Content-Type: multipart/mixed; boundary="-------------0205301103208" This is a multi-part message in MIME format. ---------------0205301103208 Content-Type: text/plain; name="02-246.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-246.keywords" Reducibility, Floquet theory, KAM theory, Linear equations, Linear quasi-periodic skew-products, Schrodinger equation with quasi-periodic potential ---------------0205301103208 Content-Type: application/postscript; name="0201puig.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="0201puig.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: qpred.dvi %%Pages: 132 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Helvetica %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips qpred %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.05.30:1737 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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Fy(.)1924 6155 y(3)p eop %%Page: 4 8 4 7 bop 28 219 a Fy(The)34 b(simplest)d(example)h(of)g(linear)f (systems)j(are)f(linear)e(equations)i(with)f(constan)m(t)h(co)s (e\016cien)m(ts)1652 437 y Fx(x)1707 395 y Ft(0)1731 437 y Fy(\()p Fx(t)p Fy(\))28 b(=)f Fx(A)33 b(x)p Fy(\()p Fx(t)p Fy(\))1570 b(\(1.3\))-118 655 y(where)30 b Fx(A)f Fy(is)g(a)f(constan)m(t)i(matrix)e(of)g(dimension)g Fx(n)p Fy(.)42 b(In)30 b(this)e(case,)j(for)d(an)m(y)i(initial)25 b(time)j Fx(t)3224 670 y FC(0)3263 655 y Fy(,)i(the)g(fundamen)m(tal) -118 775 y(matrix)h(is)h(giv)m(en)h(b)m(y)g(the)g(exp)s(onen)m(tiation) f(of)g(the)h(matrix)e(\()p Fx(t)23 b Fs(\000)f Fx(t)2345 790 y FC(0)2385 775 y Fy(\))p Fx(A)895 1048 y(X)8 b Fy(\()p Fx(t)p Fy(;)17 b Fx(t)1136 1063 y FC(0)1176 1048 y Fy(\))27 b(=)h(exp)18 b(\(\()p Fx(t)k Fs(\000)g Fx(t)1778 1063 y FC(0)1818 1048 y Fy(\))p Fx(A)p Fy(\))28 b(=)f Fx(I)8 b(d)22 b Fy(+)2320 954 y Fq(X)2325 1164 y Fr(n)p Ft(\025)p FC(1)2491 981 y Fy(\()p Fx(t)g Fs(\000)g Fx(t)2720 996 y 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5517 y Ft(j)t FC(~)-39 b Fr(x)p FC(=)p Fr(x)2445 5526 y Fp(0)-118 5709 y Fy(and)1438 5861 y Fx(A)p Fy(\()p Fx(t)p Fy(\))28 b(=)1753 5721 y Fq(\022)1836 5794 y Fx(@)5 b(f)p 1836 5839 116 4 v 1838 5930 a(@)11 b Fy(~)-55 b Fx(x)1962 5861 y Fy(\()p Fx(x)p Fy(\()p Fx(t)p Fy(;)17 b Fx(x)2227 5876 y FC(0)2267 5861 y Fy(\)\))2343 5721 y Fq(\023)2433 5861 y Fx(:)1924 6155 y Fy(4)p eop %%Page: 5 9 5 8 bop -38 1475 a currentpoint currentpoint translate 0.64568 0.64568 scale neg exch neg exch translate -38 1475 a @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 3600 @rwi @setspecial %%BeginDocument: moviment_periodic.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: moviment_periodic.eps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Wed Jan 9 17:01:47 2002 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat 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V -64 1 V -63 3 V -63 5 V -64 7 V -63 10 V -63 12 V -63 15 V -63 17 V -63 18 V -62 22 V -62 23 V -62 26 V -62 28 V -62 29 V -61 33 V -61 34 V -60 36 V -60 38 V -60 40 V -59 42 V -59 44 V -58 46 V -58 48 V -58 49 V -57 51 V -56 53 V -56 54 V -55 56 V -55 58 V -54 58 V -53 61 V -53 61 V -52 63 V -51 64 V -51 65 V -50 66 V -49 67 V -48 69 V -48 69 V -47 69 V -46 71 V -45 71 V -44 72 V -44 72 V -42 73 V -42 73 V -41 74 V -40 73 V -39 74 V -38 74 V currentpoint stroke M -37 74 V -36 74 V -35 74 V -34 73 V -33 74 V -32 73 V -31 72 V -29 72 V -29 72 V -28 70 V -27 71 V -25 69 V -25 68 V -23 68 V -22 67 V -22 65 V -20 65 V -18 63 V -18 62 V -17 61 V -15 59 V -14 58 V -13 57 V -12 55 V -11 53 V -10 52 V -8 50 V -7 48 V -6 47 V -5 44 V -3 43 V -3 41 V -1 39 V 0 37 V 1 35 V 3 33 V 3 31 V 5 28 V 6 27 V 7 24 V 8 22 V 10 20 V 11 18 V 12 15 V 13 13 V 14 11 V 15 8 V 17 7 V 18 3 V 18 2 V 20 -1 V 22 -3 V 22 -5 V 23 -8 V 25 -10 V 25 -13 V 27 -14 V 28 -17 V 29 -19 V 29 -22 V 31 -24 V 32 -25 V 33 -28 V 34 -31 V 35 -32 V 36 -34 V 37 -37 V 38 -38 V 39 -40 V 40 -43 V 41 -44 V 42 -46 V 42 -48 V 44 -49 V 44 -51 V 45 -53 V 46 -55 V 47 -56 V 48 -57 V 48 -59 V 49 -61 V 50 -61 V 51 -63 V 51 -64 V 52 -66 V 53 -66 V 53 -67 V 54 -68 V 55 -70 V 55 -69 V 56 -71 V 56 -71 V 57 -72 V 58 -73 V 58 -72 V 58 -74 V 59 -73 V 59 -74 V 60 -74 V 60 -74 V 60 -74 V 61 -73 V 61 -74 V 62 -74 V 62 -73 V 62 -73 V 62 -72 V 62 -72 V 63 -72 V 63 -70 V 63 -71 V 63 -69 V 63 -68 V 64 -68 V 63 -66 V 63 -66 V 64 -64 V 63 -63 V 64 -62 V 63 -61 V 63 -59 V 63 -58 V 63 -56 V 63 -55 V 63 -53 V 63 -52 V 62 -50 V 62 -48 V 62 -46 V 62 -45 V 61 -42 V 61 -41 V 61 -39 V 60 -37 V 60 -34 V 59 -33 V 59 -31 V 59 -28 V 58 -26 V 58 -25 V 57 -21 V 57 -20 V 56 -17 V 55 -16 V 55 -12 V 55 -11 V 53 -8 V 53 -6 V 53 -4 V 52 -1 V 51 1 V 50 3 V 49 6 V 49 8 V 48 10 V 47 13 V 47 15 V 45 17 V 45 20 V 44 21 V 43 24 V 42 26 V 41 28 V 41 31 V 39 32 V 39 35 V 37 36 V 37 39 V 35 40 V 35 43 V 33 44 V 33 46 V 31 48 V 30 50 V 30 52 V 28 53 V 27 54 V 26 57 V 26 57 V 23 59 V 23 61 V 22 62 V 21 63 V 19 64 V 18 65 V 17 67 V 16 67 V 15 68 V 14 70 V 12 69 V 12 71 V 10 72 V 9 71 V 7 73 V 7 73 V 5 73 V 5 73 V 3 74 V 1 74 V 1 74 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial -38 1475 a currentpoint currentpoint translate 1 0.64568 div 1 0.64568 div scale neg exch neg exch translate -38 1475 a 1999 1475 a currentpoint currentpoint translate 0.64568 0.64568 scale neg exch neg exch translate 1999 1475 a @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 3600 @rwi @setspecial %%BeginDocument: moviment_qperiodic.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: moviment_qperiodic.eps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Wed Jan 9 17:03:27 2002 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 1.000 UL LT0 6962 2520 M -6 222 V -17 219 V -28 216 V -40 210 V -51 202 V -62 192 V -72 182 V -83 168 V -93 155 V -103 138 V -113 122 V -122 103 V -130 85 V -139 65 V -147 45 V -154 24 V -161 3 V -167 -18 V -173 -39 V -177 -58 V -183 -79 V -186 -98 V 4366 4462 L 4174 4329 L 3980 4179 L 3784 4015 L 3588 3837 L 3392 3648 L 3197 3448 L 3003 3240 L 2812 3026 L 2622 2808 L 2437 2587 L 2255 2365 L 2078 2144 L 1905 1927 L 1739 1716 L 1579 1511 L 1426 1315 L 1280 1130 L 1142 958 L 1013 799 L 892 656 L 781 529 L 679 420 L 587 329 L 505 258 L 434 208 L 373 177 L -49 -9 V -39 12 V -27 32 V -16 53 V -4 73 V 7 92 V 19 111 V 29 128 V 42 145 V 52 160 V 63 174 V 74 186 V 84 197 V 94 205 V 104 213 V 114 217 V 123 221 V 132 222 V 139 221 V 148 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b(ha)m(v)m(e)h(already)e(giv)m(en)h(some)f(motiv) -5 b(ations)30 b(to)i(study)i(equations)f(of)f(the)h(t)m(yp)s(e)1702 1532 y Fx(x)1757 1491 y Ft(0)1809 1532 y Fy(=)27 b Fx(A)p Fy(\()p Fx(t)p Fy(\))p Fx(x;)1588 b Fy(\(1.10\))-118 1750 y(where)35 b Fx(x)30 b Fs(2)h Fw(R)413 1714 y Fr(n)500 1750 y Fy(and)j Fx(A)p Fy(\()p Fx(t)p Fy(\))g(is)f(a)h(matrix)e(dep)s (ending)i(quasi-p)s(erio)s(dically)c(on)k(time.)46 b(This)34 b(means)g(that)g(there)-118 1871 y(exists)f(a)g(frequency)h(v)m(ector)g Fx(!)d Fs(2)d Fw(R)1219 1835 y Fr(d)1298 1871 y Fy(and)33 b(a)f(lift)e(of)i Fx(A)p Fy(,)h(whic)m(h)g(w)m(e)h(will)c(denote)j(as) 3034 1846 y(~)3008 1871 y Fx(A)p Fy(,)g(suc)m(h)h(that)1653 2089 y Fx(A)p Fy(\()p Fx(t)p Fy(\))28 b(=)1994 2064 y(~)1969 2089 y Fx(A)p Fy(\()p Fx(!)t(t)p Fy(\))p Fx(;)-118 2307 y Fy(and)100 2282 y(~)74 2307 y Fx(A)36 b Fy(is)f(de\014ned)i(on)e(the) h Fx(d)p Fy(-dimensional)c(torus)k Fw(T)1871 2271 y Fr(d)1915 2307 y Fy(.)52 b(The)36 b(quasi-p)s(erio)s(dicit)m(y)d(of)i Fx(A)h Fy(mak)m(es)g(it)e(p)s(ossible)h(to)-118 2428 y FA(lift)42 b Fy(the)33 b(equation)f(\(1.10\))g(to)g(a)h(system)g(of)f (linear)f(equations)i(on)g Fw(R)2433 2392 y Fr(n)2509 2428 y Fs(\002)22 b Fw(T)2671 2392 y Fr(d)2748 2428 y Fy(simply)31 b(writing)1443 2646 y Fx(x)1498 2605 y Ft(0)1549 2646 y Fy(=)1678 2621 y(~)1653 2646 y Fx(A)p Fy(\()p Fx(\022)s Fy(\))p Fx(x;)212 b(\022)2192 2605 y Ft(0)2243 2646 y Fy(=)28 b Fx(!)t(;)1327 b Fy(\(1.11\))-118 2864 y(where)32 b(the)f(equation)g(\(1.10\))f(is)g(obtained)h(when)h(the)f (initial)c(v)-5 b(alue)30 b(for)h Fx(\022)j Fy(is)c(zero.)43 b(It)31 b(will)e(turn)i(out)g(that)f(for)-118 2985 y(most)37 b(prop)s(erties)g(of)g(the)h(quasi-p)s(erio)s(dic)d(equation)i (\(1.10\))f(w)m(e)j(will)c(ha)m(v)m(e)k(to)e(resort)g(to)g(the)h (lifted)e(system)-118 3105 y(\(1.11\).)28 3226 y(There)26 b(is)e(a)h(useful)g(formalism)c(to)j(study)i(linear)d(equations)i(with) f(quasi-p)s(erio)s(dic)f(co)s(e\016cien)m(ts.)42 b(Note)25 b(that)-118 3346 y(the)k(fundamen)m(tal)e(matrix)g(b)s(elongs)h(alw)m (a)m(ys)h(to)f(the)h(group)f Fx(GL)p Fy(\()p Fx(n;)17 b Fw(R)6 b Fy(\))34 b(of)28 b(linear)f(in)m(v)m(ertible)g (transformations)-118 3466 y(from)d Fw(R)171 3430 y Fr(n)249 3466 y Fy(to)i(itself.)39 b(This)26 b(means)f(that)h(the)g(matrix)e (generating)g(the)i(di\013eren)m(tial)e(equation,)j(in)d(our)i (notations)-118 3587 y Fx(A)p Fy(\()p Fx(t)p Fy(\),)42 b(b)s(elongs)d(to)h(the)g(in\014nitesimal)d(Lie)i(algebra)f Fx(g)t(l)r Fy(\()p Fx(n;)17 b Fw(R)5 b Fy(\),)47 b(of)40 b(linear)e(transformations)g(of)i Fw(R)3566 3551 y Fr(n)3658 3587 y Fy(to)g(itself.)-118 3707 y(See,)h(for)e(instance,)i([71])e(or)f (an)m(y)i(textb)s(o)s(ok)f(on)g(di\013eren)m(tial)f(geometry)-8 b(,)40 b(for)e(an)h(exp)s(osition)f(of)h(Lie)f(groups)-118 3828 y(and)e(Lie)e(algebras.)52 b(No)m(w)36 b(w)m(e)h(can)f(consider)g (the)g(matrix)e(equation)h(for)g(\(1.11\),)h(so)f(that)h(the)g(lifted)e (system)-118 3948 y(b)s(ecomes)1412 4166 y Fx(X)1501 4125 y Ft(0)1552 4166 y Fy(=)1681 4141 y(~)1655 4166 y Fx(A)q Fy(\()p Fx(\022)s Fy(\))p Fx(X)r(;)212 b(\022)2223 4125 y Ft(0)2274 4166 y Fy(=)27 b Fx(!)t(;)1297 b Fy(\(1.12\))-118 4385 y(where)34 b Fx(X)8 b Fy(\()p Fx(t)p Fy(\))27 b Fs(2)h Fx(GL)p Fy(\()p Fx(n;)17 b Fw(R)5 b Fy(\))39 b(and)33 b Fx(A)p Fy(\()p Fx(t)p Fy(\))28 b Fs(2)g Fx(g)t(l)r Fy(\()p Fx(n;)17 b Fw(R)t Fy(\),)39 b(so)32 b(that)h(the)g(pair)e(\()p Fx(X)r(;)17 b(\022)s Fy(\))33 b(b)s(elongs)f(to)g Fx(GL)p Fy(\()p Fx(n;)17 b Fw(R)5 b Fy(\))28 b Fs(\002)23 b Fw(T)3828 4348 y Fr(d)3872 4385 y Fy(.)-118 4611 y Fv(Remark)37 b(1.3.1)49 b FA(The)41 b(same)g(c)-5 b(onstruction)41 b(c)-5 b(an)42 b(b)-5 b(e)41 b(done)g(if)g Fx(G)g Fs(\032)f Fx(GL)p Fy(\()p Fx(n;)17 b Fw(R)5 b Fy(\))48 b FA(is)42 b(a)f(c)-5 b(ertain)41 b(Lie)h(sub)-5 b(gr)g(oup)-118 4731 y(\(for)34 b(instanc)-5 b(e)34 b Fx(S)6 b(L)p Fy(\()p Fx(n;)17 b Fw(R)5 b Fy(\))41 b FA(or)35 b Fx(S)6 b(O)s Fy(\()p Fx(n;)17 b Fw(R)t Fy(\))p FA(,)41 b(se)-5 b(e)34 b([71]\).)-118 4958 y Fv(Remark)j(1.3.2)49 b FA(A)n(lthough)42 b(we)h(wil)5 b(l)42 b(mainly)g(fo)-5 b(cus)42 b(on)g(the)h(pr)-5 b(op)g(erties)42 b(of)h(line)-5 b(ar)42 b(e)-5 b(quations)42 b(with)g(quasi-)-118 5078 y(p)-5 b(erio)g(dic)34 b(c)-5 b(o)g(e\016cients)35 b(in)g Fw(R)5 b FA(,)41 b(it)36 b(is)f(cle)-5 b(ar)35 b(that)h(al)5 b(l)35 b(the)g(ab)-5 b(ove)35 b(c)-5 b(onstruction)35 b(c)-5 b(an)35 b(b)-5 b(e)35 b(p)-5 b(erforme)g(d)34 b(to)i(de)-5 b(al)35 b(with)-118 5198 y(the)g(c)-5 b(ase)34 b(of)h(c)-5 b(omplex)33 b(c)-5 b(o)g(e\016cients.)28 5425 y Fy(Th)m(us)31 b(a)d(natural)g(question)i (arises:)41 b(whic)m(h)30 b(to)s(ols)d(ha)m(v)m(e)k(to)d(b)s(e)h(used)h (to)f(describ)s(e)h(the)f(b)s(eha)m(viour)g(of)f(linear)-118 5545 y(equations)33 b(with)f(quasi-p)s(erio)s(dic)e(co)s(e\016cien)m (ts)k(?)28 5666 y(So)45 b(far,)i(w)m(e)f(ha)m(v)m(e)g(only)e(giv)m(en)h (one)g(example)f(of)g(linear)f(equation)i(with)f(quasi-p)s(erio)s(dic)e (co)s(e\016cien)m(ts,)-118 5786 y(whic)m(h)31 b(is)f(the)h(trivial)d (case)j(of)f(a)g(linear)f(equation)h(whose)i(matrix)d(do)s(es)i(not)f (dep)s(end)i(on)e(time.)42 b(In)30 b(that)h(case,)-118 5906 y(b)s(oth)h(the)h(qualitativ)m(e)e(and)h(the)h(quan)m(titativ)m(e) f(b)s(eha)m(viour)g(of)g(the)h(system)g(could)f(b)s(e)h(obtained,)f(b)s (ecause)i(w)m(e)1924 6155 y(8)p eop %%Page: 9 13 9 12 bop -118 219 a Fy(could)38 b(directly)g(in)m(tegrate)h(the)g (system.)63 b(One)39 b(w)m(ould)f(lik)m(e)g(to)h(ha)m(v)m(e)h(alw)m(a)m (ys)f(suc)m(h)h(a)f(strong)g(kno)m(wledge)g(of)-118 339 y(the)31 b(prop)s(erties)g(of)f(the)h(solutions)f(of)g(a)h(linear)e (equation)i(with)f(quasi-p)s(erio)s(dic)f(co)s(e\016cien)m(ts,)j(but)f (this)g(seems)-118 459 y(di\016cult)i(if)f(w)m(e)j(cannot)f(in)m (tegrate)f(the)h(system)h(of)e(equations)h(\(whic)m(h)g(is)f(the)h (usual)f(situation\).)45 b(Therefore,)-118 580 y(w)m(e)36 b(w)m(ould)f(lik)m(e)f(to)h(kno)m(w)h(whic)m(h)g(are)f(the)g(systems)i (whose)f(qualitativ)m(e)d(b)s(eha)m(viour)i(can)g(b)s(e)h FA(r)-5 b(e)g(duc)g(e)g(d)45 b Fy(to)34 b(the)-118 700 y(b)s(eha)m(viour)e(of)h(a)f(system)h(with)g(constan)m(t)g(co)s (e\016cien)m(ts.)28 820 y(In)g(more)f(precise)h(terms)g(this)f (reduction)h(can)f(b)s(e)h(expressed)j(in)31 b(the)i(follo)m(wing)d(w)m (a)m(y)-118 1049 y Fv(De\014nition)36 b(1.3.3)49 b FA(A)n(n)37 b(e)-5 b(quation)36 b(\(1.10\))g(is)h(said)f(to)h(b)-5 b(e)37 b(\()16 b Fy(Ly)m(apuno)m(v-P)m(erron)p FA(\))42 b Fy(reducible)37 b FA(whenever)f(ther)-5 b(e)-118 1169 y(exists)34 b(a)h(line)-5 b(ar)34 b(time-varying)g(change)g(of)h (variables)1724 1389 y Fx(x)28 b Fy(=)g Fx(Z)7 b Fy(\()p Fx(t)p Fy(\))p Fx(y)t(;)-118 1609 y FA(c)-5 b(al)5 b(le)-5 b(d)43 b Fy(Ly)m(apuno)m(v-P)m(erron)33 b(transformation)p FA(,)f(such)h(that)h(it)g(is)f(non-singular)f(for)h(al)5 b(l)34 b Fx(t)27 b Fs(2)i Fw(R)5 b FA(,)39 b Fx(Z)7 b FA(,)34 b Fx(Z)3603 1573 y Ft(\000)p FC(1)3731 1609 y FA(and)f Fx(Z)3993 1573 y Ft(0)-118 1730 y FA(ar)-5 b(e)35 b(b)-5 b(ounde)g(d)34 b(in)g Fw(R)5 b FA(,)41 b(and)34 b(which)g(tr)-5 b(ansforms)34 b(the)h(system)g(into)g(an)f(e)-5 b(quation)35 b(like)1767 1950 y Fx(y)1819 1908 y Ft(0)1869 1950 y Fy(=)28 b Fx(B)5 b(y)t(;)-118 2170 y FA(wher)-5 b(e)34 b Fx(B)40 b FA(is)35 b(a)f(c)-5 b(onstant)35 b(matrix.)28 2398 y Fy(This)41 b(notion)e(is)h(also)g(v)-5 b(alid)38 b(for)i(general)g(linear)f(systems)j(lik)m(e)e(\(1.1\),)i(and)e(it)g (implies)e(that,)k(whenev)m(er)-118 2518 y(suc)m(h)28 b(a)e(system)i(is)e(Ly)m(apuno)m(v-P)m(erron)j(reducible)d(to)h(a)f (constan)m(t)i(co)s(e\016cien)m(ts)f(system)h(lik)m(e)e(\(1.3\),)h (then)h(man)m(y)-118 2639 y(prop)s(erties)k(of)g(the)h(original)c (system)k(\(suc)m(h)h(as)f(the)f(gro)m(wth)h(of)f(the)h(solutions)e(or) h(their)g(b)s(oundness\))i(are)e(the)-118 2759 y(same)g(as)h(those)g (of)f(the)h(reduced)i(system)e(with)f(constan)m(t)i(co)s(e\016cien)m (ts.)28 2879 y(Ly)m(apuno)m(v-P)m(erron)j(reducibilit)m(y)-8 b(,)33 b(without)h(requiring)g(additional)e(conditions)h(on)i(the)g (transformation,)-118 3000 y(is)27 b(not)h(\014ne)g(enough)g(to)g (study)h(linear)d(equations)i(with)f(quasi-p)s(erio)s(dic)f(co)s (e\016cien)m(ts,)k(b)s(ecause)f(the)f FA(r)-5 b(otational)-118 3120 y(b)g(ehaviour)42 b Fy(is)33 b(not)f(tak)m(en)i(in)m(to)e(accoun)m (t.)44 b(F)-8 b(or)32 b(instance,)h(if)f(all)e(the)j(solutions)f(of)g (\(1.10\))g(and)h(its)f(deriv)-5 b(ativ)m(es)-118 3241 y(are)41 b(b)s(ounded,)k(then)d(the)g(system)h(is)e(Ly)m(apuno)m(v-P)m (erron)i(reducible)f(and)f(an)m(y)h(t)m(w)m(o)h(suc)m(h)g(systems)g (can)e(b)s(e)-118 3361 y(reduced)d(to)e(the)i(same)e(system)i(with)e (constan)m(t)i(co)s(e\016cien)m(ts.)56 b(Therefore)38 b(additional)c(conditions)i(m)m(ust)g(b)s(e)-118 3481 y(imp)s(osed.)28 3602 y(F)-8 b(or)30 b(the)g(rest)h(of)f(the)g(surv)m (ey)-8 b(,)33 b(reducibilit)m(y)28 b(will)g(mean)i(Ly)m(apuno)m(v-P)m (erron)i(reducibilit)m(y)c(to)i(constan)m(t)h(co-)-118 3722 y(e\016cien)m(ts)e(b)m(y)f(means)g(of)f(a)g(quasi-p)s(erio)s(dic)f (transformation.)39 b(W)-8 b(e)28 b(shall)e(usually)h(imp)s(ose)g(that) g(the)h(minim)m(um)-118 3842 y(n)m(um)m(b)s(er)g(of)f(basic)g (frequencies)i(of)e(the)h(transformation)d(is)i(the)h(same)f(that)h (for)f(the)g(original)e(system,)k(but)f(not)-118 3963 y(in)g(general)g(that)g(the)i(basic)e(frequencies)i(are)f(the)g(same.) 42 b(The)30 b(t)m(w)m(o)f(follo)m(wing)d(examples)j(of)f(linear)f (equations)-118 4083 y(with)42 b(quasi-p)s(erio)s(dic)d(co)s(e\016cien) m(ts)44 b(together)e(with)g(the)g(discussion)g(of)g(their)f (reducibilit)m(y)g(problems)g(will,)-118 4204 y(hop)s(efully)-8 b(,)31 b(mak)m(e)i(subsequen)m(t)j(discussion)d(clearer.)-118 4536 y Fu(1.4)161 b(Example)37 b(1:)64 b(One)37 b(frequency)f(and)i (man)l(y)f(dimensions.)65 b(Flo-)250 4719 y(quet)53 b(theory)f(for)i(p) t(erio)t(dic)h(systems)-118 4938 y Fy(In)34 b(this)f(section)h(w)m(e)h (will)c(deal)i(with)g(linear)f(equations)i(with)g(quasi-p)s(erio)s(dic) d(co)s(e\016cien)m(ts)k(ha)m(ving)e(only)g(one)-118 5058 y(frequency)-8 b(,)35 b(in)e(the)g(previous)h(notation)e Fx(d)c Fy(=)g(1.)45 b(Therefore,)35 b(our)e(linear)e(equation)i(will)e (b)s(e)i(p)s(erio)s(dic)f(and,)h(in)-118 5179 y(this)f(case,)i(the)f (easy)g(argumen)m(ts)g(of)f(Flo)s(quet)g(theory)h(will)d(guaran)m(tee)j (reducibilit)m(y)-8 b(.)28 5299 y(Indeed,)34 b(consider)f(the)g (equation)1702 5519 y Fx(x)1757 5478 y Ft(0)1809 5519 y Fy(=)27 b Fx(A)p Fy(\()p Fx(t)p Fy(\))p Fx(x;)1588 b Fy(\(1.13\))-118 5739 y(where)34 b Fx(x)29 b Fs(2)g Fw(R)409 5703 y Fr(n)495 5739 y Fy(and)k Fx(A)g Fy(is)g(a)f(matrix)g (dep)s(ending)h(p)s(erio)s(dically)d(on)j(time)f(with)g(p)s(erio)s(d)g Fx(T)14 b Fy(.)45 b(This)33 b(means)g(that)-118 5860 y Fx(A)p Fy(\()p Fx(t)24 b Fy(+)g Fx(T)14 b Fy(\))31 b(=)h Fx(A)p Fy(\()p Fx(t)p Fy(\))k(for)e(all)f Fx(t)f Fs(2)h Fw(R)5 b Fy(.)57 b(Let)35 b Fx(X)8 b Fy(\()p Fx(t)p Fy(\))35 b(b)s(e)g(a)g(fundamen)m(tal)f(matrix)g(for)g(the)i(p)s(erio)s (dic)d(system)j(\(1.13\))1924 6155 y(9)p eop %%Page: 10 14 10 13 bop -118 219 a Fy(b)s(eing)33 b(the)h(iden)m(tit)m(y)f(at)g Fx(t)c Fy(=)h(0,)j(and)h(consider)f(the)h(map)f Fs(P)38 b Fy(=)29 b Fx(X)8 b Fy(\()p Fx(T)14 b Fy(\))33 b(\(whic)m(h)h(due)g (to)f(the)h(p)s(erio)s(dicit)m(y)-8 b(,)32 b(is)h(an)-118 339 y(endomorphism)e(of)h Fw(R)715 303 y Fr(d)762 339 y Fy(,)g(called)g(the)h FA(Poinc)-5 b(ar)n(\023)-47 b(e)33 b(map)38 b Fy(for)32 b(the)h(system)g(\(1.13\)\).)28 459 y(Let)g Fx(B)38 b Fy(b)s(e)32 b(a)h(matrix)e(satisfying)g(that)1460 677 y Fs(P)36 b Fy(=)28 b Fx(X)8 b Fy(\()p Fx(T)14 b Fy(\))27 b(=)h(exp)q(\()p Fx(T)14 b(B)5 b Fy(\))p Fx(:)-118 895 y Fy(Suc)m(h)30 b(a)f(matrix)f(can)i(b)s(e)f(alw)m(a)m(ys)h(found)g (if)e(w)m(e)i(don't)g(require)f Fx(B)35 b Fy(to)29 b(b)s(e)g(real)g (\(if)f Fs(P)37 b Fy(has)30 b(negativ)m(e)g(eigen)m(v)-5 b(alues,)-118 1015 y(then)26 b Fx(B)31 b Fy(m)m(ust)25 b(necessarily)i(b)s(e)e(complex\).)41 b(W)-8 b(e)26 b(shall)e(call)g Fx(B)31 b Fy(a)25 b FA(Flo)-5 b(quet)28 b(matrix)37 b Fy(of)25 b(\(1.13\).)41 b(Flo)s(quet)25 b(matrices)-118 1135 y(for)32 b(a)g(p)s(erio)s(dic)f(system)j(are)e FA(not)42 b Fy(uniquely)33 b(determined.)28 1256 y(Then,)h(it)e(is)g(satis\014ed) h(that)1181 1473 y Fx(X)1262 1488 y Fr(T)1317 1473 y Fy(\()p Fx(t)p Fy(\))28 b(=)g Fx(X)8 b Fy(\()p Fx(t)22 b Fy(+)g Fx(T)14 b Fy(\))27 b(=)g Fx(X)8 b Fy(\()p Fx(t)p Fy(\))17 b(exp)q(\()p Fx(T)d(B)5 b Fy(\))p Fx(;)1066 b Fy(\(1.14\))-118 1691 y(for)32 b(all)e(real)i Fx(t)p Fy(.)44 b(Indeed,)34 b(the)f(equalit)m(y)f(is)g(true)h(for)f Fx(t)c Fy(=)g(0,)k(and)h Fx(X)2344 1706 y Fr(T)2431 1691 y Fy(satis\014es)h(the)f(di\013eren)m(tial)d(equation)1084 1909 y Fx(X)1165 1924 y Fr(T)1220 1867 y Ft(0)1243 1909 y Fy(\()p Fx(t)p Fy(\))e(=)f Fx(A)p Fy(\()p Fx(t)c Fy(+)f Fx(T)14 b Fy(\))p Fx(X)8 b Fy(\()p Fx(t)22 b Fy(+)g Fx(T)14 b Fy(\))27 b(=)h Fx(A)p Fy(\()p Fx(t)p Fy(\))p Fx(X)2648 1924 y Fr(T)2703 1909 y Fy(\()p Fx(t)p Fy(\))-118 2126 y(so)e(it)f(is)h(a)g(fundamen)m(tal)f(solution)f(for)i(\(1.13\).)41 b(Therefore)27 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b(We)-118 4530 y(ne)-5 b(e)g(d)41 b(to)h(imp)-5 b(ose)40 b(that)i(the)g(fr)-5 b(e)g(quency)41 b(ve)-5 b(ctor)41 b(of)h(the)f(r)-5 b(e)g(ducing)41 b(tr)-5 b(ansformation)41 b(is)g Fx(!)t(=\037)3309 4545 y Fr(g)3390 4530 y FA(\(wher)-5 b(e)41 b Fx(\037)3773 4545 y Fr(g)3855 4530 y FA(is)g(a)-118 4651 y(c)-5 b(ertain)37 b(inte)-5 b(ger)36 b(dep)-5 b(ending)36 b(on)h(the)g(Lie)g(algebr)-5 b(a)36 b Fx(g)k FA(wher)-5 b(e)37 b(the)g(e)-5 b(quation)37 b(takes)f(plac)-5 b(e\))37 b(inste)-5 b(ad)36 b(of)h Fx(!)j FA(if)d(we)-118 4771 y(don)-10 b('t)34 b(want)h(to)g(c)-5 b(omplexify)34 b(the)h(system.)28 4997 y Fy(W)-8 b(e)33 b(ha)m(v)m(e)h(th)m(us)g(pro)m(v)m(ed)-118 5222 y Fv(Theorem)j(1.4.2)h (\(Flo)s(quet's)f(theorem\))48 b FA(L)-5 b(et)25 b(e)-5 b(quation)25 b(\(1.13\))f(b)-5 b(e)24 b(p)-5 b(erio)g(dic)24 b(with)h(p)-5 b(erio)g(d)24 b Fx(T)14 b FA(.)42 b(Then)24 b(ther)-5 b(e)-118 5343 y(exists)34 b(a)h Fy(2)p Fx(T)14 b FA(-p)-5 b(erio)g(dic)33 b(change)h(of)h(variables)e Fx(x)c Fy(=)e Fx(Z)7 b Fy(\()p Fx(t)p Fy(\))p Fx(y)38 b FA(and)c(a)h(r)-5 b(e)g(al)35 b(c)-5 b(onstant)34 b(matrix)h Fx(B)40 b FA(such)34 b(that)1767 5560 y Fx(y)1819 5519 y Ft(0)1869 5560 y Fy(=)28 b Fx(B)5 b(y)t(:)28 5786 y Fy(That)33 b(is,)f(all)e(p)s(erio)s(dic)h(systems)j(are)f(reducible)f (b)m(y)h(means)g(of)f(a)g(p)s(erio)s(dic)f(transformation)f(with)i (double)-118 5906 y(the)h(p)s(erio)s(d)e(of)h(the)h(original)d (equation.)1900 6155 y(10)p eop %%Page: 11 15 11 14 bop -118 219 a Fv(Remark)37 b(1.4.3)49 b FA(As)39 b(we)g(have)f(se)-5 b(en,)40 b(the)f(Flo)-5 b(quet)38 b(matrix)h(of)g(a)g(p)-5 b(erio)g(dic)38 b(system)h(is)g(not)g (uniquely)g(deter-)-118 339 y(mine)-5 b(d)46 b(\(b)-5 b(e)g(c)g(ause)46 b(the)h(exp)-5 b(onential)46 b(is)h(not)g (one-to-one\).)80 b(In)46 b(pr)-5 b(actic)g(al)46 b(situations,)k (however,)f(it)f(c)-5 b(an)46 b(b)-5 b(e)-118 459 y(useful)33 b(to)h(imp)-5 b(ose)33 b(extr)-5 b(a)33 b(c)-5 b(onditions)32 b(so)i(that)g(it)f(is)h(uniquely)f(determine)-5 b(d.)43 b(F)-7 b(or)33 b(instanc)-5 b(e,)33 b(it)h(c)-5 b(an)33 b(b)-5 b(e)33 b(chosen)-118 580 y(to)f(minimize)f(a)g(c)-5 b(ertain)32 b(norm)f(or,)i(if)e(the)h(system)g(dep)-5 b(ends)31 b(on)g(external)h(p)-5 b(ar)g(ameters,)32 b(to)g(b)-5 b(e)31 b(c)-5 b(ontinuous)32 b(on)-118 700 y(these)i(p)-5 b(ar)g(ameters.)44 b(A)n(l)5 b(l)35 b(these)g(choic)-5 b(es)33 b(dep)-5 b(end)34 b(on)h(the)g(pr)-5 b(op)g(erties)34 b(of)g(our)i(system.)