 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 648 208 rndmvto 160 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (T) -16 304 sh (x) 1014 304 sh (x) 1894 304 sh (f) 2513 304 sh (f) 3400 304 sh (I) 0 1208 sh (I) 544 1208 sh (T) 1656 1208 sh 224 ns (i) 168 400 sh (i) 1206 400 sh (i) 2086 400 sh (i) 2648 400 sh (i) 3535 400 sh (T) 664 1304 sh (i) 1840 1304 sh (i) 1288 1616 sh (N) 1240 776 sh 384 /Symbol f1 (=) 344 304 sh (-) 1590 304 sh (+) 3032 304 sh (\273) 240 1208 sh (=) 936 1208 sh 224 ns (+) 1286 400 sh (+) 2728 400 sh (=) 1352 1616 sh (-) 1416 776 sh 576 ns (\345) 1224 1264 sh 224 /Times-Roman f1 (1) 672 168 sh (2) 680 416 sh (1) 1398 400 sh (1) 2840 400 sh (1) 1480 1616 sh (1) 1528 776 sh 384 /Times-Roman f1 (\() 856 304 sh (\)\() 2180 304 sh (\)) 3629 304 sh end MTsave restore dMATH! T i = 12 (x i+1 -x i )(f i+1 +f i )II T =T ii=1N-1 DSET2Hbray/FlP{Ql#6*P z z6y @@dxpr CWEE6yGrphbj , Symbol `.(0 " currentpoint ",Times(Z T 8+.i `(Zf=,Times 8(81+>2"@' `(Zh)3()4f 8+!i)+)1 `(Z+)\f 8+"i `(Zh)(<I 8+T `(<r= 8(1+>2""' `(<h)3()4f 8+1 `(<+)\f 8+"N `(<X))0+)Lh)f 8+"i(i)=)(2( N),-)1 (J/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3872 div 1728 3 -1 roll exch div scale currentpoint translate 64 56 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 648 208 rndmvto 160 0 rlineto stroke 696 1112 rndmvto 160 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (T) -16 304 sh (h) 872 304 sh (f) 1284 304 sh (f) 2171 304 sh (I) 0 1208 sh (h) 920 1208 sh (f) 1332 1208 sh (f) 1995 1208 sh (h) 2832 1208 sh (f) 3550 1208 sh 224 ns (i) 168 400 sh (i) 1419 400 sh (i) 2306 400 sh (T) 120 1304 sh (N) 2130 1304 sh (i) 3685 1304 sh (i) 3078 1616 sh (N) 3054 776 sh 384 /Symbol f1 (=) 344 304 sh (+) 1803 304 sh (=) 392 1208 sh (+) 1627 1208 sh (+) 2528 1208 sh 224 ns (+) 1499 400 sh (=) 3142 1616 sh (-) 3230 776 sh 576 ns (\345) 3038 1264 sh 224 /Times-Roman f1 (1) 672 168 sh (2) 680 416 sh (1) 1611 400 sh (1) 720 1072 sh (2) 728 1320 sh (1) 1435 1304 sh (2) 3302 1616 sh (1) 3342 776 sh 384 /Times-Roman f1 (\() 1078 304 sh (\)) 2400 304 sh (\() 1126 1208 sh (\)) 2336 1208 sh end MTsave restore dMATH#C T i = 12 h(f i+1 +f i )I T = 12 h(f 1 +f N )+hf ii=2N-1 DSET2Hbtnl  $T Tel6+J?`*vvb @@dxpr. CWEE.bGrphbj , Symbol `.( " currentpoint ",Times(I)4-,Times)HI 8+T `(=)H-(V1(r12"wv_(()b)@-)Ha)0))#h 8(f2 `+0+f)0')')()z)<))# for some (z)P)X[)#a),,)b)/]),)  i.e.,(I)4-)HI 8+T `(=)H-(S1(r12"uv_(()b)@-)Ha)0))#h 8(d2 `+W+f)0')()b)0))0-)Xf)/')()$a)/)z`(8[(D](t+)HO)C()$h 8(d[4 `+'+)(^Truncation error is (^O)D()#h 8(32 `+$+))#" and the error is dependent on the( +magnitude of first derivative at the edges./MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 9120 div 3136 3 -1 roll exch div scale currentpoint translate 64 56 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1432 430 rndmvto 384 0 rlineto stroke 1432 1444 rndmvto 384 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (I) 0 526 sh (I) 496 526 sh (b) 1990 526 sh (a) 2534 526 sh (h) 2866 526 sh (f) 3282 526 sh (a) 6511 526 sh (b) 6797 526 sh (I) 0 1540 sh (I) 496 1540 sh (b) 1990 1540 sh (a) 2534 1540 sh (h) 2866 1540 sh (f) 3438 1540 sh (b) 3850 1540 sh (f) 4584 1540 sh (a) 5012 1540 sh (O) 5808 1540 sh (h) 6220 1540 sh (O) 2967 2368 sh (h) 3379 2368 sh 224 ns (T) 616 622 sh (T) 616 1636 sh 384 /Symbol f1 (-) 208 526 sh (=) 888 526 sh (-) 1176 526 sh (-) 2246 526 sh (\316) 6017 526 sh (-) 208 1540 sh (=) 888 1540 sh (-) 1176 1540 sh (-) 2246 1540 sh (-) 4232 1540 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1270 /Symbol f3 ([) 3232 1572 sh (]) 5328 1572 sh 384 /Symbol f1 (+) 5520 1540 sh 384 /Times-Roman f1 (1) 1528 288 sh (12) 1416 822 sh (1) 1528 1302 sh (12) 1416 1836 sh 224 ns (2) 3090 354 sh (2) 3090 1368 sh (4) 6444 1368 sh (2) 3603 2196 sh 384 /Times-Roman f1 (\() 1864 526 sh (\)) 2724 526 sh (\251) 3472 526 sh (\251) 3568 526 sh (\() 3664 526 sh (\)) 4012 526 sh (,) 7081 526 sh (\() 1864 1540 sh (\)) 2724 1540 sh (\251) 3628 1540 sh ,(\() 3724 1540 sh (\)) 4040 1540 sh (\251) 4774 1540 sh (\() 4870 1540 sh (\)) 5202 1540 sh (\() 6078 1540 sh (\)) 6602 1540 sh (\() 3237 2368 sh (\)) 3745 2368 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (z) 3774 526 sh (z) 5697 526 sh 384 /Times-Roman f1 ( for some ) 4154 526 sh ([) 6369 526 sh (,) 6685 526 sh (]) 6987 526 sh ( i.e.,) 7209 526 sh (Truncation error is ) 0 2368 sh ( and the error is dependent on the) 3887 2368 sh (magnitude of first derivative at the edg) -16 2944 sh (es.) show end MTsave restore dMATH I-I T =-112(b-a)h 2 f(z) for some z[a,b], i.e.,I-I T =-112(b-a)h 2 f(b)-f(a)[]+O(h 4 )Truncation error is O(h 2 ) and the error is dependent on themagnitude of first derivative at the edges.enDSETT%%%lll  &l&Truncation Error for Trapezoidal ruleDSETT܂lll @1E1p111l$-5-5 lImproving accuracy Decrease h Number of function evaluations goes up Compare calculation with h against h/2 ... Run into machine precision Unequal bins helps More bins where function is rapidly varying Adaptive algorithmDSET2Hbtnl    $T Jh*DSETTl,l$l(  l0Romberg IntegrationDSETT@@[l@l4l8l<XSS(3A67Z'Z3B5D6 Z(H4$B@ҁ0Z$4,"? \lD\Recursive method with corrections to cancel errors  Extended Simpsons rule - column 2 DSET2H HlvelHld@6*dd' @@P 8dxprB CWEEB'Grphbj P 8, MT Extra `.(P , Symbol( "P 8 currentpoint ",Times(-R)P=,Times(UR 8+21),)1 `(R 8+:2),)1 `(R 8+:2),)2 `(uR 8+:3),)1 `(uR 8+:3),)2 `(ukR 8+:3).)3 `(M)M)M)O( Recursive trapeziodal rule:( )R 8+:i),)1 `(=)LI 8+T `(P()#h 8+2i `()(d Richardson extrapolation:(d )R 8+:i)+)1),)j)+)1 `(d=)LR 8+:i)+)1),)j `(d$+()1(x4 8(j `+,+-)<1 JpJO(dR 8+:i)+)1),)j `(dI-)HR 8+:i),)j ``(t\[(t ](& Romberg integral value:(& )R 8+:N),,)N/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 9440 div 4416 3 -1 roll exch div scale currentpoint translate 64 53 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 6531 3326 rndmvto 896 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (R) 0 1152 sh (R) 648 288 sh (R) 624 864 sh (R) 1494 864 sh (R) 624 1440 sh (R) 1494 1440 sh (R) 2412 1440 sh (R) 4228 2592 sh (I) 5066 2592 sh (h) 5518 2592 sh (R) 4071 3422 sh (R) 5421 3422 sh (R) 7617 3422 sh (R) 8711 3422 sh (R) 3826 4198 sh 224 ns (i) 4460 2688 sh (T) 5186 2688 sh (i) 5717 2688 sh (i) 4303 3518 sh (j) 4701 3518 sh (i) 5653 3518 sh (j) 6051 3518 sh (j) 6835 3546 sh (i) 7849 3518 sh (j) 8247 3518 sh (i) 8943 3518 sh (j) 9117 3518 sh (N) 4058 4294 sh (N) 4296 4294 sh 384 /Symbol f1 (=) 320 1152 sh (=) 4762 2592 sh (=) 5117 3422 sh (+) 6227 3422 sh (-) 7011 3718 sh (-) 8423 3422 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1544 /Symbol f3 ([) 7475 3486 sh (]) 9227 3486 sh 224 /Symbol f1 (+) 4383 3518 sh (+) 4797 3518 sh (+) 5733 3518 sh (+) 7929 3518 sh 224 /Times-Roman f1 (1) 848 384 sh (1) 974 384 sh (2) 856 960 sh (1) 998 960 sh (2) 1726 960 sh (2) 1900 960 sh (3) 856 1536 sh (1) 998 1536 sh (3) 1726 1536 sh (2) 1900 1536 sh (3) 2644 1536 sh (3) 2812 1536 sh (1) 4554 2688 sh (1) 4495 3518 sh (1) 4909 3518 sh (1) 5845 3518 sh (1) 8041 3518 sh j384 ns (1) 6883 3184 sh (4) 6563 3718 sh (1) 7251 3718 sh 224 /Times-Roman f1 (,) 942 384 sh (,) 966 960 sh (,) 1836 960 sh (,) 966 1536 sh (,) 1836 1536 sh (.) 2756 1536 sh (,) 4522 2688 sh (,) 4589 3518 sh (,) 5939 3518 sh (,) 8135 3518 sh (,) 9005 3518 sh (,) 4232 4294 sh 384 ns (\() 5376 2592 sh (\)) 5811 2592 sh 384 /Mt-ExtraPlain f1 (M) 795 2016 sh (M) 1689 2016 sh (M) 2604 2016 sh (O) 3276 2016 sh 384 /Times-Roman f1 (Recursive trapeziodal rule:) -16 2592 sh ( ) 4132 2592 sh (Richardson extrapolation:) -16 3422 sh ( ) 3975 3422 sh (Romberg integral value:) -16 4198 sh ( ) 3730 4198 sh end MTsave restore dMATH: R=R 1,1 R 2,1 R 2,2 R 3,1 R 3,2 R 3.3 MMMORecursive trapeziodal rule: R i,1 =I T (h i )Richardson extrapolation: R i+1,j+1 =R i+1,j +14 j -1R i+1,j -R i,j []Romberg integral value: R N,N fDSET2Hbray/FlhP,ll#6*P " @@ dxpr CWEE"Grphbj  , MT Extra `.( , Symbol( " currentpoint ",Times(tf)/()(x)/))$dx 8($a(6$b (  `(,Times(dh+3"?(af)0()#a)0))0+)L4)Gf)0()#a)@+)Lh)4))0+)H2)Cf)0()#a)@+)H2)4h)3))0+)L4 )4K)l+)H2)Df)/() b)@-)D2)3h)4))0+)L4)Gf)0()b)@-)Hh)4))0+)\f)/() b)/)z`(.