-118 927 y Fv(Remark)h(1.4.4)49 b FA(F)-7 b(r)i(om)42 b(the)h(pr)-5 b(o)g(of)42 b(of)g(Flo)-5 b(quet's)43 b(the)-5 b(or)g(em,)44 b(we)f(get)g(a)f(r)-5 b(epr)g(esentation)42 b(of)h(a)g(fundamental)-118 1048 y(solution)35 b(as)1499 1168 y Fx(X)8 b Fy(\()p Fx(t)p Fy(\))28 b(=)f Fx(Z)7 b Fy(\()p Fx(t)p Fy(\))17 b(exp)q(\()p Fx(B)5 b(t)p Fy(\))p Fx(;)-118 1342 y FA(wher)-5 b(e)24 b Fx(Z)32 b FA(is)25 b(a)g Fy(2)p Fx(T)14 b FA(-p)-5 b(erio)g(dic)23 b(matrix)i(and)g Fx(B)30 b FA(is)25 b(a)f(c)-5 b(onstant)25 b(matrix.)41 b(This)25 b(is)f(c)-5 b(al)5 b(le)-5 b(d)25 b(a)32 b 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b(system)g(\(which,)g(r)-5 b(e)g(c)g(al)5 b(l,)45 b(r)-5 b(epr)g(esents)42 b(a)-118 2238 y(family)34 b(of)h(quasi-p)-5 b(erio)g(dic)33 b(line)-5 b(ar)35 b(e)-5 b(quations)34 b(p)-5 b(ar)g(ameterize)g(d)34 b(by)h(the)g(phase)f Fx(\036)p FA(\),)g(then)h(we)f(write)1137 2485 y Fx(X)8 b Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))27 b(=)1590 2460 y(~)1569 2485 y Fx(Z)1660 2374 y Fq(\020)1729 2418 y Fx(!)p 1729 2462 65 4 v 1737 2553 a Fy(2)1804 2485 y Fx(t)22 b Fy(+)g Fx(\036)2017 2374 y Fq(\021)2093 2485 y Fy(exp)q(\()p Fx(tB)5 b Fy(\))2453 2460 y(~)2432 2485 y Fx(Z)i Fy(\()p Fx(\036)p Fy(\))2640 2444 y Ft(\000)p FC(1)2734 2485 y Fx(;)-118 2752 y FA(wher)-5 b(e)181 2727 y Fy(~)161 2752 y Fx(Z)45 b FA(is)38 b(the)g(lift)h(of)f(the)g(quasi-p)-5 b(erio)g(dic)37 b(function)h Fx(Z)7 b FA(.)56 b(This)37 b(last)i(r)-5 b(epr)g(esentation)37 b(is)h(e)-5 b(quivalent)38 b(to)h(the)-118 2872 y(ful\014l)5 b(lment)34 b(of)h(the)g(fol)5 b(lowing)33 b(e)-5 b(quation)35 b(for)f Fx(Z)7 b FA(,)1047 3091 y Fx(@)1108 3079 y Fj(!)p 1109 3091 41 3 v 1114 3132 a Fp(2)1184 3066 y Fy(~)1164 3091 y Fx(Z)g Fy(\()p Fx(\022)s Fy(\))27 b(=)1519 3066 y(~)1493 3091 y Fx(A)p Fy(\()p Fx(\022)s Fy(\))1711 3066 y(~)1690 3091 y Fx(Z)7 b Fy(\()p Fx(\022)s Fy(\))22 b Fs(\000)2030 3066 y Fy(~)2010 3091 y Fx(Z)7 b Fy(\()p Fx(\022)s Fy(\))p Fx(B)e(;)216 b(\022)31 b Fs(2)d Fw(T)2763 3050 y Fr(d)2807 3091 y Fx(;)932 b Fy(\(1.15\))-118 3311 y FA(wher)-5 b(e)40 b Fx(@)214 3326 y Fr(!)264 3311 y Fs(\001)e Fy(=)g Fs(hr)566 3326 y Fr(\022)604 3311 y Fs(\001)p Fx(;)17 b(!)t Fs(i)p FA(.)60 b(A)n(n)40 b(e)-5 b(quation)40 b(of)g(this)g(typ)-5 b(e)41 b(is)f(c)-5 b(al)5 b(le)-5 b(d)50 b Fy(homological)34 b(equation)40 b FA(and)g(wil)5 b(l)40 b(b)-5 b(e)40 b(often)-118 3431 y(found)34 b(along)g(this)h(survey.)-118 3764 y Fu(1.5)161 b(Example)54 b(2:)71 b(Man)l(y)53 b(frequencies)g(and)h(one) f(dimension.)250 3946 y(Small)i(divisor)f(problems)-118 4165 y Fy(W)-8 b(e)27 b(ha)m(v)m(e)h(seen)g(that)e(p)s(erio)s(dic)f (linear)g(equations)i(are)g(reducible,)g(and)g(the)g(reason)g(is)f (that)g(the)h(\015o)m(w)h(is)e FA(exactly)-118 4286 y Fy(the)34 b(same)g(after)f(the)i(p)s(erio)s(d.)46 b(When)34 b(w)m(e)h(ha)m(v)m(e)g(more)f(than)f(t)m(w)m(o)i(frequencies,)h(the)e (\015o)m(w)g(is)g FA(never)44 b Fy(the)34 b(same)-118 4406 y(for)41 b(an)m(y)i(shift)f(of)f(time,)j(but)e(it)f(gets)i(closer) f(and)g(closer)g(to)g(previous)g(v)-5 b(alues.)72 b(In)43 b(view)f(of)g(the)g(previous)-118 4526 y(section)37 b(one)g(could)f (think)h(that)f(this)h(recurrence)i(of)d(the)h(\015o)m(w)h(is)e(enough) h(to)f(pro)m(v)m(e)j(reducibilit)m(y)-8 b(.)54 b(W)-8 b(e)37 b(will)-118 4647 y(see)d(in)d(this)i(section)f(that)h(this)f(is) g(not)g(so)h(easy)h(b)m(y)f(means)g(of)f(a)g(simple)f(example.)28 4767 y(Among)43 b(linear)e(equations)j(with)f(quasi-p)s(erio)s(dic)e (co)s(e\016cien)m(ts)k(and)e(more)g(than)g(t)m(w)m(o)h(frequencies)h (the)-118 4888 y(simplest)31 b(are)i(scalar)f(equations,)h(in)f(our)g (notations)g Fx(x)p Fy(\()p Fx(t)p Fy(\))c Fs(2)g Fw(R)5 b Fy(,)1602 5107 y Fx(x)1657 5066 y Ft(0)1681 5107 y Fy(\()p Fx(t)p Fy(\))28 b(=)f Fx(a)p Fy(\()p Fx(t)p Fy(\))p Fx(x)p Fy(\()p Fx(t)p Fy(\))p Fx(;)1488 b Fy(\(1.16\))-118 5326 y(where)38 b Fx(a)e Fy(:)g Fw(R)46 b Fs(!)36 b Fw(R)48 b Fy(is)36 b(a)h(quasi-p)s(erio)s(dic)f(function)g(with)h Fx(d)g Fy(frequencies.)59 b(This)37 b(equation)g(can)h(b)s(e)f (directly)-118 5446 y(in)m(tegrated,)29 b(but)h(as)f(w)m(e)h(w)m(an)m (t)f(to)g(illustrate)e(the)i(reducibilit)m(y)e(problem,)i(w)m(e)h(pro)s (ceed)f(in)f(a)h(more)f(qualitativ)m(e)-118 5567 y(w)m(a)m(y)-8 b(.)28 5687 y(T)g(o)33 b(render)g(equation)g(\(1.16\))e(to)i(constan)m (t)g(co)s(e\016cien)m(ts)h(w)m(e)f(merely)f(ha)m(v)m(e)i(to)f(set)1596 5906 y Fx(z)t Fy(\()p Fx(t)p Fy(\))c(=)e(exp)q(\()p Fx(g)t Fy(\()p Fx(t)p Fy(\)\))p Fx(;)1900 6155 y Fy(11)p eop %%Page: 12 16 12 15 bop -118 219 a Fy(where)1413 466 y Fx(g)t Fy(\()p Fx(t)p Fy(\))27 b(=)1706 330 y Fq(Z)1805 356 y Fr(t)1761 556 y FC(0)1852 466 y Fy(\()o Fx(a)p Fy(\()p Fx(s)p Fy(\))c Fs(\000)f Fy([)p Fx(a)p Fy(]\))17 b Fx(ds;)1298 b Fy(\(1.17\))-118 728 y(and)33 b([)p Fx(a)p Fy(])f(is)h(the)g(a)m(v)m(erage)g(of)f(the)h (quasi-p)s(erio)s(dic)e(function)h Fx(a)p Fy(.)43 b(Indeed,)35 b(if)c Fx(x)d Fy(=)g Fx(z)t Fy(\()p Fx(t)p Fy(\))p Fx(y)t Fy(,)k(then)628 937 y Fx(y)680 896 y Ft(0)730 937 y Fy(=)27 b Fx(a)p Fy(\()p Fx(t)p Fy(\))p Fx(x)17 b Fy(exp)r(\()p Fs(\000)p Fx(g)t Fy(\()p Fx(t)p Fy(\)\))22 b Fs(\000)g Fx(x)17 b Fy(exp)q(\()p Fx(g)t Fy(\()p Fx(t)p Fy(\)\))p Fx(g)2163 896 y Ft(0)2185 937 y Fy(\()p Fx(t)p Fy(\))28 b(=)f(\()p Fx(a)p Fy(\()p Fx(t)p Fy(\))c Fs(\000)f Fx(a)p Fy(\()p Fx(t)p Fy(\))h(+)f([)p Fx(a)p Fy(]\))17 b Fx(y)t(;)-118 1146 y Fy(so)33 b(the)g(transformed)f(system)h(is)1754 1266 y Fx(y)1806 1225 y Ft(0)1856 1266 y Fy(=)28 b([)p Fx(a)p Fy(])p Fx(y)t(:)-118 1436 y Fy(One)41 b(ma)m(y)f(think)g(that)g (the)h(addition)d(of)i(the)h(term)f Fs(\000)p Fy([)p Fx(a)p Fy(])h(in)f(the)h(in)m(tegrand)f(of)g(\(1.17\))f(is)h(quite)g (arti\014cial.)-118 1556 y(Ho)m(w)m(ev)m(er,)31 b(w)m(e)e(ha)m(v)m (en't)g(y)m(et)g(c)m(hec)m(k)m(ed)i(whether)e(the)f(transformation)e (de\014ned)j(b)m(y)g Fx(z)j Fy(is)c(quasi-p)s(erio)s(dic)d(or)j(not.) -118 1677 y(This)36 b(is)g(equiv)-5 b(alen)m(t)36 b(to)g(see)h(whether) h Fx(g)i Fy(is)35 b(quasi-p)s(erio)s(dic)f(or)i(not.)55 b(That)36 b(the)h(addition)d(of)i(the)h(a)m(v)m(erage)g(is)-118 1797 y(necessary)e(is)d(clear)g(from)f(the)i(theorem)g(on)f(a)m(v)m (erages,)i(b)s(ecause,)g(as)1490 2064 y(lim)1475 2124 y Fr(t)p Ft(!1)1668 1996 y Fy(1)p 1668 2041 49 4 v 1675 2132 a Fx(t)1743 1928 y Fq(Z)1843 1954 y Fr(t)1799 2154 y FC(0)1889 2064 y Fx(a)p Fy(\()p Fx(s)p Fy(\))p Fx(ds)28 b Fy(=)f([)p Fx(a)p Fy(])p Fx(;)-118 2326 y Fy(if)39 b(w)m(e)i(replace)f([)p Fx(a)p Fy(])g(b)m(y)h(an)m(y)g(other)f(constan) m(t)h(in)e(\(1.17\))g(the)i(function)e(is)h(un)m(b)s(ounded)h(and,)h (therefore,)h(not)-118 2446 y(quasi-p)s(erio)s(dic.)e(Note)33 b(that)f Fx(g)k Fy(satis\014es)d(the)g(equation)1574 2655 y Fx(g)1625 2614 y Ft(0)1648 2655 y Fy(\()p Fx(t)p Fy(\))28 b(=)f Fx(a)p Fy(\()p Fx(t)p Fy(\))c Fs(\000)f Fy([)p Fx(a)p Fy(])p Fx(;)1460 b Fy(\(1.18\))-118 2864 y(so)31 b(if)e Fx(g)k Fy(has)e(to)f(b)s(e)h(quasi-p)s(erio)s(dic)d (\(with)i(a)h(certain)f(regularit)m(y\),)f(then)i(it)f(can)h(b)s(e)f (written)g(b)m(y)i(means)e(of)g(the)-118 2985 y(F)-8 b(ourier)31 b(series)i(of)g(the)g(lift)g(~)-52 b Fx(g)s Fy(,)1028 3206 y Fx(g)t Fy(\()p Fx(t)p Fy(\))27 b(=)k(~)-52 b Fx(g)s Fy(\()p Fx(!)t(t)22 b Fy(+)g Fx(\036)p Fy(\))27 b(=)1869 3111 y Fq(X)1856 3331 y Fl(k)p Ft(2)p Fk(Z)1996 3312 y Fj(d)2046 3206 y Fy(~)-52 b Fx(g)2090 3221 y Fl(k)2153 3206 y Fy(exp)q(\()p Fx(i)p Fs(h)p Fv(k)p Fx(;)17 b(!)t(t)22 b Fy(+)g Fx(\036)p Fs(i)p Fy(\))-118 3511 y(and)33 b(also)e Fx(a)i Fy(has)g(suc)m(h)h(an)e(expression)1012 3732 y Fx(a)p Fy(\()p Fx(t)p Fy(\))c(=)g(~)-50 b Fx(a)p Fy(\()p Fx(!)t(t)22 b Fy(+)g Fx(\036)p Fy(\))27 b(=)1854 3638 y Fq(X)1841 3857 y Fl(k)p Ft(2)p Fk(Z)1981 3838 y Fj(d)2029 3732 y Fy(~)-50 b Fx(a)2079 3747 y Fl(k)2143 3732 y Fy(exp)q(\()p Fx(i)p Fs(h)p Fv(k)p Fx(;)17 b(!)t(t)k Fy(+)h Fx(\036)p Fs(i)p Fy(\))p Fx(:)-118 4051 y Fy(With)32 b(these)i(de\014nitions,)e (w)m(e)h(can)g(lift)e(equation)h(\(1.18\))g(to)g Fw(T)2222 4014 y Fr(d)2266 4051 y Fy(,)h(with)1522 4260 y Fx(@)1573 4275 y Fr(!)1627 4260 y Fy(~)-52 b Fx(g)s Fy(\()p Fx(\022)s Fy(\))28 b(=)h(~)-51 b Fx(a)q Fy(\()p Fx(\022)s Fy(\))22 b Fs(\000)h Fy([)p Fx(a)p Fy(])p Fx(;)1407 b Fy(\(1.19\))-118 4469 y(for)32 b(all)e Fx(\022)h Fs(2)d Fw(T)399 4433 y Fr(d)443 4469 y Fy(.)44 b(Note)32 b(that)1163 4589 y Fx(@)1214 4604 y Fr(!)1268 4589 y Fy(~)-53 b Fx(g)t Fy(\()p Fx(\022)s Fy(\))27 b(=)1583 4495 y Fq(X)1570 4714 y Fl(k)p Ft(2)p Fk(Z)1710 4695 y Fj(d)1761 4589 y Fy(~)-53 b Fx(g)1804 4604 y Fl(k)1851 4589 y Fy(\()p Fx(i)p Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)p Fy(\))g(exp)o(\()p Fx(i)p Fs(h)p Fv(k)p Fx(;)g(\022)s Fs(i)p Fy(\))p Fx(;)-118 4861 y Fy(so)40 b(the)h(formal)d(solution)g(of)i(the)h(homological)36 b(equation)k(\(1.19\))f(in)h(terms)g(of)f(the)i(F)-8 b(ourier)39 b(co)s(e\016cien)m(ts)i(is)-118 4981 y(giv)m(en)33 b(b)m(y)1652 5142 y(~)-52 b Fx(g)1696 5157 y Fl(k)1770 5142 y Fy(=)28 b Fs(\000)p Fx(i)2069 5075 y Fy(~)-50 b Fx(a)2119 5090 y Fl(k)p 1994 5119 245 4 v 1994 5211 a Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)-118 5375 y Fy(for)32 b Fv(k)c Fs(2)g Fw(Z)281 5339 y Fr(d)351 5375 y Fy(di\013eren)m(t)33 b(from)e(zero)i(and)1806 5496 y(~)-53 b Fx(g)1849 5511 y FC(0)1916 5496 y Fy(=)28 b(0)p Fx(;)-118 5666 y Fy(pro)m(vided)j (none)h(of)e(the)i(quotien)m(ts)f Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)30 b Fy(v)-5 b(anishes.)43 b(If)31 b(there)h(is)e(a)h (non-trivial)d(m)m(ulti-in)m(teger)g Fv(k)g Fs(2)g Fw(Z)3732 5629 y Fr(d)3770 5666 y Fy(,)j(suc)m(h)-118 5786 y(that)1736 5906 y Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)27 b Fy(=)g(0)1900 6155 y(12)p eop %%Page: 13 17 13 16 bop -118 219 a Fy(then)35 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y(quite)c(lo)s(ose)g(in)g(the)h(smo)s(othness)g (prop)s(erties,)g(but)g(no)m(w)g(w)m(e)g(will)e(ha)m(v)m(e)j(to)e(b)s (e)h(more)e(accurate.)28 1862 y(Assume)38 b(that)g(~)-50 b Fx(a)37 b Fy(is)g(of)f(class)h Fx(C)1226 1826 y Fr(r)1264 1862 y Fy(\()p Fw(T)1365 1826 y Fr(d)1409 1862 y Fy(\),)h(where)g Fx(r)i Fy(can)d(b)s(e)g(a)g(\014nite)g(n)m(um)m(b)s(er)g(\(satisfying)f (a)h(certain)g(lo)m(w)m(er)-118 1983 y(b)s(ound)c(that)f(w)m(e)i(will)c (giv)m(e)i(in)g(a)h(momen)m(t\))e(or)h Fs(1)p Fy(.)43 b(W)-8 b(e)33 b(kno)m(w)h(that)e(writing)1063 2260 y Fs(D)1140 2275 y Fr(\013)1217 2260 y Fy(=)1321 2120 y Fq(\032)1395 2260 y Fx(f)39 b Fs(2)28 b Fx(L)1642 2219 y FC(2)1682 2260 y Fy(\()p Fw(T)1783 2219 y Fr(d)1826 2260 y Fy(\);)h(sup)1908 2350 y Fl(k)p Ft(2)p Fk(Z)2048 2331 y Fj(d)2095 2260 y Fs(j)p Fv(k)p Fs(j)2210 2219 y Fr(\013)2259 2260 y Fs(j)p Fx(f)2335 2275 y Fl(k)2382 2260 y Fs(j)e Fx(<)h Fy(+)p Fs(1)2717 2120 y Fq(\033)2808 2260 y Fx(;)-118 2541 y Fy(then)1448 2661 y Fs(D)1525 2676 y Fr(r)r FC(+)p Fr(d)p FC(+1)1772 2661 y Fs(\032)g Fx(C)1954 2620 y Fr(r)1992 2661 y Fy(\()p Fw(T)2093 2620 y Fr(d)2137 2661 y Fy(\))f Fs(\032)i(D)2385 2676 y Fr(r)2422 2661 y Fx(:)-118 2836 y Fy(The)k(analytic)f(case)h(is)f(treated)h (considering)953 3107 y Fs(A)1033 3122 y Fr(\032)1101 3107 y Fy(=)1204 2967 y Fq(\032)1279 3107 y Fx(f)39 b Fs(2)28 b Fx(L)1526 3066 y FC(2)1565 3107 y Fy(\()p Fw(T)1666 3066 y Fr(d)1710 3107 y Fy(\);)h(sup)1792 3197 y Fl(k)p Ft(2)p Fk(Z)1932 3178 y Fj(d)1979 3107 y Fy(exp)q(\()p Fx(\032)p Fs(j)p Fv(k)p Fs(j)p Fy(\))p Fs(j)p Fx(f)2445 3122 y Fl(k)2491 3107 y Fs(j)f Fx(<)f Fy(+)p Fs(1)2826 2967 y Fq(\033)2917 3107 y Fx(:)-118 3388 y Fy(The)33 b(v)-5 b(alue)32 b Fx(\032)h Fy(app)s(earing)e(in)h(the)g(de\014nition) g(of)g(the)g(set)h Fx(A)2087 3403 y Fr(\032)2160 3388 y Fy(is)f(the)h(width)f(of)g(the)h(analyticit)m(y)e(strip)g(that)i(w)m (e)-118 3508 y(are)f(considering,)h(so)f(functions)h(in)f(this)g(set)h (are)g(analytic)e(in)h(the)h(set)g Fs(j)p Fy(Im)f Fx(\022)s Fs(j)27 b Fx(<)h(\032)p Fy(.)28 3629 y(T)-8 b(o)22 b(measure)g(the)h (quotien)m(ts)f(whic)m(h)h(app)s(ear)f(in)f(the)h(de\014nition)f(of)h Fx(g)2512 3644 y Fl(k)2580 3629 y Fy(w)m(e)h(shall)e(imp)s(ose)g(that)g (the)i(frequency)-118 3749 y(v)m(ector)34 b Fx(!)h Fy(is)d(not)h(to)s (o)f(close)g(to)g(resonances.)46 b(Of)32 b(course)i(that)e(w)m(e)i (cannot)e(exp)s(ect)i(that)f(the)g(quotien)m(ts)1924 3941 y(1)p 1826 3985 245 4 v 1826 4076 a Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)-118 4292 y Fy(are)31 b(uniformly)f(b)s(ounded)i (for)f Fv(k)d Fs(2)g Fw(Z)1282 4255 y Fr(d)1320 4292 y Fy(,)k(b)s(ecause)h(the)f(v)-5 b(alues)31 b Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)p Fy(,)31 b(for)g Fv(k)d Fs(2)g Fw(Z)2897 4255 y Fr(d)2966 4292 y Fy(are)k(dense)h(in)e Fw(R)42 b Fy(\(pro)m(vided)-118 4412 y(that)35 b(at)g(least)g(t)m(w)m (o)h(comp)s(onen)m(ts)f(of)g Fx(!)k Fy(are)c(rationally)d(indep)s (enden)m(t\).)53 b(W)-8 b(e)36 b(shall)e(only)g(imp)s(ose)g(that)h (these)-118 4532 y(quotien)m(ts)44 b(can)f(b)s(e)h(con)m(trolled)e(b)m (y)i(p)s(o)m(w)m(ers)h(of)d Fs(j)p Fv(k)p Fs(j)k Fy(=)f Fs(j)p Fx(k)2074 4547 y FC(1)2113 4532 y Fs(j)29 b Fy(+)g Fs(\001)17 b(\001)g(\001)28 b Fy(+)h Fs(j)p Fx(k)2605 4547 y Fr(d)2645 4532 y Fs(j)p Fy(.)75 b(This)44 b(is)e(called)g(a)h FA(Diophantine)-118 4653 y(c)-5 b(ondition)34 b(on)g Fx(!)t Fy(:)43 b(w)m(e)34 b(will)c(assume)j(that)f(there)i(exist)f(p)s (ositiv)m(e)e(constan)m(ts)j Fx(\024)f Fy(and)g Fx(\034)44 b Fy(suc)m(h)34 b(that)1622 4890 y Fs(jh)p Fv(k)p Fx(;)17 b(!)t Fs(ij)25 b(\025)2115 4823 y Fx(\024)p 2065 4867 158 4 v 2065 4959 a Fs(j)p Fv(k)p Fs(j)2180 4930 y Fr(\034)2232 4890 y Fx(;)1507 b Fy(\(1.20\))-118 5174 y(for)33 b(all)f Fv(k)e Fs(2)g Fw(Z)423 5138 y Fr(d)460 5174 y Fy(,)35 b Fv(k)29 b Fs(6)p Fy(=)h(0.)47 b(It)33 b(can)h(b)s(e)g(sho)m(wn)h (that)f(almost)e(all)g(frequency)j(v)m(ectors)h Fx(!)d Fs(2)d Fw(R)3253 5138 y Fr(d)3333 5174 y Fy(\(with)j(resp)s(ect)i(to) -118 5294 y(the)30 b(Leb)s(esgue)i(measure\))e(satisfy)g(some)g 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Fx(d)f Fy(+)h(1,)36 b(and)f Fx(!)h Fs(2)d Fx(D)s(C)7 b Fy(\()p Fx(\024;)17 b(\034)11 b Fy(\),)35 b Fx(g)-118 916 y Fy(is)e(quasi-p)s(erio)s(dic)f(and)i(its)f(lift)f(if)g(of)i (class)g Fx(C)1603 880 y Fr(r)r Ft(\000)p FC(1)p Ft(\000)p Fr(\034)8 b Ft(\000)p Fr(d)1916 916 y Fy(\()p Fw(T)2017 880 y Fr(d)2061 916 y Fy(\).)47 b(If)34 b(~)-50 b Fx(a)34 b Fy(is)g(analytic,)e(then)j(w)m(e)g(can)f(pro)s(ceed)g(in)f(the)-118 1037 y(same)e(w)m(a)m(y)-8 b(,)32 b(but)f(using)g(the)g(sets)h Fs(A)1206 1052 y Fr(\032)1277 1037 y Fy(instead)e(of)h Fs(D)1798 1052 y Fr(r)1866 1037 y Fy(and)g(reducing)g Fx(\032)g Fy(instead)g(of)f Fx(r)s Fy(,)h(whic)m(h)h(means)f(reducing) -118 1157 y(the)i(domain)e(of)h(analyticit)m(y)-8 b(.)28 1278 y(W)g(e)27 b(ha)m(v)m(e)g(therefore)g(sho)m(wn)h(that,)f(ev)m(en)h (in)d(the)i(simple)e(case)i(of)e(one-dimensional)f(linear)h(equations)h (with)-118 1398 y(quasi-p)s(erio)s(dic)39 b(co)s(e\016cien)m(ts,)44 b(the)d(question)g(of)f(reducibilit)m(y)f(is)h(not)h(trivial.)65 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(of)f(the)g(homolo)-5 b(gic)g(al)47 b(e)-5 b(quation)-118 2456 y(\(1.15\))29 b(do)h(not)h(c)-5 b(ommute)30 b(b)-5 b(etwe)g(en)29 b(them,)i(so)f(we)g(c)-5 b(annot)30 b(solve)g(the)g(e)-5 b(quation)30 b(dir)-5 b(e)g(ctly.)43 b(This)30 b(is)g(also)g(r)-5 b(elate)g(d)-118 2577 y(to)40 b(the)h(fact)f(that,)i(when)e(dimension)e (gr)-5 b(ows,)41 b(ther)-5 b(e)41 b(app)-5 b(e)g(ar)39 b(mor)-5 b(e)40 b(fr)-5 b(e)g(quencies)40 b(in)g(the)g(system,)i(which) d(ar)-5 b(e)-118 2697 y(c)g(al)5 b(le)-5 b(d)50 b Fy(in)m(ternal)38 b(frequencies)k FA(of)f(the)g(system)f(\(in)h(opp)-5 b(osition)39 b(to)i(the)48 b Fy(external)39 b(frequencies)k Fx(!)t FA(\),)e(and)f(that)-118 2818 y(c)-5 b(an)33 b(b)-5 b(e)33 b(r)-5 b(esonant)33 b(with)g(the)g(external)g(fr)-5 b(e)g(quencies)33 b(and)g(b)-5 b(etwe)g(en)32 b(them.)44 b(These)33 b(fr)-5 b(e)g(quencies)32 b(ar)-5 b(e)33 b(not)h(known)-118 2938 y(a)47 b(priori,)i(but)f(if)e(the)h(system)g(is)g(r)-5 b(e)g(ducible,)49 b(they)e(ar)-5 b(e)47 b(r)-5 b(elate)g(d)47 b(to)g(the)g(imaginary)f(p)-5 b(arts)47 b(of)f(the)h(r)-5 b(e)g(duc)g(e)g(d)-118 3058 y(\(Flo)g(quet\))34 b(matrix.)1900 6155 y Fy(14)p eop %%Page: 15 19 15 18 bop -118 883 a Fz(Chapter)78 b(2)-118 1298 y(Reducibilit)-6 b(y)76 b(through)i(the)g(Sac)-6 b(k)g(er-Sell)-118 1547 y(sp)6 b(ectrum)-118 1999 y Fy(This)33 b(c)m(hapter)g(is)g(dev)m(oted)h (to)e(the)h(theory)h(dev)m(elop)s(ed)f(b)m(y)h(Johnson,)f(Sac)m(k)m(er) i(and)d(Sell)g(to)g(study)i(systems)g(of)-118 2120 y(linear)j (di\013eren)m(tial)g(equations)i(with)f(quasi-p)s(erio)s(dic)f(co)s (e\016cien)m(ts.)63 b(The)39 b(idea)f(is)h(to)f(in)m(tro)s(duce)h(a)f (sp)s(ectral)-118 2240 y(theory)f(v)-5 b(alid)34 b(for)i(a)g(large)f(v) -5 b(ariet)m(y)36 b(of)f(linear)g(di\013eren)m(tial)f(systems)k(and)e (that)g(generalizes)g(man)m(y)g(concepts)-118 2360 y(from)31 b(the)i(theory)g(of)g(linear)d(equations)j(with)f(constan)m(t)i(co)s (e\016cien)m(ts.)28 2481 y(The)28 b(sp)s(ectral)f(in)m(v)-5 b(arian)m(ts)27 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(osition)g(con\014rms)j(the)g(naturalit)m(y)e(of)h(suc)m(h)i(ob)5 b(jects:)1900 6155 y(19)p eop %%Page: 20 24 20 23 bop -118 219 a Fv(Prop)s(osition)35 b(2.1.15)j(\([78]\))49 b FA(L)-5 b(et)38 b Fx(P)48 b Fy(:)34 b Fx(Z)41 b Fs(!)34 b Fx(Z)45 b FA(b)-5 b(e)38 b(a)h(pr)-5 b(oje)g(ction)37 b(on)h(a)g(ve)-5 b(ctor)39 b(bund)5 b(le,)38 b(as)g(ab)-5 b(ove.)55 b(Then)-118 339 y(the)35 b(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties)35 b(hold:)-33 519 y(\(i\))49 b(The)34 b(sets)h Fs(R)g FA(and)f Fs(N)50 b FA(ar)-5 b(e)34 b(subbund)5 b(les)34 b(of)h Fx(Z)42 b FA(and)34 b(they)h(ar)-5 b(e)35 b(c)-5 b(omplementary,)34 b(that)h(is,)1482 711 y Fs(R)22 b(\010)h(N)42 b Fy(=)27 b Fx(Z)35 b FA(\(Whitney)g(sum\))-62 941 y(\(ii\))48 b(Given)38 b(any)h(two)f(c)-5 b(omplementary)38 b(subbund)5 b(les)37 b Fx(V)2016 956 y FC(1)2094 941 y FA(and)h Fx(V)2344 956 y FC(2)2384 941 y FA(,)h(ther)-5 b(e)39 b(is)f(a)g(unique)h(pr)-5 b(oje)g(ctor)38 b(such)g(that)126 1061 y(its)d(r)-5 b(ange)34 b(is)h Fx(V)690 1076 y 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b(bund)5 b(le)37 b Fy(\()p Fx(Z)r(;)17 b Fy(\012)p Fx(;)g(\031)t Fy(\))p FA(.)50 b(We)37 b(shal)5 b(l)-118 2056 y(say)36 b(that)h(a)f(LSPF)g Fy(\010)g FA(\()h(wher)-5 b(e)35 b Fy(\010\()p Fx(t)p Fy(;)17 b Fx(x;)g(\036)p Fy(\))31 b(=)f(\()p Fx(M)10 b Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))p Fx(x;)g(\033)t Fy(\()p Fx(t)p Fy(;)g Fx(\036)p Fy(\)\))36 b FA(\))g(de\014ne)-5 b(d)35 b(on)h Fx(Z)43 b FA(admits)36 b(an)43 b Fy(exp)s(onen)m(tial)-118 2177 y(dic)m(hotom)m(y)32 b(o)m(v)m(er)i Fx(N)45 b FA(if)35 b(ther)-5 b(e)35 b(is)f(a)h(pr)-5 b(oje)g(ctor)1523 2369 y Fx(P)41 b Fy(:)28 b Fx(Z)7 b Fy(\()p Fx(N)j Fy(\))28 b Fs(\000)-16 b(!)27 b Fx(Z)7 b Fy(\()p Fx(N)j Fy(\))-118 2561 y FA(and)34 b(p)-5 b(ositive)34 b(c)-5 b(onstants)35 b Fx(K)42 b FA(and)34 b Fx(\013)i FA(such)e(that)i(the)e(fol)5 b(lowing)34 b(ine)-5 b(qualities)34 b(hold)1028 2669 y Fq(\014)1028 2729 y(\014)1061 2754 y Fx(M)10 b Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))p Fx(P)d Fy(\()p Fx(\036)p Fy(\))p Fx(M)1693 2713 y Ft(\000)p FC(1)1787 2754 y Fy(\()p Fx(s)p Fy(;)j Fx(\036)p Fy(\))2011 2669 y Fq(\014)2011 2729 y(\014)2072 2754 y Fs(\024)28 b Fx(K)7 b(e)2312 2713 y Ft(\000)p Fr(\013)p FC(\()p Fr(t)p Ft(\000)p Fr(s)p FC(\))2584 2754 y Fx(;)45 b(s)27 b Fs(\024)i Fx(t)887 2875 y Fq(\014)887 2935 y(\014)920 2960 y Fx(M)10 b Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))g(\()p Fx(I)30 b Fs(\000)23 b Fx(P)14 b Fy(\()p Fx(\036)p Fy(\)\))h Fx(M)1833 2919 y Ft(\000)p FC(1)1928 2960 y Fy(\()p Fx(s)p Fy(;)i Fx(\036)p Fy(\))2152 2875 y Fq(\014)2152 2935 y(\014)2212 2960 y Fs(\024)29 b Fx(K)7 b(e)2453 2919 y Ft(\000)p Fr(\013)p FC(\()p Fr(s)p Ft(\000)p Fr(t)p FC(\))2725 2960 y Fx(;)45 b(s)27 b Fs(\025)h Fx(t)-118 3122 y FA(for)35 b(al)5 b(l)34 b Fx(\036)27 b Fs(2)i Fx(N)10 b FA(.)45 b(If)34 b Fx(N)45 b FA(is)35 b(al)5 b(l)35 b Fx(Z)7 b FA(,)34 b(we)h(shal)5 b(l)34 b(simply)g(say)h(that)g Fy(\010)h FA(admits)e(an)42 b Fy(exp)s(onen)m(tial)32 b(dic)m(hotom)m(y)p FA(.)28 3319 y Fy(By)25 b(the)h(ab)s(o)m(v)m(e)f FA(e)-5 b(quivalenc)g(e)31 b Fy(b)s(et)m(w)m(een)c(pro)5 b(jectors)26 b(and)e(complemen)m(tary)g(in)m(v)-5 b(arian)m(t)24 b(subbundles,)k (there)d(is)-118 3439 y(an)j(equiv)-5 b(alen)m(t)28 b(form)m(ulation)d (of)i(the)i(concept)g(of)e(exp)s(onen)m(tial)h(dic)m(hotom)m(y)f(using) h(complemen)m(tary)f(in)m(v)-5 b(arian)m(t)-118 3560 y(subbundles.)-118 3756 y Fv(Example)36 b(2.1.17)50 b FA(Assume)34 b(that)i(we)e(have)g(a)h(LSPF)f Fy(\010)h FA(on)g Fx(Z)7 b FA(.)44 b(F)-7 b(or)34 b(e)-5 b(ach)34 b Fx(\025)28 b Fs(2)g Fw(R)5 b FA(,)41 b(we)34 b(c)-5 b(an)34 b(de\014ne)g(another)-118 3877 y(\015ow)h Fy(\010)159 3892 y Fr(\025)239 3877 y FA(on)g Fx(Z)42 b FA(by)1194 3997 y Fy(\010)1264 4012 y Fr(\025)1310 3997 y Fy(\()p Fx(t)p Fy(;)17 b Fx(x;)g(\036)p Fy(\))27 b(=)1753 3917 y Fq(\000)1799 3997 y Fx(e)1844 3956 y Ft(\000)p Fr(\025t)1970 3997 y Fx(M)10 b Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))p Fx(x;)g(\033)t Fy(\()p Fx(t)p Fy(;)g Fx(\036)p Fy(\))2658 3917 y Fq(\001)-118 4160 y FA(which)34 b(is)g(also)h(a)f(LSPF)h(on)f Fx(Z)7 b FA(.)45 b(F)-7 b(urthermor)i(e,)34 b(the)h(sets)1001 4404 y Fs(B)1066 4419 y Fr(\025)1139 4404 y Fy(=)1243 4263 y Fq(\032)1318 4404 y Fy(\()p Fx(x;)17 b(\036)p Fy(\))27 b Fs(2)h Fx(Z)7 b Fy(;)45 b(sup)1831 4485 y Fr(t)p Ft(2)p Fk(R)1981 4319 y Fq(\014)1981 4379 y(\014)2014 4404 y Fx(e)2059 4363 y Ft(\000)p Fr(\025t)2185 4404 y Fx(M)10 b Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))p Fx(x)2557 4319 y Fq(\014)2557 4379 y(\014)2619 4404 y Fx(<)27 b Fs(1)2822 4263 y Fq(\033)992 4724 y Fs(S)1052 4739 y Fr(\025)1125 4724 y Fy(=)1229 4584 y Fq(\032)1303 4724 y Fy(\()p Fx(x;)17 b(\036)p Fy(\))28 b Fs(2)g Fx(Z)7 b Fy(;)87 b(lim)1803 4784 y Fr(t)p Ft(!)p FC(+)p Ft(1)2041 4640 y Fq(\014)2041 4699 y(\014)2074 4724 y Fx(e)2119 4683 y Ft(\000)p Fr(\025t)2245 4724 y Fx(M)10 b Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))p Fx(x)2617 4640 y Fq(\014)2617 4699 y(\014)2679 4724 y Fy(=)27 b(0)2831 4584 y Fq(\033)991 5014 y Fs(U)1053 5029 y Fr(\025)1126 5014 y Fy(=)1230 4873 y Fq(\032)1304 5014 y Fy(\()p Fx(x;)17 b(\036)p Fy(\))28 b Fs(2)g Fx(Z)7 b Fy(;)87 b(lim)1804 5074 y Fr(t)p Ft(!\0001)2042 4929 y Fq(\014)2042 4989 y(\014)2075 5014 y Fx(e)2120 4973 y Ft(\000)p Fr(\025t)2246 5014 y Fx(M)10 b Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))p Fx(x)2618 4929 y Fq(\014)2618 4989 y(\014)2680 5014 y Fy(=)27 b(0)2832 4873 y Fq(\033)-118 5228 y FA(ar)-5 b(e)31 b(cle)-5 b(arly)32 b(invariant)f(subsets)h(of)f Fx(Z)39 b FA(under)32 b(b)-5 b(oth)31 b(the)h(\015ows)g Fy(\010)2262 5243 y Fr(\025)2339 5228 y FA(and)f Fy(\010)p FA(.)45 b(F)-7 b(or)30 b(every)i(\014xe)-5 b(d)31 b Fx(\036)d Fs(2)g Fy(\012)p FA(,)k(the)g(\014b)-5 b(ers)-118 5348 y Fs(B)-53 5363 y Fr(\025)-7 5348 y Fy(\()p Fx(\036)p Fy(\))p FA(,)34 b Fs(S)251 5363 y Fr(\025)296 5348 y Fy(\()p Fx(\036)p Fy(\))g FA(and)g Fs(U)715 5363 y Fr(\025)761 5348 y Fy(\()p Fx(\036)p Fy(\))f FA(ar)-5 b(e)35 b(line)-5 b(ar)33 b(subsp)-5 b(ac)g(es)33 b(of)i Fx(Z)1980 5363 y Fr(\036)2060 5348 y FA(and)e(for)i Fx(\026)27 b Fs(\024)h Fx(\025)34 b FA(we)g(have)g(the)g(inclusions)f Fs(S)3730 5363 y Fr(\026)3805 5348 y Fs(\032)28 b(S)3970 5363 y Fr(\025)-118 5469 y FA(and)34 b Fs(U)133 5484 y Fr(\025)207 5469 y Fs(\032)28 b(U)374 5484 y Fr(\026)421 5469 y FA(.)28 5666 y Fy(The)f(question)g(of)f(whether)h Fs(S)1136 5681 y Fr(\025)1208 5666 y Fy(and)f Fs(U)1453 5681 y Fr(\025)1525 5666 y Fy(are)g(complemen)m(tary)f(in)m(v)-5 b(arian)m(t)25 b(subbundles)j(under)f(\010)3599 5681 y Fr(\025)3670 5666 y Fy(or)f(not)g(is)-118 5786 y(one)32 b(of)g(the)g(motiv)-5 b(ations)30 b(to)h(in)m(tro)s(duce)h(the)h(Sac)m(k)m(er-Sell)f(sp)s (ectrum,)h(whic)m(h)f(will)e(b)s(e)i(done)h(in)e(the)h(follo)m(wing) -118 5906 y(section.)1900 6155 y(20)p eop %%Page: 21 25 21 24 bop -118 219 a Fu(2.2)161 b(The)53 b(Sac)l(k)l(er-Sell)g(sp)t (ectrum)-118 438 y Fy(W)-8 b(e)33 b(\014rst)g(form)m(ulate)e(the)i (main)e(to)s(ol)g(for)h(reducibilit)m(y)e(that)j(w)m(e)g(shall)f(use)h (in)f(the)h(presen)m(t)h(c)m(hapter.)-118 658 y Fv(De\014nition)i (2.2.1)49 b FA(L)-5 b(et)28 b Fy(\010)915 673 y Fr(\025)988 658 y FA(b)-5 b(e)27 b(a)g(LSPF)g(on)g Fy(\()p Fx(Z)r(;)17 b Fy(\012)p Fx(;)g(\031)t Fy(\))p FA(,)28 b(a)g(ve)-5 b(ctor)27 b(bund)5 b(le,)28 b(and)f(assume)g(that)g Fy(\012)h FA(is)f(a)h(c)-5 b(omp)g(act)-118 778 y(Hausdor\013)39 b(sp)-5 b(ac)g(e.)55 b(Consider,)39 b(for)f(al)5 b(l)39 b Fx(\025)c Fs(2)g Fw(R)5 b FA(,)45 b(the)39 b(\015ow)g Fy(\010)2175 793 y Fr(\025)2260 778 y FA(\(de\014ne)-5 b(d)37 b(in)i(the)f(pr)-5 b(evious)39 b(se)-5 b(ction\).)55 b(F)-7 b(or)38 b(al)5 b(l)-118 898 y Fx(\036)27 b Fs(2)h Fy(\012)p FA(,)36 b(we)e(de\014ne)g(the)42 b Fy(resolv)m(en)m(t)34 b(of)e(\010)h(at)f Fx(\036)p FA(,)j Fx(\032)p Fy(\()p Fx(\036;)17 b Fy(\010\))p FA(,)35 b(as)544 1111 y Fx(\032)p Fy(\()p Fx(\036;)17 b Fy(\010\))28 b(=)g Fs(f)o Fx(\025)g Fs(2)g Fw(R)5 b Fy(;)50 b(\010)1415 1126 y Fr(\025)1496 1111 y FA(admits)34 b(an)h(exp)-5 b(onential)33 b(dichotomy)i(over)f Fs(f)p Fx(\036)p Fs(gg)-118 1324 y FA(and)g(its)43 b Fy(sp)s(ectrum)33 b(at)g Fx(\036)h FA(as)h Fy(\006\()p Fx(\036;)17 b Fy(\010\))28 b(=)f Fw(R)33 b Fs(\000)23 b Fx(\032)p Fy(\()p Fx(\036;)17 b Fy(\010\))p FA(.)45 b(In)34 b(gener)-5 b(al,)34 b(we)g(c)-5 b(al)5 b(l)35 b(the)42 b Fy(resolv)m(en)m(t)34 b(of)e(\010)p FA(,)j Fx(\032)p Fy(\(\010\))p FA(,)g(by)1467 1554 y Fx(\032)28 b Fy(=)f Fx(\032)p Fy(\(\010\))i(=)1991 1459 y Fq(\\)1976 1671 y Fr(\036)p Ft(2)p FC(\012)2133 1554 y Fx(\032)p Fy(\()p Fx(\036;)17 b Fy(\010\))-118 1864 y FA(and)34 b(the)43 b Fy(Sac)m(k)m(er-Sell)32 b(sp)s(ectrum)h(of)f(\010)j FA(by)g(its)g(c)-5 b(omplementary)34 b(over)h Fw(R)5 b FA(.)-118 2084 y Fv(Example)36 b(2.2.2)50 b FA(A)-5 b(c)g(c)g(or)g(ding)34 b(to)h(the)g(de\014nition)f(ab)-5 b(ove,)34 b Fx(\025)g FA(is)h(in)f(the)h(r)-5 b(esolvent)35 b(set)g(of)f(the)h(e)-5 b(quation)1603 2297 y Fx(x)1658 2256 y Ft(0)1709 2297 y Fy(=)1839 2272 y(~)1813 2297 y Fx(A)p Fy(\()p Fx(!)t(t)22 b Fy(+)g Fx(\036)p Fy(\))p Fx(x)-118 2510 y FA(if,)34 b(and)h(only)f(if,)h(the)g(shifte)-5 b(d)34 b(e)-5 b(quation)1420 2744 y Fx(x)1475 2703 y Ft(0)1527 2744 y Fy(=)1630 2634 y Fq(\020)1716 2719 y Fy(~)1690 2744 y Fx(A)p Fy(\()p Fx(!)t(t)22 b Fy(+)g Fx(\036)p Fy(\))f Fs(\000)i Fx(\025I)2346 2634 y Fq(\021)2422 2744 y Fx(x)-118 2979 y FA(admits)34 b(exp)-5 b(onential)34 b(dichotomy.)28 3199 y Fy(The)g(follo)m(wing)c(prop)s(osition)g (supplies)j(us)g(with)f(basic)h(prop)s(erties)f(of)g(the)h(sp)s(ectrum) g(de\014ned)h(ab)s(o)m(v)m(e.)-118 3419 y Fv(Prop)s(osition)h(2.2.3)j (\([78]\))134 b FA(\(i\))48 b(If)f Fy(\010)1494 3434 y Fr(\025)1586 3419 y FA(admits)g(an)f(exp)-5 b(onential)46 b(dichotomy)g(over)h Fx(\036)j Fs(2)g Fy(\012)p FA(,)g(then)d Fy(\010)3970 3434 y Fr(\025)126 3539 y FA(admits)35 b(an)h(exp)-5 b(onential)34 b(dichotomy)h(over)h(the)43 b Fy(h)m(ull)35 b FA(of)h Fx(\036)f FA(in)h Fy(\012)p FA(,)g(de\014ne)-5 b(d)35 b(as)h(the)g(fol)5 b(lowing)34 b(subset)i(of)126 3660 y Fy(\012)1558 3780 y Fx(H)8 b Fy(\()p Fx(\036)p Fy(\))27 b(=)p 1912 3693 672 4 v 28 w Fs(f)o Fx(\033)t Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\);)44 b Fx(t)28 b Fs(2)g Fw(R)5 b Fs(g)-62 3992 y FA(\(ii\))48 b(As)35 b(a)g(c)-5 b(onse)g(quenc)g(e,)1719 4112 y Fy(\006\()p Fx(\036)p Fy(\))28 b(=)f(\006\()p Fx(H)8 b Fy(\()p Fx(\036)p Fy(\)\))126 4283 y FA(for)31 b(al)5 b(l)32 b Fx(\036)27 b Fs(2)h Fy(\012)p FA(.)44 b(In)31 b(p)-5 b(articular,)33 b(if)e Fx(H)8 b Fy(\()p Fx(\036)p Fy(\))27 b(=)h(\012)k FA(for)f(al)5 b(l)32 b Fx(\036)27 b Fs(2)h Fy(\012)k FA(\(that)g(is,)g(if)g(the)g(\015ow)f(is)40 b Fy(minimal)p FA(\),)28 b(then)126 4404 y Fy(\006)g(=)f(\006\()p Fx(\036)p Fy(\))35 b FA(do)-5 b(es)35 b(not)f(dep)-5 b(end)34 b(on)h(the)g (chosen)e Fx(\036)p FA(.)28 4624 y Fy(The)f(de\014nition)f(of)g(the)g (h)m(ull)f(leads)h(us)h(to)f(another)h(imp)s(ortan)m(t)d(de\014nition.) 42 b(A)31 b(subset)i Fx(N)42 b Fy(of)31 b(\012)g(is)g(said)g(to)-118 4744 y(b)s(e)36 b FA(minimal)46 b Fy(with)36 b(resp)s(ect)h(to)f(the)h (\015o)m(w)g(\010)g(if)e Fx(H)8 b Fy(\()p Fx(N)i Fy(\))34 b(=)g(\012.)55 b(In)37 b(general,)g(the)g(\015o)m(w)g(is)f(said)f(to)h (b)s(e)h FA(minimal)-118 4864 y Fy(whenev)m(er)e Fx(H)8 b Fy(\()p Fs(f)p Fx(\036)p Fs(g)p Fy(\))26 b(=)i(\012,)33 b(for)f(all)f Fx(\036)c Fs(2)h Fy(\012.)-118 5084 y Fv(Example)36 b(2.2.4)50 b FA(Consider)40 b(the)h(ab)-5 b(ove)40 b(example)g(of)h(a)g (\015ow)g(induc)-5 b(e)g(d)40 b(by)i(a)f(line)-5 b(ar)40 b(e)-5 b(quation)41 b(with)g(quasi-)-118 5205 y(p)-5 b(erio)g(dic)34 b(c)-5 b(o)g(e\016cients.)45 b(L)-5 b(et)35 b Fx(!)k FA(b)-5 b(e)35 b(the)g(fr)-5 b(e)g(quency)35 b(ve)-5 b(ctor.)45 b(If)35 b(ther)-5 b(e)35 b(ar)-5 b(e)35 b Fx(s)g FA(c)-5 b(omp)g(onents)34 b(of)g(the)i(ve)-5 b(ctor)35 b(which)-118 5325 y(ar)-5 b(e)27 b(r)-5 b(ational)5 b(ly)26 b(indep)-5 b(endent,)28 b(then)e(for)h(any)g Fx(\036)h Fs(2)g Fw(T)1814 5289 y Fr(d)1857 5325 y FA(,)h(the)e(hul)5 b(l)27 b Fx(H)8 b Fy(\()p Fx(\036)p Fy(\))26 b FA(is)h(home)-5 b(omorphic)25 b(to)i(a)g Fx(s)p FA(-dimensional)-118 5446 y(torus)40 b(in)f Fw(T)324 5409 y Fr(d)408 5446 y FA(and)f(the)i(Sacker-Sel)5 b(l)38 b(sp)-5 b(e)g(ctrum)39 b Fy(\006\()p Fx(\036)p Fy(\))h FA(is)f(c)-5 b(onstant)40 b(over)f(this)g Fx(s)p FA(-dimensional)f(torus.)59 b(If)39 b(the)-118 5566 y(fr)-5 b(e)g(quencies)34 b(ar)-5 b(e)35 b(not)h(c)-5 b(ommensur)g(able,)33 b(then)i(the)h(Sacker-Sel)5 b(l)33 b(sp)-5 b(e)g(ctrum)35 b(do)-5 b(es)35 b(not)g(dep)-5 b(end)35 b(on)f(the)i(chosen)-118 5686 y Fx(\036)27 b Fs(2)h Fw(T)124 5650 y Fr(d)168 5686 y FA(.)28 5906 y Fy(Some)k(equiv)-5 b(alen)m(t)33 b(conditions)e(to)i(ha)m(v)m(e)g(an)g (exp)s(onen)m(tial)f(dic)m(hotom)m(y)g(can)h(b)s(e)g(found)f(easily)-8 b(,)1900 6155 y(21)p eop %%Page: 22 26 22 25 bop -118 219 a Fv(Prop)s(osition)35 b(2.2.5)j(\([78]\))48 b FA(L)-5 b(et)33 b Fx(N)43 b FA(b)-5 b(e)31 b(a)h(c)-5 b(omp)g(act)32 b(invariant)f(set)h(and)g(assume)f(that)i Fx(N)42 b FA(is)32 b(a)g(minimal)f(set)-118 339 y(in)j(the)h(\015ow)g Fx(\033)k FA(on)34 b Fy(\012)i FA(\(this)e(me)-5 b(ans)34 b(that)i Fx(H)8 b Fy(\()p Fx(N)i Fy(\))27 b(=)h(\012)p FA(\).)45 b(Then)34 b(the)h(fol)5 b(lowing)33 b(statements)i(ar)-5 b(e)34 b(e)-5 b(quivalent)-33 542 y(\(i\))49 b Fx(\025)35 b FA(is)f(in)h(the)g(r)-5 b(esolvent)34 b(set)h(of)f(the)h(Sacker-Sel)5 b(l)34 b(sp)-5 b(e)g(ctrum,)34 b Fx(\032)p Fy(\()p Fx(N)10 b Fy(\))p FA(.)-62 746 y(\(ii\))48 b(The)34 b(\015ow)h Fy(\010)602 761 y Fr(\025)682 746 y FA(has)g(an)f(exp)-5 b(onential)34 b(dichotomy)g(over)g Fx(N)10 b FA(.)-92 949 y(\(iii\))48 b Fs(B)191 964 y Fr(\025)273 949 y Fy(=)37 b Fs(f)p Fy(0)p Fs(g)25 b(\002)h Fy(\012)p FA(,)42 b(the)d(zer)-5 b(o)40 b(se)-5 b(ction)39 b(of)g Fx(Z)7 b Fy(\()p Fx(N)j Fy(\))p FA(.)60 b(That)40 b(is,)g(the)g(only)f(elements)g(of)h Fx(Z)7 b Fy(\()p Fx(N)j Fy(\))40 b FA(whose)f(orbit)126 1069 y(under)34 b(the)h(\015ow)g Fy(\010)842 1084 y Fr(\025)923 1069 y FA(is)f(b)-5 b(ounde)g(d)34 b(ar)-5 b(e)35 b(the)g(trivial)g (ones.)28 1298 y Fy(The)f(follo)m(wing)c(theorem)i(describ)s(es)i(the)f (sp)s(ectrum:)-118 1526 y Fv(Theorem)k(2.2.6)h(\(Sp)s(ectral)e (Theorem,)h([78)q(]\))48 b FA(L)-5 b(et)25 b Fy(\010)h FA(b)-5 b(e)24 b(a)h(LSPF)g(on)f(a)h(ve)-5 b(ctor)25 b(bund)5 b(le)24 b Fx(Z)32 b FA(with)25 b(c)-5 b(omp)g(act)-118 1646 y(Hausdor\013)44 b(b)-5 b(ase)43 b Fy(\012)p FA(.)71 b(L)-5 b(et)45 b Fx(n)f Fy(=)g(dim)15 b Fx(Z)51 b FA(and)43 b Fx(N)54 b Fs(\032)45 b Fy(\012)f FA(an)g(invariant)f(c)-5 b(omp)g(act)43 b(c)-5 b(onne)g(cte)g(d)42 b(set.)72 b(Then)43 b(the)-118 1767 y(Sacker-Sel)5 b(l)33 b(sp)-5 b(e)g(ctrum)35 b Fy(\006\()p Fx(N)10 b Fy(\))36 b FA(is)e(the)h(union)1327 1987 y Fy(\006\()p Fx(N)10 b Fy(\))29 b(=)e([)p Fx(a)1771 2002 y FC(1)1811 1987 y Fx(;)17 b(b)1896 2002 y FC(1)1936 1987 y Fy(])22 b Fs([)g(\001)17 b(\001)g(\001)k([)h Fy([)p Fx(a)2378 2002 y Fr(p)2418 1987 y Fx(;)17 b(b)2503 2002 y Fr(p)2543 1987 y Fy(])-118 2207 y FA(of)33 b Fx(p)g FA(non-overlapping)e(c)-5 b(omp)g(act)33 b(intervals)g(with)g Fx(p)28 b Fs(\024)g Fx(n)33 b FA(and)g Fx(a)2282 2222 y FC(1)2350 2207 y Fs(\024)28 b Fx(b)2496 2222 y FC(1)2563 2207 y Fx(<)g(a)2718 2222 y FC(2)2785 2207 y Fs(\024)g Fx(b)2931 2222 y FC(2)2999 2207 y Fx(<)f Fs(\001)17 b(\001)g(\001)26 b Fx(<)i(a)3401 2222 y Fr(p)3468 2207 y Fs(\024)h Fx(b)3615 2222 y Fr(p)3655 2207 y FA(,)k(and)g(we)-118 2327 y(shal)5 b(l)39 b(c)-5 b(al)5 b(l)40 b(the)g(ab)-5 b(ove)39 b(expr)-5 b(ession)39 b(the)47 b Fy(sp)s(ectral)38 b(decomp)s(osition)f(of)g(\()p Fx(Z)r(;)17 b Fy(\010\))p FA(.)60 b(F)-7 b(urthermor)i(e,)41 b(if)e Fx(\025)3628 2342 y FC(0)3668 2327 y Fx(;)17 b(:)g(:)g(:)f(;)h (\025)3944 2342 y Fr(k)3986 2327 y FA(,)-118 2448 y(with)35 b Fx(k)30 b Fs(\024)f Fx(p)p FA(,)34 b(ar)-5 b(e)35 b(chosen)f(in)g (the)h(r)-5 b(esolvent)34 b(set)h(so)g(that)1106 2668 y Fx(\025)1163 2683 y FC(0)1230 2668 y Fx(<)28 b(a)1385 2683 y FC(1)1452 2668 y Fs(\024)g Fx(b)1598 2683 y FC(1)1666 2668 y Fx(<)f(\025)1826 2683 y FC(1)1894 2668 y Fx(<)g(a)2048 2683 y FC(2)2115 2668 y Fs(\024)i Fx(b)2262 2683 y FC(2)2329 2668 y Fx(<)f(\025)2490 2683 y FC(2)2557 2668 y Fx(<)f(:)17 b(:)g(:)-118 2888 y FA(then,)34 b(for)h Fy(1)28 b Fs(\024)g Fx(i)g Fs(\024)g Fx(k)s FA(,)35 b(the)g(set)1367 3008 y Fx(V)1424 3023 y Fr(i)1452 3008 y Fy(\()p Fx(N)10 b Fy(\))28 b(=)f Fs(S)1807 3023 y Fr(\025)1848 3033 y Fj(i)1879 3008 y Fy(\()p Fx(N)10 b Fy(\))23 b Fs(\\)f(U)2216 3023 y Fr(\025)2257 3033 y Fj(i)p Fi(\000)p Fp(1)2367 3008 y Fy(\()p Fx(N)10 b Fy(\))-118 3182 y FA(is)31 b(an)g(invariant)g (subbund)5 b(le)31 b(of)g Fx(Z)7 b Fy(\()p Fx(N)j Fy(\))32 b FA(c)-5 b(al)5 b(le)-5 b(d)31 b(the)39 b Fy(sp)s(ectral)29 b(subbundle)j FA(asso)-5 b(ciate)g(d)31 b(to)h(the)f(sp)-5 b(e)g(ctr)g(al)31 b(interval)-118 3303 y Fy([)p Fx(a)-40 3318 y Fr(i)-12 3303 y Fx(;)17 b(b)73 3318 y Fr(i)102 3303 y Fy(])p FA(.)45 b(Its)34 b(dimension)f(is)i Fy(1)27 b Fs(\024)i Fx(n)1161 3318 y Fr(i)1217 3303 y Fs(\024)f Fx(n)35 b FA(and)f Fx(n)1662 3318 y FC(1)1724 3303 y Fy(+)22 b Fx(:)17 b(:)g(:)f(n)2011 3318 y Fr(k)2082 3303 y Fy(=)27 b Fx(n)p FA(.)45 b(Mor)-5 b(e)g(over)35 b(we)f(have)g(that) 967 3523 y Fx(Z)7 b Fy(\()p Fx(N)j Fy(\))28 b(=)g Fx(V)1394 3538 y FC(1)1433 3523 y Fy(\()p Fx(N)10 b Fy(\))22 b Fs(\010)h(\001)17 b(\001)g(\001)j(\010)j Fx(V)2014 3538 y Fr(p)2054 3523 y Fy(\()p Fx(N)10 b Fy(\))35 b FA(\(Whitney)g(sum\)) -118 3743 y(Final)5 b(ly,)34 b(the)h(sp)-5 b(e)g(ctrum)34 b Fy(\006)888 3758 y Fr(i)917 3743 y Fy(\()p Fx(N)10 b Fy(\))35 b FA(of)g Fy(\()p Fx(V)1326 3758 y Fr(i)1354 3743 y Fy(\()p Fx(N)10 b Fy(\))p Fx(;)17 b Fy(\010)1632 3758 y Ft(j)p Fr(V)1693 3768 y Fj(i)1720 3758 y FC(\()p Fr(N)7 b FC(\))1842 3743 y Fy(\))34 b FA(is)h Fy(\006)2089 3758 y Fr(i)2118 3743 y Fy(\()p Fx(N)10 b Fy(\))28 b(=)f([)p Fx(a)2491 3758 y Fr(i)2520 3743 y Fx(;)17 b(b)2605 3758 y Fr(i)2633 3743 y Fy(])p FA(.)28 3971 y Fy(The)29 b(pro)s(of)e(is)h (also)f(found)h(in)f([78)o(].)42 b(It)28 b(uses)i(the)e(follo)m(wing)d FA(uniformization)30 b(lemma)k Fy(whic)m(h)29 b(is)e(in)m(teresting) -118 4091 y(for)32 b(its)g(o)m(wn)h(sak)m(e)-118 4320 y Fv(Lemma)k(2.2.7)h(\([78]\))48 b FA(L)-5 b(et)45 b Fx(N)55 b FA(b)-5 b(e)44 b(a)g(c)-5 b(omp)g(act)44 b(invariant)f(set)i (in)f Fy(\012)h FA(and)f Fx(\025)h Fs(2)h Fw(R)5 b FA(.)79 b(Then)44 b(the)g(fol)5 b(lowing)-118 4440 y(statements)35 b(ar)-5 b(e)34 b(valid:)-33 4644 y(\(i\))49 b(If)38 b Fs(j)p Fy(\010)330 4659 y Fr(\025)375 4644 y Fy(\()p Fx(t)p Fy(;)17 b Fx(z)t Fy(\))p Fs(j)35 b(!)f Fy(0)39 b FA(as)f Fx(t)d Fs(!)f Fy(+)p Fs(1)k FA(for)h(e)-5 b(ach)38 b Fx(z)h Fs(2)c Fx(N)10 b FA(,)40 b(then)f Fx(\025)34 b Fs(2)h Fx(\032)p Fy(\()p Fx(N)10 b Fy(\))p FA(,)41 b Fy(\006\()p Fx(N)10 b Fy(\))35 b Fs(\032)g Fy(\()p Fs(\0001)p Fx(;)17 b(\025)p Fy(\))38 b FA(and)g Fs(S)3860 4659 y Fr(\026)3946 4644 y FA(is)126 4764 y(the)d(whole)f(bund)5 b(le)34 b Fx(Z)7 b Fy(\()p Fx(N)j Fy(\))35 b FA(for)g(al)5 b(l)34 b Fx(\026)28 b Fs(\025)g Fx(\025)p FA(.)