[( ]/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 13024 div 1088 3 -1 roll exch div scale currentpoint translate 64 48 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1864 496 rndmvto 256 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (f) 400 592 sh (x) 748 592 sh (dx) 1080 592 sh (h) 1896 354 sh (f) 2374 592 sh (a) 2706 592 sh (f) 3678 592 sh (a) 4010 592 sh (h) 4570 592 sh (f) 5526 592 sh (a) 5858 592 sh (h) 6608 592 sh (f) 8508 592 sh (b) 8824 592 sh (h) 9558 592 sh (f) 10546 592 sh (b) 10862 592 sh (h) 11406 592 sh (f) 12172 592 sh (b) 12488 592 sh 224 ns (a) 80 992 sh (b) 80 168 sh 384 /Times-Roman f1 (\() 590 592 sh (\)) 938 592 sh (\() 2564 592 sh (\)) 2896 592 sh (\() 3868 592 sh (\)) 4776 592 sh (\() 5716 592 sh (\)) 6814 592 sh (\() 8698 592 sh (\)) 9764 592 sh (\() 10736 592 sh (\)) 11612 592 sh (\() 12362 592 sh (\)) 12678 592 sh 576 /Symbol f1 (\362) 64 656 sh 384 ns (\273) 1560 592 sh (+) 3088 592 sh (+) 4266 592 sh (+) 4968 592 sh (+) 6114 592 sh (+) 7006 592 sh (+) 7950 592 sh (-) 9080 592 sh (+) 9956 592 sh (-) 11118 592 sh (+) 11804 592 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1270 /Symbol f3 ([) 2168 624 sh (]) 12804 624 sh 384 /Times-Roman f1 (3) 1912 888 sh (4) 3392 592 sh (2) 5256 592 sh (2) 6402 592 sh (4) 7310 592 sh (2) 8238 592 sh (2) 9352 592 sh (4) 10260 592 sh 9384 /Mt-ExtraPlain f1 (K) 7518 592 sh end MTsave restore dMATH2 f(x)dx ab  h3f(a)+4f(a+h)+2f(a+2h)+4K+2f(b-2h)+4f(b-h)+f(b)[]DSET2H lp   $S! 9t@8 lt#G0O"?`*  @@PxdxprR CWEERGrphbj Px, Symbol `.( "Px currentpoint ",Times(S!Trapezoidal rule truncation error(LE 8+>T `(()$h 8+1n `(6))4=,Times)LI)4-)HI 8+T `(()$h 8+2n `())4=)H-(1(h12"_((h 8+2n(`2 `+W+f)0')()b)0))0-)Xf)/')()$a)/)z`(&[(&](+)HO)C()$h 8+1n(4 `+'+)(E 8+>T `(()$h 8+1n)$+)1 `(6))4=)LI)4-)HI 8+T `(()$h 8+2n)$+)1 `(4))4=)H-(1(f48"k(lh 8+2n(2 `+W+f)0')()b)0))0-)Xf)/')()$a)/)z`($[($](+)HO)C()$h 8+1n(4 `+'+)(!I)4-)HR 8+:n)$+)1),)2 `(!j=)LI)4-)wI 8+T `(!()#h 8+2n)$+)1 `(!d))0+(1+3"7(!GI 8+T `(!()#h 8+2n)$+)1 `(!I))0-)HI 8+T `(!()#h 8+2n `(!)z`()$[()](. *D *F ( *D *F (@j=)H-(#1( 48"&k(@+(1+3"&7(Z1(D48"&h.*DSETT-,ll l}@  l Three Point Gauss-Legendre RuleDSETTWX.,Xl(ll l${}D{0AAA0X2[WXJDJ0Z2PG_K]N_O]PPS]T_UbV]WZl/(\tE@7A@00(*{ RW1(=d<V Yl,YThree points - cubic  Roots of the polynomial  Weights, considering f(x)=1,x and x3 DSET2H/ l0".l4@6*zzt @@Xdxpr7 CWEE7tGrphbj X, Symbol `.(  "X currentpoint ",Times(] q)/()(x)/))4=,Times)Hc 8+00 `(]r+)Hc 8+(1 `(]x)@+)Hc 8+02 `(]x 8(22 `+4++)Hc 8+03 `(]x 8(23 `(9`q)/()(x)/))$dx 8(-)1($1 (I  `(9~=)xq)Z()'x)0))#dx 8(-)1(1 (I `(9j=)x 8(B2 `++q)0()'x)0))#dx 8(-)1(1 (I `(9|=)H0(P )t2)/c 8+00 `(P+(c2+3"6_7(Pc 8+02 `(P =([2+3"6W7(Pc 8+(1 `(P+(C2*5"6?7(Pc 8+03 `(P=(;2+3"677(P{c 8+00 `(P+(/2*5"6+7(Poc 8+02 `(P=)H0(N q)/()(x)/))4=(5*2"47(N^x 8(#3 `+4+-(3(2"47(NZx/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6496 div 3712 3 -1 roll exch div scale currentpoint translate 64 32 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1342 2242 rndmvto 224 0 rlineto stroke 2334 2242 rndmvto 224 0 rlineto stroke 3262 2242 rndmvto 224 0 rlineto stroke 4254 2242 rndmvto 224 0 rlineto stroke 5230 2242 rndmvto 224 0 rlineto stroke 1034 3256 rndmvto 224 0 rlineto stroke 2042 3256 rndmvto 224 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (q) -16 340 sh (x) 332 340 sh (c) 1018 340 sh (c) 1706 340 sh (x) 1994 340 sh (c) 2538 340 sh (x) 2874 340 sh (c) 3578 340 sh (x) 3914 340 sh (q) 320 1220 sh (x) 668 1220 sh (dx) 1000 1220 sh (xq) 2120 1220 sh (x) 2638 1220 sh (dx) 2970 1220 sh (x) 4090 1220 sh (q) 4410 1220 sh (x) 4758 1220 sh (dx) 5090 1220 sh (c) 638 2338 sh (c) 1614 2338 sh (c) 2606 2338 sh (c) 3534 2338 sh (c) 4526 2338 sh (c) 5502 2338 sh (q) -16 3352 sh (x) 332 3352 sh (x) 1338 3352 sh (x) 2346 3352 sh 384 /Times-Roman f1 (\() 174 340 sh (\)) 522 340 sh (\() 510 1220 sh (\)) 858 1220 sh (\() 2480 1220 sh (\)) 2828 1220 sh (\() 4600 1220 sh (\)) 4948 1220 sh (\() 174 3352 sh (\)) 522 3352 sh 384 /Symbol f1 (=) 730 340 sh (+) 1418 340 sh (+) 2250 340 sh (+) 3290 340 sh (=) 1464 1220 sh (=) 3434 1220 sh (=) 5554 1220 sh (\336) -16 2338 sh (+) 1038 2338 sh (=) 2030 2338 sh (+) 2958 2338 sh (=) 3950 2338 sh (+) 4926 2338 sh @(=) 5918 2338 sh (=) 730 3352 sh (-) 1754 3352 sh 224 ns (-) 27 1620 sh (-) 1795 1620 sh (-) 3765 1620 sh 576 ns (\362) 64 1284 sh (\362) 1832 1284 sh (\362) 3802 1284 sh 224 /Times-Roman f1 (0) 1210 436 sh (1) 1866 436 sh (2) 2730 436 sh (2) 3082 168 sh (3) 3770 436 sh (3) 4122 168 sh (1) 149 1620 sh (1) 80 796 sh (1) 1917 1620 sh (1) 1848 796 sh (2) 4298 1048 sh (1) 3887 1620 sh (1) 3818 796 sh (0) 830 2434 sh (2) 1806 2434 sh (1) 2766 2434 sh (3) 3726 2434 sh (0) 4718 2434 sh (2) 5694 2434 sh (3) 1546 3180 sh 384 ns (0) 5842 1220 sh (2) 448 2338 sh (2) 1358 2100 sh (3) 1374 2634 sh (2) 2350 2100 sh (3) 2366 2634 sh (2) 3278 2100 sh (5) 3278 2634 sh (2) 4270 2100 sh (3) 4286 2634 sh (2) 5246 2100 sh (5) 5246 2634 sh (0) 6206 2338 sh (5) 1050 3114 sh (2) 1050 3648 sh (3) 2074 3114 sh (2) 2058 3648 sh end MTsave restore dFMATH:Z q(x)=c 0 +c 1 x+c 2 x 2 +c 3 x 3 q(x)dx -11  =xq(x)dx -11  =x 2 q(x)dx -11  =02c 0 +23c 2 =23c 1 +25c 3 =23c 0 +25c 2 =0q(x)=52x 3 -32xDSET2H0bray/Fl8.P*l<#6*P   { @@dxpr CWEE {Grphbj , Symbol `.( " currentpoint ",Times(x 8+(1 `(p=)H-,Times(d83(45"07"#b (#D(p,)$x 8+02 `(=)H0)+,)$x 8+03 `(=(d3(5"7"G#b k#DG/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3936 div 1024 3 -1 roll exch div scale currentpoint translate 64 34 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1154 510 rndmvto 224 0 rlineto stroke /sqr { 1.5 mul 3 index div 6 1 roll gsave translate dup dup neg scale 4 -1 roll 2 div rnddist 2 mul /thick exch def dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .34 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .34 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 516 934 384 894 934 16 sqr 16 th 3554 510 rndmvto 224 0 rlineto stroke 516 934 384 3294 934 16 sqr /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (x) 16 606 sh (x) 1552 606 sh (x) 2574 606 sh 224 /Times-Roman f1 (1) 176 702 sh (2) 1744 702 sh (3) 2766 702 sh 384 ns (3) 1186 368 sh (5) 1170 902 sh (0) 2256 606 sh (3) 3586 368 sh (5) 3570 902 sh 384 /Symbol f1 (=) 384 606 sh (-) 672 606 sh (=) 1968 606 sh (=) 2990 606 sh 384 /Times-Roman f1 (,) 1408 606 sh (,) 2430 606 sh end MTsave restore dMATH x 1 =- 35  ,x 2 =0,x 3 = 35 DSET2H1 l@.P{lD#6*Pffd @@ dxpr/ CWEE/dGrphbj  , Symbol `.( "  currentpoint ",Times(Pw 8+<1 `(P|+,Times)Lw 8+D2 `(P@+)Lw 8+D3 `(P=)H2( - 8(3*>5"'"C#: g|#4 `(w 8+<1 `( + 8(3*>5"'"l#: #4 `(w 8+D3 `(Y=)H0(3(5"7(Xw 8+<1 `(+(3(5"7(Xw 8+D3 `(=($2+3" 7( i.e.,) )w 8+<1 `()=(y5*9"u7(,) w 8+D2 `(Q=(8*9"7(,) w 8+D3 `(y=(5*9"7/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4128 div 3200 3 -1 roll exch div scale currentpoint translate 64 34 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 466 864 rndmvto 160 0 rlineto stroke /sqr { 1.5 mul 3 index div 6 1 roll gsave translate dup dup neg scale 4 -1 roll 2 div rnddist 2 mul /thick exch def dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .34 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .34 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 452 528 384 206 1104 16 sqr 16 th 1654 864 rndmvto 160 0 rlineto stroke 452 528 384 1394 1104 16 sqr 16 th 0 1726 rndmvto 224 0 rlineto stroke 1024 1726 rndmvto 224 0 rlineto stroke 2112 1726 rndmvto 224 0 rlineto stroke 1428 2740 rndmvto 224 0 rlineto stroke 2612 2740 rndmvto 224 0 rlineto stroke 3796 2740 rndmvto 224 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (w) 0 288 sh (w) 736 288 sh (w) 1520 288 sh (w) 658 960 sh (w) 1846 960 sh (w) 288 1822 sh (w) 1312 1822 sh (w) 676 2836 sh (w) 1812 2836 sh (w) 2996 2836 sh 224 /Times-Roman f1 (1) 240 384 sh (2) 1008 384 sh (3) 1792 384 sh (3) 498 824 sh (5) 498 1072 sh (1) 898 1056 sh (3) 1686 824 sh (5) 1686 1072 sh (3) 2118 1056 sh (1) 528 1918 sh (3) 1584 1918 sh (1) 916 2932 sh (2) 2084 2932 sh (3) 3268 2932 sh 384 ns (2) 2304 288 sh (0) 2630 960 sh (3) 32 1584 sh (5) 16 2118 sh (3) 1056 1584 sh (5) 1040 2118 sh (2) 2128 1584 sh (3) 2144 2118 sh (5) 1444 2598 sh (9) 1444 3132 sh (8) 2628 2598 sh (9) 2628 3132 sh (5) 3812 2598 sh (9) 3812 3132 sh 384 /Symbol f1 (+) 432 288 sh (+) 1216 288 sh (=) 2016 288 sh (-) -16 960 sh (+) 1090 960 sh (=) 2342 960 sh (+) 720 1822 sh (=) 1808 1822 sh (=) 1124 2836 sh (=) 2308 2836 sh (=) 3492 2836 sh 384 /Times-Roman f1 (i.e.,) -16 2836 sh ( ) 580 2836 sh 384 /Times-Roman f1 (,) 1684 2836 sh (,) 2868 2836 sh end MTsave restore dMATH~0 w 1 +w 2 +w 3 =2-  35 w 1 +  35 w 3 =035w 1 +35w 3 =23i.e., w 1 =59,w 2 =89,w 3 =59DSET2H2lH  $T3 533k4*DSETT32lTlLlP}H  lXImproper IntegralsDSETToo42 lhl\l`ldX}L P(<+IsKp!CqCRV'"( ;8</H9IwK-pnonpnEuESrtlI5jj4t%m6(bK7r7(F6r llWhat if either or both the limits are Change of variables a=- or b=+  a=- and b=+, or range includes x=0 Divide the integral into two ranges: -0 and 0 Integrands with power law singularities at limits  Integrands falls off exponentially DSET2H5uatilp4Pu4lt#6*P~~%t @@(dxpr CWEE%tGrphbj (, Symbol `.( "( currentpoint ",Times(tf)/()(x)/))$dx 8($a(6 +z `(=,Times(dD1(:t 8(^2"2S `(f)/((d1+ t"+(())$dt ((1+*a" 8(60 (/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3712 div 1184 3 -1 roll exch div scale currentpoint translate 64 48 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2184 496 rndmvto 336 0 rlineto stroke 2980 496 rndmvto 176 0 rlineto stroke 8 th 1934 944 rndmvto 100 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (f) 400 592 sh (x) 748 592 sh (dx) 1080 592 sh (t) 2216 888 sh (f) 2648 592 sh (t) 3012 888 sh (dt) 3312 592 sh 224 ns (a) 80 992 sh 160 ns (a) 1952 1088 sh 384 /Times-Roman f1 (\() 590 592 sh (\)) 938 592 sh (\() 2838 592 sh (\)) 3170 592 sh 224 /Symbol f1 (\245) 56 168 sh 576 ns (\362) 64 656 sh (\362) 1912 656 sh 384 ns (=) 1544 592 sh 384 /Times-Roman f1 (1) 2256 354 sh (1) 2972 354 sh 224 ns (2) 2360 716 sh (0) 1936 168 sh 160 ns (1) 1944 920 sh end MTsave restore dMATHa f(x)dx a  =1t 2 f(1t)dt  1a 0 (DSET2H6bray/Flx4Pbl|#6*PK @@Xdxpr+ CWEE+KGrphbj X, Symbol `.( "X currentpoint ",Times(c f)/()(x)/))8,Times)H()(x)@-)Ha)/) 8(8%-) g `+#+,)  )0)D<)Hg)D<)@1( then let,(H )t)4=)H()(x)@-)Ha)/) 8(1)-) g `+$+,)  for )g)D= 8(1+>2"' `(tf)/()(x)/))$dx)t=)2)4tf)J()#a)@+)Lt 8(2 `+$+))#dt 8(O0(b) -)$a"p# " #i ( 8(O$a($b ( /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4832 div 2400 3 -1 roll exch div scale currentpoint translate 64 56 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 4554 872 rndmvto 160 0 rlineto stroke /sqr { 1.5 mul 3 index div 6 1 roll gsave translate dup dup neg scale 4 -1 roll 2 div rnddist 2 mul /thick exch def dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .34 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .34 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 566 232 224 1848 1504 9 sqr /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (f) 64 340 sh (x) 412 340 sh (x) 1272 340 sh (a) 1816 340 sh (t) 1346 968 sh (x) 2000 968 sh (a) 2544 968 sh (f) 400 1910 sh (x) 748 1910 sh (dx) 1080 1910 sh (tf) 2668 1910 sh (a) 3106 1910 sh (t) 3666 1910 sh (dt) 4094 1910 sh 224 ns (b) 2012 1486 sh (a) 2284 1486 sh (a) 80 2310 sh (b) 80 1486 sh 384 /Times-Roman f1 (\() 254 340 sh (\)) 602 340 sh (\() 1114 340 sh (\)) 2006 340 sh (\() 1842 968 sh (\)) 2734 968 sh (,) 3212 968 sh (\() 590 1910 sh (\)) 938 1910 sh (\() 2964 1910 sh (\)) 3952 1910 sh 384 /Symbol f1 (\273) 826 340 sh (-) 1528 340 sh (<) 2912 340 sh (<) 3472 340 sh (=) 1554 968 sh ](-) 2256 968 sh (=) 4250 968 sh (=) 1544 1910 sh (+) 3362 1910 sh 224 ns (-) 2134 168 sh (-) 2942 796 sh (-) 2140 1486 sh 576 ns (\362) 2059 1974 sh (\362) 64 1974 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 224 /Symbol f2 (g) 2260 168 sh (g) 3070 796 sh 384 ns (g) 3200 340 sh (g) 3978 968 sh 384 /Times-Roman f1 (,) 2402 340 sh ( ) 2530 340 sh (then let,) -16 968 sh ( ) 1250 968 sh ( for ) 3340 968 sh 384 /Times-Roman f1 (0) 2640 340 sh (1) 3728 340 sh (2) 2462 1910 sh 224 ns (1) 2846 796 sh (1) 4578 832 sh (2) 4586 1080 sh (2) 3810 1738 sh (0) 2083 2310 sh end MTsave restore dJMATH>A f(x)(x-a) -g , 0<g<1then let, t=(x-a) 1-g , for g= 12 f(x)dx=2tf(a+t 2 )dt 0 b-a   ab DSET2H7bray/Fl4PFl@6*P6 @@dxpr CWEE6Grphbj , Symbol `.(0 " currentpoint ",Times(Sx)D=)H-,Times)<log)t(<tf)/()(x)/))$dx)t=)f)0()-)<log)t))(dt+t""[ 8(0(e ((-)a (L 8($a( +z/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4192 div 1728 3 -1 roll exch div scale currentpoint translate 64 46 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3706 1124 rndmvto 368 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (x) 16 288 sh (t) 1354 288 sh (f) 400 1220 sh (x) 748 1220 sh (dx) 1080 1220 sh (f) 2310 1220 sh (t) 3404 1220 sh (dt) 3738 982 sh (t) 3834 1516 sh 224 ns (e) 1848 796 sh (a) 80 1620 sh 160 ns (a) 2086 696 sh 384 /Symbol f1 (=) 288 288 sh (-) 576 288 sh (=) 1544 1220 sh (-) 2626 1220 sh 160 ns (-) 1976 696 sh 576 ns (\362) 1943 1284 sh (\362) 64 1284 sh 224 ns (\245) 56 796 sh 384 /Times-Roman f1 (log) 816 288 sh (\() 590 1220 sh (\)) 938 1220 sh (\() 2500 1220 sh (log) 2866 1220 sh (\)) 3530 1220 sh 224 /Times-Roman f1 (0) 1967 1620 sh end MTsave restore dMATH  x=-logtf(x)dx=f(-logt)dtt 0e -a  a jDSET2H8sg9l  $S9 A"k:`*DSETT98lll}P  lQuadrature summaryDSETTBf:8 #lllI}Tp'CScCCfBhghhh fvY" 'E8\9E>a?E@`B`CES$cEt\uEzc~E$E\EdEdeE$iBmfjgk l g  l Weight function, r(x)=1 Gauss-Legendre Weight function, r(x)=xae-x for (0 < x < ) Gauss-Leguerre Weight function, r(x)=e-x2 for (- < x < ) Gauss-Hermite Weight function, r(x)=(1-x)a(1+x)b for (-1 < x < 1) Gauss-Jacobi Use tables of roots and associated weights to evaluate integrals using appropriate weight function. Use Romberg for polynomial functions Use Gauss-Legendre for most other functions Use Gauss-Leguerre for exponentially damped functions Use Gauss-Hermite for peaked functions with tails on both ends... DSET2H;  $T< :f|ff=Ơ::#*uud HHd ddHHHHhVff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""dd%IYIdemotJcdlmopu  )*GHef89VWtuKbcmopqLbdklopMlmOkmop)*GHefVWtuPhjmopqQdehjlpS  89`u{au{vu{wu{7L078KLS,018KRSWu{(,-1RVW[u{%()-VZ[^"%&)Z]^a "#&]`ad !#`cdf!cefheghjgijliklnkmnpmopqopqsprstrstvsuvwuvwx 2Cvwxy  -23BCHwxyz   *-.3BGHLxyz{    '*+.GKLNyz{}    %'(+KMNPz|}~  #%&(MOPR|}u{ !#$&4?OQRT~u{   !"$/45>?DQSTV}~  !",/05>CDGSUVW~!),-0CFGJUVWY')*-1AFIJLVXYZ&'(*.12@AEIKLNXYZ[$&'(+./DEGKMNOYZ[\#$%')+,236;?@CDFGIMNOQZ[\]"#$%()*/01367:;?ABCHIKNPQR[\]^!"#$&(),-.0127:@ABDEFJKLPQRS\] !"#%&'*+,-./2@CDEFGHIJKLNQRST]_ !"$%&'()+,-/CEFHIJKMNORSTU_` !#$%(+-EFGHJLMNST^_ "#$%'(*+FHJKLMOPUV`a#$%&'()*HIJLNOPQTUVW_`!"$%()IJKLMNOPQRUVab"$%&'(JKLMNOPQWX`a !$&LNRSVW!"&'KLOPQRbc"#$%MOabbflp #$%&LMXYcefgmp !STWXacefmpuRSmopquKLbclmop !abefkl #$%&RScefghjlp"#$%LMSTbfhjkqJKMOWX!"$%&'OPXY !#$%&'(IJLNQR"#%'()HIKLNORSVW!"#%&()*GHLMPQWXab "#$&')*+FGJLMNOPQRUVbc !#$%'(*+,DFIKLMNOPQVW !"$%&()*,.CDEFHJKLOPTU`a !"#%&'*.0@CEFGIJKMNSTUVab !"#$&'(,04<@ABEFGHIJKMNORSTU_`  "#$&')*.234<=>ABCDEFGIJKLNST`a #%&')+,./123=>?BCDEGHJKLPR^_  %&')+./12>?BCEFHIKOPQS]^_`     &()+.12>?CDFGINOPQ^_     "(*+.1?ADEGLNOP\]   !"$*-.3<ABEKLMO[\]^  !#$&-23<=BIKLMZ[\] #%&(2=FIJLYZ[\u{%'(+DFGJXYZ[u{'*+/@DEGWXYZ*./@AEUWXY.ATUVXRTUVPRSUOPQSLOPQ JLMP #GJKM"#'CGHK"&',>CDH&+,>?D+?u{u{~}~~|}{|}~z{|}yz{|xyz{wxyzvwxyuvwxtuvwstuvrstuqrstu{pqrsu{npqrmnoqlmnoklmniklmhijlfhijefgicefgbcdf`bcd_`ac]_`a []^` Y[\^ WYZ\ VWXZ TVWXRTUWORSUMOPSu{KMNPu{HKLNFHILCFGIACDG>ABD;>?B8;<?589<1569.126*./2&*+/!&'+!"' "!" #  $ %bflp&cefgmopq'acefkmop(ud)bclmuTde*abefklopITUe+cefghjmopq@IJU,bcefhilmop9@AJ-ceijmo29:A.,23:/',-30"'(-1"#(2#3456 7  8  9 :;<=>u{?u{@ABCDEFGHIJKLMNOPQRSTu{Uu{VWXYZ[\]^_`abcdefghiju{ku{lmnopqrstuv"w!"%x!$%(y$'(*z')*,{)+,.|+-./}-./1~.0120124u{1345u{3456!4567 !#5678 "#%6789"$%&789:$%&'89:;%&'(9:!&'();< !"'()*:;<= !"#()*+;<!"#$)*+,=> "#$%*+,-<=>? !#$%&+,=> !"$%-.!"#&',-?@!"%&./>? !#$'(-.@A "#&'?@/0bflp !"()./cefgmp !$%'(acefmp#$ABmopqu@AY`bclmopuY`abkl#$cfhjlp !$%'(bceghikmoq !"()ceijmo./ "#/0 "#$&' "#%&'(-.@A !&'./AB !"$% !#$%&,- $%+,-.?@"#,-@A   "#$*+   !#)*+,>? !()*+?@ '()*=>&'()>?%&'(<=#%&'=>"#$&;<u{ "#$<=u{ !#:;!9:;<89:;789:6789567845673456234512340123.012-./1+-./)+,.()*,&()*#&')!#$'!"$"u{u{    u{u{u{u{  89celpbcmopqbdkmop)*GHefVWtuY`Y`lmophjklmopqhilmopu  )*GHef89VWtudeijmotsu| 15;?rs{|}  2456<?rt|}0245<?<>?@ip12;<>?ip|}01:<xz{|}  2579:;<?yz|}   124678;<>@tuxy}  2489<>_bcbb*+""!" &+*+/48AFDEA;@@:=?<;CGFOJMG?;?A ABACAGD<83./*&&&!$$$&$#$"$'&*/-48>FCD AB>>? ?<?