-62 4967 y(\(ii\))48 b(If)38 b Fs(j)p Fy(\010)330 4982 y Fr(\025)375 4967 y Fy(\()p Fx(t)p Fy(;)17 b Fx(z)t Fy(\))p Fs(j)35 b(!)f Fy(0)k FA(as)h Fx(t)34 b Fs(!)h(\0001)j FA(for)h(e)-5 b(ach)38 b Fx(z)h Fs(2)c Fx(N)10 b FA(,)40 b(then)e Fx(\025)d Fs(2)f Fx(\032)p Fy(\()p Fx(N)10 b Fy(\))p FA(,)41 b Fy(\006\()p Fx(N)10 b Fy(\))35 b Fs(\032)g Fy(\()p Fx(\025;)17 b Fy(+)p Fs(1)p Fy(\))37 b FA(and)h Fs(U)3860 4982 y Fr(\026)3946 4967 y FA(is)126 5088 y(the)d(whole)f(bund)5 b(le)34 b Fx(Z)7 b Fy(\()p Fx(N)j Fy(\))35 b FA(for)g(al)5 b(l)34 b Fx(\026)28 b Fs(\024)g Fx(\025)p FA(.)28 5316 y Fy(If)23 b(nothing)g(is)f(said)h(explicitly)-8 b(,)23 b(w)m(e)h(shall)e(consider)i(the)f(sp)s(ectral)g(decomp)s(osition)e (with)i(resp)s(ect)i(to)d Fx(N)39 b Fy(=)27 b(\012.)-118 5544 y Fv(Remark)37 b(2.2.8)49 b FA(L)-5 b(et)42 b Fx(V)818 5559 y Fr(i)888 5544 y FA(b)-5 b(e)41 b(a)h(sp)-5 b(e)g(ctr)g(al)41 b(subbund)5 b(le.)65 b(The)41 b(unique)g(pr)-5 b(oje)g(ctor)42 b Fx(P)2974 5559 y Fr(i)3042 5544 y Fy(:)e Fx(Z)48 b Fs(!)39 b Fx(Z)49 b FA(such)42 b(that)g(its)-118 5665 y(r)-5 b(ange)34 b(is)h Fx(V)309 5680 y Fr(i)372 5665 y FA(is)f(c)-5 b(al)5 b(le)-5 b(d)34 b(the)43 b Fy(sp)s(ectral)32 b(pro)5 b(jector)35 b FA(and)g(its)g(kernel)f(is)g(the)h(Whitney)h(sum) 3184 5590 y Fq(L)3295 5694 y Fr(i)p Ft(6)p FC(=)p Fr(j)3427 5665 y Fx(V)3484 5680 y Fr(j)3520 5665 y FA(.)1900 6155 y Fy(22)p eop %%Page: 23 27 23 26 bop -118 219 a Fv(Remark)37 b(2.2.9)49 b FA(If)42 b(the)h(matric)-5 b(es)41 b(of)i(the)f(\015ow)h Fx(M)10 b Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))43 b FA(b)-5 b(elong)41 b(to)i(some)f(sub)-5 b(gr)g(oup)42 b(of)g Fx(GL)p Fy(\()p Fx(n)p Fy(\))h FA(then)g(the)-118 339 y(pr)-5 b(op)g(erties)38 b(of)h(this)g(gr)-5 b(oup)39 b(wil)5 b(l)39 b(imply)f(that)i(for)f(the)g(Sacker-Sel)5 b(l)38 b(sp)-5 b(e)g(ctrum)38 b(has)h(some)f(sp)-5 b(e)g(cial)38 b(pr)-5 b(op)g(erties.)-118 459 y(F)e(or)38 b(instanc)-5 b(e,)39 b(if)g(the)g(gr)-5 b(oup)39 b(is)g Fx(S)6 b(L)p Fy(\(2)p Fx(;)17 b Fw(R)t Fy(\))p FA(,)46 b(then)39 b(if)g Fx(\025)g FA(b)-5 b(elongs)38 b(to)h(the)g(sp)-5 b(e)g(ctrum,)40 b(then)f(also)g Fs(\000)p Fx(\025)g FA(is)g(in)g(the)-118 580 y(sp)-5 b(e)g(ctrum.)28 768 y Fy(F)d(ollo)m(wing)44 b(the)j(example)f(from)f(the)i(previous)f(section)h(of)f(a)g(\015o)m(w) h(generated)h(b)m(y)f(a)f(linear)f(equation)-118 888 y(with)h(constan)m(t)h(co)s(e\016cien)m(ts,)52 b(all)44 b(the)j(sp)s(ectral)f(in)m(terv)-5 b(als)46 b(reduce)i(to)e(p)s(oin)m (ts)g(whic)m(h)h(are)g(the)g(real)e(parts)-118 1008 y(of)40 b(the)g(eigen)m(v)-5 b(alues)41 b(of)e(the)i(matrix.)65 b(The)41 b(corresp)s(onding)f(sp)s(ectral)g(in)m(v)-5 b(arian)m(t)39 b(subbundles)j(are)e(also)f(the)-118 1129 y(subbundles)34 b(corresp)s(onding)f(to)f(eigen)m(v)-5 b(alues)33 b(with)f(equal)h(real)e(parts.)44 b(The)34 b(analysis)e(for)g(p)s(erio)s(dic)f(systems)-118 1249 y(is)36 b(analogous)g(to)h(case)h(of)e(constan)m(t)i(co)s(e\016cien)m (ts)g(b)s(ecause)h(of)d(the)i(reducibilit)m(y)d(giv)m(en)i(b)m(y)h(Flo) s(quet)e(theory)-8 b(.)-118 1369 y(In)33 b(b)s(oth)f(cases)i(all)d(sp)s (ectral)h(in)m(terv)-5 b(als)32 b(reduce)i(to)e(p)s(oin)m(ts.)43 b(This)33 b(concept)g(deserv)m(es)j(a)c(de\014nition.)-118 1557 y Fv(De\014nition)k(2.2.10)49 b FA(We)38 b(shal)5 b(l)37 b(say)h(that)g(a)f(LSPF)h Fy(\010)g FA(on)f(a)h(ve)-5 b(ctor)37 b(bund)5 b(le)37 b Fx(Z)45 b FA(has)g Fy(pure)37 b(p)s(oin)m(t)d(sp)s(ectrum)-118 1678 y FA(if)42 b(al)5 b(l)43 b(sp)-5 b(e)g(ctr)g(al)42 b(intervals)g(de)-5 b(gener)g(ate)42 b(to)h(p)-5 b(oints.)68 b(If,)44 b(mor)-5 b(e)g(over,)44 b Fx(n)f Fy(=)f Fx(p)p FA(,)i(we)f(shal)5 b(l)42 b(say)h(that)g Fy(\010)g FA(has)51 b Fy(full)-118 1798 y(sp)s(ectrum)p FA(.)42 b(If)24 b(ther)-5 b(e)25 b(is)g(an)g(interval)f(in)h(the)g(sp)-5 b(e)g(ctrum,)27 b(we)d(shal)5 b(l)25 b(sp)-5 b(e)g(ak)24 b(of)46 b Fy(absolutely)21 b(con)m(tin)m(uous)h(sp)s(ectrum)p FA(.)28 1986 y Fy(The)34 b(follo)m(wing)c(prop)s(osition)g(clari\014es)i(the)h(second)h(h)m(yp)s (othesis)g(in)d(the)i(de\014nition.)-118 2174 y Fv(Prop)s(osition)i (2.2.11)j(\([78]\))49 b FA(Consider)43 b(the)h(e)-5 b(quation)44 b Fx(x)2170 2138 y Ft(0)2239 2174 y Fy(=)h Fx(A)p Fy(\()p Fx(t)p Fy(\))p Fx(x)p FA(,)j(with)c(quasi-p)-5 b(erio)g(dic)43 b(c)-5 b(o)g(e\016cients.)-118 2294 y(If)39 b(one)g(sp)-5 b(e)g(ctr)g(al)39 b(subbund)5 b(le)38 b(is)i(one-dimensional,)d(then)j (the)f(c)-5 b(orr)g(esp)g(onding)38 b(sp)-5 b(e)g(ctr)g(al)39 b(interval)g(is)g(a)g(p)-5 b(oint.)-118 2415 y(Henc)g(e,)34 b(if)h(ther)-5 b(e)35 b(ar)-5 b(e)34 b(exactly)h Fx(n)g FA(sp)-5 b(e)g(ctr)g(al)35 b(intervals,)f(then)h(e)-5 b(ach)34 b(interval)g(is)h(de)-5 b(gener)g(ate.)28 2603 y Fy(W)d(e)26 b(no)m(w)g(w)m(an)m(t)g(to)f(further)g(explore)h(these)g (concepts,)i(fo)s(cusing)d(mainly)e(on)i(linear)f(equations)h(with)g (quasi-)-118 2723 y(p)s(erio)s(dic)j(co)s(e\016cien)m(ts.)43 b(In)30 b(the)g(follo)m(wing)d(section)i(w)m(e)i(relate)e(the)h(sp)s (ectrum)g(of)f(a)g(linear)f(sk)m(ew-pro)s(duct)j(\015o)m(w)-118 2843 y(to)25 b(the)i(existence)g(of)e(Ly)m(apuno)m(v)j(exp)s(onen)m (ts.)43 b(In)26 b(the)g(other)g(t)m(w)m(o)h(sections)f(w)m(e)h(pro)m(v) m(e)g(a)f(reducibilit)m(y)e(theorem.)-118 3171 y Fu(2.3)161 b(Relation)55 b(with)e(Ly)l(apuno)l(v)g(exp)t(onen)l(ts)-118 3390 y Fy(It)35 b(turns)h(out)f(that)h(the)f(Sac)m(k)m(er-Sell)h(sp)s (ectrum)f(is)g(strongly)g(connected)i(to)e(the)h(theory)g(of)f(Ly)m (apuno)m(v)i(c)m(ha-)-118 3510 y(racteristic)32 b(exp)s(onen)m(ts,)j (whic)m(h)e(w)m(e)h(presen)m(t)h(in)d(this)g(section.)45 b(F)-8 b(or)32 b(this)g(purp)s(ose)i(consider)f(a)g(LSPF)g(\010)g(on)g (a)-118 3631 y(v)m(ector)i(bundle)g(\()p Fx(Z)r(;)17 b Fy(\012\))34 b(and)h(assume)g(that)f(\012)h(is)f(compact)g(and)g(in)m (v)-5 b(arian)m(tly)33 b(connected.)51 b(This)34 b(means)h(that)-118 3751 y(\012)40 b(cannot)g(b)s(e)g(written)g(as)g(the)g(union)f(of)h(t)m (w)m(o)g(disjoin)m(t)f(nonempt)m(y)h(compact)g(in)m(v)-5 b(arian)m(t)38 b(sets,)43 b(and)d(this)f(is)-118 3871 y(true)33 b(whenev)m(er)i(\012)28 b(=)g Fw(T)781 3835 y Fr(d)857 3871 y Fy(and)33 b(the)g(\015o)m(w)g(is)f(linear)f(with)h (rationally)e(indep)s(enden)m(t)k(frequencies.)28 3992 y(Let)1232 4112 y(\006)29 b(=)e(\006\(\012\))i(=)e([)p Fx(a)1860 4127 y FC(1)1900 4112 y Fx(;)17 b(b)1985 4127 y FC(1)2025 4112 y Fy(])22 b Fs([)g(\001)17 b(\001)g(\001)k([)h Fy([)p Fx(a)2467 4127 y Fr(k)2510 4112 y Fx(;)17 b(b)2595 4127 y Fr(k)2638 4112 y Fy(])-118 4271 y(and)1145 4392 y Fx(Z)35 b Fy(=)28 b Fx(V)1408 4407 y FC(1)1469 4392 y Fs(\010)23 b(\001)17 b(\001)g(\001)j(\010)j Fx(V)1864 4407 y Fr(k)2101 4392 y Fy(\(Whitney)33 b(sum\))-118 4551 y(b)s(e)e(the)g(sp)s(ectral)f(decomp)s(osition)e(of)i(\()p Fx(Z)r(;)17 b Fy(\010\))30 b(giv)m(en)h(b)m(y)h(theorem)e(2.2.6,)g(the) h(in)m(terv)-5 b(als)30 b([)p Fx(a)3236 4566 y Fr(i)3264 4551 y Fx(;)17 b(b)3349 4566 y Fr(i)3378 4551 y Fy(])30 b(b)s(eing)g(ordered)-118 4671 y(so)j(that)f Fx(b)254 4686 y Fr(i)310 4671 y Fx(<)c(a)465 4686 y Fr(i)p FC(+1)583 4671 y Fy(,)33 b(and)g(let)f Fx(P)1037 4686 y Fr(i)1092 4671 y Fy(:)c Fx(Z)35 b Fs(!)27 b Fx(V)1433 4686 y Fr(i)1494 4671 y Fy(b)s(e)32 b(the)h(sp)s(ectral)g(pro)5 b(jectors.)28 4792 y(Giv)m(en)33 b(a)g(p)s(oin)m(t)f(\()p Fx(x;)17 b(\036)p Fy(\))28 b Fs(2)g Fx(Z)7 b Fy(,)33 b(with)g Fx(x)28 b Fs(6)p Fy(=)h(0,)j(the)i(usual)e(de\014nition)g(of)h(the)g (four)f(Ly)m(apuno)m(v)j(c)m(haracteristic)-118 4912 y(n)m(um)m(b)s(ers)e(is)1049 5116 y Fx(\014)1110 5075 y FC(+)1104 5141 y Fr(sup)1218 5116 y Fy(\()p Fx(x;)17 b(\036)p Fy(\))27 b(=)h(lim)17 b(sup)1620 5195 y Fr(t)p Ft(!)p FC(+)p Ft(1)1907 5049 y Fy(1)p 1907 5093 49 4 v 1914 5185 a Fx(t)1983 5116 y Fy(log)f Fs(j)p Fy(\010\()p Fx(t)p Fy(;)h(\()p Fx(x;)g(\036)p Fy(\)\))p Fs(j)p Fx(;)1149 b Fy(\(2.6\))1079 5377 y Fx(\014)1140 5336 y FC(+)1134 5405 y Fr(inf)1246 5377 y Fy(\()p Fx(x;)17 b(\036)p Fy(\))28 b(=)f(lim)17 b(inf)1635 5437 y Fr(t)p Ft(!)p FC(+)p Ft(1)1907 5310 y Fy(1)p 1907 5354 V 1914 5445 a Fx(t)1983 5377 y Fy(log)f Fs(j)p Fy(\010\()p Fx(t)p Fy(;)h(\()p Fx(x;)g(\036)p Fy(\)\))p Fs(j)p Fx(;)1149 b Fy(\(2.7\))1049 5619 y Fx(\014)1110 5577 y Ft(\000)1104 5643 y Fr(sup)1218 5619 y Fy(\()p Fx(x;)17 b(\036)p Fy(\))27 b(=)h(lim)17 b(sup)1620 5698 y Fr(t)p Ft(!\0001)1907 5551 y Fy(1)p 1907 5596 V 1914 5687 a Fx(t)1983 5619 y Fy(log)f Fs(j)p Fy(\010\()p Fx(t)p Fy(;)h(\()p Fx(x;)g(\036)p Fy(\)\))p Fs(j)p Fx(;)1149 b Fy(\(2.8\))1096 5879 y Fx(\014)1157 5838 y Ft(\000)1151 5907 y Fr(inf)1263 5879 y Fy(\()p Fx(x;)17 b(\036)p Fy(\))28 b(=)f(lim)17 b(inf)1651 5939 y Fr(t)p Ft(!\0001)1924 5812 y Fy(1)p 1924 5857 V 1931 5948 a Fx(t)1999 5879 y Fy(log)g Fs(j)p Fy(\010\()p Fx(t)p Fy(;)g(\()p Fx(x;)g(\036)p Fy(\)\))p Fs(j)p Fx(:)1132 b Fy(\(2.9\))1900 6155 y(23)p eop %%Page: 24 28 24 27 bop 28 219 a Fy(Using)30 b(the)g(c)m(haracterization)f(of)g(the)i (sp)s(ectral)e(in)m(terv)-5 b(als)30 b(and)g(the)g(asso)s(ciated)g (subbundles)h(w)m(e)g(ha)m(v)m(e)g(the)-118 339 y(follo)m(wing)-118 526 y Fv(Theorem)37 b(2.3.1)h(\([78]\))48 b FA(If)26 b Fy(\()p Fx(x;)17 b(\036)p Fy(\))28 b Fs(2)g Fx(V)1441 541 y Fr(i)1469 526 y FA(,)g(b)-5 b(eing)27 b Fx(V)1826 541 y Fr(i)1881 526 y FA(the)g Fx(i)p FA(-th)g(sp)-5 b(e)g(ctr)g(al)27 b(subbund)5 b(le)26 b(asso)-5 b(ciate)g(d)26 b(to)h(the)g(sp)-5 b(e)g(ctr)g(al)-118 647 y(interval)42 b Fy([)p Fx(a)325 662 y Fr(i)354 647 y Fx(;)17 b(b)439 662 y Fr(i)467 647 y Fy(])p FA(,)45 b(and)e Fx(x)g Fs(6)p Fy(=)f(0)p FA(,)j(then)e(the)g(four)g(Lyapunov)g(char)-5 b(acteristic)42 b(numb)-5 b(ers)42 b(de\014ne)-5 b(d)42 b(ab)-5 b(ove)42 b(lie)h(in)-118 767 y Fy([)p Fx(a)-40 782 y Fr(i)-12 767 y Fx(;)17 b(b)73 782 y Fr(i)102 767 y Fy(])p FA(.)43 b(In)29 b(p)-5 b(articular,)31 b(if)f Fx(a)937 782 y Fr(i)993 767 y Fy(=)e Fx(b)1138 782 y Fr(i)1166 767 y FA(,)j(then)f(for)g(al)5 b(l)30 b Fy(\()p Fx(x;)17 b(\036)p Fy(\))27 b Fs(2)h Fx(V)2136 782 y Fr(i)2194 767 y FA(and)i Fx(x)e Fs(6)p Fy(=)f(0)j FA(the)g(four)g(char)-5 b(acteristic)30 b(exp)-5 b(onents)-118 887 y(agr)g(e)g(e)34 b(and)g(the)h(limits)947 1111 y Fy(lim)904 1170 y Fr(t)p Ft(!)p FC(+)p Ft(1)1152 1043 y Fy(1)p 1152 1088 49 4 v 1159 1179 a Fx(t)1227 1111 y Fy(log)17 b Fs(j)p Fy(\010\()p Fx(t)p Fy(;)g(\()p Fx(x;)g(\036)p Fy(\)\))p Fs(j)27 b Fy(=)70 b(lim)2014 1170 y Fr(t)p Ft(!\0001)2263 1043 y Fy(1)p 2263 1088 V 2270 1179 a Fx(t)2338 1111 y Fy(log)16 b Fs(j)p Fy(\010\()p Fx(t)p Fy(;)h(\()p Fx(x;)g(\036)p Fy(\)\))p Fs(j)-118 1328 y FA(exist)35 b(and)f(e)-5 b(qual)34 b Fx(a)595 1343 y Fr(i)624 1328 y FA(.)28 1516 y Fy(Therefore,)e (considering)d(this)h(result)g(for)f(all)f(the)j(sp)s(ectral)e (subbundles,)k(together)d(with)f(some)h(w)m(ork,)i(w)m(e)-118 1636 y(obtain)f(the)i(follo)m(wing,)-118 1824 y Fv(Theorem)k(2.3.2)h (\([78]\))48 b FA(F)-7 b(or)25 b(al)5 b(l)26 b Fy(\()p Fx(x;)17 b(\036)p Fy(\))27 b Fs(2)h Fx(Z)7 b FA(,)28 b(with)d Fx(x)k Fs(6)p Fy(=)e(0)p FA(,)h(the)e(four)g(Lyapunov)f(char) -5 b(acteristic)26 b(exp)-5 b(onents)-118 1944 y(lie)34 b(in)h(the)g(sp)-5 b(e)g(ctrum)35 b Fy(\006)p FA(.)45 b(Mor)-5 b(e)35 b(pr)-5 b(e)g(cisely,)34 b(let)h Fy(\()p Fx(x;)17 b(\036)p Fy(\))27 b Fs(2)h Fx(Z)7 b FA(,)35 b(with)g Fx(x)28 b Fs(6)p Fy(=)f(0)p FA(,)35 b(and)f(de\014ne)g Fx(q)39 b FA(and)34 b Fx(q)3461 1908 y Ft(0)3519 1944 y FA(by)1418 2128 y Fx(q)d Fy(=)d(max)o Fs(f)p Fx(i)g Fy(:)g Fx(P)2006 2143 y Fr(i)2034 2128 y Fy(\()p Fx(\036)p Fy(\))p Fx(x)g Fs(6)p Fy(=)f(0)p Fs(g)p Fx(;)1416 2312 y(q)1463 2271 y Ft(0)1513 2312 y Fy(=)h(min)o Fs(f)p Fx(i)f Fy(:)h Fx(P)2008 2327 y Fr(i)2036 2312 y Fy(\()p Fx(\036)p Fy(\))p Fx(x)g Fs(6)p Fy(=)f(0)p Fs(g)p Fx(:)-118 2471 y FA(Then)34 b(one)g(has)h(the)f(ine)-5 b(qualities)1251 2656 y Fx(a)1302 2671 y Fr(q)1368 2656 y Fs(\024)28 b Fx(\014)1534 2614 y FC(+)1528 2683 y Fr(inf)1641 2656 y Fy(\()p Fx(x;)17 b(\036)p Fy(\))27 b Fs(\024)h Fx(\014)2067 2614 y FC(+)2061 2680 y Fr(sup)2174 2656 y Fy(\()p Fx(x;)17 b(\036)p Fy(\))28 b Fs(\024)g Fx(b)2581 2671 y Fr(q)2619 2656 y Fx(;)1229 2850 y(a)1280 2865 y Fr(q)1314 2846 y Fi(0)1368 2850 y Fs(\024)g Fx(\014)1534 2809 y Ft(\000)1528 2877 y Fr(inf)1641 2850 y Fy(\()p Fx(x;)17 b(\036)p Fy(\))27 b Fs(\024)h Fx(\014)2067 2809 y Ft(\000)2061 2874 y Fr(sup)2174 2850 y Fy(\()p Fx(x;)17 b(\036)p Fy(\))28 b Fs(\024)g Fx(b)2581 2865 y Fr(q)2615 2846 y Fi(0)2642 2850 y Fx(:)28 3037 y Fy(Finally)-8 b(,)46 b(w)m(e)g(also)f(ha)m(v)m(e)h(some)f (information)d(on)k(the)f(amoun)m(t)g(of)g(elemen)m(ts)g(of)g Fx(Z)52 b Fy(ha)m(ving)45 b(Ly)m(apuno)m(v)-118 3158 y(exp)s(onen)m(t)34 b(one)f(endp)s(oin)m(t)f(of)g(a)h(sp)s(ectral)f(in) m(terv)-5 b(al.)-118 3345 y Fv(Theorem)37 b(2.3.3)h(\([78]\))48 b FA(Assume)41 b(that)g Fx(M)49 b Fs(\032)38 b Fy(\012)j FA(is)g(a)f(minimal)f(set)i(with)f(r)-5 b(esp)g(e)g(ct)41 b(to)g(the)f(\015ow)h Fx(\033)j FA(on)c Fy(\012)p FA(.)-118 3465 y(Then)34 b(the)h(set)g Fx(G)g FA(of)f(al)5 b(l)35 b Fx(\036)27 b Fs(2)h Fx(M)46 b FA(such)34 b(that)i(ther)-5 b(e)35 b(exist)f Fx(x)2080 3424 y Ft(\000)2080 3490 y FC(1)2140 3465 y Fx(;)17 b(:)g(:)g(:)e(;)i(x)2413 3429 y Ft(\000)2413 3490 y Fr(p)2473 3465 y Fx(;)g(x)2572 3424 y FC(+)2572 3490 y(1)2631 3465 y Fx(;)g(:)g(:)g(:)f(;)h(x)2905 3429 y FC(+)2905 3490 y 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Fy(+)g Fx(!)t(t)p Fy(\))p Fx(x)-118 4749 y Fy(where)195 4723 y(~)169 4749 y Fx(A)38 b Fy(:)f Fw(T)407 4712 y Fr(d)488 4749 y Fs(!)f Fx(L)p Fy(\()p Fw(R)795 4712 y Fr(n)848 4749 y Fy(\))i(is)f(a)h(mapping)f(from)g Fw(T)1820 4712 y Fr(d)1902 4749 y Fy(to)g Fx(L)p Fy(\()p Fw(R)2197 4712 y Fr(n)2250 4749 y Fy(\),)i(the)g(space)g(of)f(linear)e (op)s(erators,)k(with)d(the)-118 4869 y(suprem)m(um)c(norm.)42 b(W)-8 b(e)33 b(assume)g(that)1391 4844 y(~)1366 4869 y Fx(A)27 b Fs(2)i Fx(C)1638 4833 y Fr(\013)1687 4869 y Fy(\()p Fw(T)1788 4833 y Fr(d)1832 4869 y Fy(\),)j(for)g Fx(\013)d Fy(=)e(0)p Fx(;)17 b(:)g(:)g(:)f(;)h Fs(1)p Fx(;)g(a)p Fy(.)-118 5056 y Fv(Theorem)37 b(2.4.1)h(\(Smo)s(othness)f (of)h(sp)s(ectral)f(subbundles,)i([49]\))48 b FA(In)33 b(the)g(ab)-5 b(ove)33 b(situation,)h(let)f Fx(!)k FA(b)-5 b(e)-118 5177 y(an)40 b(irr)-5 b(ational)40 b(fr)-5 b(e)g(quency)40 b(ve)-5 b(ctor)40 b(on)g Fw(R)1405 5141 y Fr(d)1451 5177 y FA(.)61 b(L)-5 b(et)41 b Fx(V)1773 5192 y Fr(i)1801 5177 y FA(,)g(for)f Fx(i)e Fy(=)g(1)p Fx(;)17 b(:)g(:)g(:)e(;)i(p)p FA(,)42 b(denote)d(the)i(sp)-5 b(e)g(ctr)g(al)39 b(subbund)5 b(les)40 b(of)-118 5297 y(the)34 b(asso)-5 b(ciate)g(d)34 b(\015ow)g(and)g Fx(V)951 5312 y Fr(i)979 5297 y Fy(\()p Fx(\036)p Fy(\))h FA(denote)f(its)g(sp)-5 b(e)g(ctr)g(al)34 b(subsp)-5 b(ac)g(es,)34 b(for)g Fx(\036)28 b Fs(2)g Fw(T)2801 5261 y Fr(d)2845 5297 y FA(.)44 b(Assume)35 b(that)3508 5272 y Fy(~)3482 5297 y Fx(A)28 b Fs(2)g Fx(C)3754 5261 y Fr(\013)3803 5297 y Fy(\()p Fw(T)3904 5261 y Fr(d)3948 5297 y Fy(\))p FA(,)-118 5418 y(for)44 b Fx(\013)h Fy(=)g(0)p Fx(;)17 b(:)g(:)g(:)e(;)i Fs(1)p Fx(;)g(a)p FA(.)72 b(Then)43 b(we)g(c)-5 b(an)44 b(cho)-5 b(ose)43 b(a)h(lo)-5 b(c)g(al)43 b(b)-5 b(asis)43 b(in)h Fx(V)2513 5433 y Fr(i)2541 5418 y Fy(\()p Fx(\036)p Fy(\))g FA(that)g(it)h(is)e(a)h Fx(C)3318 5381 y Fr(\013)3412 5418 y FA(function)f(of)h Fx(\036)p FA(.)-118 5538 y(Equivalently)35 b(the)g(sp)-5 b(e)g(ctr)g(al)34 b(pr)-5 b(oje)g(ctors)1215 5722 y Fx(P)1278 5737 y Fr(i)1334 5722 y Fy(:)28 b Fw(T)1452 5681 y Fr(d)1524 5722 y Fs(\000)-17 b(!)1712 5641 y Fq(\010)1770 5722 y Fx(P)41 b Fs(2)28 b Fx(L)p Fy(\()p Fw(R)2138 5681 y Fr(n)2191 5722 y Fy(\);)45 b Fx(P)2378 5681 y FC(2)2444 5722 y Fy(=)28 b Fx(P)2625 5641 y Fq(\011)-118 5906 y FA(ar)-5 b(e)35 b(of)f(class)g Fx(C)473 5870 y Fr(\013)523 5906 y FA(.)1900 6155 y Fy(24)p eop %%Page: 25 29 25 28 bop 28 219 a Fy(In)33 b(order)g(to)f(pro)m(v)m(e)h(this)f (theorem)h(it)e(su\016ces)k(to)d(pic)m(k)g(an)m(y)i Fx(\036)2352 234 y FC(0)2418 219 y Fs(2)28 b Fw(T)2575 182 y Fr(d)2652 219 y Fy(and)k(restrict)h(the)g(v)-5 b(alues)32 b(of)g Fx(\036)g Fy(that)-118 339 y(w)m(e)c(consider)g(to)g(a)f(suitable)f (small)g(neigh)m(b)s(ourho)s(o)s(d)g(of)h Fx(\036)2020 354 y FC(0)2060 339 y Fy(.)41 b(No)m(w)28 b(let)f([)p Fx(a;)17 b(b)p Fy(])28 b(denote)h(a)e(giv)m(en)h(sp)s(ectral)f(in)m (terv)-5 b(al)-118 459 y(and)36 b(let)g Fx(V)58 b Fy(b)s(e)36 b(the)h(corresp)s(onding)f(sp)s(ectral)g(subbundle.)56 b(Next)37 b(let)e Fx(\026;)17 b(\025)36 b Fy(b)s(e)g(c)m(hosen)i(in)e (the)h(resolv)m(en)m(t)g(set)-118 580 y(so)c(that)f(\()p Fx(\026;)17 b(\025)p Fy(\))k Fs(\\)i Fy(\006)28 b(=)g([)p Fx(a;)17 b(b)p Fy(].)44 b(This)32 b(means)h(that)f(the)h(shifted)g (equations)617 827 y Fx(x)672 786 y Ft(0)724 827 y Fy(=)827 716 y Fq(\020)912 802 y Fy(~)887 827 y Fx(A)p Fy(\()p Fx(\036)22 b Fy(+)g Fx(!)t(t)p Fy(\))f Fs(\000)i Fx(\026I)1545 716 y Fq(\021)1621 827 y Fx(x)195 b Fy(and)g Fx(x)2278 786 y Ft(0)2330 827 y Fy(=)2433 716 y Fq(\020)2519 802 y Fy(~)2493 827 y Fx(A)p Fy(\()p Fx(\036)22 b Fy(+)g Fx(!)t(t)p Fy(\))f Fs(\000)i Fx(\025I)3149 716 y Fq(\021)3225 827 y Fx(x)-118 1069 y Fy(admit)49 b(exp)s(onen)m(tial)i(dic)m (hotomies.)98 b(The)52 b(stable)f(and)g(unstable)g(subbundles)i(asso)s (ciated)e(to)g(these)h(di-)-118 1189 y(c)m(hotomies)32 b(are,)h(for)f Fx(\027)i Fy(=)27 b Fx(\026;)17 b(\025)p Fy(,)549 1409 y Fs(S)609 1424 y Fr(\027)680 1409 y Fy(=)28 b Fs(f)o Fy(\()p Fx(x;)17 b(\036)p Fy(\);)45 b Fx(P)1201 1424 y Fr(\027)1244 1409 y Fy(\()p Fx(\036)p Fy(\))p Fx(x)27 b Fy(=)h Fx(x)p Fs(g)212 b Fy(and)195 b Fs(U)2295 1424 y Fr(\027)2366 1409 y Fy(=)28 b Fs(f)p Fy(\()p Fx(x;)17 b(\036)p Fy(\);)44 b Fx(P)2887 1424 y Fr(\027)2930 1409 y Fy(\()p Fx(\036)p Fy(\))p Fx(x)28 b Fy(=)f(0)p Fs(g)-118 1629 y Fy(and,)32 b(b)m(y)i(the)e(sp)s(ectral)g(theorem,)g(the)h(sp)s (ectral)f(subbundle)h Fx(V)54 b Fy(is)32 b(giv)m(en)g(b)m(y)h Fx(V)49 b Fy(=)28 b Fs(S)3041 1644 y Fr(\025)3108 1629 y Fs(\\)22 b(U)3258 1644 y Fr(\026)3305 1629 y Fy(;)32 b(that)g(is,)g Fx(V)22 b Fy(\()p Fx(\036)p Fy(\))27 b(=)-118 1750 y Fs(S)-58 1765 y Fr(\025)-12 1750 y Fy(\()p Fx(\036)p Fy(\))e Fs(\\)i(U)302 1765 y Fr(\026)349 1750 y Fy(\()p Fx(\036)p Fy(\))37 b(for)h(all)e Fx(\036)i Fs(2)g Fw(T)1079 1714 y Fr(d)1122 1750 y Fy(.)61 b(The)39 b(bundles)g Fs(S)1840 1765 y Fr(\025)1924 1750 y Fy(and)g Fs(U)2182 1765 y Fr(\026)2267 1750 y Fy(are)f(transv)m(ersal)h(in)f(a)g(neigh)m (b)s(ourho)s(o)s(d)f(of)h Fx(\036)p Fy(.)-118 1870 y(The)43 b(transv)m(ersalit)m(y)g(\(i.e.,)i(for)d(all)f Fx(\036)h Fy(in)g(a)g(neigh)m(b)s(ourho)s(o)s(d)f(of)h Fx(\036)2420 1885 y FC(0)2502 1870 y Fy(the)h(v)m(ector)h(spaces)g Fs(S)3352 1885 y Fr(\025)3397 1870 y Fy(\()p Fx(\036)p Fy(\))f(and)f Fs(U)3835 1885 y Fr(\026)3882 1870 y Fy(\()p Fx(\036)p Fy(\))-118 1990 y(generate)e(all)e Fx(Z)7 b Fy(\()p Fx(\036)p Fy(\))39 b(\))g(comes)h(from)e(the)i(fact)f(that)h Fx(Z)7 b Fy(\()p Fx(\036)p Fy(\))39 b(=)g Fs(S)2293 2005 y Fr(\026)2340 1990 y Fy(\()p Fx(\036)p Fy(\))27 b Fs(\010)g(U)2667 2005 y Fr(\026)2714 1990 y Fy(\()p Fx(\036)p Fy(\))39 b(and)h(w)m(e)g(ha)m(v)m(e)h(the)f(inclusion)-118 2111 y Fs(S)-58 2126 y Fr(\026)-11 2111 y Fy(\()p Fx(\036)p Fy(\))27 b Fs(\032)h(S)315 2126 y Fr(\025)361 2111 y Fy(\()p Fx(\036)p Fy(\))k(\(b)s(ecause)i Fx(\026)27 b(<)h(\025)p Fy(\).)28 2231 y(In)k(order)f(to)f(pro)m(v)m(e)j(that)d Fx(V)53 b Fy(dep)s(ends)32 b(smo)s(othly)e(on)h Fx(\036)g Fy(in)f(a)h(neigh)m(b)s(ourho)s(o)s(d)f(of)g Fx(\036)3144 2246 y FC(0)3183 2231 y Fy(,)i(it)e(su\016ces)j(to)d(pro)m(v)m(e)-118 2352 y(that)36 b(b)s(oth)f Fs(S)390 2367 y Fr(\025)472 2352 y Fy(and)h Fs(U)727 2367 y Fr(\026)810 2352 y Fy(dep)s(end)h(smo)s (othly)e(on)g Fx(\036)h Fy(in)f(a)h(neigh)m(b)s(ourho)s(o)s(d)f(of)h Fx(\036)2856 2367 y FC(0)2895 2352 y Fy(,)h(due)f(to)g(the)g(transv)m (ersalit)m(y)-118 2472 y(condition.)55 b(Without)36 b(loss)g(of)h (generalit)m(y)f(\(replacing)2002 2447 y(~)1976 2472 y Fx(A)h Fy(b)m(y)2252 2447 y(~)2226 2472 y Fx(A)25 b Fs(\000)h Fx(\027)6 b(I)i Fy(,)38 b(if)e(necessary\))j(w)m(e)e(can)h (assume)f(that)-118 2592 y Fx(\027)d Fy(=)28 b(0.)28 2713 y(Belo)m(w)33 b(w)m(e)g(state)g(a)g(theorem)f(that)g(will)f(imply) f(the)j(previous)g(one)-118 2941 y Fv(Theorem)k(2.4.2)h(\([49]\))48 b FA(L)-5 b(et)39 b Fx(U)50 b FA(b)-5 b(e)38 b(an)h(op)-5 b(en)38 b(set)h(in)f Fw(R)2068 2905 y Fr(k)2156 2941 y FA(and)g(let)h Fx(A)p Fy(\()p Fs(\001)p Fx(;)17 b Fs(\001)p Fy(\))34 b(:)h Fx(U)h Fs(\002)25 b Fw(R)46 b Fs(!)35 b 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b(follo)m(wing)c(prop)s (osition)g(relates)j(the)g(full)e(sp)s(ectrum)i(prop)s(ert)m(y)g(with)g (the)g(Lillo)d(prop)s(ert)m(y)-8 b(.)43 b(It)31 b(is)f(an)h(easy)-118 459 y(consequence)36 b(of)c(the)h(sp)s(ectral)f(theorem)g(and)h(of)f (the)h(exp)s(onen)m(tial)f(dic)m(hotom)m(y)g(in)g(the)h(resolv)m(en)m (t)h(set.)-118 688 y Fv(Prop)s(osition)h(2.5.5)j(\([49]\))48 b FA(The)35 b(fol)5 b(lowing)33 b(two)i(statements)g(ar)-5 b(e)34 b(e)-5 b(quivalent:)-75 891 y(\(A\))49 b(Equation)35 b(\(2.13\))e(has)i(ful)5 b(l)35 b(sp)-5 b(e)g(ctrum)34 b(with)h Fy(\006)28 b(=)g Fs(f)p Fx(\013)2116 906 y FC(1)2155 891 y Fx(;)17 b(:)g(:)g(:)f(;)h(\013)2436 906 y Fr(n)2483 891 y Fs(g)p FA(.)-71 1094 y(\(B\))48 b(Equation)36 b(\(2.13\))f(has)h (the)h(Lil)5 b(lo)36 b(pr)-5 b(op)g(erty.)49 b(In)36 b(this)g(c)-5 b(ase)36 b(one)g(also)f(has)h(the)h(ine)-5 b(qualities)35 b Fx(\025)3654 1109 y Fr(i)3713 1094 y Fx(<)30 b(\013)3881 1109 y Fr(i)3940 1094 y Fx(<)126 1215 y(\025)183 1230 y Fr(i)p FC(+1)336 1215 y FA(for)35 b Fx(i)28 b Fy(=)f(1)p Fx(;)17 b(:)g(:)g(:)f(;)h(n)p FA(.)28 1443 y Fy(There)36 b(is)f(a)g(w)m(eak)m(er)i(concept)f(than)f (Lillo)d(prop)s(ert)m(y)k(whic)m(h)f(w)m(e)h(shall)e(also)g(use)i (later.)50 b(W)-8 b(e)35 b(form)m(ulate)f(it)-118 1563 y(no)m(w.)-118 1792 y Fv(De\014nition)i(2.5.6)49 b FA(Equation)36 b(\(2.13\))f(is)h(said)f(to)h(satisfy)g(the)44 b Fy(Bylo)m(v)34 b(prop)s(ert)m(y)i FA(if)g(ther)-5 b(e)36 b(exist)g Fx(n)g FA(solutions)-118 1912 y Fx(x)-63 1927 y FC(1)-23 1912 y Fy(\()p Fx(t)p Fy(\))p Fx(;)17 b(:)g(:)g(:)f(;)h(x)362 1927 y Fr(n)409 1912 y Fy(\()p Fx(t)p Fy(\))35 b FA(with)f(the)h(pr)-5 b(op)g(erty)35 b(that)1582 2160 y Fy(inf)1581 2222 y Fr(t)p Ft(2)p Fk(R)1718 2045 y Fq(\014)1718 2105 y(\014)1718 2165 y(\014)1751 2160 y Fy(det)1932 2134 y(~)1903 2160 y Fx(X)8 b Fy(\()p Fx(t)p Fy(\))2103 2045 y Fq(\014)2103 2105 y(\014)2103 2165 y(\014)2164 2160 y Fx(>)28 b Fy(0)-118 2435 y FA(wher)-5 b(e)185 2410 y Fy(~)157 2435 y Fx(X)8 b Fy(\()p Fx(t)p Fy(\))35 b 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b(principle,)g(it)f(could)g(happ)s(en)i (that)-118 4371 y(the)35 b(pre-image)f(of)g Fx(W)710 4386 y Fr(i)774 4371 y Fy(under)i Fx(\033)28 b Fs(\002)c Fx(I)8 b(d)35 b Fy(is)f(comp)s(osed)h(of)g(sev)m(eral)h(connected)h (comp)s(onen)m(ts.)51 b(Let)3607 4346 y(~)3579 4371 y Fx(W)3671 4386 y Fr(i)3734 4371 y Fy(denote)-118 4492 y(one)34 b(of)g(these)h(connected)h(comp)s(onen)m(ts,)f(for)e Fx(i)e Fy(=)f(1)p Fx(;)17 b(:)g(:)g(:)e(;)i(n)p Fy(.)48 b(Since)35 b(\012)2496 4507 y FC(2)2569 4492 y Fy(is)f(a)g(co)m(v)m (ering)g(space)h(for)f(eac)m(h)h Fx(W)3861 4507 y Fr(i)3889 4492 y Fy(,)g(it)-118 4612 y(follo)m(ws)i(that)h(eac)m(h)679 4587 y(~)651 4612 y Fx(W)743 4627 y Fr(i)809 4612 y Fy(is)g(a)g(1-co)m (v)m(er)i(of)e(\012)1527 4627 y FC(2)1566 4612 y Fy(.)62 b(That)38 b(is,)i(eac)m(h)2291 4587 y(~)2262 4612 y Fx(W)2354 4627 y Fr(i)2421 4612 y Fy(can)f(b)s(e)f(represen)m(ted)j(as)e(the)g (graph)f(of)g(a)-118 4732 y(function)32 b Fx(f)312 4747 y Fr(i)368 4732 y Fy(:)c(\012)493 4747 y FC(2)560 4732 y Fs(!)g Fx(S)754 4696 y Fr(n)p Ft(\000)p FC(1)890 4732 y Fy(.)44 b(The)33 b(regularit)m(y)f(of)g(eac)m(h)h Fx(f)1984 4747 y Fr(i)2045 4732 y Fy(is)f Fx(C)2220 4696 y Fr(\013)2302 4732 y Fy(since,)h(lo)s(cally)-8 b(,)30 b(w)m(e)k(can)f(write)f Fx(f)3516 4747 y Fr(i)3572 4732 y Fy(=)27 b Fx(\021)3727 4691 y Ft(\000)p FC(1)3723 4758 y Fr(i)3844 4732 y Fs(\016)22 b Fx(\033)t Fy(.)28 4853 y(Because)46 b(of)f(de\014nition,)h(for)f(eac) m(h)1418 4826 y(~)1405 4853 y Fx(\036)j Fs(2)h Fy(\012)1696 4868 y FC(2)1780 4853 y Fy(and)c Fx(i)k Fy(=)f(1)p Fx(;)17 b(:)g(:)g(:)f(;)h(n)p Fy(,)47 b Fx(f)2636 4868 y Fr(i)2664 4853 y Fy(\()2715 4826 y(~)2702 4853 y Fx(\036)p Fy(\))e(is)f(a)g(unit) g(v)m(ector)i(in)e Fw(R)3761 4817 y Fr(n)3859 4853 y Fy(and)-118 4973 y Fs(f)p Fx(f)-20 4988 y FC(1)19 4973 y Fy(\()70 4947 y(~)57 4973 y Fx(\036)p Fy(\))p Fx(;)17 b(:)g(:)g(:)f(;)h(f)420 4988 y Fr(n)466 4973 y Fy(\()517 4947 y(~)504 4973 y Fx(\036)p Fy(\))p Fs(g)34 b Fy(is)f(a)h(v)m(ector)h (frame)f(for)f 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Fy(,)i(eac)m(h)f(of)e(the)i Fx(\025)3685 248 y Fr(i)3750 233 y Fy(is)f(also)-118 381 y(of)46 b(class)h Fx(C)327 345 y Fr(\013)376 381 y Fy(.)86 b(W)-8 b(e)48 b(apply)e(the)h(results)g(on)g(diagonal)d(systems)k(\(and)f(the)g (Diophan)m(tine)f(assumption)g(on)-118 502 y Fx(!)t Fy(!\))69 b(to)41 b(deduce)i(the)e(existence)i(of)e(a)g(transformation)e Fx(Q)p Fy(\()p Fx(t)p Fy(;)2233 475 y(~)2220 502 y Fx(\036)p Fy(\))i(that)g(renders)i(the)f(system)g(to)f(a)g(constan)m(t)-118 622 y(co)s(e\016cien)m(t)28 b(system)h(with)f(diagonal)d(matrix.)40 b(Finally)-8 b(,)26 b(due)j(to)e(the)i(construction)f(of)f(the)h (di\013eren)m(t)g(\015o)m(ws,)i(the)-118 742 y(comp)s(osition)g(of)i (transformations)f(can)i(b)s(e)g(written)f(as)1261 962 y Fx(x)c Fy(=)1470 937 y(~)1448 962 y Fx(P)i Fy(\()p Fx(\036)1637 977 y FC(1)1698 962 y Fy(+)g(\026)-57 b Fx(!)1857 977 y FC(1)1896 962 y Fx(t;)17 b(:)g(:)g(:)f(;)h(\036)2208 977 y Fr(d)2270 962 y Fy(+)30 b(\026)-56 b Fx(!)2430 977 y Fr(d)2470 962 y Fx(t)p Fy(\))16 b Fx(z)t(:)-118 1182 y Fh(\003)1900 6155 y Fy(30)p eop %%Page: 31 35 31 34 bop -118 883 a Fz(Chapter)78 b(3)-118 1298 y(Reducibilit)-6 b(y)76 b(in)h(Sc)-6 b(hr\177)-116 b(odinger)78 b(equation)-118 1547 y(with)f(quasi-p)6 b(erio)g(dic)78 b(p)6 b(oten)-6 b(tial)-118 1999 y Fy(In)43 b(this)g(c)m(hapter)h(w)m(e)g(will)d(study) j(the)f(reducibilit)m(y)f(problem)f(in)i(a)f(sp)s(ecial)g(linear)g (equation)h(with)f(quasi-)-118 2120 y(p)s(erio)s(dic)31 b(co)s(e\016cien)m(ts,)j(namely)1442 2294 y Fs(\000)1547 2226 y Fx(d)1598 2190 y FC(2)p 1529 2271 126 4 v 1529 2362 a Fx(dt)1615 2333 y FC(2)1665 2294 y Fx(x)p Fy(\()p Fx(t)p Fy(\))22 b(+)g Fx(q)t Fy(\()p Fx(t)p Fy(\))p Fx(x)p Fy(\()p Fx(t)p Fy(\))28 b(=)g(0)-118 2492 y(where)g Fx(x)p Fy(\()p Fx(t)p Fy(\))g Fs(2)g Fw(R)38 b Fy(and)27 b Fx(q)t Fy(\()p Fx(t)p Fy(\))g(=)h Fx(Q)p Fy(\()p Fx(!)t(t)10 b Fy(+)g Fx(\036)p Fy(\))27 b(is)f(a)g(quasi-p)s(erio)s(dic)f (function.)41 b(This)27 b(equation)g(is)f(usually)g(referred)-118 2613 y(in)g(the)i(literature)e(as)i(the)f(one-dimensional)e(Sc)m (hr\177)-49 b(odinger)27 b(equation)h(with)e(a)h(quasi-p)s(erio)s(dic)f (p)s(oten)m(tial.)40 b(The)-118 2733 y(sp)s(ectral)h(theory)g(dev)m (elop)s(ed)h(for)f(second-order)h(op)s(erators)f(in)f(Hilb)s(ert)g (spaces,)45 b(together)c(with)f(the)i(to)s(ols)-118 2854 y(in)m(tro)s(duced)e(b)m(y)g(ergo)s(dic)f(theory)-8 b(,)42 b(enables)e(us)g(to)f(mak)m(e)h(a)f(more)g(precise)h(description)f(of)g (the)h(reducibilit)m(y)-118 2974 y(problem.)50 b(All)33 b(these)k(theories)e(ha)m(v)m(e)h(b)s(een)g(studied)g(in)e(the)i(last)e (thirt)m(y)h(y)m(ears)h(from)e(a)h(dynamical)e(systems)-118 3094 y(p)s(oin)m(t)f(of)g(view)h(and,)f(esp)s(ecially)-8 b(,)32 b(KAM)h(tec)m(hniques)h(ha)m(v)m(e)g(pro)m(v)m(ed)g(to)e(b)s(e)h (v)m(ery)h(useful)f(for)f(this)g(study)-8 b(.)28 3215 y(This)42 b(c)m(hapter)h(is)e(the)h(longest)f(in)g(the)h(presen)m(t)h (surv)m(ey)-8 b(.)73 b(This)42 b(is)f(due)h(to)g(the)g(fact)f(that)g (Sc)m(hr\177)-49 b(odinger)-118 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b Fx(q)t Fy(\()p Fx(t)p Fy(\))p Fx(x)28 b Fy(=)g(0)p Fx(;)1516 b Fy(\(3.1\))-118 4775 y(where)35 b Fx(x)c Fs(2)g Fw(R)414 4739 y Fr(d)461 4775 y Fy(,)j(and)h Fx(q)j Fy(is)c(a)g(quasi-p)s(erio)s(dic)e(function) h(with)h(frequency)j Fx(!)d Fs(2)d Fw(R)2915 4739 y Fr(d)2995 4775 y Fy(whic)m(h,)k(in)f(the)h(con)m(text)g(of)-118 4895 y(Sc)m(hr\177)-49 b(odinger)33 b(equation,)h(is)f(called)f(the)i (p)s(oten)m(tial.)44 b(This)33 b(means)h(that)f(there)h(exists)g(a)f (con)m(tin)m(uous)h(function)-118 5016 y Fx(Q)28 b Fy(:)g Fw(T)105 4980 y Fr(d)176 5016 y Fs(!)f Fw(R)5 b Fy(,)39 b(b)s(eing)32 b Fw(T)f Fy(=)c Fw(R)5 b Fx(=)p Fy(\(2)p Fx(\031)t Fw(Z)p Fy(\),)36 b(suc)m(h)e(that)e Fx(q)t Fy(\()p Fx(t)p Fy(\))c(=)f Fx(Q)p Fy(\()p Fx(!)t(t)p Fy(\).)28 5136 y(Related)32 b(to)g(equation)h(\(3.1\),)f(it)f(will)g(b) s(e)i(useful)f(to)g(study)i(the)f(follo)m(wing)d(family)g(of)i (equations)1551 5341 y Fs(\000)p Fx(x)1683 5300 y Ft(0)q(0)1748 5341 y Fy(+)22 b Fx(q)t Fy(\()p Fx(t)p Fy(\))p Fx(x)28 b Fy(=)g Fx(\025x;)1485 b Fy(\(3.2\))-118 5545 y(b)s(eing)28 b Fx(\025)g Fy(a)h(real)f(\(or)g(sometimes)g(complex\))g(parameter)g (whic)m(h)h(w)m(e)h(shall)d(denote)j(as)f(the)g(sp)s(ectral)f (parameter)-118 5666 y(or)k(the)h(energy)-8 b(.)28 5786 y(Equation)48 b(\(3.1\))g(app)s(ears)g(in)f(man)m(y)h(branc)m(hes)i(of) e(mathematics)e(and)i(ph)m(ysics.)92 b(F)-8 b(or)47 b(instance,)52 b(the)-118 5906 y(Sc)m(hr\177)-49 b(odinger)41 b(equation)f(with)h(p)s (erio)s(dic)e(p)s(oten)m(tial)g(arises)i(in)f(a)g(natural)g(w)m(a)m(y)i (in)e(the)h(quan)m(tum)g(theory)g(of)1900 6155 y(31)p eop %%Page: 32 36 32 35 bop -118 219 a Fy(solids,)50 b(to)c(b)s(e)h(more)g(precise,)k(in) 46 b(the)i(quan)m(tum)f(theory)g(of)g(crystals,)k(for)46 b(example)h(metals.)85 b(The)48 b(ions)-118 339 y(forming)34 b(a)i(crystal)h(lattice)d(actually)h(generate)i(a)f(p)s(erio)s(dic)f (\014eld)h(and)h(one)f(can)h(examine)f(the)h(motion)d(of)i(a)-118 459 y(free)31 b(electron)h(in)e(this)h(\014eld.)43 b(One)31 b(can,)h(in)e(a)h(n)m(um)m(b)s(er)h(of)f(cases,)h(b)m(y)h(in)m(tro)s (ducing)c(comp)s(ensating)i(additional)-118 580 y(terms)g(to)g(the)h (ion)f(p)s(oten)m(tial,)f(disregard)h(the)h(in)m(teraction)f(of)g(the)g (free)h(electrons.)44 b(When)32 b(more)f(frequencies)-118 700 y(are)37 b(added)h(to)e(this)h(problem)f(\(coming)g(from)f (considering)i(more)f(general)h(lattices)f(or)h(applying)e(a)i(\014eld) g(to)-118 820 y(p)s(erio)s(dic)d(lattices\),)i(Sc)m(hr\177)-49 b(odinger)36 b(equation)f(with)h(quasi-p)s(erio)s(dic)e(p)s(oten)m (tial)g(turns)i(to)g(b)s(e)g(a)f(useful)h(mo)s(del)-118 941 y(\(see)e([5)o(],)f([61],)f([3],)h([81],)f([82])h(and)f(references) j(therein\).)28 1061 y(A)49 b(useful)f(ob)5 b(ject)50 b(to)e(study)i(the)f(solutions)e(of)h(a)g(linear)f(system)j(is)e(the)h (W)-8 b(ronskian.)91 b(In)49 b(our)f(t)m(w)m(o-)-118 1182 y(dimensional)43 b(setting,)48 b(the)d(W)-8 b(ronskian)45 b(of)g(t)m(w)m(o)h(di\013eren)m(tiable)e(complex)g(functions)h Fx(u)g Fy(and)g Fx(v)k Fy(tak)m(es)d(the)-118 1302 y(form)1236 1422 y Fx(W)14 b Fy(\()p Fx(u;)j(v)t Fy(\)\()p Fx(t)p Fy(\))26 b(=)i Fx(u)p Fy(\()p Fx(t)p Fy(\))p Fx(v)2028 1381 y Ft(0)2050 1422 y Fy(\()p Fx(t)p Fy(\))22 b Fs(\000)h Fx(u)2339 1381 y Ft(0)2362 1422 y Fy(\()p Fx(t)p Fy(\))p Fx(v)t Fy(\()p Fx(t)p Fy(\))p Fx(:)28 1581 y Fy(It)39 b(follo)m(ws)f(from)f(Liouville)g(theorem)h(that)h(the)g(W)-8 b(ronskian)39 b(of)g(t)m(w)m(o)g(solutions)f(of)g(equation)h(\(3.1\))g (is)f(a)-118 1701 y(constan)m(t)33 b(function)f(whic)m(h)i(is)e (nonzero)h(if,)e(and)i(only)f(if,)f(the)i(t)m(w)m(o)h(solutions)d(are)i (linearly)d(indep)s(enden)m(t.)28 1822 y(More)j(general)f(equations)h (suc)m(h)h(as)1450 2004 y Fx(y)1502 1963 y Ft(0)o(0)1566 2004 y Fy(+)22 b Fx(a)p Fy(\()p Fx(t)p Fy(\))p Fx(y)1878 1963 y Ft(0)1922 2004 y Fy(+)g Fx(b)p Fy(\()p Fx(t)p Fy(\))p Fx(y)32 b Fy(=)27 b(0)p Fx(;)1384 b Fy(\(3.3\))-118 2187 y(can)25 b(b)s(e)f(transformed)g(to)g(the)h(form)e(of)h(equation)g (\(3.1\))g(using)g(the)h(tec)m(hniques)h(\(and)f(therefore)g(the)g (conditions)-118 2308 y(on)32 b(the)h(p)s(oten)m(tial)e(and)i(the)g (frequency\))i(in)m(tro)s(duced)d(in)g(the)h(\014rst)g(c)m(hapter)h (\([31)o(]\).)28 2428 y(Sc)m(hr\177)-49 b(odinger)42 b(equation)g(with)f(b)s(ounded)i(p)s(oten)m(tial)d(has)j(some)e(in)m (teresting)h(prop)s(erties)g(whic)m(h)g(will)d(b)s(e)-118 2548 y(useful)c(in)g(the)g(sequel.)53 b(One)35 b(of)g(the)h(most)e(imp) s(ortan)m(t)g(is)g(the)i(transformation)d(of)i(equation)g(\(3.1\))f(to) h(p)s(olar)-118 2669 y(co)s(ordinates)h(with)g(the)h(reduction)g(imp)s (osed)f(b)m(y)h(the)g(symmetries)g(that)f(exist)h(in)f(the)h(problem.) 55 b(This)37 b(also)-118 2789 y(has)c(to)f(do)g(with)h(the)f(linear)f (c)m(haracter)j(of)e(the)h(\015o)m(w.)44 b(As)33 b(it)e(will)g(b)s(e)h (necessary)j(later)d(on,)g(w)m(e)i(describ)s(e)f(b)s(oth)-118 2909 y(the)j(situation)d(in)i(the)h(complex)f(case)h(\(assume)g(that)f Fx(Q)g Fy(is)g(complex)g(in)g(equation)g(\(3.1\))f(or)h(simply)f(that)h Fx(Q)-118 3030 y Fy(is)d(real)g(and)g Fx(\025)h Fy(is)f(complex)g(in)g (equation)g(\(3.2\)\).)28 3150 y(F)-8 b(or)32 b(eac)m(h)h Fx(\036)28 b Fs(2)g Fw(T)665 3114 y Fr(d)709 3150 y Fy(,)k(the)h (equation)1468 3333 y Fs(\000)p Fx(x)1600 3292 y Ft(0)q(0)1666 3333 y Fy(+)22 b Fx(Q)p Fy(\()p Fx(!)t(t)f Fy(+)h Fx(\036)p Fy(\))p Fx(x)28 b Fy(=)g(0)1385 b(\(3.4\))-118 3516 y(is)29 b(linear)e(and)i(the)h(fundamen)m(tal)e(matrix)g(solution)g(\010\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))29 b(\(with)f(\010\(0;)17 b Fx(\036)p Fy(\))28 b(=)f Fx(I)8 b Fy(\))29 b(maps)g(complex)g(lines)f (\(that)-118 3636 y(is,)41 b(1-dimensional)36 b(subspaces)41 b(of)e Fw(C)1277 3600 y FC(2)1323 3636 y Fy(\))g(to)g(complex)g(lines.) 63 b(If)39 b Fx(l)j Fy(is)c(a)h(complex)g(line)f(in)h Fw(C)3330 3600 y FC(2)3375 3636 y Fy(,)i(let)e Fx(l)r Fy(\()p Fx(t)p Fy(\))h(b)s(e)f(its)-118 3757 y(ev)m(olution)29 b(b)m(y)h(the)g(\015o)m(w)h(so)f(that)f Fx(l)r Fy(\()p Fx(t)p Fy(\))f(=)g(\010\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))p Fx(l)r Fy(.)43 b(Letting)28 b Fw(P)2181 3720 y FC(1)2223 3757 y Fy(\()p Fw(C)20 b Fy(\))35 b(b)s(e)30 b(the)g(usual)g(space)g(of)g(all)d(complex)j(lines)-118 3877 y(in)i Fw(C)62 3841 y FC(2)107 3877 y Fy(,)h(w)m(e)g(de\014ne)h(a) e(\015o)m(w)i(\011)e(on)h(\006)28 b(=)f Fw(P)1381 3841 y FC(1)1423 3877 y Fy(\()p Fw(C)19 b Fy(\))28 b Fs(\002)23 b Fw(T)1755 3841 y Fr(d)1831 3877 y Fy(as)33 b(follo)m(ws)730 4060 y(\011\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))27 b(=)g(\()p Fx(!)t(t)22 b Fy(+)g Fx(\036;)17 b(l)r Fy(\()p Fx(t)p Fy(\)\))f Fx(;)50 b Fy(for)32 b Fx(\036)27 b Fs(2)h Fw(T)2173 4019 y Fr(d)2217 4060 y Fx(;)44 b(l)30 b Fs(2)e Fw(P)2500 4019 y FC(1)2542 4060 y Fy(\()p Fw(C)20 b Fy(\))38 b(and)33 b Fx(t)28 b Fs(2)g Fw(R)5 b Fx(:)-118 4243 y Fy(This)36 b(description)f(b)s(ecomes)i(easier)f(in)f(the)h(case)h(when)g Fx(Q)f Fy(is)f(real.)53 b(Let)36 b(then)h Fw(P)2948 4207 y FC(1)2989 4243 y Fy(\()p Fw(R)5 b Fy(\))42 b(b)s(e)36 b(the)g(space)h(of)f(real)-118 4363 y(one-dimensional)e(subspaces)40 b(of)c Fw(R)1234 4327 y FC(2)1279 4363 y Fy(,)j(whose)f(elemen)m(ts)f (w)m(e)h(shall)e(call)f(lines)h(\(opp)s(osed)h(to)g(complex)f(lines\),) -118 4483 y(and)h(\006)146 4498 y Fk(R)235 4483 y Fy(=)e Fw(P)405 4447 y FC(1)446 4483 y Fy(\()p Fw(R)5 b Fy(\))32 b Fs(\002)25 b Fw(T)785 4447 y Fr(d)866 4483 y Fy(the)38 b(real)f(bundle)g(on)g(whic)m(h)h(the)g(equation)f(\(3.4\))f(de\014nes) j(a)e(\015o)m(w.)59 b(Then)38 b(w)m(e)g(can)-118 4604 y(view)43 b Fw(P)176 4568 y FC(1)217 4604 y Fy(\()p Fw(R)5 b Fy(\))48 b(as)43 b(a)f(subset)i(of)d Fw(P)1116 4568 y FC(1)1158 4604 y Fy(\()p Fw(C)20 b Fy(\))48 b(using)42 b(the)h(usual)f(iden)m(ti\014cation)f(of)g(the)i(Riemann)e(sphere)j Fw(S)3750 4568 y FC(2)3826 4604 y Fy(with)-118 4724 y Fw(P)-59 4688 y FC(1)-18 4724 y Fy(\()p Fw(C)20 b Fy(\))35 b(and)29 b(the)h(iden)m(ti\014cation)e(of)g Fw(P)1261 4688 y FC(1)1303 4724 y Fy(\()p Fw(R)t Fy(\))35 b(with)29 b Fw(R)20 b Fs([)c(f1g)33 b(\032)28 b Fw(P)2258 4688 y FC(1)2299 4724 y Fy(\()p Fw(C)20 b Fy(\).)49 b(Th)m(us,)31 b(if)d(w)m(e)i(w)m(an)m(t)g(to)f(use)h(co)s(ordinates)-118 4845 y(for)37 b(the)h(real)e(case,)k(w)m(e)e(can)g(use)h(this)e(last)f (remark,)j(parameterizing)c Fw(P)2630 4808 y FC(1)2672 4845 y Fy(\()p Fw(R)5 b Fy(\))43 b(with)37 b(an)h(angle)e Fx(')h Fy(b)s(et)m(w)m(een)j(0)-118 4965 y(and)d Fx(\031)t Fy(.)55 b(Once)37 b(an)g(initial)c(condition)i(is)h(\014xed,)j(whic)m (h)e(amoun)m(ts)f(to)g(an)h(initial)c(line)i(\(angle\))h(in)f Fw(P)3626 4929 y FC(1)3668 4965 y Fy(\()p Fw(R)5 b Fy(\),)43 b(the)-118 5085 y(ev)m(olution)32 b(is)g(giv)m(en)g(b)m(y)i(the)f (follo)m(wing)d(equation)1433 5268 y Fx(')1497 5227 y Ft(0)1548 5268 y Fy(=)d(cos)1782 5227 y FC(2)1838 5268 y Fx(')22 b Fs(\000)h Fx(q)t Fy(\()p Fx(t)p Fy(\))17 b(sin)2318 5227 y FC(2)2374 5268 y Fx(':)1350 b Fy(\(3.5\))-118 5451 y(Assuming)28 b(the)i(kno)m(wledge)g(of)e(the)h(ev)m(olution)f(of) h(the)g(line)f(giv)m(en)h(b)m(y)h Fx(')p Fy(\()p Fx(t)p Fy(\),)g(then)f(the)g(radius)g(can)g(b)s(e)g(directly)-118 5571 y(in)m(tegrated)j(as)h(follo)m(ws)938 5812 y Fx(r)s Fy(\()p Fx(t)p Fy(\))27 b(=)h Fx(r)s Fy(\(0\))17 b(exp)1580 5671 y Fq(\022)1663 5744 y Fy(1)p 1663 5789 49 4 v 1663 5880 a(2)1739 5676 y Fq(Z)1838 5702 y Fr(t)1794 5902 y FC(0)1885 5812 y Fy(\()p Fx(q)t Fy(\()p Fx(s)p Fy(\))k Fs(\000)i Fy(1\))16 b(sin\(2)p Fx(')p Fy(\()p Fx(s)p Fy(\)\))p Fx(ds)2844 5671 y Fq(\023)2933 5812 y Fx(:)855 b Fy(\(3.6\))1900 6155 y(32)p eop %%Page: 33 37 33 36 bop -118 219 a Fy(If)36 b(w)m(e)h(w)m(an)m(t)f(to)g(stress)h(the) f(dep)s(endence)j(on)c(the)i(sp)s(ecial)d(parameter)i Fx(\025)f Fy(in)g(the)i(case)f(of)g(equation)f(\(3.2\))h(w)m(e)-118 339 y(will)30 b(write)i Fx(')p Fy(\()p Fs(\001)p Fy(;)17 b Fx(\025)p Fy(\))32 b(and)h Fx(r)s Fy(\()p Fs(\001)p Fy(;)17 b Fx(\025)p Fy(\).)28 459 y(Before)39 b(ending)e(the)i(in)m (tro)s(duction)d(let)i(us)g(outline)f(the)i(con)m(ten)m(ts)g(of)f(this) g(c)m(hapter.)60 b(In)39 b(section)f(3.2)g(w)m(e)-118 580 y(sk)m(etc)m(h)31 b(the)e(sp)s(ectral)g(theory)g(for)f(Sc)m(hr\177) -49 