<CIKKJLFJ@<;B@=BDEEA93/+*+%"""""'*bbfcc4JJRSTGN%!PS-Adobe-3.0 EPSF-3.0 %%Title: (cornu.pdf) %%Version: 1 2 %%CreationDate: (D:20010917132303) %%DocumentData: Clean7Bit %%LanguageLevel: 2 %%BoundingBox: 0 0 792 612 %%Pages: 0 %%DocumentProcessColors: Black %%DocumentSuppliedResources: %%+ procset (Adobe Acrobat - PDF operators) 1.2 0 %%EndComments 0 0 moveto 792 0 lineto 792 612 lineto 0 612 lineto closepath clip newpath %%BeginProlog %%EndProlog %%BeginSetup %%BeginResource: l2check %%Copyright: Copyright 1993 Adobe Systems Incorporated. All Rights Reserved. systemdict /languagelevel known { systemdict /languagelevel get 1 eq } { true } ifelse { initgraphics /Helvetica findfont 18 scalefont setfont 72 600 moveto (Error: Your printer driver needs to be configured) dup show 72 580 moveto (for printing to a PostScript Language Level 1 printer.) dup show exch = = /Helvetica-Bold findfont 16 scalefont setfont 72 520 moveto (Windows and Unix) show /Times-Roman findfont 16 scalefont setfont 72 500 moveto (Select \252Language Level 1\272 in the PostScript options section) show 72 480 moveto (of the Acrobat print dialog.) show /Helvetica-Bold findfont 16 scalefont setfont 72 440 moveto (Macintosh) show /Times-Roman findfont 16 scalefont setfont 72 420 moveto (In the Chooser, select your printer driver.) show 72 400 moveto (Then select your printer and click the Setup button.) show 72 380 moveto (Follow any on-screen dialogs that may appear.) show showpage quit } if %%EndResource /currentpacking where{pop currentpacking true setpacking}if %%BeginResource: procset pdfvars %%Copyright: Copyright 1987-1999 Adobe Systems Incorporated. All Rights Reserved. %%Version: 4.0 2 %%Title: definition of dictionary of variables used by PDF & PDFText procsets userdict /PDF 160 dict put userdict /PDFVars 86 dict dup begin put /_save 0 def /_cshow 0 def /InitAll 0 def /TermAll 0 def /DocInitAll 0 def /DocTermAll 0 def /_lp /none def /_doClip 0 def /sfc 0 def /_sfcs 0 def /_sfc 0 def /ssc 0 def /_sscs 0 def /_ssc 0 def /_fcs 0 def /_scs 0 def /_fp 0 def /_sp 0 def /AGM_MAX_CS_COMPONENTS 10 def /_fillColors [ 0 1 AGM_MAX_CS_COMPONENTS { array } for ] def /_strokeColors [ 0 1 AGM_MAX_CS_COMPONENTS { array } for ] def /_fc null def /_sc null def /DefaultGray [/DeviceGray] def /DefaultRGB [/DeviceRGB] def /DefaultCMYK [/DeviceCMYK] def /_inT false def /_tr -1 def /_rise 0 def /_ax 0 def /_cx 0 def /_ld 0 def /_tm matrix def /_ctm matrix def /_mtx matrix def /_hy (-) def /_fScl 0 def /_hs 1 def /_pdfEncodings 2 array def /_baselineadj 0 def /_fTzero false def /_Tj 0 def /_italMtx [1 0 .212557 1 0 0] def /_italMtx_WMode1 [1 -.212557 0 1 0 0] def /_italMtxType0 [1 0 .1062785 1 0 0] def /_italMtx_WMode1Type0 [1 -.1062785 0 1 0 0] def /_basefont 0 def /_basefonto 0 def /_pdf_oldCIDInit null def /_pdf_FontDirectory 30 dict def /_categories 10 dict def /_sa? true def /_op? false def /_OP? false def /_opmode 0 def /_ColorSep5044? false def /_tmpcolr? [] def /_tmpop? {} def /_processColors 0 def /_defaulttransfer currenttransfer def /_defaultflatness currentflat def /_defaulthalftone null def /_defaultcolortransfer null def /_defaultblackgeneration null def /_defaultundercolorremoval null def /_defaultcolortransfer null def end %%EndResource PDFVars begin PDF begin %%BeginResource: procset pdfutil %%Copyright: Copyright 1993-1999 Adobe Systems Incorporated. All Rights Reserved. %%Version: 4.0 2 %%Title: Basic utilities used by other PDF procsets /bd {bind def} bind def /ld {load def} bd /bld { dup length dict begin { null def } forall bind end def } bd /dd { PDFVars 3 1 roll put } bd /xdd { exch dd } bd /Level2? systemdict /languagelevel known { systemdict /languagelevel get 2 ge } { false } ifelse def /Level3? systemdict /languagelevel known {systemdict /languagelevel get 3 eq } { false } ifelse def /getifknown { 2 copy known { get true } { pop pop false } ifelse } bd /here { currentdict exch getifknown } bd /isdefined? { where { pop true } { false } ifelse } bd /StartLoad { dup dup not { /_save save dd } if } bd /EndLoad { if not { _save restore } if } bd %%EndResource %%BeginResource: procset pdf %%Version: 4.0 3 %%Copyright: Copyright 1998-1999 Adobe Systems Incorporated. All Rights Reserved. %%Title: General operators for PDF, common to all Language Levels. [/b/B/b*/B*/BDC/BI/BMC/BT/BX/c/cm/cs/CS/d/d0/d1/Do/DP/EI/EMC/ET/EX/f/f*/g/G/gs /h/i/j/J/k/K/l/m/M/MP/n/q/Q/re/rg/RG/ri/s/S/sc/SC/scn/SCN/sg/Tc/Td/TD/Tf/Tj/TJ /TL/Tm/Tr/Ts/Tw/Tz/T*/v/w/W/W*/y/'/" /applyInterpFunc/applystitchFunc/domainClip/EF/encodeInput/gsDI/ilp/icl /initgs/int/limit/PS/rangeClip/RC/rf/makePat/csfamily /? /! /| /: /+ /GetGlyphDirectory ] {null def} bind forall /v { currentpoint 6 2 roll c } bd /y { 2 copy c } bd /h/closepath ld /d/setdash ld /j/setlinejoin ld /J/setlinecap ld /M/setmiterlimit ld /w/setlinewidth ld /i { dup 0 eq { pop _defaultflatness } if setflat } bd /gsDI { begin /OP here { /_OP? xdd } if /op here { /_op? xdd } { /OP here { /_op? xdd } if } ifelse /OPM here { /_opmode xdd } if /Font here { aload pop Tf } if /LW here { w } if /LC here { J } if /LJ here { j } if /ML here { M } if /D here { aload pop d } if end } bd /ilp { /_lp /none dd } bd /icl { /_doClip 0 dd } bd /W { /_doClip 1 dd } bd /W* { /_doClip 2 dd } bd /n { {{} {clip} {eoclip}} _doClip get exec icl newpath } bd /s { h S } bd /B { q f Q S } bd /B* { q f* Q S } bd /b { h B } bd /b* { h B* } bd /q/save ld /Q { restore ilp } bd /GetCSFamily { dup type /arraytype eq {0 get} if } bd /GetCompsDict 11 dict begin /DeviceGray { pop 1 } bd /DeviceRGB { pop 3 } bd /DeviceCMYK { pop 4 } bd /CIEBasedA { pop 1 } bd /CIEBasedABC { pop 3 } bd /CIEBasedDEF { pop 3 } bd /CIEBasedDEFG { pop 4 } bd /DeviceN { 1 get length } bd /Separation { pop 1 } bd /Indexed { pop 1 } bd /Pattern { pop 0 } bd currentdict end def /GetComps { GetCompsDict 1 index GetCSFamily get exec } bd /cs { dup _fcs eq { pop } { dup /_fcs xdd GetComps _fillColors exch get /_fc xdd /_fp null dd } ifelse } bd /CS { dup _scs eq { pop } { dup /_scs xdd GetComps _strokeColors exch get /_sc xdd /_sp null dd } ifelse } bd /sc { _fc astore pop ilp } bd /SC { _sc astore pop ilp } bd /g { DefaultGray cs sc } bd /rg { DefaultRGB cs sc } bd /k { DefaultCMYK cs sc } bd /G { DefaultGray CS SC } bd /RG { DefaultRGB CS SC } bd /K { DefaultCMYK CS SC } bd /cm { _mtx astore concat } bd /re { 4 2 roll m 1 index 0 rlineto 0 exch rlineto neg 0 rlineto h } bd /RC/rectclip ld /EF/execform ld /PS { cvx exec } bd /initgs { /DefaultGray [/DeviceGray] dd /DefaultRGB [/DeviceRGB] dd /DefaultCMYK [/DeviceCMYK] dd 0 g 0 G [] 0 d 0 j 0 J 10 M 1 w true setSA /_op? false dd /_OP? false dd /_opmode 0 dd /_defaulttransfer load settransfer 0 i /RelativeColorimetric ri newpath } bd /int { dup 2 index sub 3 index 5 index sub div 6 -2 roll sub mul exch pop add exch pop } bd /limit { dup 2 index le { exch } if pop dup 2 index ge { exch } if pop } bd /domainClip { Domain aload pop 3 2 roll limit } [/Domain] bld /applyInterpFunc { 0 1 DimOut 1 sub { dup C0 exch get exch dup C1 exch get exch 3 1 roll 1 index sub 3 index N exp mul add exch currentdict /Range_lo known { dup Range_lo exch get exch Range_hi exch get 3 2 roll limit } { pop } ifelse exch } for pop } [/DimOut /C0 /C1 /N /Range_lo /Range_hi] bld /encodeInput { NumParts 1 sub 0 1 2 index { dup Bounds exch get 2 index gt { exit } { dup 3 index eq { exit } { pop } ifelse } ifelse } for 3 2 roll pop dup Bounds exch get exch dup 1 add Bounds exch get exch 2 mul dup Encode exch get exch 1 add Encode exch get int } [/NumParts /Bounds /Encode] bld /rangeClip { exch dup Range_lo exch get exch Range_hi exch get 3 2 roll limit } [/Range_lo /Range_hi] bld /applyStitchFunc { Functions exch get exec currentdict /Range_lo known { 0 1 DimOut 1 sub { DimOut 1 add -1 roll rangeClip } for } if } [/Functions /Range_lo /DimOut] bld %%EndResource %%BeginResource: procset pdflev2 %%Version: 4.0 5 %%Copyright: Copyright 1987-1999 Adobe Systems Incorporated. All Rights Reserved. %%LanguageLevel: 2 %%Title: PDF operators, with code specific for Level 2 /_defaulthalftone currenthalftone dd /_defaultblackgeneration currentblackgeneration dd /_defaultundercolorremoval currentundercolorremoval dd /_defaultcolortransfer [currentcolortransfer] dd /initialize { _defaulthalftone sethalftone /_defaultblackgeneration load setblackgeneration /_defaultundercolorremoval load setundercolorremoval _defaultcolortransfer aload pop setcolortransfer false setoverprint <> setuserparams } bd /terminate { } bd /m/moveto ld /l/lineto ld /c/curveto ld /setSA/setstrokeadjust ld /defineRes/defineresource ld /findRes/findresource ld currentglobal true systemdict /setglobal get exec [/Function /ExtGState /Form /Shading /FunctionDictionary /MadePattern /PatternPrototype /DataSource] { /Generic /Category findresource dup length dict copy /Category defineresource pop } forall systemdict /setglobal get exec /ri { /findcolorrendering isdefined? { mark exch findcolorrendering counttomark 2 eq { type /booleantype eq { dup type /nametype eq { dup /ColorRendering resourcestatus { pop pop dup /DefaultColorRendering ne { /ColorRendering findresource setcolorrendering } if } if } if } if } if cleartomark } { pop } ifelse } bd /_sfcs {_fcs setcolorspace} bind dd /_sscs {_scs setcolorspace} bind dd /_sfc { _fc aload pop _fp null eq { setcolor } { _fp setpattern } ifelse } bind dd /_ssc { _sc aload pop _sp null eq { setcolor} { _sp setpattern } ifelse } bind dd /scn { dup type /dicttype eq { dup /_fp xdd /PaintType get 1 eq { /_fc _fillColors 0 get dd ilp } { /_fc _fillColors _fcs 1 get GetComps get dd sc } ifelse } { sc } ifelse } bd /SCN { dup type /dicttype eq { dup /_sp xdd /PaintType get 1 eq { /_sc _strokeColors 0 get dd ilp } { /_sc _strokeColors _scs 1 get GetComps get dd SC } ifelse } { SC } ifelse } bd /gs { begin /SA here { setstrokeadjust } if /BG here { setblackgeneration } if /UCR here { setundercolorremoval } if /FL here { i } if /RI here { ri } if /TR here { dup xcheck { settransfer } { aload pop setcolortransfer } ifelse } if /sethalftonephase isdefined? { /HTP here { sethalftonephase } if } if /HT here { sethalftone } if currentdict gsDI end } bd /sfc { _op? setoverprint _lp /fill ne { _sfcs _sfc /_lp /fill dd } if } dd /ssc { _OP? setoverprint _lp /stroke ne { _sscs _ssc /_lp /stroke dd } if } dd /f { { { sfc fill } {gsave sfc fill grestore clip newpath icl ilp} {gsave sfc fill grestore eoclip newpath icl ilp} } _doClip get exec } bd /f* { { { sfc eofill } {gsave sfc eofill grestore clip newpath icl ilp} {gsave sfc eofill grestore eoclip newpath icl ilp} } _doClip get exec } bd /S { { { ssc stroke } {gsave ssc stroke grestore clip newpath icl ilp} {gsave ssc stroke grestore eoclip newpath icl ilp} } _doClip get exec } bd /rf { { { sfc rectfill } {gsave sfc rectfill grestore clip newpath icl ilp} {gsave sfc rectfill grestore eoclip newpath icl ilp} } _doClip get exec } bd /knownColorants? { pop false } bd /makePat { gsave dup /Matrix get concat matrix makepattern grestore /MadePattern defineRes pop } bd %%EndResource currentdict readonly pop end end /currentpacking where {pop setpacking}if PDFVars/DocInitAll{[]{/docinitialize get exec}forall }put PDFVars/InitAll{[PDF]{/initialize get exec}forall initgs}put PDFVars/TermAll{[PDF]{/terminate get exec}forall}put PDFVars begin PDF begin PDFVars/DocInitAll get exec PDFVars/InitAll get exec PDFVars/TermAll get exec end end %%EndSetup PDFVars begin PDF begin PDFVars/InitAll get exec 0 0 792 612 re W n [ 0 1 -1 0 792 0 ] concat /N23 << /SA false /TR {} /SM 0.02 >> /ExtGState defineRes pop 0 G 0 J 0 j 0.283 w 2.5 M []0 d /N23 /ExtGState findRes gs 1 i 93.912 675.071 m 93.912 72.87801 l 93.912 675.071 m 102.784 675.071 l 81.92101 686.66499 m 81.92101 680.71201 l 84.586 677.395 m 84.89799 676.715 l 85.89 675.72301 l 78.94501 675.72301 l 79.597 671.10201 m 79.285 671.414 l 78.94501 671.10201 l 79.285 670.76199 l 79.597 671.10201 l 85.89 666.453 m 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535.181 l 223.597 534.897 l 223.68201 534.642 l 223.767 534.35899 l S 223.767 534.35899 m 223.937 533.79201 l 223.994 533.508 l 224.164 532.94099 l 224.22099 532.658 l 224.306 532.375 l 224.362 532.091 l 224.44701 531.808 l 224.561 531.241 l 224.618 530.929 l 224.731 530.362 l 224.78799 530.05 l 224.81599 529.767 l 224.873 529.483 l 224.929 529.17101 l 224.95799 528.888 l 225.01401 528.576 l 225.043 528.293 l 225.071 527.981 l 225.099 527.69701 l 225.12801 527.386 l 225.15601 527.10201 l 225.18401 526.78999 l 225.213 526.507 l 225.241 526.19501 l 225.241 525.912 l 225.269 525.60001 l 225.269 525.28799 l 225.298 525.005 l 225.298 522.595 l 225.269 522.283 l 225.269 522 l 225.241 521.688 l 225.241 521.405 l 225.18401 520.78101 l 225.15601 520.498 l 225.12801 520.186 l 225.099 519.90199 l 225.071 519.59 l 225.043 519.30701 l 225.01401 518.995 l 224.95799 518.71201 l 224.929 518.39999 l 224.873 518.116 l 224.84399 517.83299 l 224.78799 517.521 l 224.674 516.95399 l 224.618 516.642 l 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193.521 l 343.33299 193.29401 l 343.474 193.039 l 343.588 192.784 l 343.73 192.55701 l 343.843 192.302 l 343.985 192.047 l 344.09801 191.82001 l 344.24001 191.565 l 344.353 191.31 l 344.495 191.08299 l 344.63699 190.828 l 344.75 190.60201 l 344.892 190.34599 l 345.005 190.12 l 345.289 189.60899 l 345.40199 189.383 l 345.54401 189.12801 l 345.685 188.901 l 345.827 188.646 l 345.94099 188.41901 l 346.082 188.192 l 346.224 187.937 l 346.366 187.71001 l 346.479 187.455 l 346.621 187.228 l 346.763 186.97301 l 347.04601 186.52 l 347.159 186.265 l 347.44299 185.811 l 347.58501 185.556 l 348.00999 184.87601 l 348.15199 184.62 l 348.577 183.94 l 348.69 183.685 l 348.974 183.231 l 349.144 183.005 l 349.569 182.32401 l 349.711 182.069 l 350.561 180.709 l 350.731 180.48199 l 351.298 179.575 l 351.468 179.34801 l 351.61 179.121 l 351.752 178.923 l 351.89301 178.696 l 352.063 178.46899 l 352.489 177.789 l 352.659 177.591 l 352.942 177.13699 l 353.112 176.91 l 353.254 176.71201 l 353.424 176.485 l 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161.291 l 366.17999 161.121 l 366.37801 160.951 l 366.548 160.75301 l 366.71899 160.582 l 366.91701 160.412 l 367.79601 159.534 l 367.994 159.364 l 368.164 159.19299 l 368.36301 159.02299 l 368.533 158.853 l 368.731 158.683 l 368.901 158.513 l 369.10001 158.343 l 369.27 158.173 l 369.468 158.00301 l 369.638 157.83299 l 369.83701 157.66299 l 370.007 157.493 l 370.205 157.323 l 370.375 157.181 l 370.57401 157.011 l 370.744 156.841 l 370.942 156.67101 l 371.14101 156.52901 l 371.311 156.35899 l 371.509 156.189 l 371.679 156.047 l 372.076 155.707 l 372.246 155.565 l 372.44501 155.395 l 372.64301 155.25301 l 372.813 155.08299 l 373.011 154.94099 l 373.21001 154.771 l 373.408 154.63 l 373.578 154.459 l 373.77699 154.31799 l 373.97501 154.14799 l 374.174 154.006 l 374.34399 153.864 l 374.54201 153.694 l 375.13699 153.269 l 375.30701 153.099 l 376.3 152.39 l 376.47 152.248 l 376.668 152.078 l 377.263 151.653 l 377.46201 151.53999 l 378.65199 150.689 l 378.851 150.576 l 379.446 150.151 l 379.645 150.037 l 380.041 149.754 l 380.24001 149.64 l 380.438 149.49899 l 380.63699 149.38499 l 380.83501 149.244 l 381.034 149.13 l 381.23199 148.98801 l 381.459 148.875 l 381.657 148.733 l 381.856 148.62 l 382.054 148.478 l 382.649 148.138 l 382.87601 147.996 l 383.47099 147.65601 l 383.67 147.51401 l 383.897 147.401 l 384.095 147.28799 l 384.492 147.061 l 384.71899 146.948 l 385.314 146.60699 l 385.541 146.494 l 385.938 146.267 l 386.164 146.18201 l 386.75999 145.842 l 386.98599 145.729 l 387.185 145.644 l 387.383 145.53 l 387.61 145.41701 l 387.808 145.332 l 388.035 145.218 l 388.23399 145.133 l 388.43201 145.02 l 388.659 144.935 l 388.85699 144.82201 l 389.084 144.737 l 389.282 144.623 l 389.481 144.53799 l 389.70799 144.425 l 389.90601 144.34 l 390.133 144.255 l 390.33099 144.17 l 390.558 144.056 l 390.757 143.97099 l 390.983 143.886 l 391.38 143.716 l 391.60699 143.603 l 391.80499 143.51801 l 392.032 143.433 l 392.231 143.34801 l 392.457 143.26199 l 392.68401 143.177 l 392.883 143.092 l 393.10899 143.007 l 393.308 142.951 l 393.534 142.866 l 393.733 142.78101 l 393.96001 142.69501 l 394.158 142.61 l 394.38499 142.554 l 394.58299 142.46899 l 394.81 142.384 l 395.037 142.327 l 395.235 142.242 l 395.46201 142.157 l 395.66 142.10001 l 395.88699 142.015 l 396.114 141.95799 l 396.312 141.873 l 396.539 141.817 l 396.76601 141.73199 l 396.964 141.675 l 397.19099 141.618 l 397.39 141.533 l 397.616 141.47701 l 397.843 141.42 l 398.04201 141.36301 l 398.26801 141.278 l 398.495 141.22099 l 398.694 141.16499 l 398.92 141.108 l 399.147 141.05099 l 399.34599 140.995 l 399.57201 140.938 l 399.799 140.881 l 400.026 140.825 l 400.224 140.76801 l 400.67799 140.655 l 400.87601 140.59801 l 401.103 140.569 l 401.33 140.513 l 401.55701 140.45599 l 401.755 140.399 l 401.98199 140.371 l 402.435 140.258 l 402.634 140.229 l 402.86099 140.173 l 403.08701 140.144 l 403.314 140.088 l 403.51199 140.05901 l 403.739 140.00301 l 403.966 139.974 l 404.19299 139.946 l 404.39101 139.88901 l 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151.256 l 446.939 151.42599 l 447.138 151.56799 l 447.336 151.73801 l 447.535 151.88 l 447.705 152.05 l 447.903 152.192 l 448.10201 152.362 l 448.272 152.532 l 448.47 152.674 l 448.64 152.84399 l 449.037 153.18401 l 449.207 153.326 l 449.40601 153.496 l 449.576 153.666 l 449.774 153.836 l 449.944 154.006 l 450.14301 154.17599 l 450.483 154.51601 l 450.681 154.686 l 450.851 154.856 l 451.05 155.02699 l 451.22 155.22501 l 451.39 155.395 l 451.588 155.565 l 451.758 155.735 l 451.92799 155.93401 l 452.127 156.104 l 452.297 156.274 l 452.467 156.472 l 453.006 157.011 l 453.17599 157.209 l 453.34599 157.379 l 453.51601 157.578 l 453.884 157.946 l 454.22501 158.343 l 454.395 158.513 l 455.075 159.30701 l 455.245 159.47701 l 455.58501 159.87399 l 455.72701 160.07201 l 456.067 160.46899 l 456.237 160.696 l 456.577 161.093 l 456.71899 161.291 l 457.05901 161.688 l 457.229 161.91499 l 457.371 162.11301 l 457.541 162.312 l 457.711 162.53799 l 457.853 162.737 l 458.02299 162.935 l 458.16499 163.162 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l 468.37 181.672 l 468.483 181.927 l 468.56799 182.183 l 468.681 182.466 l 468.795 182.72099 l 468.88 183.005 l 468.993 183.25999 l 469.078 183.543 l 469.192 183.798 l 469.27699 184.082 l 469.362 184.33701 l 469.47501 184.62 l 469.56 184.90401 l 469.645 185.159 l 469.759 185.442 