b(odinger)29 b(equation)g(with)f(quasi-p)s(erio)s(dic)f(p)s(oten)m (tial.)40 b(There)30 b(are)f(three)-118 700 y(steps)44 b(in)e(this)g(approac)m(h:)64 b(w)m(e)44 b(\014rst)f(study)h(the)f(sp)s (ectral)g(theory)g(for)f(general)g(self-adjoin)m(t)f(op)s(erators)i(on) -118 820 y(Hilb)s(ert)26 b(spaces,)k(w)m(e)e(then)g(pass)g(to)f(Sc)m (hr\177)-49 b(odinger)28 b(equation)f(with)g(a)g(time-dep)s(enden)m(t)h (p)s(oten)m(tial)d(and,)k(\014nally)-8 b(,)-118 941 y(w)m(e)30 b(mak)m(e)g(all)d(the)j(previous)f(theory)h(a)f(bit)g(more)g(precise)h (in)e(the)i(case)g(of)f(quasi-p)s(erio)s(dic)e(p)s(oten)m(tials.)41 b(In)30 b(this)-118 1061 y(case)e(w)m(e)h(can)f(c)m(haracterize)g(the)g (sp)s(ectrum)g(of)f(Sc)m(hr\177)-49 b(odinger)28 b(equation)g (according)f(to)g(a)g(dynamical)f(criterion,)-118 1182 y(whic)m(h)33 b(uses)h(the)f FA(dynamic)-5 b(al)42 b Fy(notion)31 b(of)h(exp)s(onen)m(tial)g(dic)m(hotom)m(y)h(that)f(w)m(e) i(sa)m(w)f(in)f(the)h(previous)g(c)m(hapter.)28 1302 y(The)40 b(close)e(relation)e(of)i(the)h(dynamical)e(b)s(eha)m(viour)h (of)g(the)h(solutions)e(of)h(Sc)m(hr\177)-49 b(odinger)39 b(equation)f(with)-118 1422 y(quasi-p)s(erio)s(dic)32 b(p)s(oten)m(tial)g(with)h(the)i(sp)s(ectral)e(analysis)h(approac)m(h)g (is)f(made)h(more)f(eviden)m(t)i(in)e(section)h(3.3,)-118 1543 y(where)j(some)f(of)g(the)g(links)g(b)s(et)m(w)m(een)i(the)e(prop) s(erties)g(of)g(the)g(rotation)f(n)m(um)m(b)s(er)h(and)h(the)f(sp)s (ectrum)h(of)e(the)-118 1663 y(Sc)m(hr\177)-49 b(odinger)25 b(op)s(erator)g(are)g(describ)s(ed.)42 b(A)25 b(study)i(of)d(the)i (prop)s(erties)f(of)g(the)h(rotation)d(n)m(um)m(b)s(er)j(is)f(imp)s (ortan)m(t,)-118 1783 y(b)s(ecause)39 b(in)d(section)i(3.4)f(it)f(is)h (used)h(to)f(con)m(trol)g(the)h(imaginary)d(part)i(of)g(the)h(eigen)m (v)-5 b(alues)37 b(of)g(the)h(Flo)s(quet)-118 1904 y(matrix,)26 b(whic)m(h)g(is)f(one)h(of)g(the)g(imp)s(ortan)m(t)e(p)s(oin)m(ts)h(in) g(the)h(KAM)g(approac)m(h)g(to)g(reducibilit)m(y)-8 b(.)39 b(W)-8 b(e)26 b(also)f(include)-118 2024 y(some)35 b(theory)h(on)g(the) g(Ly)m(apuno)m(v)h(exp)s(onen)m(ts)g(for)e(Sc)m(hr\177)-49 b(odinger)36 b(equation,)g(sho)m(wing)g(its)f(relation)e(with)i(the) -118 2145 y(sp)s(ectral)d(ob)5 b(jects)34 b(de\014ned)g(in)e(section)h (3.2.)28 2265 y(Section)k(3.4)f(is)g(the)h(cen)m(tral)g(section)g(in)f (the)h(c)m(hapter,)i(b)s(ecause)f(w)m(e)g(state)f(a)f(theorem)h(b)m(y)g (H.)g(Eliasson,)-118 2385 y(on)j(the)h(almost)e(ev)m(erywhere)44 b(reducibilit)m(y)38 b(of)i(Sc)m(hr\177)-49 b(odinger)41 b(equation)f(with)h(quasi-p)s(erio)s(dic)d(p)s(oten)m(tial)h(on)-118 2506 y(the)c(assumption)g(of)f(analyticit)m(y)-8 b(,)34 b(strong)h(non-resonance)h(and)f(closeness)i(to)d(constan)m(t)i(co)s (e\016cien)m(ts.)52 b(Some)-118 2626 y(previous)40 b(reducibilit)m(y)e (results)i(are)g(also)f(discussed.)67 b(The)41 b(steps)g(of)f(the)g 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y(the)e(reducible)f(and)h(non-reducible)f(zones.)-118 3800 y Fu(3.2)161 b(Some)52 b(sp)t(ectral)i(theory)-118 4048 y Fg(3.2.1)136 b(Statemen)l(t)46 b(of)g(the)f(problem)-118 4233 y Fy(The)40 b(classical)d(sp)s(ectral)i(theory)h(of)e(Sc)m(hr\177) -49 b(odinger)39 b(op)s(erators)g(with)g(b)s(ounded)g(p)s(oten)m(tial)f (pro)m(vides)h(us)h(with)-118 4353 y(useful)29 b(to)s(ols)f(to)g(study) i(the)g(ultimate)c(b)s(eha)m(viour)j(of)g(solutions)e(of)i(equation)g (\(3.1\).)41 b(As)30 b(w)m(e)g(are)f(dealing)e(with)-118 4473 y(a)32 b(linear)f(case,)j(the)f(ultimate)d(b)s(eha)m(viour)i(is)h (quite)f(de\014nitiv)m(e)h(for)f(the)h(classi\014cation)e(of)h(these)i (solutions,)d(as)-118 4594 y(not)h(all)f(kinds)i(of)f(b)s(eha)m(viour)g (are)h(p)s(ossible)f(in)g(a)g(linear)f(system.)28 4714 y(This)36 b(theory)g(w)m(as)h(\014rst)f(dev)m(elop)s(ed)g(b)m(y)h(H.)f (W)-8 b(eyl)35 b(in)g(1910.)52 b(The)36 b(idea)f(is)g(the)i(follo)m (wing.)49 b(Assume)37 b(that)-118 4835 y(\003)f Fs(\032)h Fw(R)48 b Fy(is)38 b(a)f(certain)g(in)m(terv)-5 b(al,)38 b(and)g(to)f(\014x)i(ideas,)g(tak)m(e)f(it)f(\003)f(=)g(\()p Fs(\0001)p Fx(;)17 b Fs(1)p Fy(\).)58 b(Then)39 b(on)f(the)g(set)g Fx(C)3681 4798 y Ft(1)3674 4859 y Fr(c)3756 4835 y Fy(\(\003\))f(of) -118 4955 y(in\014nitely)31 b(di\013eren)m(tiable)g(functions)i(with)f (compact)g(supp)s(ort)h(on)g(\003)f(w)m(e)i(can)e(consider)h(the)g(op)s (erator)1308 5163 y Fx(H)8 b(f)38 b Fy(=)27 b Fx(H)1667 5178 y FC(0)1706 5163 y Fx(f)33 b Fy(+)22 b Fx(V)g(f)38 b Fy(=)28 b Fs(\000)p Fx(f)2290 5122 y Ft(00)2355 5163 y Fy(+)22 b Fx(V)f(f)1236 b Fy(\(3.7\))-118 5370 y(for)32 b(a)g(certain)g(real)g(function)g Fx(V)54 b Fy(and)33 b(the)g(related)f(eigen)m(v)-5 b(alue)32 b(problem)1752 5578 y Fx(H)8 b(f)38 b Fy(=)27 b Fx(\025f)1680 b Fy(\(3.8\))-118 5786 y(for)34 b Fx(\025)e Fs(2)g Fw(C)20 b Fy(.)56 b(As)35 b Fx(C)592 5750 y Ft(1)585 5811 y Fr(c)667 5786 y Fy(\(\003\))f(is)h (dense)h(in)e Fx(L)1396 5750 y FC(2)1436 5786 y Fy(\(\003\))g(\(the)i (space)g(of)e(square)i(in)m(tegrable)e(functions)h(on)f(\003\))h(it)f (w)m(ould)-118 5906 y(b)s(e)c(nice)h(to)f(de\014ne)h(a)f(natural)f(and) i(self-adjoin)m(t)e(extension)i(of)f(the)g(ab)s(o)m(v)m(e)h(op)s (erator)f(on)g Fx(L)3280 5870 y FC(2)3320 5906 y Fy(\(\003\),)h(b)s (ecause)g(the)1900 6155 y(33)p eop %%Page: 34 38 34 37 bop -118 219 a Fy(Hilb)s(ert)31 b(space)j(structure)g(mak)m(es)g (the)f(eigen)m(v)-5 b(alue)32 b(problem)g(\(3.8\))g(m)m(uc)m(h)i (clearer.)44 b(Ho)m(w)m(ev)m(er)35 b(it)d(is)h(not)f(clear)-118 339 y(a)39 b(priori)f(whic)m(h)i(conditions)f(should)g(b)s(e)h(imp)s (osed)f(on)g(the)h(quasi-p)s(erio)s(dic)e(p)s(oten)m(tial)g(\(or)h (more)g(generally)-118 459 y(b)s(ounded\))33 b(so)g(that)f(suc)m(h)i(a) e(natural)f(extension)i(exists)g(\(and)g(has)f(useful)h(prop)s (erties\))f(and)h(ho)m(w)g(to)f(p)s(erform)-118 580 y(this)g (extension.)44 b(This)33 b(is)f(the)h(purp)s(ose)h(of)e(the)h(follo)m (wing)d(section.)-118 865 y Fg(3.2.2)136 b(Some)45 b(sp)t(ectral)h 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b(for)h(our)g(purp)s(oses.)48 b(It)33 b(is)g(imp)s(ortan) m(t)e(to)i(note)h(that)f(the)h(initial)29 b(op)s(erator)k(is)g (de\014ned)i(b)m(y)f(b)s(oth)f(the)-118 2695 y(action)f(in)f(\(3.7\))h (and)h(the)g(domain)e Fs(D)s Fy(\()p Fx(H)8 b Fy(\).)28 2815 y(The)30 b FA(gr)-5 b(aph)36 b Fy(of)29 b(an)g(op)s(erator)f Fx(H)36 b Fy(as)30 b(ab)s(o)m(v)m(e)g(will)c(b)s(e)k(the)f(follo)m (wing)d(subset)31 b(of)e Fs(H)16 b(\002)f(H)30 b Fy(\(whic)m(h)g(is)e (naturally)-118 2936 y(a)k(Hilb)s(ert)f(space\))1287 3056 y Fx(g)t(r)s Fy(\()p Fx(H)8 b Fy(\))26 b(=)h Fs(f)p Fy(\()p Fx(f)5 b(;)17 b(H)8 b(f)j Fy(\);)17 b Fx(f)37 b Fs(2)28 b(D)s Fy(\()p Fx(H)8 b Fy(\))p Fs(g)16 b Fx(;)-118 3221 y Fy(and)34 b(the)h(op)s(erator)e Fx(H)42 b Fy(is)33 b(said)h(to)g(b)s(e)g FA(close)-5 b(d)43 b Fy(if)33 b(its)h(graph)g Fx(g)t(r)s Fy(\()p Fx(H)8 b Fy(\))32 b(is)i(a)g(closed)g(subset)i(of)d Fs(H)24 b(\002)g(H)q Fy(.)48 b(Another)-118 3341 y(op)s(erator)32 b Fx(H)356 3356 y FC(1)429 3341 y Fy(on)h Fs(H)h 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35 39 35 38 bop 28 219 a Fy(W)-8 b(e)39 b(can)f(naturally)e(de\014ne)k(the)e (op)s(erator)g Fx(z)t(H)46 b Fy(on)38 b Fs(H)h Fy(for)e(an)m(y)i(op)s (erator)f Fx(H)45 b Fy(on)38 b Fs(H)h Fy(and)f Fx(z)k Fs(2)37 b Fw(C)64 b Fy(b)m(y)39 b(the)-118 339 y(de\014nitions)28 b Fs(D)s Fy(\()p Fx(z)t(H)8 b Fy(\))27 b(=)h Fs(D)s Fy(\()p Fx(H)8 b Fy(\))28 b(and)h(\()p Fx(z)t(H)8 b Fy(\))p Fx(f)38 b Fy(=)28 b Fx(z)t Fy(\()p Fx(H)8 b(f)j Fy(\))28 b(for)g(all)f Fx(f)38 b Fs(2)28 b(D)s Fy(\()p Fx(H)8 b Fy(\).)42 b(It)28 b(is)h(also)e(c)m(hec)m(k)m(ed)32 b(that)d(\()p Fx(z)t(H)8 b Fy(\))3873 303 y Ft(\003)3940 339 y Fy(=)-112 459 y(\026)-55 b Fx(z)5 b(H)21 423 y Ft(\003)60 459 y Fy(.)43 b(If)31 b(w)m(e)i(w)m(an)m(t)g(to)e(de\014ne)i(the)f(sum)g(of)f(t)m(w)m(o)h(op) s(erators)g(on)f Fs(H)i Fy(w)m(e)g(m)m(ust)f(b)s(e)g(a)f(bit)g(more)g (careful,)g(b)s(ecause)-118 580 y(it)h(migh)m(t)f(happ)s(en)j(that)f Fs(D)s Fy(\()p Fx(H)8 b Fy(\))21 b Fs(\\)i(D)s Fy(\()p Fx(K)7 b Fy(\))28 b(=)h(0)k(ev)m(en)h(if)e Fs(D)s Fy(\()p Fx(H)8 b Fy(\))32 b(and)h Fs(D)s Fy(\()p Fx(K)7 b Fy(\))33 b(are)g(the)g(domains)f(of)h(t)m(w)m(o)h(densely)-118 700 y(de\014ned)29 b(op)s(erators.)41 b(An)m(yw)m(a)m(y)-8 b(,)31 b(if)c(w)m(e)h(de\014ne)h Fs(D)s Fy(\()p Fx(H)19 b Fy(+)12 b Fx(K)7 b Fy(\))27 b(=)g Fs(D)s Fy(\()p Fx(H)8 b Fy(\))k Fs(\\)g(D)s Fy(\()p Fx(K)7 b Fy(\))26 b(and)i(\()p Fx(H)19 b Fy(+)12 b Fx(K)7 b Fy(\))p Fx(f)38 b Fy(=)28 b(\()p Fx(H)8 b(f)j Fy(\))h(+)g(\()p Fx(K)7 b(f)k Fy(\))-118 820 y(w)m(e)46 b(can)f(sp)s(eak)h(of)f(the)g(op)s(erator)g Fx(H)38 b Fy(+)30 b Fx(K)52 b Fy(ev)m(en)47 b(if)d(it)g(is)g(not)h (densely)h(de\014ned.)83 b(W)-8 b(e)45 b(alw)m(a)m(ys)h(ha)m(v)m(e)g (that)-118 941 y(\()p Fx(H)29 b Fy(+)21 b Fx(K)7 b Fy(\))255 905 y Ft(\003)322 941 y Fy(=)28 b Fx(H)515 905 y Ft(\003)575 941 y Fy(+)21 b Fx(K)762 905 y Ft(\003)834 941 y Fy(if)31 b(either)h Fx(H)39 b Fy(or)32 b Fx(K)39 b Fy(is)32 b(b)s(ounded.)44 b(As)33 b(an)f(imp)s(ortan)m(t)e(example)i(w)m(e)h(ha)m(v)m(e)h(that)e (giv)m(en)-118 1061 y(an)37 b(op)s(erator)g Fx(H)45 b Fy(and)38 b(a)f(complex)g(n)m(um)m(b)s(er)h Fx(z)t Fy(,)i(w)m(e)e(can)g (de\014ne)h(the)f(shifted)f(op)s(erator)g Fx(H)3283 1076 y Fr(z)3359 1061 y Fy(=)f Fx(H)d Fs(\000)26 b Fx(z)t(I)46 b Fy(with)-118 1182 y(domain)31 b Fs(D)s Fy(\()p Fx(H)428 1197 y Fr(z)467 1182 y Fy(\))d(=)f Fs(D)s Fy(\()p Fx(H)8 b Fy(\).)28 1302 y(The)40 b(previous)f(example)f(leads)h(us)g(to)g(the) g(core)g(de\014nitions)f(of)h(the)g(sp)s(ectral)f(theory)i(that)e(w)m (e)i(are)f(in-)-118 1422 y(terested)44 b(in.)73 b(The)44 b(set)f(of)g(complex)f(n)m(um)m(b)s(ers)h Fx(z)48 b Fy(for)42 b(whic)m(h)h(the)g(op)s(erator)f Fx(H)37 b Fs(\000)29 b Fx(z)t(I)52 b Fy(is)42 b(one-to-one)g(from)-118 1543 y Fs(D)s Fy(\()p Fx(H)8 b Fy(\))40 b(on)m(to)h Fs(H)h Fy(\(in)f(whic)m(h)g(case)h(the)g(op)s(erator)e(\()p Fx(H)c Fs(\000)28 b Fx(z)t(I)8 b Fy(\))2160 1507 y Ft(\000)p FC(1)2296 1543 y Fy(is)41 b(b)s(ounded)h(b)m(y)g(the)f(Closed)h(Graph)e (Theo-)-118 1663 y(rem,)e([73]\))f(is)g(denoted)h(b)m(y)g Fx(\032)p Fy(\()p Fx(H)8 b Fy(\))37 b(and)g(it)f(is)h(called)f(the)i FA(r)-5 b(esolvent)38 b(set)47 b Fy(of)37 b Fx(H)8 b Fy(.)57 b(In)37 b(this)g(case,)j(the)d(op)s(erator)-118 1783 y Fx(R)q Fy(\()p Fx(z)t(;)17 b(H)8 b Fy(\))35 b(=)h(\()p Fx(H)c Fs(\000)26 b Fx(z)t(I)8 b Fy(\))754 1747 y Ft(\000)p FC(1)886 1783 y Fy(is)37 b(called)f(the)i FA(r)-5 b(esolvent)38 b(op)-5 b(er)g(ator)48 b Fy(at)36 b Fx(z)t Fy(.)59 b(The)38 b(set)g Fx(\032)p Fy(\()p Fx(H)8 b Fy(\))35 b Fs(\032)h Fw(C)63 b Fy(is)37 b(op)s(en)g(and)g(the)-118 1904 y(map)g Fx(z)42 b Fs(2)37 b Fx(\032)p Fy(\()p Fx(H)8 b Fy(\))37 b Fs(7!)g Fx(R)q Fy(\()p Fx(z)t(;)17 b(H)8 b Fy(\))36 b Fs(2)i Fx(L)p Fy(\()p Fs(H)q Fy(\))g(is)g(strongly)f(analytic.)59 b(Moreo)m(v)m(er,)41 b(the)d(follo)m(wing)e(iden)m(tit)m(y)-8 b(,)39 b(called)-118 2024 y(the)33 b FA(r)-5 b(esolvent)34 b(identity)p Fy(,)f(is)f(true:)881 2194 y Fx(R)q Fy(\()p Fx(z)1039 2209 y FC(1)1079 2194 y Fx(;)17 b(H)8 b Fy(\))21 b Fs(\000)i Fx(R)q Fy(\()p Fx(z)1529 2209 y FC(2)1569 2194 y Fx(;)17 b(H)8 b Fy(\))26 b(=)i(\()p Fx(z)1953 2209 y FC(1)2015 2194 y Fs(\000)22 b Fx(z)2159 2209 y FC(2)2199 2194 y Fy(\))p Fx(R)q Fy(\()p Fx(z)2395 2209 y FC(1)2435 2194 y Fx(;)17 b(H)8 b Fy(\))p Fx(R)q Fy(\()p Fx(z)2764 2209 y FC(2)2803 2194 y Fx(;)17 b(H)8 b Fy(\))p Fx(;)814 b Fy(\(3.9\))-118 2363 y(for)30 b(all)e Fx(z)207 2378 y FC(1)246 2363 y Fx(;)17 b(z)335 2378 y FC(2)405 2363 y Fy(in)30 b(the)g(resolv)m(en)m(t)i(set)e(of)g Fx(H)8 b Fy(.)42 b(W)-8 b(e)31 b(will)d(come)i(bac)m(k)h(to)f(these)h (notions)f(later)f(on,)i(when)g(fo)s(cusing)-118 2484 y(to)h(our)h(particular)d(examples.)28 2604 y(The)f(complemen)m(t)d Fw(C)32 b Fs(\000)13 b Fx(\032)p Fy(\()p Fx(H)8 b Fy(\))33 b(of)27 b(the)h(resolv)m(en)m(t)h(set)f(is)f(called)g(the)h FA(sp)-5 b(e)g(ctrum)35 b Fy(of)27 b Fx(H)8 b Fy(,)28 b(and)g(will)d(b)s(e)j(denoted)-118 2724 y(b)m(y)i Fx(\033)t Fy(\()p Fx(H)8 b Fy(\).)42 b(The)30 b(op)s(erator)e Fx(H)37 b Fy(is)28 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Fr(A)2209 5001 y Fx(f)5 b(;)17 b(g)t Fs(i)26 b Fy(=)2526 4865 y Fq(Z)2581 5091 y Fr(A)2655 5001 y Fx(f)14 b Fy(\026)-52 b Fx(g)s(d\026)-118 5255 y FA(thus)35 b(we)g(have)f(that)1525 5375 y Fx(\036)p Fy(\()p Fx(E)6 b Fy(\))p Fx(f)38 b Fy(=)27 b Fx(\036f)38 b Fy(=)28 b Fx(M)2268 5390 y Fr(\036)2314 5375 y Fx(f)-118 5544 y FA(wher)-5 b(e)34 b Fx(M)251 5559 y Fr(\036)332 5544 y FA(is)h(the)g(multiplic)-5 b(ation)34 b(op)-5 b(er)g(ator)35 b(on)f(the)h(domain)f Fs(D)2325 5559 y Fr(\036)2406 5544 y FA(given)g(by)961 5812 y Fs(D)1038 5827 y Fr(\036)1112 5812 y Fy(=)1216 5671 y Fq(\032)1290 5812 y Fx(f)39 b Fs(2)28 b(H)q Fy(;)1600 5676 y Fq(Z)1655 5902 y FC(\012)1727 5812 y Fs(j)p Fx(\036)p Fy(\()p Fx(w)s Fy(\))p Fs(j)1990 5771 y FC(2)2027 5812 y Fs(j)p Fx(f)11 b Fy(\()p Fx(w)s Fy(\))p Fs(j)2291 5771 y FC(2)2329 5812 y Fx(\026)p Fy(\()p Fx(dw)s Fy(\))27 b Fx(<)g Fs(1)2818 5671 y Fq(\033)2909 5812 y Fx(:)1900 6155 y Fy(37)p eop %%Page: 38 42 38 41 bop 28 219 a Fy(This)36 b(example)f(sho)m(ws)i(that,)f(when)h (\012)e(is)g(a)h(Borel)e(subset)j(of)e Fw(C)62 b Fy(and)35 b Fs(B)k Fy(is)c(the)h(set)g(of)f(Borel)g(subsets)i(of)-118 339 y(\012,)g(then)g(w)m(e)f(can)g(view)h(the)f(op)s(erator)f Fx(\036)p Fy(\()p Fx(E)6 b Fy(\))36 b(as)g(the)g(function)f Fx(\036)h Fy(of)f(the)h(op)s(erator)g Fx(H)k Fy(=)33 b Fx(\036)3357 354 y FC(0)3397 339 y Fy(\()p Fx(E)6 b Fy(\),)36 b(b)s(eing)f(the)-118 459 y(function)i Fx(\036)327 474 y FC(0)367 459 y Fy(\()p Fx(w)s Fy(\))f(=)h Fx(w)s Fy(,)i(for)e(all)f Fx(w)k Fs(2)d Fy(\012.)60 b(In)39 b(some)f(sense,)j(theorem)d(3.2.4)f(pro)m(vides)i(us)g(with)f(a)f (functional)-118 580 y(calculus)45 b(for)f(the)i(self-adjoin)m(t)e(op)s (erators)h(of)f(the)i(form)e Fx(H)57 b Fy(=)49 b Fx(\036)2433 595 y FC(0)2472 580 y Fy(\()p Fx(E)6 b Fy(\).)82 b(W)-8 b(e)45 b(shall)f(later)g(see)j(that)e(ev)m(ery)-118 700 y(self-adjoin)m(t)31 b(op)s(erator)h(on)g(the)h(real)f(line)f(is)h(of)g (this)g(form.)28 820 y(Numerous)45 b(concepts)g(from)e(classical)f (scalar)i(v)-5 b(alued)43 b(measures)i(remain)e(meaningful)e(for)j(pro) 5 b(jection)-118 941 y(v)-5 b(alued)32 b(measures.)44 b(F)-8 b(or)31 b(example,)h(if)f Fx(E)38 b Fy(is)31 b(a)h(resolution)f (of)g(the)i(iden)m(tit)m(y)f(of)f Fx(H)8 b Fy(,)32 b(w)m(e)h(de\014ne)g (the)g Fx(E)6 b FA(-essential)-118 1061 y(r)-5 b(ange)48 b Fy(of)41 b(a)h(measurable)f(function)g Fx(\036)g Fy(as)h(the)g (complemen)m(t)f(of)g(the)h(largest)f(op)s(en)h(subset)h Fx(U)54 b Fs(\032)44 b Fw(C)67 b Fy(whic)m(h)-118 1182 y(satis\014es)38 b(that)e Fx(E)6 b Fy(\()p Fx(\036)635 1145 y Ft(\000)p FC(1)729 1182 y Fy(\()p Fx(U)k Fy(\)\))36 b(=)f(0.)57 b(The)38 b(function)e Fx(\036)h Fy(is)g(said)f(to)h(b)s(e)g FA(essential)5 b(ly)38 b(b)-5 b(ounde)g(d)47 b Fy(if)36 b(its)g Fx(E)6 b Fy(-essen)m(tial)-118 1302 y(range)38 b(is)h(b)s(ounded,)i(the)e Fx(E)6 b FA(-essential)39 b(supr)-5 b(emum)46 b Fy(of)38 b Fx(\036)g Fy(b)s(eing)g(de\014ned)i (as)f(the)g(suprem)m(um)f(of)h(the)g Fs(j)p Fx(z)t Fs(j)f Fy(for)-118 1422 y Fx(z)j Fy(in)35 b(the)i Fx(E)6 b Fy(-essen)m(tial)35 b(range.)55 b(It)36 b(is)f(v)m(eri\014ed)i(that)f(the)h(sp)s(ectrum)f (of)g(the)h(op)s(erator)e Fx(\036)p Fy(\()p Fx(E)6 b Fy(\),)37 b Fx(\033)t Fy(\()p Fx(\036)p Fy(\()p Fx(E)6 b Fy(\)\),)36 b(for)g(a)-118 1543 y(measurable)c(function)g Fx(\036)g Fy(and)h(a)f(resolution)g(of)g(the)h(iden)m(tit)m(y)-8 b(,)32 b(is)g(nothing)g(but)h(the)g Fx(E)6 b Fy(-essen)m(tial)32 b(range)g(of)g Fx(\036)p Fy(.)-118 1771 y Fv(De\014nition)k(3.2.6)49 b FA(If)38 b Fx(\026)g FA(is)h(a)f(non-ne)-5 b(gative)37 b Fx(\033)t FA(-\014nite)h(me)-5 b(asur)g(e)38 b(on)h Fy(\(\012)p Fx(;)17 b Fs(B)s Fy(\))p FA(,)39 b(and)f(if)h Fx(E)44 b FA(is)39 b(a)f(subr)-5 b(esolution)-118 1891 y(of)42 b(the)g(identity)h(of)f Fs(H)i FA(on)e Fy(\(\012)p Fx(;)17 b Fs(B)s Fy(\))p FA(,)45 b(by)d(the)h Fx(\026)p FA(-essential)e(supp)-5 b(ort)42 b(of)g Fx(E)49 b FA(we)42 b(wil)5 b(l)42 b(r)-5 b(efer)42 b(to)h(any)f(set)g Fx(A)g Fs(2)g(B)-118 2012 y FA(satisfying)36 b(the)h(fol)5 b(lowing)35 b(pr)-5 b(op)g(erty:)48 b(Ther)-5 b(e)36 b(exists)g Fx(A)1943 1976 y Ft(0)1998 2012 y Fs(2)31 b(B)40 b FA(with)d Fx(\026)p Fy(\()p Fx(A)2584 1976 y Ft(0)2607 2012 y Fy(\))30 b(=)h(0)37 b FA(and)f Fx(E)6 b Fy(\(\()p Fx(A)23 b Fs([)h Fx(A)3472 1976 y Ft(0)3495 2012 y Fy(\))3533 1976 y Fr(c)3568 2012 y Fy(\))31 b(=)g(0)36 b FA(such)-118 2132 y(that)f(for)g(al)5 b(l)34 b Fx(A)449 2096 y Ft(0)q(0)520 2132 y Fs(2)28 b(B)s FA(,)35 b(we)f(have)g(that)i Fx(E)6 b Fy(\()p Fx(A)1504 2096 y Ft(00)1568 2132 y Fs(\\)23 b Fx(A)p Fy(\))28 b(=)f(0)35 b FA(if,)f(and)h(only)f(if,)h Fx(\026)p Fy(\()p Fx(A)22 b Fs(\\)g Fx(A)2986 2096 y Ft(0)q(0)3029 2132 y Fy(\))28 b(=)f(0)p FA(.)28 2360 y Fy(Note)43 b(that)f(all)e(the)j Fx(\026)p Fy(-essen)m(tial)f(supp)s(orts)h(di\013er)f(only)g(b)m(y)h Fx(\026)p Fy(-negligible)38 b(sets)44 b(so)f(w)m(e)g(will)d(talk)i(of)g FA(the)-118 2481 y Fx(\026)p Fy(-essen)m(tial)27 b(supp)s(ort,)i(ev)m (en)g(if)d(the)j(latter)d(is)h(not)h(uniquely)f(de\014ned.)44 b(Note)27 b(also)g(that)h(this)f(supp)s(ort)h(do)s(es)g(not)-118 2601 y(c)m(hange)39 b(when)g(w)m(e)g(replace)f Fx(\026)f Fy(b)m(y)i(an)f(equiv)-5 b(alen)m(t)38 b(measure,)h(\014nite)f(for)f (example.)59 b(Finally)36 b(w)m(e)j(will)c(simply)-118 2722 y(talk)c(ab)s(out)g(the)h(essen)m(tial)g(supp)s(ort,)g(without)f (ev)m(en)i(men)m(tioning)d Fx(\026)p Fy(,)i(whenev)m(er)i(\012)e(is)f (the)h(real)f(line)f(and)i Fx(\026)f Fy(is)-118 2842 y(the)i(Leb)s(esgue)h(measure.)28 2962 y(W)-8 b(e)45 b(no)m(w)g(w)m(an)m(t)g(to)f(further)g(explore)h(the)f(notion)g(of)f (the)i(resolutions)f(of)f(the)i(iden)m(tit)m(y)f(b)m(y)h(using)f(the) -118 3083 y(prop)s(erties)31 b(of)g(scalar)f(v)-5 b(alued)31 b(measures.)44 b(More)31 b(concretely)-8 b(,)32 b(w)m(e)g(will)d(try)j (to)e(\014nd)i(the)f(righ)m(t)g(analogy)f(of)g(the)-118 3203 y(Leb)s(esgue)g(decomp)s(osition)e(of)h(complex-v)-5 b(alued)27 b(measures)j(in)f(the)h(case)g(of)e(resolutions)h(of)g(the)g (iden)m(tit)m(y)-8 b(.)42 b(W)-8 b(e)-118 3324 y(b)s(egin)32 b(with)g(a)g(useful)h(remark.)28 3444 y(Let)e Fx(E)273 3459 y FC(1)344 3444 y Fy(and)g Fx(E)604 3459 y FC(1)675 3444 y Fy(b)s(e)g(t)m(w)m(o)h(subresolutions)f(of)f(the)i(iden)m(tit)m (y)f(of)f Fs(H)i Fy(on)f(\(\012)p Fx(;)17 b Fs(B)s Fy(\),)32 b(and)f(let)g(us)g(assume)h(that)f Fs(H)-118 3564 y Fy(is)36 b(the)g(orthogonal)f(direct)h(sum)g(of)f Fs(H)1335 3579 y FC(1)1409 3564 y Fy(=)e Fx(E)1590 3579 y FC(1)1630 3564 y Fy(\(\012\))p Fs(H)38 b Fy(and)e Fs(H)2175 3579 y FC(2)2248 3564 y Fy(=)e Fx(E)2430 3579 y FC(2)2470 3564 y Fy(\(\012\))p Fs(H)q Fy(.)54 b(Then)38 b(the)e(function)g Fx(E)42 b Fy(de\014ned)-118 3685 y(b)m(y)e Fx(E)6 b Fy(\()p Fx(A)p Fy(\))40 b(=)f Fx(E)478 3700 y FC(1)518 3685 y Fy(\()p Fx(A)p Fy(\))27 b(+)g Fx(E)869 3700 y FC(2)908 3685 y Fy(\()p Fx(A)p Fy(\))40 b(for)f(all)e Fx(A)j Fs(2)g(B)j Fy(is)c(a)g(resolution)g(of)g(the)h(iden)m(tit)m(y)f(of)g Fs(H)i Fy(on)e(\(\012)p Fx(;)17 b Fs(B)s Fy(\))40 b(and)g(b)m(y)-118 3805 y(theorem)32 b(3.2.4,)g(for)g(eac)m(h)i(measurable)e(function)g Fx(\036)g Fy(on)h(\012,)g(and)f(for)g Fx(j)i Fy(=)28 b(1)p Fx(;)17 b Fy(2)31 b(w)m(e)j(ha)m(v)m(e)970 4025 y Fx(E)1042 4040 y Fr(j)1079 4025 y Fy(\(\012\))p Fs(D)s Fy(\()p Fx(\036)p Fy(\()p Fx(E)6 b Fy(\)\))27 b Fs(\032)h(D)s Fy(\()p Fx(\036)p Fy(\()p Fx(E)6 b Fy(\)\))32 b(and)h Fx(\036)p Fy(\()p Fx(E)6 b Fy(\))p Fs(H)2611 4040 y Fr(j)2675 4025 y Fs(\032)28 b(H)2864 4040 y Fr(j)2900 4025 y Fx(;)-118 4245 y Fy(whic)m(h)g(is)e(another)h(w)m(a)m(y)i(of)d(sa)m(ying)i(that)f (the)g(decomp)s(osition)e(reduces)k(the)f(op)s(erator,)g(since)f Fx(\036)p Fy(\()p Fx(E)6 b Fy(\))27 b(comm)m(utes)-118 4365 y(with)32 b(the)h(pro)5 b(jections)33 b Fx(E)846 4380 y Fr(j)883 4365 y Fy(\(\012\))f(on)h(the)g(spaces)h Fs(H)1748 4380 y Fr(j)1817 4365 y Fy(and)698 4585 y Fs(D)s Fy(\()p Fx(\036)p Fy(\()p Fx(E)984 4600 y Fr(j)1019 4585 y Fy(\)\))28 b(=)g Fs(D)s Fy(\()p 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Fx(\026)2028 5281 y Fr(ac)2122 5266 y Fy(+)g Fx(\026)2279 5281 y Fr(sc)2346 5266 y Fx(;)-118 5486 y Fy(b)s(eing)27 5690 y Fs(\017)49 b Fx(\026)185 5705 y Fr(pp)292 5690 y Fy(a)32 b FA(pur)-5 b(e)35 b(p)-5 b(oint)35 b(me)-5 b(asur)g(e)39 b Fy(\(i.e.)k(a)33 b(sum)f(of)g(Dirac)f(p)s(oin)m(t)h(masses\).)27 5893 y Fs(\017)49 b Fx(\026)185 5908 y Fr(ac)289 5893 y Fy(an)33 b FA(absolutely)h(c)-5 b(ontinuous)35 b(me)-5 b(asur)g(e)40 b Fy(with)32 b(resp)s(ect)i(to)e(the)h(Leb)s(esgue)h(measure.)1900 6155 y(38)p eop %%Page: 39 43 39 42 bop 27 219 a Fs(\017)49 b Fx(\026)185 234 y Fr(sc)286 219 y Fy(a)35 b FA(singular)h(c)-5 b(ontinuous)36 b(me)-5 b(asur)g(e)42 b Fy(\()p FA(c)-5 b(ontinuous)42 b Fy(means)35 b(that)f(it)g(has)h(no)f(p)s(oin)m(t)g(masses)i(and)e FA(sin-)126 339 y(gular)43 b Fy(means)33 b(that)f(it)f(is)i(carried)f (b)m(y)h(a)g(set)g(of)f(zero)h(Leb)s(esgue)h(measure\).)