l 469.84399 185.726 l 469.929 185.981 l 470.01401 186.26401 l 470.127 186.548 l 470.21201 186.80299 l 470.467 187.653 l 470.552 187.909 l 470.97701 189.326 l 471.034 189.60899 l 471.119 189.89301 l 471.20399 190.14799 l 471.289 190.431 l 471.34599 190.715 l 471.51601 191.282 l 471.573 191.565 l 471.743 192.132 l 471.799 192.444 l 471.884 192.72701 l 471.998 193.29401 l 472.08299 193.578 l 472.13901 193.86099 l 472.22501 194.145 l 472.338 194.71201 l 472.395 195.02299 l 472.48 195.30701 l 472.64999 196.157 l 472.70599 196.46899 l 472.82001 197.036 l 472.87601 197.34801 l 473.047 198.198 l 473.103 198.50999 l 473.16 198.793 l 473.188 199.077 l 473.245 199.38901 l 473.302 199.672 l 473.358 199.98399 l 473.38699 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426.416 259.369 l 426.189 259.341 l 425.73599 259.284 l 425.509 259.256 l 425.311 259.228 l 425.084 259.19901 l 424.85699 259.14301 l 424.63 259.114 l 424.43201 259.086 l 423.978 258.972 l 423.752 258.944 l 423.55299 258.88699 l 423.326 258.83099 l 423.10001 258.774 l 422.901 258.717 l 422.674 258.661 l 422.448 258.576 l 422.24899 258.519 l 422.022 258.46201 l 421.79601 258.377 l 421.597 258.321 l 421.37 258.235 l 421.172 258.14999 l 420.94501 258.065 l 420.74699 257.98 l 420.52 257.895 l 420.32201 257.81 l 420.095 257.72501 l 419.896 257.64 l 419.67 257.52699 l 419.47099 257.442 l 419.244 257.328 l 419.04601 257.243 l 418.84801 257.13 l 418.621 257.017 l 418.026 256.677 l 417.799 256.563 l 417.40199 256.336 l 417.203 256.19501 l 417.005 256.08099 l 416.80701 255.94 l 416.608 255.826 l 414.82201 254.55099 l 414.65199 254.38 l 414.45399 254.239 l 414.255 254.069 l 414.08501 253.899 l 413.88699 253.757 l 413.717 253.58701 l 413.51801 253.41701 l 413.34801 253.24699 l 413.14999 253.076 l 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247.00999 l 439.711 247.17999 l 439.541 247.351 l 439.342 247.521 l 439.172 247.69099 l 438.974 247.832 l 438.77499 248.00301 l 438.605 248.144 l 438.407 248.314 l 437.01801 249.30701 l 436.819 249.42 l 436.621 249.562 l 436.224 249.78799 l 436.026 249.92999 l 435.827 250.04401 l 435.60001 250.157 l 435.20399 250.384 l 434.97701 250.46899 l 434.778 250.582 l 434.58 250.66701 l 434.353 250.78 l 434.155 250.86501 l 433.92799 250.951 l 433.73 251.036 l 433.50301 251.121 l 433.304 251.20599 l 433.078 251.291 l 432.879 251.37601 l 432.65199 251.43201 l 432.42599 251.517 l 432.22701 251.57401 l 431.774 251.688 l 431.575 251.744 l 431.12199 251.858 l 430.923 251.914 l 430.69701 251.94299 l 430.47 251.99899 l 430.243 252.028 l 430.045 252.056 l 429.81799 252.084 l 429.591 252.11301 l 429.364 252.14101 l 429.13699 252.16901 l 428.911 252.198 l 428.71201 252.226 l 428.259 252.226 l 428.032 252.254 l 427.153 252.254 l 426.92599 252.226 l 426.7 252.226 l 426.47301 252.198 l 426.274 252.198 l 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207.439 l 410.967 207.21201 l 411.10899 206.957 l 411.959 205.59599 l 412.13 205.37 l 412.271 205.17101 l 412.41299 204.944 l 412.58299 204.718 l 412.72501 204.519 l 412.895 204.321 l 413.065 204.09399 l 413.235 203.896 l 413.377 203.69701 l 414.05701 202.90401 l 414.59599 202.36501 l 414.76601 202.166 l 414.936 201.996 l 415.134 201.826 l 415.304 201.65601 l 415.50301 201.48599 l 415.673 201.31599 l 415.871 201.146 l 416.041 200.976 l 416.24001 200.834 l 416.438 200.664 l 416.608 200.522 l 417.005 200.239 l 417.203 200.069 l 417.40199 199.927 l 417.60001 199.814 l 417.99699 199.53 l 418.196 199.41701 l 418.394 199.27499 l 418.791 199.048 l 419.01801 198.935 l 419.216 198.793 l 419.41499 198.70799 l 419.61301 198.595 l 419.84 198.481 l 420.03799 198.396 l 420.265 198.283 l 420.662 198.11301 l 420.88901 198 l 421.08701 197.91499 l 421.314 197.858 l 421.541 197.77299 l 421.739 197.688 l 421.966 197.631 l 422.164 197.54601 l 422.618 197.433 l 422.81599 197.37601 l 423.043 197.319 l 423.27 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409.35201 223.993 l S 409.35201 223.993 m 409.35201 223.681 l 409.38 223.369 l 409.38 222.774 l 409.408 222.491 l 409.408 222.179 l 409.437 221.867 l 409.437 221.584 l 409.465 221.272 l 409.493 220.98801 l 409.522 220.67599 l 409.55 220.39301 l 409.578 220.08099 l 409.63499 219.798 l 409.664 219.48599 l 409.72 219.202 l 409.74899 218.89101 l 409.862 218.32401 l 409.89 218.01199 l 409.94701 217.728 l 410.032 217.44501 l 410.089 217.162 l 410.145 216.85001 l 410.202 216.56599 l 410.287 216.283 l 410.34399 215.99899 l 410.51401 215.43201 l 410.571 215.149 l 410.65601 214.866 l 410.741 214.61 l 410.826 214.327 l 410.939 214.043 l 411.024 213.78799 l 411.10899 213.505 l 411.22301 213.22099 l 411.308 212.966 l 411.42101 212.711 l 411.534 212.42799 l 411.619 212.173 l 411.96001 211.407 l 412.215 210.897 l 412.328 210.642 l 412.47 210.38699 l 412.58299 210.132 l 412.72501 209.905 l 412.838 209.64999 l 412.98 209.423 l 413.12199 209.168 l 413.547 208.48801 l 413.689 208.23199 l 413.83 208.006 l 413.972 207.80701 l 414.142 207.58 l 414.284 207.354 l 414.42599 207.155 l 414.59599 206.929 l 414.76601 206.73 l 414.908 206.50301 l 415.758 205.511 l 415.92799 205.341 l 416.09801 205.14301 l 416.26801 204.97301 l 416.438 204.774 l 416.63699 204.604 l 416.80701 204.43401 l 417.005 204.26401 l 417.175 204.09399 l 417.37399 203.924 l 417.54401 203.754 l 417.742 203.612 l 417.94099 203.442 l 418.11099 203.3 l 419.103 202.591 l 419.30099 202.478 l 419.5 202.336 l 419.698 202.22301 l 419.925 202.08099 l 420.52 201.741 l 420.74699 201.65601 l 420.94501 201.543 l 421.144 201.45799 l 421.371 201.34399 l 421.569 201.259 l 421.79601 201.174 l 421.994 201.089 l 422.22099 201.004 l 422.41901 200.91901 l 422.646 200.862 l 422.873 200.77699 l 423.071 200.72099 l 423.52499 200.60699 l 423.72301 200.55099 l 423.95 200.494 l 424.177 200.466 l 424.40401 200.409 l 424.60201 200.38 l 424.82899 200.35201 l 425.056 200.295 l 425.282 200.295 l 425.509 200.267 l 425.70799 200.239 l 425.93401 200.21001 l 426.161 200.21001 l 426.388 200.239 l 426.61501 200.21001 l 427.267 200.21001 l 427.493 200.239 l 427.72 200.239 l 427.94701 200.267 l 428.145 200.295 l 428.37199 200.32401 l 428.599 200.35201 l 429.052 200.409 l 429.25101 200.466 l 429.478 200.522 l 429.70399 200.55099 l 429.903 200.60699 l 430.13 200.664 l 430.356 200.74899 l 430.58299 200.806 l 430.782 200.862 l 431.008 200.94701 l 431.207 201.032 l 431.43401 201.117 l 431.632 201.202 l 431.85899 201.28799 l 432.05701 201.373 l 432.284 201.45799 l 432.681 201.68401 l 432.908 201.769 l 433.30499 201.996 l 433.50301 202.138 l 433.73 202.25101 l 433.92799 202.36501 l 434.325 202.64799 l 434.52299 202.761 l 434.92 203.045 l 435.09 203.215 l 435.487 203.49899 l 435.686 203.66901 l 435.856 203.81 l 436.054 203.98 l 436.224 204.151 l 436.423 204.321 l 436.593 204.491 l 436.791 204.661 l 436.961 204.85899 l 437.131 205.02901 l 437.30099 205.228 l 437.47099 205.39799 l 438.32201 206.39 l 438.492 206.617 l 438.634 206.815 l 438.804 207.04201 l 438.946 207.24001 l 439.116 207.467 l 440.108 209.05499 l 440.25 209.31 l 440.39101 209.536 l 440.505 209.79201 l 440.646 210.047 l 440.75999 210.27299 l 440.901 210.52901 l 441.468 211.804 l 441.582 212.088 l 441.69501 212.343 l 441.78 212.59801 l 441.894 212.881 l 441.979 213.136 l 442.092 213.42 l 442.177 213.703 l 442.26199 213.95799 l 442.687 215.37601 l 442.744 215.659 l 442.82899 215.94299 l 442.942 216.50999 l 443.02699 216.793 l 443.084 217.077 l 443.14101 217.388 l 443.19701 217.672 l 443.226 217.955 l 443.282 218.267 l 443.339 218.55099 l 443.367 218.862 l 443.424 219.146 l 443.453 219.429 l 443.509 220.05299 l 443.53799 220.336 l 443.56599 220.64799 l 443.59399 220.93201 l 443.59399 221.243 l 443.623 221.52699 l 443.623 221.839 l 443.651 222.14999 l 443.651 223.936 l 443.623 224.248 l 443.623 224.532 l 443.59399 224.843 l 443.59399 225.155 l 443.56599 225.439 l 443.53799 225.75 l 443.509 226.034 l 443.481 226.34599 l 443.453 226.629 l 443.396 226.94099 l 443.367 227.224 l 443.339 227.536 l 443.226 228.103 l 443.16901 228.41499 l 443.056 228.98199 l 442.99899 229.29401 l 442.886 229.86099 l 442.80099 230.144 l 442.744 230.42799 l 442.659 230.711 l 442.54601 230.994 l 442.517 231.278 l 442.347 231.845 l 442.26199 232.10001 l 442.149 232.383 l 442.064 232.66701 l 441.979 232.922 l 441.86501 233.205 l 441.78 233.461 l 441.66701 233.744 l 440.873 235.53 l 440.731 235.785 l 440.618 236.03999 l 440.476 236.267 l 440.33501 236.522 l 440.22099 236.74899 l 440.07899 237.004 l 439.513 237.911 l 439.342 238.138 l 439.05901 238.591 l 438.88901 238.78999 l 438.74699 239.01601 l 438.407 239.41299 l 438.265 239.64 l 437.925 240.037 l 437.755 