-118 542 y(This)42 b(Leb)s(esgue)h(decomp)s(osition)c(of)i(the)h(scalar)f (measures)i Fx(\026)e Fy(suggests)i(that,)h(giv)m(en)d(a)h(resolution)e (of)h(the)-118 663 y(iden)m(tit)m(y)32 b(of)g Fx(H)40 b Fy(on)33 b(\()p Fw(R)5 b Fx(;)17 b Fs(B)823 678 y Fk(R)881 663 y Fy(\),)33 b(sa)m(y)g(E,)g(w)m(e)h(ma)m(y)e(set)126 866 y Fs(H)210 881 y Fr(pp)285 866 y Fy(\()p Fx(E)6 b Fy(\))28 b(=)f Fs(H)654 881 y Fr(pp)757 866 y Fy(=)h Fs(f)p Fx(f)38 b Fs(2)28 b(H)q Fy(;)17 b Fx(E)1292 881 y Fr(f)s(;f)1427 866 y Fy(is)32 b(pure)h(p)s(oin)m(t)f Fs(g)p Fy(.)126 1069 y Fs(H)210 1084 y Fr(ac)282 1069 y Fy(\()p Fx(E)6 b Fy(\))28 b(=)f Fs(H)651 1084 y Fr(ac)751 1069 y Fy(=)h Fs(f)p Fx(f)38 b Fs(2)28 b(H)q Fy(;)17 b Fx(E)1286 1084 y Fr(f)s(;f)1421 1069 y Fy(is)32 b(absolutely)g(con)m (tin)m(uous)h Fs(g)p Fy(.)126 1273 y Fs(H)210 1288 y Fr(sc)277 1273 y Fy(\()p Fx(E)6 b Fy(\))28 b(=)f Fs(H)646 1288 y Fr(sc)742 1273 y Fy(=)g Fs(f)p Fx(f)38 b Fs(2)28 b(H)q Fy(;)17 b Fx(E)1276 1288 y Fr(f)s(;f)1411 1273 y Fy(is)32 b(singular)f(con)m(tin)m(uous)j Fs(g)o Fy(.)-118 1476 y(These)44 b(three)f(subsets)i(are)d(m)m(utually)f(orthogonal)g (closed)i(subspaces)i(that)d(generate)h(the)g(whole)f(Hilb)s(ert)-118 1597 y(space)34 b Fs(H)q Fy(,)1472 1717 y Fs(H)29 b Fy(=)e Fs(H)1772 1732 y Fr(pp)1870 1717 y Fs(\010)22 b(H)2053 1732 y Fr(ac)2148 1717 y Fs(\010)g(H)2331 1732 y Fr(sc)2399 1717 y Fx(:)-118 1891 y Fy(In)41 b(fact)f(w)m(e)h(ha)m(v)m(e)h(that)e (if)f Fx(f)52 b Fs(2)41 b(H)q Fy(,)h(b)s(eing)e Fx(f)52 b Fy(=)40 b Fx(f)1808 1906 y FC(1)1875 1891 y Fy(+)28 b Fx(f)2027 1906 y FC(2)2093 1891 y Fy(+)g Fx(f)2245 1906 y FC(3)2325 1891 y Fy(is)39 b(the)i(decomp)s(osition)e(of)h Fx(f)51 b Fy(in)39 b(the)i(ab)s(o)m(v)m(e)-118 2012 y(direct)35 b(sum,)i(and)e(if)g Fx(E)756 2027 y Fr(f)s(;f)891 2012 y Fy(=)d Fx(\026)1058 2027 y FC(1)1122 2012 y Fy(+)24 b Fx(\026)1281 2027 y FC(2)1344 2012 y Fy(+)g Fx(\026)1503 2027 y FC(3)1578 2012 y Fy(is)35 b(the)h(Leb)s(esgue)h(decomp)s (osition)c(of)i(the)h(measure)g Fx(E)3660 2027 y Fr(f)s(;f)3763 2012 y Fy(,)g(then)-118 2132 y(necessarily)d Fx(E)440 2147 y Fr(f)474 2157 y Fj(j)507 2147 y Fr(;f)561 2157 y Fj(j)624 2132 y Fy(=)28 b Fx(\026)787 2147 y Fr(j)823 2132 y Fy(,)33 b(for)f Fx(j)i Fy(=)27 b(1)p Fx(;)17 b Fy(2)p Fx(;)g Fy(3.)28 2253 y(If)33 b Fx(\013)g Fy(stands)g(for)f (either)g Fx(pp)p Fy(,)h Fx(ac)p Fy(,)f(or)h Fx(sc)p Fy(,)f(w)m(e)h(denote)h(b)m(y)f Fx(\031)2175 2268 y Fr(\013)2257 2253 y Fy(the)g(pro)5 b(jection)32 b(of)g Fs(H)i Fy(on)m(to)e Fs(H)3420 2268 y Fr(\013)3502 2253 y Fy(and)h(if,)e(for)h Fx(f)-118 2373 y Fy(and)h Fx(g)i Fy(in)d Fs(H)i Fy(and)e Fx(A)c Fs(2)g(B)835 2388 y Fk(R)888 2373 y Fy(,)33 b(w)m(e)g(set:)1392 2593 y Fs(h)p Fx(E)1503 2608 y Fr(\013)1552 2593 y Fy(\()p Fx(A)p Fy(\))p Fx(f)5 b(;)17 b(g)t Fs(i)27 b Fy(=)h(\()o Fx(E)2128 2608 y Fr(f)s(;g)2226 2593 y Fy(\))2264 2625 y Fr(\013)2330 2593 y Fy(\()p Fx(A)p Fy(\))p Fx(;)-118 2813 y Fy(where)k(the)g(left)e(hand)h(side)g(is)g(de\014ned)h(as)f(the) h(righ)m(t)e(hand)h(side,)h(then)f(one)h(readily)e(c)m(hec)m(ks)j(that) e(collections)-118 2933 y Fs(f)p Fx(E)4 2948 y Fr(\013)53 2933 y Fy(\()p Fx(A)p Fy(\);)17 b Fx(A)32 b Fs(2)h(B)515 2948 y Fk(R)567 2933 y Fs(g)i Fy(are)g(resolutions)g(of)f(the)i(iden)m (tit)m(y)f(of)g Fx(E)2142 2948 y Fr(\013)2191 2933 y Fy(\()p Fw(R)5 b Fy(\))p Fs(H)39 b Fy(=)32 b Fs(H)2648 2948 y Fr(\013)2698 2933 y Fy(,)k(for)e Fx(\013)f Fy(=)f Fx(pp;)17 b(ac;)g(sc)35 b Fy(resp)s(ectiv)m(ely)-118 3054 y(and)e(that)1495 3174 y Fx(E)h Fy(=)27 b Fx(E)1776 3189 y Fr(pp)1874 3174 y Fy(+)22 b Fx(E)2044 3189 y Fr(ac)2138 3174 y Fy(+)g Fx(E)2308 3189 y Fr(sc)2376 3174 y Fx(:)-118 3348 y(E)-46 3363 y Fr(pp)51 3348 y Fy(\(resp.)41 b Fx(E)402 3363 y Fr(ac)474 3348 y Fy(,)24 b Fx(E)597 3363 y Fr(sc)665 3348 y Fy(\))e(is)f(a)h(pure)g(p)s(oin)m(t)f(\(resp.)41 b(absolutely)21 b(con)m(tin)m(uous,)k(singular)c(con)m(tin)m(uous\))h (subresolution)-118 3469 y(of)32 b(the)h(iden)m(tit)m(y)f(of)g Fs(H)h Fy(in)f(the)h(sense)h(that,)e(as)h Fx(f)43 b Fy(v)-5 b(aries)32 b(in)g Fs(H)q Fy(,)g(all)e(the)j(scalar)f(measures)h Fs(h)p Fx(E)3371 3484 y Fr(pp)3446 3469 y Fy(\()p Fs(\001)p Fy(\))p Fx(f)5 b(;)17 b(f)11 b Fs(i)32 b Fy(\(resp.)-118 3589 y Fs(h)p Fx(E)-7 3604 y Fr(ac)65 3589 y Fy(\()p Fs(\001)p Fy(\))p Fx(f)5 b(;)17 b(f)11 b Fs(i)39 b Fy(and)h Fs(h)p Fx(E)711 3604 y Fr(sc)779 3589 y Fy(\()p Fs(\001)p Fy(\))p Fx(f)5 b(;)17 b(f)11 b Fs(i)p Fy(\))39 b(are)h(pure)h(p)s(oin)m (t)e(\(resp.)67 b(absolutely)39 b(con)m(tin)m(uous,)k(singular)c(con)m (tin)m(uous\).)-118 3710 y(F)-8 b(or)30 b(this)g(reason,)i(the)f(ab)s (o)m(v)m(e)g(decomp)s(osition)e(is)h(called)g(the)h FA(L)-5 b(eb)g(esgue)33 b(de)-5 b(c)g(omp)g(osition)31 b(of)i(the)g(r)-5 b(esolution)33 b(of)-118 3830 y(the)i(identity)g Fx(E)6 b Fy(.)28 3950 y(Consequen)m(tly)-8 b(,)35 b(for)c(eac)m(h)i (measurable)e(function)h Fx(\036)p Fy(,)g(the)g(op)s(erator)g Fx(\036)p Fy(\()p Fx(E)6 b Fy(\))31 b(can)i(b)s(e)f(decomp)s(osed)h(in) e(three)-118 4071 y(parts,)42 b Fx(\036)p Fy(\()p Fx(E)336 4086 y Fr(pp)411 4071 y Fy(\),)g Fx(\036)p Fy(\()p Fx(E)686 4086 y Fr(ac)758 4071 y Fy(\))e(and)h Fx(\036)p Fy(\()p Fx(E)1202 4086 y Fr(sc)1269 4071 y Fy(\).)66 b(In)40 b(the)h(particular)d(case)j(when)g Fx(H)49 b Fy(=)40 b Fx(\036)2942 4086 y FC(0)2981 4071 y Fy(\()p Fx(E)6 b Fy(\),)42 b(w)m(e)f(ha)m(v)m(e)h(three)e(self-)-118 4191 y(adjoin)m(t)23 b(op)s(erators)h Fx(H)708 4206 y Fr(pp)811 4191 y Fy(=)k Fx(\036)973 4206 y FC(0)1012 4191 y Fy(\()p Fx(E)1122 4206 y Fr(pp)1197 4191 y Fy(\),)e Fx(H)1369 4206 y Fr(ac)1469 4191 y Fy(=)h Fx(\036)1630 4206 y FC(0)1670 4191 y Fy(\()p Fx(E)1780 4206 y Fr(ac)1852 4191 y Fy(\),)f(and)e Fx(H)2205 4206 y Fr(sc)2300 4191 y Fy(=)k Fx(\036)2462 4206 y FC(0)2501 4191 y Fy(\()p Fx(E)2611 4206 y Fr(sc)2678 4191 y Fy(\))d(whic)m(h)f(are)h(called)e (the)i FA(pur)-5 b(e)27 b(p)-5 b(oint)-118 4312 y(p)g(art)p Fy(,)40 b FA(absolutely)h(c)-5 b(ontinuous)40 b(p)-5 b(art)49 b Fy(and)39 b FA(singular)h(c)-5 b(ontinuous)40 b(p)-5 b(art)49 b Fy(of)38 b(the)h(op)s(erator)f Fx(H)8 b Fy(.)62 b(Their)39 b(sp)s(ectra)-118 4432 y(are)j(denoted)i(b)m(y)f Fx(\033)633 4447 y Fr(pp)709 4432 y Fy(\()p Fx(H)8 b Fy(\),)44 b Fx(\033)1000 4447 y Fr(ac)1073 4432 y Fy(\()p Fx(H)8 b Fy(\))42 b(and)g Fx(\033)1534 4447 y Fr(sc)1602 4432 y Fy(\()p Fx(H)8 b Fy(\),)45 b(and)d(are)h(called)e(the)i FA(pur)-5 b(e)44 b(p)-5 b(oint)44 b(sp)-5 b(e)g(ctrum)p Fy(,)45 b FA(absolutely)-118 4552 y(c)-5 b(ontinuous)35 b(sp)-5 b(e)g(ctrum)39 b Fy(and)33 b(the)g FA(singular)h(c)-5 b(ontinuous)35 b(sp)-5 b(e)g(ctrum)40 b Fy(of)32 b Fx(H)8 b Fy(.)43 b(Notice)32 b(that)1210 4772 y Fx(\033)t Fy(\()p Fx(H)8 b Fy(\))27 b(=)h Fx(\033)1620 4787 y Fr(pp)1695 4772 y Fy(\()p Fx(H)8 b Fy(\))22 b Fs([)g Fx(\033)2025 4787 y Fr(ac)2098 4772 y Fy(\()p Fx(H)8 b Fy(\))22 b Fs([)g Fx(\033)2428 4787 y Fr(sc)2496 4772 y Fy(\()p Fx(H)8 b Fy(\))p Fx(;)-118 4992 y Fy(but)32 b(these)g(sets)h(need)f (not)f(to)g(b)s(e)h(disjoin)m(t.)42 b(Sometimes)30 b Fx(\033)2073 5007 y Fr(c)2108 4992 y Fy(\()p Fx(H)8 b Fy(\))27 b(=)h Fx(\033)2459 5007 y Fr(ac)2531 4992 y Fy(\()p Fx(h)p Fy(\))20 b Fs([)g Fx(\033)2824 5007 y Fr(sc)2892 4992 y Fy(\()p Fx(H)8 b Fy(\))31 b(will)e(b)s(e)i(referred)i (as)e(the)-118 5113 y FA(c)-5 b(ontinuous)35 b(p)-5 b(art)35 b(of)f(the)h(sp)-5 b(e)g(ctrum)p Fy(.)-118 5341 y Fv(Remark)37 b(3.2.7)49 b FA(In)25 b(many)g(texts)h Fx(\033)1236 5356 y Fr(pp)1312 5341 y Fy(\()p Fx(H)8 b Fy(\))25 b FA(stands)g(for)g(the)h (set)f(of)g(eigenvalues)g(of)g Fx(H)8 b FA(,)27 b(while)e(for)g(us)h (it)f(actual)5 b(ly)-118 5461 y(denotes)36 b(the)g(closur)-5 b(e)36 b(of)h(this)f(set.)50 b(The)36 b(pr)-5 b(oblem)35 b(with)i(de\014ning)e Fx(\033)2435 5476 y Fr(pp)2547 5461 y FA(as)h(the)h(set)g(of)f(eigenvalues)f Fx(eig)t(en)p Fy(\()p Fx(H)8 b Fy(\))-118 5582 y FA(is)34 b(that)i(it)f(is)f(not)h (true)h(in)e(gener)-5 b(al)34 b(that)i Fx(eig)t(en)p Fy(\()p Fx(H)8 b Fy(\))21 b Fs([)i Fx(\033)1981 5597 y Fr(c)2016 5582 y Fy(\()p Fx(H)8 b Fy(\))27 b(=)h Fx(\033)t Fy(\()p Fx(H)8 b Fy(\))p FA(.)1900 6155 y Fy(39)p eop %%Page: 40 44 40 43 bop 28 219 a Fy(Finally)-8 b(,)28 b(w)m(e)j(in)m(tro)s(duce)e (another)h(distinction)e(b)s(et)m(w)m(een)k(the)e(elemen)m(ts)g(of)f (the)h(sp)s(ectrum)g(of)g(an)f(op)s(erator)-118 339 y Fx(H)8 b Fy(.)74 b(The)44 b FA(essential)g(sp)-5 b(e)g(ctrum)50 b Fy(of)43 b(an)g(op)s(erator)f Fx(H)8 b Fy(,)45 b Fx(\033)2001 354 y Fr(ess)2104 339 y Fy(\()p Fx(H)8 b Fy(\))42 b(is)h(de\014ned)h (as)g(the)f(\(closed\))g(subset)i(of)d(the)-118 459 y(sp)s(ectrum)27 b(whose)g(elemen)m(ts)g(are)f(the)h(real)f(n)m(um)m(b)s(ers)h Fx(\025)f Fy(suc)m(h)i(for)e(whic)m(h)h(the)f(rank)h(of)f(the)h(sp)s (ectral)f(pro)5 b(jection)-118 580 y Fx(E)h Fy(\(\()p Fx(\025)24 b Fs(\000)h Fx(\017;)17 b(\025)24 b Fy(+)g Fx(\017)p Fy(\)\))36 b(is)g(in\014nite)e(for)i(all)d Fx(\017)h(>)f Fy(0.)53 b(The)37 b(complemen)m(tary)e(of)g(this)g(set)i (in)e(the)h(sp)s(ectrum,)h(whic)m(h)-118 700 y(w)m(e)d(shall)d(call)g (the)i FA(discr)-5 b(ete)34 b(sp)-5 b(e)g(ctrum)p Fy(,)32 b(will)f(b)s(e)h(denoted)i(b)m(y)f Fx(\033)2268 715 y Fr(disc)2397 700 y Fy(\()p Fx(H)8 b Fy(\))28 820 y(Up)26 b(to)f(no)m(w)h(w)m(e)g(ha)m(v)m(e)h(seen)f(that)f(w)m(e)i(could)d (asso)s(ciate)h(a)g(self-adjoin)m(t)f(op)s(erator)g Fx(\036)3055 835 y FC(0)3095 820 y Fy(\()p Fx(E)6 b Fy(\))25 b(to)g(eac)m(h)h (resolution)-118 941 y(of)36 b(the)i(iden)m(tit)m(y)e Fx(E)43 b Fy(of)37 b Fs(H)h Fy(on)e(\()p Fw(R)5 b Fx(;)17 b Fs(B)1239 956 y Fk(R)1298 941 y Fy(\).)56 b(No)m(w)37 b(w)m(e)h(seek)h(for)d(a)h(con)m(v)m(erse)i(of)e(this)f(result.)57 b(It)37 b(go)s(es)g(under)g(the)-118 1061 y(name)32 b(of)g(the)h FA(sp)-5 b(e)g(ctr)g(al)35 b(the)-5 b(or)g(em)34 b(for)h(self-adjoint)e (op)-5 b(er)g(ators)p Fy(.)-118 1290 y Fv(Theorem)37 b(3.2.8)h(\(Sp)s(ectral)e(Theorem)h(for)h(self-adjoin)m(t)f(op)s (erators,)h([14]\))49 b FA(A)26 b(densely)f(de\014ne)-5 b(d)24 b(op-)-118 1410 y(er)-5 b(ator)25 b Fx(H)33 b FA(on)25 b Fs(H)h FA(is)f(self-adjoint)f(if,)j(and)e(only)g(if,)i(ther) -5 b(e)25 b(exists)g(a)g(r)-5 b(esolution)25 b(of)h(the)f(identity)g (of)g Fs(H)i FA(on)e Fy(\()p Fw(R)5 b Fx(;)17 b Fs(B)3890 1425 y Fk(R)3948 1410 y Fy(\))p FA(,)-118 1530 y(say)35 b Fx(E)6 b FA(,)35 b(such)f(that)h Fx(H)h Fy(=)27 b Fx(\036)895 1545 y FC(0)934 1530 y Fy(\()p Fx(E)6 b Fy(\))p FA(.)28 1759 y Fv(Pro)s(of:)44 b Fy(W)-8 b(e)33 b(ha)m(v)m(e)i(already)d(pro)m (v)m(ed)j(one)e(implication.)40 b(The)34 b(pro)s(of)e(for)g(the)i (other)f(one)g(go)s(es)g(as)g(follo)m(ws.)-118 1879 y(F)-8 b(or)32 b(eac)m(h)h Fx(f)39 b Fs(2)28 b(H)q Fy(,)k(the)h(function)1358 1999 y(\010)1428 2014 y Fr(f)1474 1999 y Fy(\()p Fx(z)t Fy(\))28 b(=)g Fs(\000h)p Fy(\()p Fx(H)i Fs(\000)22 b Fx(z)t(I)8 b Fy(\))2234 1951 y Ft(\000)p FC(1)2345 1999 y Fx(f)d(;)17 b(f)11 b Fs(i)-118 2174 y Fy(is)35 b(de\014ned)j(and)e (analytic)e(outside)i(the)g(sp)s(ectrum)h(of)e Fx(H)44 b Fy(\(whic)m(h)36 b(is)f(a)h(closed)g(subset)i(of)d(the)h(real)f (line,)h(due)-118 2294 y(to)44 b(the)h(self-adjoin)m(tness)f(of)f Fx(H)8 b Fy(\),)47 b(and)e(in)e(particular)g(it)g(is)h(analytic)f(on)h (\005)2815 2309 y FC(+)2874 2294 y Fy(,)j(the)e(upp)s(er)g(op)s(en)f (complex)-118 2414 y(half-plane.)d(The)34 b(resolv)m(en)m(t)g(iden)m (tit)m(y)e(giv)m(es)878 2649 y Fs(\000)p Fy(2Im)g Fx(z)1220 2565 y Fq(\015)1220 2625 y(\015)1275 2649 y Fy(\()p Fx(H)e Fs(\000)22 b Fx(z)t(I)8 b Fy(\))1662 2601 y Ft(\000)p FC(1)1773 2649 y Fx(f)1832 2565 y Fq(\015)1832 2625 y(\015)1887 2585 y FC(2)1954 2649 y Fy(=)28 b(2)p Fx(i)p Fy(Im)k Fs(h)p Fy(\()o Fx(H)e Fs(\000)23 b Fx(z)t(I)8 b Fy(\))2714 2601 y Ft(\000)p FC(1)2825 2649 y Fx(f)d(;)17 b(f)11 b Fs(i)-118 2870 y Fy(whic)m(h)45 b(sho)m(ws)h(that)f(the)g(imaginary)d (part)j(of)f(\010)1756 2885 y Fr(f)1846 2870 y Fy(is)g(non-negativ)m(e) h(in)f(the)h(upp)s(er)g(half-plane)e(and,)48 b(con-)-118 2990 y(sequen)m(tly)-8 b(,)49 b(it)44 b(is)g(a)g(Herglotz)g(function.) 79 b(The)45 b(represen)m(tation)h(theorem)e(for)g(these)i(functions)e (giv)m(es)h(the)-118 3110 y(existence)34 b(of)e(a)g(\014nite)h (non-negativ)m(e)f(measure)h Fx(\026)1752 3125 y Fr(f)1830 3110 y Fy(on)f Fw(R)44 b Fy(suc)m(h)34 b(that)959 3382 y Fs(h)p Fy(\()p Fx(H)29 b Fs(\000)23 b Fx(z)t(I)8 b Fy(\))1385 3333 y Ft(\000)p FC(1)1495 3382 y Fx(f)d(;)17 b(f)11 b Fs(i)28 b Fy(=)1821 3246 y Fq(Z)1877 3472 y Fk(R)2045 3314 y Fy(1)p 1955 3359 229 4 v 1955 3450 a Fx(\025)22 b Fs(\000)h Fx(z)2194 3382 y(d\026)2304 3397 y Fr(f)2348 3382 y Fy(\()p Fx(\025)p Fy(\))p Fx(;)100 b(z)32 b Fs(2)c Fy(\005)2852 3397 y FC(+)2912 3382 y Fx(:)-118 3649 y Fy(No)m(w,)33 b(for)f Fx(f)43 b Fy(and)33 b Fx(g)j Fy(in)c Fs(H)q Fy(,)g(w)m(e)i(set:)1066 3908 y Fx(\026)1125 3923 y Fr(f)s(;g)1250 3908 y Fy(=)1363 3840 y(1)p 1363 3885 49 4 v 1363 3976 a(4)1438 3908 y(\()p Fx(\026)1535 3923 y Fr(f)7 b FC(+)p Fr(g)1693 3908 y Fs(\000)23 b Fx(\026)1852 3923 y Fr(f)7 b Ft(\000)p Fr(g)2010 3908 y Fy(+)22 b Fx(i)17 b Fy(\()p Fx(\026)2255 3923 y Fr(f)7 b FC(+)p Fr(ig)2437 3908 y Fs(\000)23 b Fx(\026)2596 3923 y Fr(f)7 b Ft(\000)p Fr(ig)2756 3908 y Fy(\)\))-118 4153 y(and)33 b(w)m(e)g(get)937 4316 y Fs(h)p Fy(\()p Fx(H)d Fs(\000)22 b Fx(z)t(I)8 b Fy(\))1363 4268 y Ft(\000)p FC(1)1474 4316 y Fx(f)d(;)17 b(g)t Fs(i)27 b Fy(=)1791 4181 y Fq(Z)1846 4406 y Fk(R)2015 4249 y Fy(1)p 1925 4293 229 4 v 1925 4385 a Fx(\025)22 b Fs(\000)h Fx(z)2164 4316 y(d\026)2274 4331 y Fr(f)s(;g)2370 4316 y Fy(\()p Fx(\025)p Fy(\))p Fx(;)99 b(z)33 b Fs(2)28 b Fy(\005)2874 4331 y FC(+)2933 4316 y Fx(:)-118 4538 y Fy(No)m(w)33 b(the)g(uniqueness)h(part)e(of)g(the)g(represen)m(tation)h(theorem)g (for)e(Herglotz)h(functions)g(giv)m(es)h(that)f(for)g(eac)m(h)-118 4658 y(Borel)37 b(set)i Fx(A)f Fy(in)f Fw(R)49 b Fy(the)38 b(function)f(\()p Fx(f)5 b(;)17 b(g)t Fy(\))36 b Fs(7!)h Fx(\026)1660 4673 y Fr(f)s(;g)1757 4658 y Fy(\()p Fx(A)p Fy(\))g(is)h(linear)e(in)h Fx(f)49 b Fy(and)38 b(an)m(ti-linear)d(in)i Fx(g)t Fy(,)i(and)f(therefore)-118 4779 y(there)g(is)e(a)h(linear)e (map)h Fx(E)6 b Fy(\()p Fx(A)p Fy(\))37 b(from)f Fs(H)i Fy(in)m(to)e(itself)g(suc)m(h)j(that)d Fs(h)p Fx(E)6 b Fy(\()p Fx(A)p Fy(\))p Fx(f)f(;)17 b(g)t Fs(i)34 b Fy(=)h Fx(\026)2981 4794 y Fr(f)s(;g)3078 4779 y Fy(\()p Fx(A)p Fy(\))i(for)g(all)d Fx(f)48 b Fy(and)37 b Fx(g)j Fy(in)-118 4899 y Fs(H)q Fy(.)53 b(The)37 b(map)e Fx(E)42 b Fy(de\014ned)37 b(in)e(this)g(w)m(a)m(y)i(satis\014es)g(that)e Fx(E)6 b Fy(\()p Fx(A)p Fy(\))36 b(is)g(an)f(orthogonal)f(pro)5 b(jection)36 b(and)g(that)f(the)-118 5019 y(collection)30 b Fs(f)p Fx(E)6 b Fy(\()p Fx(A)p Fy(\);)17 b Fx(A)28 b Fs(2)g(B)896 5034 y Fk(R)948 5019 y Fs(g)33 b Fy(is)f(a)g(resolution) g(of)g(the)h(iden)m(tit)m(y)f(of)g Fs(H)i Fy(suc)m(h)g(that)e Fx(\036)3018 5034 y FC(0)3057 5019 y Fy(\()p Fx(E)6 b Fy(\))28 b(=)f Fx(H)8 b Fy(.)43 b Fh(\003)-118 5308 y Fg(3.2.3)136 b(Some)45 b(Sp)t(ectral)g(theory)h(of)f(1D)g(Sc)l(hr\177) -67 b(odinger)45 b(op)t(erators)-118 5493 y Fy(W)-8 b(e)37 b(shall)d(no)m(w)j(apply)f(the)h(de\014nition)e(and)i(results)f(of)g (the)h(previous)f(section)h(\(and)f(in)m(tro)s(duce)g(some)g(more)-118 5613 y(sp)s(eci\014c)d(ones\))g(to)g(the)g(one-dimensional)d(Sc)m (hr\177)-49 b(odinger)32 b(equation,)h(whic)m(h)g(w)m(e)g(recall)1575 5833 y Fs(\000)p Fx(x)1707 5792 y Ft(0)q(0)1772 5833 y Fy(+)22 b Fx(V)g Fy(\()p Fx(t)p Fy(\))p Fx(x)28 b Fy(=)g(0)p Fx(:)1900 6155 y Fy(40)p eop %%Page: 41 45 41 44 bop -118 219 a Fy(In)30 b(our)h(case)g(the)f(p)s(oten)m(tial)f (function)h(is)f Fx(V)22 b Fy(\()p Fx(t)p Fy(\))28 b(=)f Fx(Q)p Fy(\()p Fx(!)t(t)17 b Fy(+)g Fx(\036)p Fy(\),)31 b(but)f(the)h(discussion)f(in)g(this)g(subsection)h(holds)-118 339 y(for)k(m)m(uc)m(h)h(more)f(general)g(functions.)53 b(W)-8 b(e)36 b(reserv)m(e,)i(th)m(us,)g(the)e(notation)e Fx(q)39 b Fy(for)d(a)f(quasi-p)s(erio)s(dic)e(p)s(oten)m(tial)-118 459 y(and)g Fx(V)54 b Fy(for)32 b(a)g(general)g(one-dimensional)e(p)s (oten)m(tial.)42 b(Then)33 b(the)g(op)s(erator)1308 679 y Fx(H)8 b(f)38 b Fy(=)27 b Fx(H)1667 694 y FC(0)1706 679 y Fx(f)33 b Fy(+)22 b Fx(V)g(f)38 b Fy(=)28 b Fs(\000)p Fx(f)2290 638 y Ft(00)2355 679 y Fy(+)22 b Fx(V)f(f)-118 899 y Fy(can)37 b(b)s(e)g(de\014ned)h(on)f Fx(C)759 863 y Ft(1)752 924 y Fr(c)833 899 y Fy(\()p Fw(R)5 b Fy(\))43 b(b)m(y)37 b(means)g(of)f(the)i(usual)e(deriv)-5 b(ativ)m(e.)55 b(This)37 b(op)s(erator)f(is)h(clearly)e(symmetric.)-118 1020 y(Indeed,)266 1283 y Fs(h)p Fx(H)8 b(f)d(;)17 b(g)t Fs(i)26 b Fy(=)711 1147 y Fq(Z)766 1373 y Fk(R)835 1283 y Fy(\()p Fs(\000)p Fx(f)1009 1241 y Ft(00)1052 1283 y Fy(\()p Fx(t)p Fy(\))c(+)g Fx(V)f Fy(\()p Fx(t)p Fy(\))p Fx(f)11 b Fy(\()p Fx(t)p Fy(\)\))17 b Fx(g)t Fy(\()p Fx(t)p Fy(\))p Fx(dt)27 b Fy(=)g Fs(\000)2169 1147 y Fq(Z)2225 1373 y Fk(R)2294 1283 y Fx(f)2353 1241 y Ft(0)o(0)2395 1283 y Fy(\()p Fx(t)p Fy(\))p Fx(g)t Fy(\()p Fx(t)p Fy(\))p Fx(dt)21 b Fy(+)2873 1147 y Fq(Z)2929 1373 y Fk(R)2997 1283 y Fx(V)h Fy(\()p Fx(t)p Fy(\))p Fx(f)11 b Fy(\()p Fx(t)p Fy(\))p Fx(g)t Fy(\()p Fx(t)p Fy(\))p Fx(dt:)-118 1550 y Fy(On)32 b(the)h(other)g(hand)889 1558 y Fq(Z)944 1783 y Fk(R)1013 1694 y Fy(\()p Fx(f)1110 1652 y Ft(00)1152 1694 y Fy(\()p Fx(t)p Fy(\))p Fx(g)t Fy(\()p Fx(t)p Fy(\))22 b Fs(\000)h Fx(f)11 b Fy(\()p Fx(t)p Fy(\))p Fx(g)1768 1652 y Ft(0)o(0)1809 1694 y Fy(\()p Fx(t)p Fy(\)\))28 b(=)2089 1558 y Fq(Z)2145 1783 y Fk(R)2213 1694 y Fy(\()p Fx(f)2310 1652 y Ft(0)2333 1694 y Fx(g)e Fs(\000)c Fx(f)11 b(g)2615 1652 y Ft(0)2638 1694 y Fy(\))2676 1642 y Ft(0)2716 1694 y Fx(dt)27 b Fy(=)h(0)p Fx(;)-118 1915 y Fy(b)s(ecause)34 b(b)s(oth)e Fx(f)43 b Fy(and)33 b Fx(g)j Fy(are)c(of)h(compact)f(supp)s (ort.)28 2035 y(Here)h(w)m(e)f(ha)m(v)m(e)h(c)m(hosen)h Fs(H)f Fy(to)e(b)s(e)h Fx(L)1369 1999 y FC(2)1409 2035 y Fy(\()p Fw(R)t Fx(;)18 b(dt)p Fy(\),)37 b(but)32 b(this)f(c)m(hoice)h (is)g(b)m(y)g(no)g(means)g(irrelev)-5 b(an)m(t.)42 b(Indeed,)33 b(the)-118 2156 y(existence)c(of)d(self-adjoin)m(t)g(extensions)i(and)f (the)h(sp)s(ectrum)g(dep)s(end)g(strongly)f(on)g(the)g(Hilb)s(ert)f (space)i(c)m(hosen.)