240.207 l 437.41499 240.604 l 436.508 241.511 l 436.338 241.653 l 436.13901 241.823 l 435.96899 241.993 l 435.771 242.16299 l 435.57201 242.30499 l 435.40199 242.446 l 435.20399 242.617 l 434.608 243.04201 l 434.41 243.155 l 434.21201 243.297 l 434.013 243.41 l 433.815 243.552 l 433.588 243.66499 l 432.993 244.006 l 432.76601 244.091 l 432.567 244.20399 l 432.341 244.289 l 431.944 244.459 l 431.717 244.54401 l 431.49001 244.629 l 431.29201 244.714 l 431.065 244.771 l 430.867 244.856 l 430.41299 244.96899 l 430.215 245.026 l 429.98801 245.08299 l 429.761 245.13901 l 429.534 245.168 l 429.336 245.224 l 429.10899 245.25301 l 428.882 245.28101 l 428.65601 245.30901 l 428.429 245.338 l 428.23 245.366 l 428.004 245.366 l 427.77699 245.394 l 426.89799 245.394 l 426.67101 245.366 l 426.44501 245.366 l 425.991 245.30901 l 425.793 245.30901 l 425.56599 245.25301 l 425.339 245.224 l 425.112 245.196 l 424.886 245.13901 l 424.687 245.11099 l 424.23399 244.998 l 424.035 244.94099 l 423.808 244.884 l 423.582 244.799 l 423.383 244.742 l 423.15601 244.657 l 422.95799 244.601 l 422.731 244.51601 l 422.533 244.431 l 422.306 244.317 l 422.108 244.23199 l 421.881 244.147 l 421.48399 243.92 l 421.257 243.80701 l 420.662 243.467 l 420.435 243.325 l 420.237 243.21201 l 419.64101 242.787 l 419.47099 242.645 l 419.07401 242.36099 l 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411.22301 223.00101 l 411.25101 222.689 l 411.25101 222.40601 l 411.27901 222.09399 l 411.27901 221.782 l 411.308 221.49899 l 411.336 221.187 l 411.364 220.903 l 411.39301 220.591 l 411.42101 220.308 l 411.478 219.996 l 411.506 219.713 l 411.563 219.429 l 411.591 219.117 l 411.64799 218.834 l 411.70399 218.522 l 411.931 217.388 l 412.01601 217.077 l 412.073 216.793 l 412.243 216.226 l 412.3 215.94299 l 412.38499 215.688 l 412.55499 215.121 l 412.668 214.83701 l 412.75301 214.582 l 412.838 214.299 l 412.952 214.043 l 413.037 213.75999 l 413.263 213.25 l 413.377 212.966 l 413.83 211.946 l 413.972 211.71899 l 414.086 211.464 l 414.22701 211.209 l 414.369 210.98199 l 414.48199 210.72701 l 414.908 210.047 l 415.049 209.79201 l 415.19099 209.565 l 415.36099 209.366 l 415.645 208.91299 l 415.815 208.714 l 415.95599 208.48801 l 416.12601 208.289 l 416.297 208.062 l 416.467 207.864 l 416.608 207.66499 l 416.778 207.467 l 417.147 207.099 l 417.317 206.89999 l 417.657 206.56 l 417.856 206.39 l 418.026 206.19099 l 418.224 206.021 l 418.394 205.88 l 418.593 205.71001 l 418.763 205.539 l 420.15199 204.547 l 420.35001 204.43401 l 420.549 204.29201 l 420.94501 204.065 l 421.172 203.952 l 421.371 203.839 l 421.569 203.754 l 421.79601 203.64 l 421.994 203.55499 l 422.22099 203.47 l 422.41901 203.38499 l 422.646 203.3 l 422.845 203.215 l 423.071 203.13 l 423.27 203.073 l 423.49699 202.98801 l 423.72301 202.93201 l 423.922 202.875 l 424.149 202.81799 l 424.375 202.78999 l 424.60201 202.733 l 424.80099 202.705 l 425.02699 202.64799 l 425.254 202.62 l 425.481 202.592 l 425.679 202.563 l 425.90601 202.563 l 426.133 202.535 l 427.465 202.535 l 427.91901 202.592 l 428.145 202.592 l 428.34399 202.62 l 428.571 202.677 l 429.024 202.733 l 429.22301 202.78999 l 429.67599 202.903 l 429.903 202.96001 l 430.101 203.017 l 430.328 203.073 l 430.52699 203.158 l 430.75301 203.215 l 430.98 203.3 l 431.17799 203.38499 l 431.405 203.47 l 431.604 203.584 l 431.802 203.66901 l 432.02901 203.782 l 432.22701 203.867 l 432.42599 203.98 l 432.653 204.09399 l 433.049 204.321 l 433.248 204.46201 l 433.446 204.576 l 434.438 205.284 l 434.63699 205.45399 l 434.80701 205.59599 l 435.005 205.76601 l 435.20399 205.908 l 435.37399 206.078 l 435.57201 206.248 l 435.742 206.418 l 435.912 206.617 l 436.11099 206.787 l 436.28101 206.985 l 436.451 207.155 l 437.131 207.94901 l 437.27299 208.147 l 437.44299 208.37399 l 437.61301 208.573 l 437.755 208.771 l 437.925 208.998 l 438.35001 209.67799 l 438.52 209.905 l 438.662 210.132 l 438.776 210.358 l 438.91701 210.614 l 439.05901 210.84 l 439.201 211.095 l 439.314 211.32201 l 439.45599 211.577 l 439.79601 212.343 l 439.938 212.59801 l 440.02299 212.853 l 440.136 213.108 l 440.25 213.39101 l 440.36301 213.647 l 440.448 213.90199 l 440.561 214.185 l 440.646 214.44 l 440.731 214.724 l 440.845 215.007 l 440.92999 215.291 l 441.015 215.54601 l 441.071 215.82899 l 441.242 216.396 l 441.298 216.67999 l 441.383 216.991 l 441.61 218.125 l 441.66701 218.437 l 441.78 219.004 l 441.808 219.31599 l 441.83701 219.599 l 441.89301 219.911 l 441.922 220.19501 l 441.95 220.506 l 441.979 220.78999 l 442.007 221.10201 l 442.007 221.38499 l 442.064 222.009 l 442.064 224.106 l 442.035 224.418 l 442.035 224.702 l 442.007 225.013 l 441.979 225.297 l 441.95 225.60899 l 441.922 225.892 l 441.89301 226.20399 l 441.86501 226.487 l 441.83701 226.799 l 441.78 227.08299 l 441.752 227.394 l 441.638 227.961 l 441.582 228.27299 l 441.355 229.407 l 441.27 229.71899 l 441.185 230.002 l 441.12801 230.286 l 441.043 230.541 l 440.703 231.675 l 440.59 231.92999 l 440.505 232.213 l 440.39101 232.468 l 440.306 232.752 l 439.62601 234.283 l 439.48399 234.53799 l 439.371 234.793 l 439.229 235.048 l 439.116 235.27499 l 438.974 235.53 l 438.407 236.437 l 438.265 236.692 l 438.12399 236.89101 l 437.953 237.117 l 437.812 237.34399 l 437.67 237.54201 l 437.5 237.769 l 437.16 238.166 l 437.01801 238.36501 l 436.67799 238.761 l S 436.67799 238.761 m 436.508 238.96001 l 435.601 239.867 l 435.42999 240.009 l 435.23199 240.179 l 435.062 240.349 l 434.864 240.519 l 434.467 240.802 l 434.297 240.944 l 433.50301 241.511 l 433.276 241.625 l 432.681 241.965 l 432.45399 242.078 l 432.05701 242.30499 l 431.83 242.39 l 431.632 242.47501 l 431.405 242.56 l 431.207 242.645 l 430.98 242.73 l 430.782 242.815 l 430.328 242.92799 l 430.13 243.013 l 429.67599 243.127 l 429.478 243.155 l 429.25101 243.21201 l 429.024 243.24001 l 428.797 243.269 l 428.599 243.297 l 428.37199 243.325 l 428.145 243.354 l 427.91901 243.382 l 426.586 243.382 l 426.36 243.354 l 426.133 243.354 l 425.93401 243.325 l 425.70799 243.297 l 425.481 243.269 l 425.254 243.21201 l 425.056 243.183 l 424.375 243.013 l 424.177 242.957 l 423.95 242.89999 l 423.752 242.843 l 423.298 242.673 l 423.10001 242.588 l 422.873 242.50301 l 422.674 242.418 l 422.448 242.33299 l 422.24899 242.22 l 422.05099 242.13499 l 421.82401 242.021 l 421.427 241.795 l 421.229 241.653 l 421.03 241.539 l 420.804 241.39799 l 420.605 241.284 l 420.407 241.14301 l 420.237 241.00101 l 420.03799 240.85899 l 419.84 240.689 l 419.44299 240.40601 l 419.27299 240.23599 l 419.07401 240.065 l 418.90401 239.895 l 418.70599 239.72501 l 418.536 239.55499 l 418.366 239.35699 l 417.827 238.81799 l 417.487 238.42101 l 417.345 238.22301 l 417.175 237.996 l 417.005 237.798 l 416.86301 237.599 l 416.69299 237.37199 l 416.552 237.146 l 416.382 236.94701 l 415.95599 236.267 l 415.815 236.01199 l 415.53101 235.558 l 415.418 235.30299 l 415.276 235.076 l 415.16299 234.821 l 415.021 234.56599 l 414.341 233.035 l 414.22701 232.752 l 414.142 232.49699 l 414.02901 232.213 l 413.944 231.95799 l 413.774 231.39101 l 413.66 231.108 l 413.604 230.853 l 413.34801 230.002 l 413.29201 229.69099 l 413.235 229.407 l 413.14999 229.12399 l 413.037 228.55701 l 412.98 228.245 l 412.952 227.96201 l 412.895 227.67799 l 412.838 227.366 l 412.81 227.08299 l 412.782 226.771 l 412.75301 226.48801 l 412.72501 226.17599 l 412.69701 225.892 l 412.668 225.58 l 412.64 225.297 l 412.612 224.985 l 412.612 224.39 l 412.58299 224.078 l 412.58299 223.483 l 412.612 223.17101 l 412.612 222.576 l 412.64 222.26401 l 412.64 221.98 l 412.668 221.66901 l 412.69701 221.38499 l 412.72501 221.073 l 412.75301 220.78999 l 412.782 220.478 l 412.838 220.19501 l 412.867 219.883 l 412.923 219.599 l 412.952 219.31599 l 413.008 219.004 l 413.12199 218.437 l 413.17799 218.125 l 413.263 217.842 l 413.377 217.27499 l 413.717 216.14101 l 413.802 215.886 l 413.972 215.319 l 414.086 215.064 l 414.17101 214.78101 l 414.284 214.52499 l 414.369 214.27 l 414.48199 213.987 l 414.823 213.22099 l 414.964 212.966 l 415.078 212.711 l 415.21899 212.48399 l 415.33299 212.229 l 415.474 211.974 l 415.588 211.74699 l 415.73 211.521 l 415.871 211.26601 l 416.155 210.812 l 416.325 210.58501 l 416.467 210.358 l 416.608 210.16 l 416.778 209.933 l 416.92 209.735 l 417.09 209.508 l 417.94099 208.51601 l 418.11099 208.34599 l 418.28101 208.147 l 418.451 207.97701 l 418.649 207.80701 l 418.819 207.836 l 419.01801 207.467 l 419.188 207.297 l 419.386 207.127 l 419.556 206.985 l 419.755 206.843 l 419.953 206.673 l 420.548 206.248 l 420.74699 206.13499 l 420.94501 205.993 l 421.342 205.76601 l 421.569 205.653 l 421.966 205.42599 l 422.19299 205.341 l 422.39101 205.228 l 422.589 205.14301 l 422.81599 205.058 l 423.015 204.97301 l 423.241 204.888 l 423.468 204.83099 l 423.66701 204.746 l 423.89301 204.689 l 424.12 204.633 l 424.319 204.576 l 424.772 204.46201 l 424.97099 204.43401 l 425.19701 204.377 l 425.424 204.349 l 425.651 204.321 l 425.87801 204.321 l 426.076 204.29201 l 426.30299 204.26401 l 426.756 204.26401 l S PDFVars/TermAll get exec end end %%PageTrailer %%EndPage %%Trailer %%EOF DSETT <; l4   Cornu SpiralDSETTHI=;IИXBwBH'Z" H0I0$H>(Lh| JJFresnel Diffraction Diffraction froma orifice without circular symmetry DSET2H>l@=P{Ll8#6*P @@dxprW CWEEWGrphbj , Symbol `.(  " currentpoint ",Times( Electric field element,(c )dE 8+vp `(-=,Times(R}K)H()q)8))e 8+,0 `(r)5r)%l sys(cos)k)/()$r)H+)Lr)'))0-)@w)Btz`([(](8dS(After change of variables,( (" (E 8+Dp `(=( (E  8+B0 `(e 8(i)p)u ((s2 8+ /)2 `+ +du 8(Ru (+1 8(EPu (+2 (X `(e 8(Mi)p)v ((s2 8+ /)2 `+ +dv 8(v (+1 8(Ev (+2 ( `( )Express in the complex plane in terms of (CC)C()$w)C),)@S)3()$w)C)(} C)C()$w)C))4=)cos()p)5x 8(R2 `+8+/)(2)/))$dx 8(L0(Fw *z `(S)3()$w)C))4=)sin()p)5x 8(zr2 `+8+/)(2)/))$dx 8( @0(;:w *z `( Must be calculated numerically(-)4 spiral chart/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8864 div 5760 3 -1 roll exch div scale currentpoint translate 64 41 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 4518 430 rndmvto 1080 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (Electric field element,) -16 526 sh ( ) 3406 526 sh (After change of variables,) 0 1718 sh ( ) 4010 1718 sh (E) 4106 1718 sh (Express in the complex plane in terms of) -16 2654 sh ( ) show (Must be calculated numerically) -16 5598 sh (-) 4867 5598 sh ( spiral chart) 5075 5598 sh 224 ns (p) 4378 1814 sh 384 /Times-Italic f1 (dE) 3502 526 sh (K) 4534 288 sh (r) 4985 822 sh (k) 6292 526 sh (r) 7216 526 sh (t) 8085 526 sh (dS) 8353 526 sh (E) 4922 1718 sh (e) 5664 1718 sh (du) 6504 1718 sh (e) 7222 1718 sh (dv) 8062 1718 sh (C) 6348 2654 sh (w) 6760 2654 sh (S) 7285 2654 sh (w) 7633 2654 sh (C) -16 3534 sh (w) 396 3534 sh (x) 2345 3534 sh (dx) 3269 3534 sh (S) 0 4718 sh (w) 348 4718 sh (x) 2233 4718 sh (dx) 3157 4718 sh 224 ns (p) 3974 622 sh (i) 5856 1546 sh (u) 6040 1546 sh (u) 5384 2118 sh (u) 5376 1262 sh (i) 7414 1546 sh (v) 7598 1546 sh (v) 6942 2118 sh (v) 6934 1262 sh (w) 1242 3110 sh (w) 1194 4294 sh 384 /Symbol f1 (=) 4214 526 sh (+) 6912 526 sh (-) 7566 526 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1270 /Symbol f3 ([) 6150 558 sh (]) 8211 558 sh 384 /Symbol f1 (=) 4602 1718 sh (=) 874 3534 sh (=) 826 4718 sh 576 ns (\362) 5408 1782 sh (\362) 6966 1782 sh (\362) 1242 3598 sh (\362) 1194 4782 sh 384 /Times-Roman f1 (\() 4820 288 sh (\)) 5152 288 sh (cos) 5646 526 sh (\() 6482 526 sh (\)) 7374 526 sh (\304) 4170 1622 sh (\304) 5002 1622 sh (\() 6618 2654 sh (\),) 7030 2654 sh (\() 7491 2654 sh (\)) 7903 2654 sh (\() 254 3534 sh (\)) 666 3534 sh (cos\() 1498 3534 sh (/) 2777 3534 sh (\)) 3127 3534 sh (\() 206 4718 sh (\)) 618 4718 sh (sin\() 1450 4718 sh (/) 2665 4718 sh (\)) 3015 4718 sh 224 ns (/) 6296 1546 sh (/) 7854 1546 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (q) 4930 288 sh (e) 5262 288 sh (r) 4775 822 sh (l) 5134 822 sh (r) 6624 526 sh (w) 7822 526 sh (p) 2135 3534 sh (p) 2023 4718 sh 224 ns (p) 5918 1546 sh (p) 7476 1546 sh 224 /Times-Roman f1 (0) 5438 384 sh (0) 5186 1814 sh (2) 6376 1546 sh (2) 7934 1546 sh (2) 2553 3362 sh (0) 1266 3934 sh (2) 2441 4546 sh (0) 1218 5118 sh 160 ns (2) 6168 1446 sh (1) 5480 2174 sh (2) 5488 1318 sh (2) 7726 1446 sh (1) 7038 2174 sh (2) 7046 1318 sh 384 ns (2) 2937 3534 sh (2) 2825 4718 sh end MTsave restore dMATHG Electric field element, dE p =K(q)e 0 rrlcosk(r+r)-wt[]dSAfter change of variables, "E p =2E 0 e ipu 2 /2 du u 1 u 2  e ipv 2 /2 dv v 1 v 2  Express in the complex plane in terms of C(w),S(w)C(w)=cos(px 2 /2)dx 0w  S(w)=sin(px 2 /2)dx 0w  Must be calculated numerically- spiral chart(DSET2H?l  $T@ 8ffiAgyC*DSETT@?vp|||  |Specific Heat ProblemDSETT??A??||||X{lP{  %2'l "  %{' ?.$%BHJ%%x @|@Debye theory Heat capcity of a solid  Molar specific heat vs TDSET2HB~AP%|#6*P b b% @@(dxpr CWEE%Grphbj (, Symbol `.( "( currentpoint ",Times( C 8+>V `(=,Times)H9)3kN(|T 8(Q3 `(q 8+5D(3"l `(|x 8(Q4 `+ +e 8(QGx `({() e 8(x `+4+-)<1)') 8(2 w `(dx 8(,0 ((,q+ D+(T"cE (& `(,)  where (TN)@ is number of atomsU/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8768 div 1184 3 -1 roll exch div scale currentpoint translate 64 56 translate /rndpos { transform round exch round exch itransform } def /rndvec { dtransform round exch round exch idtransform } def /rnddist { 0 rndvec pop } def /rndmvto { rndpos moveto } def /rndlnto { rndpos lineto } def /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1520 584 rndmvto 439 0 rlineto stroke 2463 584 rndmvto 1244 0 rlineto stroke 8 th /stb { newpath rndmvto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -199 280 2381 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (C) -16 680 sh (kN) 1014 680 sh (T) 1539 442 sh (x) 2765 442 sh (e) 3101 442 sh (e) 2605 976 sh (dx) 3771 680 sh (N) 5393 680 sh 224 ns (V) 232 776 sh (D) 1735 1072 sh (x) 3293 270 sh (x) 2797 804 sh 160 ns (D) 2127 160 sh (T) 2287 280 sh 384 /Symbol f1 (=) 520 680 sh (-) 3005 976 sh 576 ns (\362) 2139 744 sh 384 /Times-Roman f1 (9) 808 680 sh (1) 3245 976 sh 224 ns (3) 1795 270 sh (3) 1760 804 sh (4) 2973 270 sh (2) 3547 804 sh (0) 2163 1080 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (q) 1520 976 sh 160 ns (q) 2007 120 sh 384 /Times-Roman f1 (\() 2479 976 sh (\)) 3403 976 sh (,) 4137 680 sh 384 /Times-Roman f1 ( where ) 4265 680 sh ( is number of atoms) 5649 680 sh end MTsave restore d9MATH-4. C V =9kNT 3 q D3 x 4 e x (e x -1) 2 dx 0) qDT  , where N is number of atomsQDSETT܂C?l`ldlhwyyPy]yaybypy~yyyyyy yMy]ymy}yyy$ v  lLfunction integrand, x f = (x * x * x * x * exp(x))/((exp(x)-1.)*(exp(x)-1.)) return, f end k = 1.381E-23 debyeT = 309. T = 350. xT = 0. yS = 0. while(T le 1083.) do begin uLimit = debyeT / T integral = qromb('integrand', 0., uLimit, /DOUBLE,EPS=0.001) S = (9. * k * T * T * T / (debyeT * debyeT * debyeT)) * integral xT = [xT, T] yS = [yS, S] T = T + 100. Endwhile plot,xT,yS DSET2HDl &iE(@|@E`*`PGOB DSETTEDlll<<(&&&&& * lType notes here DSET2HFl  &U'@|@G`*YPGOB DSETTGFlll<<(  lType notes here DSET2HHl &U0@|@I`*YPGOBDSETTIHlll<<(  lType notes here DSET2HJlD &izx@|@K`*YPGOBDSETTKJlll<<(  lType notes here DSET2HLl  &U @|@M`*YPGOBDSETTMLlll<<(  l Type notes here DSET2HNl  &@|@O`*YPGOB DSETTONlll<<(  l Type notes here DSET2HP8l$ &U@|@Q`*YPGOB#DSETTQPl4l(l0<<(  l8Type notes here DSET2HR@lX &U9`@|@S`*YPGOB'DSETTSRldl\l`<<(  lhType notes here DSET2HT{ll{ &iP@|@U`*YPGOB,DSETTUTlxlplt<<(  l|Type notes here DSET2HVl &TĠ@|@W`*YPGOB2DSETTWVlll<<(  lType notes here DSET2HX@l &T@|@Y`*YPGOB8DSETTYXlll<<(  lType notes here DSET2HZo &/(@>@[`*lPGOB;DSETT[Zl@l(l,<(  l8Type notes here DSET2H\l< &iy(@|@]`*lPGOB?DSETT]\|||<l<(  |Type notes here DSUMHDNISTYL1{JSTYLllllllllll;                        ,  "!   1 51 & ,*&     .!8"+  #9+ $:! %)  &<)'>) )( ) *$+- K,;-.  /%*0J1 ;1 2 3  4#5'  6(' 70' 8/+  92  :32  ;7+ <4  ==4 >6??@@ AA  BBA CDA,DC EECFF GG HHIK  JMKKL  LOL !MINN OQNPTN QX  $RZX% S[  &T]['U_['Vb['WJAXP['YR "ZSR [V2  \U #]WU  ^^U _Y `\CaaCb`CccCddCeeCff (ggf)hif )ih *jjh /klh0lkhmmhnn! oo! pp! qrR 7rtR sq +tuq%us  ,vws!^w{2 xxyy xzvyrHASH $ Xz + . +4 ,@ ;~ 8<^./|XyY2,)/X+Wx+Y\,,a . - ,^.r/S2,s2/ Q;,Y;/ 5W/ <`,\`/ 9|/ I,N,K/ u.A,D.A:?A<]`,)`. ",. /T/$UV,f*/R2/r;+.;,i;,(;. W/w/X/C1B/t / /E 1 3b c/q/ Z16/`1_3ad>/=F/[F/ ]F1:X/O]/Je/vj/Pk/L/W/ eF/#F/ +F18G7 ;. %2/!2/-e/$e/ e1nh1oh1p/ g/ h/M/m/k/ l133 F7 H G!/ j/&/'-+~>/.9;|;.hjazo}Bod CHARl 3  "        + &   HASH    '+    އ       > CELLlzHASH  GRPHl~+lz#_y3LK"$[n3&Y++&+HASH tr55R   p24v & L(2(4PU    R f -l RULRlC  $   l $$ HHl HH $ $H l$ HH  llll ll $ H$$ $$ l-HASH' %%- JR ovw!R&-$-+6 6"R[ [#ww , )-  [[ (666*6  2 KSENlzHASH|zLKUP- 01().234*/56%"+ 879:<>;!#$,&='?@ABDCEFGHMWIKJNLXOYZP\[]Q_RS`T^UbaVcdefgihjlkmnopsqurtzvxyw$NAMEDefault Default SS Default TBHeaderBodyFooter(@  #ffffff5b6dffffffcbddffcbdddDFNTM HelveticaGeneva Lucida GrandeSymbolMonacoMCROMCROoBlNMARKMRKS MOBJWMBTSNAP/// HH HH:@{o{o|oo|o{o|o{sNs!!c"Bcc  CBC!!c c!?                         scRJR^!Ns!! !)!B dd5L)cC c!d * d C + +dc  c,g9RZ^^Js!"BBC d1[1dCB""!9  L + +  mM++, m !m$c$% ((%% !!) !%M%*(!(!%%M% %m%%(%!n,$%+!%%L%+%L% %n) !(,<vRquNs!e  ,!o>voBp  d!H  +,+ BL n!*M(!(!+(%*)+)+-*- - )-m-M-M1L-n)1+1+1L1 -m8-)-<4<@41+5 1,4)5*-,4-n-m@<@<Hu~}~eNs!"d#!"BBC d+)!+cC""CD" B ! 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