-118 2276 y(More)40 b(precisely)h(it)e(dep)s(ends)i (on)f(the)g(existence)i(and)e(prop)s(erties)g(of)g(the)g Fx(L)2801 2240 y FC(2)2841 2276 y Fy(-solutions)e(of)i(the)g(eigen)m(v) -5 b(alue)-118 2396 y(equation)1552 2616 y Fs(\000)p Fx(f)1688 2575 y Ft(00)1753 2616 y Fy(+)22 b Fx(V)f Fy(\()p Fx(t)p Fy(\))p Fx(f)39 b Fy(=)27 b Fx(\025f)1431 b Fy(\(3.10\))-118 2836 y(with)37 b Fx(\025)f Fs(2)g Fw(C)20 b Fy(.)64 b(W)-8 b(e)37 b(dev)m(ote)i(this)e(section)h(to)f(in)m(v)m(estigate)g(the)h (existence)h(of)d(these)j(solutions,)f(a)f(kno)m(wledge)-118 2957 y(that)f(will)e(supply)k(us)f(with)f(the)h(essen)m(tial)g (self-adjoin)m(tness)f(of)g(the)h(Sc)m(hr\177)-49 b(odinger)37 b(op)s(erator)f(with)g(b)s(ounded)-118 3077 y(p)s(oten)m(tial.)-118 3337 y Fv(First)g(prop)s(erties.)50 b(The)38 b(limit)33 b(p)s(oin)m(t)k(and)h(the)g(limit)33 b(circle)j(case.)50 b(Essen)m(tial)36 b(self-adjoin)m(tness)-118 3522 y Fy(The)d(\014rst)g (imp)s(ortan)m(t)e(result)i(will)d(b)s(e)j(the)g(follo)m(wing)-118 3750 y Fv(Lemma)k(3.2.9)h(\([14],)f([17]\))49 b FA(If)e(Im)34 b Fx(\025)52 b Fs(6)p Fy(=)g(0)c FA(then)g(at)g(le)-5 b(ast)48 b(one)g(solution)f(of)h(the)g(eigenvalue)f(e)-5 b(quation)-118 3870 y(\(3.10\))48 b(is)h(squar)-5 b(e)48 b(inte)-5 b(gr)g(able)49 b(ne)-5 b(ar)48 b Fy(+)p Fs(1)h FA(and)f(at)h(le)-5 b(ast)49 b(one)g(solution)f(is)h(squar)-5 b(e)49 b(inte)-5 b(gr)g(able)48 b(ne)-5 b(ar)49 b Fs(\0001)p FA(.)-118 3991 y(Mor)-5 b(e)g(over,)44 b(if)f(for)f(some)g Fx(\025)g Fs(2)h Fw(C)69 b FA(two)42 b(line)-5 b(arly)43 b(indep)-5 b(endent)41 b(solutions)h(of)h(\(3.10\))f(ar)-5 b(e)42 b(squar)-5 b(e)43 b(inte)-5 b(gr)g(able)-118 4111 y(ne)g(ar)39 b Fy(+)p Fs(1)g FA(\(r)-5 b(esp.)57 b Fs(\0001)p FA(\),)41 b(then)e(for)g(al)5 b(l)39 b Fx(\025)d Fs(2)g Fw(C)20 b FA(,)47 b(al)5 b(l)39 b(the)g(solutions)g(of)g(\(3.10\))f(ar) -5 b(e)39 b(squar)-5 b(e)40 b(inte)-5 b(gr)g(able)38 b(ne)-5 b(ar)-118 4232 y Fy(+)p Fs(1)34 b FA(\(r)-5 b(esp.)44 b Fs(\0001)p FA(\))28 4460 y Fv(Pro)s(of:)k Fy(The)35 b(pro)s(of)f(of)g(the)h(\014rst)g(statemen)m(t)g(will)e(b)s(e)i(pro)m (v)m(ed)h(b)s(elo)m(w,)f(in)f(the)h(discussion)g(of)f(W)-8 b(eyl's)35 b Fx(m)p Fy(-)-118 4580 y(functions.)45 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Fx(V)44 b Fs(\000)22 b Fx(\025)1896 5230 y FC(0)1936 5075 y Fq(\023)2026 5215 y Fx(')27 b Fy(=)h(\()p Fx(\025)2316 5230 y FC(1)2377 5215 y Fs(\000)23 b Fx(\025)2534 5230 y FC(0)2573 5215 y Fy(\))p Fx(')-118 5487 y Fy(and)33 b(th)m(us,)415 5764 y Fx(')p Fy(\()p Fx(t)p Fy(\))28 b(=)g Fx(c)764 5779 y FC(1)803 5764 y Fx(')867 5779 y FC(1)906 5764 y Fy(\()p Fx(t)p Fy(\))23 b(+)f Fx(c)1180 5779 y FC(2)1219 5764 y Fx(')1283 5779 y FC(2)1322 5764 y Fy(\()p Fx(t)p Fy(\))h(+)f(\()p Fx(\025)1649 5779 y FC(1)1710 5764 y Fs(\000)h Fx(\025)1867 5779 y FC(0)1906 5764 y Fy(\))1961 5629 y Fq(Z)2060 5655 y Fr(t)2016 5854 y(c)2106 5764 y Fy(\()p Fx(')2208 5779 y FC(1)2248 5764 y Fy(\()p Fx(t)p Fy(\))p Fx(')2423 5779 y FC(2)2462 5764 y Fy(\()p Fx(s)p Fy(\))f Fs(\000)h Fx(')2770 5779 y FC(1)2809 5764 y Fy(\()p Fx(s)p Fy(\))p Fx(')2995 5779 y FC(2)3034 5764 y Fy(\()p Fx(t)p Fy(\)\))17 b Fx(')p Fy(\()p Fx(s)p Fy(\))p Fx(ds)1900 6155 y Fy(41)p eop %%Page: 42 46 42 45 bop -118 219 a Fy(for)32 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Fs(k)p Fx(')p Fs(k)1566 1822 y FC([)p Fr(c;t)p FC(])1712 1806 y Fs(\024)e Fy(2)17 b(\()o Fs(j)p Fx(c)1991 1821 y FC(1)2030 1806 y Fs(j)22 b Fy(+)g Fs(j)p Fx(c)2248 1821 y FC(2)2287 1806 y Fs(j)p Fy(\))16 b Fx(M)5 b(;)-118 1981 y Fy(pro)m(vided)27 b(that)f Fx(c)h Fy(is)f(c)m(hosen)i(large)e(enough)h(so)f(that)h(4)p Fx(M)1984 1945 y FC(2)2024 1981 y Fs(j)p Fx(\025)2109 1996 y FC(1)2158 1981 y Fs(\000)10 b Fx(\025)2302 1996 y FC(0)2341 1981 y Fs(j)28 b Fx(<)f Fy(1.)42 b(This)26 b(completes)h(the)g(pro)s(of,)g(lea)m(ving)-118 2101 y(the)33 b(\014rst)g(part)f(for)g(later)g(on.)43 b Fh(\003)28 2222 y Fy(This)33 b(result)f(leads)h(us)g(to)f(an)h(imp)s(ortan)m(t)d (de\014nition:)-118 2450 y Fv(De\014nition)36 b(3.2.10)49 b FA(The)40 b(p)-5 b(otential)40 b(function)g Fx(V)62 b FA(is)40 b(said)g(to)h(b)-5 b(e)40 b(in)g(the)48 b Fy(limit)35 b(circle)j(case)h(at)g(+)p Fs(1)f Fy(\(resp.)-118 2570 y Fs(\0001)p Fy(\))33 b FA(if)f(for)h(some)f(\(and)g(henc)-5 b(e)32 b(for)h(al)5 b(l\))32 b Fx(\025)c 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5754 a Fx(s)1751 5725 y FC(2)1864 5686 y Fx(ds)21 b Fs(\000)i Fy(2)2148 5550 y Fq(Z)2247 5577 y Fr(t)2203 5776 y(t)2228 5785 y Fp(1)2303 5619 y Fx( )2370 5582 y Ft(0)2394 5619 y Fy(\()p Fx(s)p Fy(\))p Fx( )t Fy(\()p Fx(s)p Fy(\))p 2303 5663 401 4 v 2461 5754 a Fx(s)2507 5725 y FC(3)2714 5686 y Fx(ds)k Fs(\024)i Fx(c)2986 5645 y Ft(0)1900 6155 y Fy(42)p eop %%Page: 43 47 43 46 bop -118 219 a Fy(for)27 b(some)h(constan)m(t)h Fx(c)697 182 y Ft(0)748 219 y Fx(>)e Fy(0.)42 b(Using)27 b(Sc)m(h)m(w)m(artz)j(inequalit)m(y)d(to)g(con)m(trol)h(the)g(righ)m(t) f(most)g(term)h(of)f(the)h(previous)-118 339 y(equation)k(w)m(e)i(get) 1225 509 y Fs(\000)1312 441 y Fx( )1379 405 y Ft(0)1402 441 y Fy(\()p Fx(t)p Fy(\))p Fx( )t Fy(\()p Fx(t)p Fy(\))p 1312 486 380 4 v 1464 577 a Fx(t)1499 548 y FC(2)1724 509 y Fy(+)22 b Fx(h)p Fy(\()p Fx(t)p Fy(\))g Fs(\000)h Fx(c)2153 524 y FC(2)2192 419 y Fq(p)p 2292 419 168 4 v 90 x Fx(h)p Fy(\()p Fx(t)p Fy(\))28 b Fs(\024)g Fx(c)2634 524 y FC(1)-118 708 y Fy(if)j(w)m(e)j(set)1515 867 y Fx(h)p Fy(\()p Fx(t)p Fy(\))28 b(=)1813 731 y Fq(Z)1913 758 y Fr(t)1869 957 y(t)1894 966 y Fp(1)1969 800 y Fs(j)p Fx( )2064 763 y Ft(0)2087 800 y Fy(\()p Fx(s)p Fy(\))p Fs(j)2237 763 y FC(2)p 1969 844 307 4 v 2080 935 a Fx(s)2126 907 y FC(2)2286 867 y Fx(ds)-118 1103 y Fy(and)35 b(b)s(eing)g Fx(c)382 1118 y FC(1)457 1103 y Fy(and)h Fx(c)692 1118 y FC(2)767 1103 y Fy(t)m(w)m(o)g(p)s(ositiv)m(e)f(constan)m(ts.)53 b(W)-8 b(e)36 b(no)m(w)g(conclude)g(that)f Fx(h)p Fy(\()p Fx(t)p Fy(\))h(cannot)g(con)m(v)m(erge)h(to)e(+)p Fs(1)-118 1223 y Fy(when)d Fx(t)c Fs(!)g Fy(+)p Fs(1)p Fy(,)j(b)s(ecause)i(this)e (w)m(ould)g(imply)f(that)h Fx( )1942 1187 y Ft(0)1965 1223 y Fy(\()p Fx(t)p Fy(\))p Fx( )t Fy(\()p Fx(t)p Fy(\))d Fs(\025)g Fx(t)2422 1187 y FC(2)2462 1223 y Fx(h)p Fy(\()p Fx(t)p Fy(\))p Fx(=)p Fy(2)j(for)g Fx(t)g Fy(large)g(enough.)43 b(The)33 b(latter)-118 1344 y(w)m(ould)k(mean)f(that)h Fx( )t Fy(\()p Fx(t)p Fy(\))g(and)g Fx( )1127 1308 y Ft(0)1151 1344 y Fy(\()p Fx(t)p Fy(\))g(ha)m(v)m(e)h(the)f(same)g(sign) g(for)f Fx(t)h Fy(large,)g(and)g(this)g(is)g(a)f(con)m(tradiction)g (with)-118 1464 y(the)d(square)h(in)m(tegrabilit)m(y)c(of)i Fx( )t Fy(.)43 b Fh(\003)p Fy(.)28 1585 y(As)33 b(an)g(imp)s(ortan)m(t) d(corollary)h(w)m(e)j(ha)m(v)m(e)-118 1813 y Fv(Corollary)i(3.2.12)49 b FA(Assumption)41 b(\(3.11\))f(implies)g(that)h(for)g(any)g Fx(\025)e Fs(2)h Fw(C)67 b FA(one)40 b(c)-5 b(annot)41 b(have)f(two)h(line)-5 b(arly)-118 1933 y(indep)g(endent)33 b(solutions)i(which)f(ar)-5 b(e)34 b(b)-5 b(oth)35 b(squar)-5 b(e)35 b(inte)-5 b(gr)g(able)34 b(ne)-5 b(ar)34 b Fy(+)p Fs(1)h FA(\(r)-5 b(esp.)44 b Fs(\0001)p FA(\).)28 2162 y Fv(Pro)s(of:)f Fy(Let)32 b Fx(')g Fy(and)g Fx( )k Fy(b)s(e)d (linearly)d(indep)s(enden)m(t)j(solutions)e(of)g(\(3.10\).)43 b(W)-8 b(e)32 b(assume)h(that)f(their)f(W)-8 b(ron-)-118 2282 y(skian)32 b(is)h(1.)43 b(Then)1402 2465 y(1)p 1402 2509 49 4 v 1409 2600 a Fx(t)1488 2532 y Fy(=)28 b Fx(')p Fy(\()p Fx(t)p Fy(\))1777 2465 y Fx( )1844 2429 y Ft(0)1867 2465 y Fy(\()p Fx(t)p Fy(\))p 1777 2509 202 4 v 1860 2600 a Fx(t)2010 2532 y Fs(\000)23 b Fx( )t Fy(\()p Fx(t)p Fy(\))2298 2465 y Fx(')2362 2429 y Ft(0)2385 2465 y Fy(\()p Fx(t)p Fy(\))p 2298 2509 199 4 v 2379 2600 a Fx(t)3766 2532 y Fy(\(3.12\))-118 2783 y(and)30 b(if)e(w)m(e)j(assume)f(that)g Fx(')f Fy(and)h Fx( )k Fy(are)c(linearly)d(indep)s(enden)m(t)k (solutions)e(of)g(\(3.10\))g(b)s(oth)h(square)h(in)m(tegrable)-118 2903 y(at)i(+)p Fs(1)p Fy(,)h(then)g(this)g(is)f(a)g(con)m(tradiction,) g(b)s(ecause)j(of)d(the)h(lemma,)e(that)i(implies)d(that)i(the)i(righ)m (t)e(hand)h(side)-118 3024 y(\(and)f(therefore)g(also)f(the)h(left)f (one\))h(of)f(equation)g(\(3.12\))g(is)g(in)m(tegrable)g(b)m(y)i(Sc)m (h)m(w)m(artz)g(inequalit)m(y)-8 b(.)43 b(This)33 b(is)f(a)-118 3144 y(con)m(tradiction,)f(since)i(1)p Fx(=t)g Fy(is)f(not)g(in)m (tegrable)g(around)g(+)p Fs(1)p Fy(.)43 b Fh(\003)28 3265 y Fy(This)29 b(last)e(corollary)f(directly)i(implies)e(the)i 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b(or)g(dinacy)34 b(of)h(a)f(solution)h(at)g Fs(\0001)g FA(is)g(similarly)f(de\014ne)-5 b(d.)1900 6155 y Fy(43)p eop %%Page: 44 48 44 47 bop -118 219 a Fv(Green's)38 b(and)g(W)-9 b(eyl's)37 b Fx(m)h Fv(functions)-118 403 y Fy(This)h(subsection)h(is)f(dev)m (oted)i(to)d(the)i(in)m(tro)s(duction)e(of)g(t)m(w)m(o)i(ob)5 b(jects,)42 b(the)e(so-called)e(W)-8 b(eyl's)39 b Fx(m)h Fy(functions)-118 524 y(\(also)d(called)g(W)-8 b(eyl-Titc)m(hmarsc)m (h\))38 b(and)g(Green's)h(functions)f(for)f(the)h(Sc)m(hr\177)-49 b(odinger)39 b(op)s(erator)e Fx(H)45 b Fy(on)38 b(a)g(half)-118 644 y(axis)32 b(or)g(on)h(the)g(whole)f(real)g(line)f Fw(R)5 b Fy(.)28 764 y(Recall)31 b(that)i(w)m(e)g(still)d(m)m(ust)j (pro)m(v)m(e)h(that)e(if)g(Im)g Fx(\025)27 b Fs(6)p Fy(=)h(0,)k(then)h (the)g(equation)1461 977 y Fs(\000)p Fx(x)1593 936 y Ft(0)q(0)1659 977 y Fy(+)22 b(\()p Fx(V)f Fy(\()p Fx(t)p Fy(\))h Fs(\000)h Fx(\025)p Fy(\))p Fx(x)28 b Fy(=)g(0)1329 b(\(3.13\))-118 1189 y(has)24 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Fx(dt)22 b Fy(+)g Fx(W)14 b Fy(\()p Fx(\037;)28 b Fy(\026)-60 b Fx(\037)o Fy(\)\(0\))p Fx(:)-118 4040 y Fy(Since)33 b(b)m(y)g(de\014nition)f(of)g Fx(m)p Fy(,)h Fx(W)14 b Fy(\()p Fx(\037;)j(\037)p Fy(\)\(0\))26 b(=)i Fs(\000)p Fy(2)p Fx(i)p Fy(Im)33 b Fx(m)p Fy(,)f(the)h(last)f (equation)h(reads)1046 4305 y Fx(W)14 b Fy(\()p Fx(\037;)28 b Fy(\026)-60 b Fx(\037)p Fy(\)\()p Fx(b)p Fy(\))28 b(=)f(2)p Fx(i)p Fy(\(Im)32 b Fx(\025)p Fy(\))2023 4169 y Fq(Z)2122 4196 y Fr(b)2078 4395 y FC(0)2173 4305 y Fs(j)p Fx(\037)p Fs(j)2290 4264 y FC(2)2329 4305 y Fx(dt)22 b Fy(+)g(2)p Fx(i)p Fy(Im)32 b Fx(m)-118 4549 y Fy(and)h(the)g(equation)f(for)g(the) h(in)m(terior)e(of)h(the)h(circle)f(b)s(ecomes)1546 4673 y Fq(Z)1646 4699 y Fr(b)1602 4899 y FC(0)1697 4809 y Fs(j)p Fx(\037)p Fs(j)1814 4768 y FC(2)1853 4809 y Fx(dt)27 b(<)2080 4741 y Fy(Im)32 b Fx(m)p 2080 4786 235 4 v 2094 4877 a Fy(Im)g Fx(\025)2324 4809 y(;)-118 5053 y Fy(pro)m(vided)h(Im)f Fx(\025)c Fs(6)p Fy(=)f(0.)43 b(Therefore)34 b(p)s(oin)m(ts)e Fx(m)h Fy(are)g(on)f Fx(C)1960 5068 y Fr(b)2027 5053 y Fy(if,)f(and)i(only)f(if,)1546 5177 y Fq(Z)1646 5203 y Fr(b)1602 5402 y FC(0)1697 5313 y Fs(j)p Fx(\037)p Fs(j)1814 5271 y FC(2)1853 5313 y Fx(dt)27 b Fy(=)2080 5245 y(Im)32 b Fx(m)p 2080 5290 V 2094 5381 a Fy(Im)g Fx(\025)2324 5313 y(;)-118 5557 y Fy(and)h(under)g(the)g(same)f(h)m(yp) s(othesis)i(the)f(radius)f Fx(r)1746 5572 y Fr(b)1813 5557 y Fy(is)g(giv)m(en)h(b)m(y)1511 5749 y(1)p 1496 5793 79 4 v 1496 5885 a Fx(r)1540 5900 y Fr(b)1612 5816 y Fy(=)27 b(2Im)32 b Fx(\025)1987 5681 y Fq(Z)2086 5707 y Fr(b)2042 5906 y FC(0)2137 5816 y Fs(j)p Fx( )t Fs(j)2260 5775 y FC(2)2299 5816 y Fx(dt:)1354 b Fy(\(3.18\))1900 6155 y(45)p eop %%Page: 46 50 46 49 bop -118 219 a Fy(No)m(w)33 b(let)f(0)27 b Fx(<)h(a)g(<)f(b)i(<)e Fy(+)p Fs(1)p Fy(.)43 b(Then)34 b(if)d Fx(m)i Fy(is)f(inside)g(or)g(on) h Fx(C)2188 234 y Fr(b)1337 344 y Fq(Z)1436 371 y Fr(a)1392 570 y FC(0)1494 480 y Fs(j)p Fx(\037)p Fs(j)1611 439 y FC(2)1650 480 y Fx(dt)28 b(<)1867 344 y Fq(Z)1967 371 y Fr(b)1923 570 y FC(0)2018 480 y Fs(j)p Fx(\037)p Fs(j)2135 439 y FC(2)2202 480 y Fs(\024)2317 413 y Fy(Im)k Fx(m)p 2317 457 235 4 v 2331 548 a Fy(Im)g Fx(\025)-118 730 y Fy(whic)m(h)c(implies)e(that)h Fx(m)h Fy(is)g(inside)f Fx(C)1238 745 y Fr(a)1307 730 y Fy(and)h(therefore)g Fx(C)1966 745 y Fr(a)2036 730 y Fy(con)m(tains)f(the)i(in)m(terior)d (of)h Fx(C)3092 745 y Fr(b)3154 730 y Fy(if)g Fx(a)h(<)f(b)p Fy(.)43 b(This)28 b(means)-118 851 y(that)k(k)m(eeping)i Fx(\025)e Fy(\014xed,)i(with)e(Im)g Fx(\025)c(>)f Fy(0,)33 b(and)f(letting)f Fx(b)e Fs(!)e(1)p Fy(,)32 b(the)h(circles)g Fx(C)2852 866 y Fr(b)2918 851 y Fy(con)m(v)m(erge)i(either)d(to)g(a)h (circle)-118 971 y Fx(C)-48 986 y Ft(1)59 971 y Fy(or)f(to)g(a)h(p)s (oin)m(t)e Fx(m)718 986 y Ft(1)793 971 y Fy(.)44 b(W)-8 b(e)33 b(no)m(w)g(discuss)h(these)g(t)m(w)m(o)f(p)s(ossibilities.)28 1091 y(If)41 b(the)g Fx(C)380 1106 y Fr(b)455 1091 y Fy(con)m(v)m(erge)i(to)d(a)h(circle,)h(then)g(its)e(radius)g Fx(r)2091 1106 y Ft(1)2208 1091 y Fy(=)i(lim)14 b Fx(r)2522 1106 y Fr(b)2597 1091 y Fy(is)40 b(strictly)g(p)s(ositiv)m(e)h(and,)i (from)c(the)-118 1212 y(expression)31 b(of)d(the)i(radius)f(\(3.18\),)h (w)m(e)g(get)f(that)h Fx( )h Fs(2)d Fx(L)1980 1176 y FC(2)2020 1212 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\).)41 b(If)48 b(^)-67 b Fx(m)2613 1227 y Ft(1)2717 1212 y Fy(is)29 b(an)m(y)h(p)s(oin)m(t)e(on)i(this)f(limit)d(circle)-118 1332 y Fx(C)-48 1347 y Ft(1)26 1332 y Fy(,)33 b(then,)g(as)g(sho)m(w)m (ed)i(ab)s(o)m(v)m(e,)51 b(^)-67 b Fx(m)1180 1347 y Ft(1)1288 1332 y Fy(is)32 b Fx(C)1456 1347 y Fr(b)1522 1332 y Fy(for)g(an)m(y)i Fx(b)28 b(>)f Fy(0,)33 b(hence)1339 1463 y Fq(Z)1439 1489 y Fr(b)1395 1688 y FC(0)1490 1598 y Fs(j)o Fx(')23 b Fy(+)40 b(^)-67 b Fx(m)1787 1613 y Ft(1)1862 1598 y Fx( )t Fs(j)1956 1550 y FC(2)2012 1598 y Fx(dt)28 b(<)2239 1531 y Fy(Im)51 b(^)-68 b Fx(m)2473 1546 y Ft(1)p 2239 1575 310 4 v 2291 1667 a Fy(Im)32 b Fx(\025)-118 1858 y Fy(whic)m(h,)h(letting)d Fx(b)e Fs(!)g(1)p Fy(,)j(it)g(implies)f (that)i Fx(\036)21 b Fy(+)39 b(^)-67 b Fx(m)1751 1873 y Ft(1)1826 1858 y Fx( )36 b Fy(b)s(elongs)c(to)f Fx(L)2459 1822 y FC(2)2499 1858 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\).)42 b(The)33 b(same)f(argumen)m(t)g(holds)-118 1979 y(if)50 b(^)-68 b Fx(m)56 1994 y Ft(1)164 1979 y Fy(reduces)34 b(to)e(the)h(p)s(oin)m(t)f Fx(m)1138 1994 y Ft(1)1246 1979 y Fy(\(that)g(is,)g(if)g(w)m(e)h(are)g(in)f(the)h (second)h(of)e(the)h(alternativ)m(es\).)28 2099 y(Therefore,)f(if)e(Im) i Fx(\025)27 b Fs(6)p Fy(=)h(0,)i(there)i(alw)m(a)m(ys)f(exists)g(a)f (solution)f(of)h Fx(H)8 b(f)38 b Fy(=)27 b Fx(\025f)41 b Fy(whic)m(h)31 b(is)f(of)g(class)h Fx(L)3605 2063 y FC(2)3645 2099 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\).)-118 2219 y(The)35 b(di\013erence)g(b)s(et)m(w)m(een)h(these)f(cases)g(is)f (that)g(when)h Fx(C)2036 2234 y Fr(b)2100 2219 y Fs(!)30 b Fx(C)2300 2234 y Ft(1)2409 2219 y Fy(con)m(v)m(erge)35 b(to)f(a)g(circle)f(then)i FA(al)5 b(l)44 b Fy(solutions)-118 2340 y(are)33 b(of)f(class)h Fx(L)451 2304 y FC(2)491 2340 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))31 b(for)h(Im)g Fx(\025)c Fs(6)p Fy(=)g(0,)33 b(b)s(ecause)h(b)s(oth)e Fx( )37 b Fy(and)c Fx(')22 b Fy(+)40 b(^)-67 b Fx(m)2613 2355 y Ft(1)2688 2340 y Fx( )37 b Fy(are)32 b(so,)i(and)e(this)h(is)f (the)h(reason)-118 2460 y(for)g(the)h(name)f FA(limit)j(cir)-5 b(cle)35 b(c)-5 b(ase)p Fy(.)46 b(On)34 b(the)g(other)f(hand,)i(when)g Fx(C)2391 2475 y Fr(b)2454 2460 y Fs(!)29 b Fx(m)2668 2475 y Ft(1)2743 2460 y Fy(,)34 b(the)g(result)g(lim)14 b Fx(r)3442 2475 y Fr(b)3506 2460 y Fs(!)29 b Fy(0)k(implies)-118 2581 y(that)d Fx( )j Fy(is)d(not)g(of)f(class)h Fx(L)854 2544 y FC(2)894 2581 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))29 b(and,)h(therefore,)h(in)f(this)f(case)i(there)g(is)e(only)g (one)i(linearly)c(indep)s(enden)m(t)-118 2701 y(solution)f(b)s (elonging)f(to)i Fx(L)862 2665 y FC(2)902 2701 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))26 b(for)g(Im)32 b Fx(\025)c Fs(6)p Fy(=)f(0)g(whic)m(h)h(is)f(precisely)g Fx(\036)11 b Fy(+)g Fx(m)2830 2716 y Ft(1)2905 2701 y Fx( )t Fy(.)42 b(This)28 b(is)e(the)i(justi\014cation)-118 2821 y(for)k(the)h(name)f FA(limit)j(p)-5 b(oint)34 b(c)-5 b(ase)p Fy(.)28 2942 y(Summing)31 b(up,)i(w)m(e)g(ha)m(v)m(e)h(pro)m(v)m(en)g(the)f(follo)m (wing)-118 3151 y Fv(Theorem)k(3.2.15)h(\([17]\))48 b FA(If)42 b(Im)34 b Fx(\025)42 b Fs(6)p Fy(=)g(0)g FA(and)g(if)h Fx(')f FA(and)g Fx( )47 b FA(ar)-5 b(e)42 b(the)h(line)-5 b(arly)42 b(indep)-5 b(endent)41 b(solutions)h(of)-118 3271 y Fx(H)8 b(f)38 b Fy(=)27 b Fx(\025f)40 b FA(satisfying)29 b(the)h(c)-5 b(onditions)28 b(\(3.14\),)h(then)h(the)f(solution)g Fx(\037)f Fy(=)f Fx(')10 b Fy(+)g Fx(m )34 b FA(satis\014es)28 b(the)i(r)-5 b(e)g(al)29 b(b)-5 b(oundary)-118 3392 y(c)g(ondition)34 b(\(3.16\))g(if,)g(and)h(only)f(if,)h Fx(m)g FA(lies)f(on)h(a)f(cir)-5 b(cle)35 b Fx(C)2101 3407 y Fr(b)2170 3392 y FA(in)f(the)h(c)-5 b(omplex)34 b(plane)g(whose)g(e)-5 b(quation)34 b(is)1613 3595 y Fx(W)14 b Fy(\()p Fx(\037;)28 b Fy(\026)-60 b Fx(\037)o Fy(\)\()p Fx(b)p Fy(\))28 b(=)g(0)p Fx(:)-118 3798 y FA(As)49 b Fx(b)55 b Fs(!)e(1)c FA(either)g Fx(C)803 3813 y Fr(b)891 3798 y Fs(!)54 b Fx(C)1115 3813 y Ft(1)1189 3798 y FA(,)f(a)c(limit)f(cir)-5 b(cle,)52 b(or)d Fx(C)2134 3813 y Fr(b)2222 3798 y Fs(!)54 b Fx(m)2461 3813 y Ft(1)2536 3798 y FA(,)f(a)48 b(limit)h(p)-5 b(oint.)87 b(A)n(l)5 b(l)49 b(the)g(solutions)-118 3918 y(of)f Fx(H)8 b(f)63 b Fy(=)52 b Fx(\025f)59 b FA(b)-5 b(elong)47 b(to)i Fx(L)1006 3882 y FC(2)1046 3918 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))47 b FA(in)h(the)g(former)g(c)-5 b(ase,)51 b(and)d(if)g(Im)34 b Fx(\025)52 b Fs(6)p Fy(=)h(0)p FA(,)e(exactly)d(one)g(line)-5 b(arly)-118 4039 y(indep)g(endent)31 b(solution)i(is)g Fx(L)956 4003 y FC(2)996 4039 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))31 b FA(in)i(the)g(latter)g(c)-5 b(ase.)44 b(Mor)-5 b(e)g(over,)33 b(in)f(the)h(limit-cir)-5 b(cle)32 b(c)-5 b(ase,)33 b(a)f(p)-5 b(oint)33 b(is)-118 4159 y(on)h(the)h(limit)g(cir)-5 b(cle)34 b Fx(C)745 4174 y Ft(1)820 4159 y Fy(\()p Fx(\025)p Fy(\))g FA(if,)h(and)f(only)h(if,)f Fx(W)14 b Fy(\()p Fx(\037;)28 b Fy(\026)-60 b Fx(\037)p Fy(\)\()p Fs(1)p Fy(\))27 b(=)h(0)p FA(.)44 b Fh(\003)p FA(.)28 4368 y Fy(In)31 b(the)g(limit-p)s(oin)m(t)26 b(case,)32 b(if)d Fx(m)i Fy(is)f(an)m(y)h(p)s(oin)m(t)f(on)g Fx(C)1969 4383 y Fr(b)2004 4368 y Fy(,)g(then)i Fx(m)c Fs(!)f Fx(m)2607 4383 y Ft(1)2682 4368 y Fy(,)k(the)g(limit)c(p)s(oin)m (t,)j(and)h(this)f(holds)-118 4489 y(indep)s(enden)m(tly)39 b(of)f(the)h(c)m(hoice)g(of)f Fx(\014)44 b Fy(in)38 b(the)h(b)s (oundary)g(condition)e(\(3.16\).)61 b(In)39 b(particular,)f(this)g (will)e(hold)-118 4609 y(when)e Fx(\014)f Fy(=)27 b(0,)33 b(and)f(th)m(us)i(the)f(limit)c(p)s(oin)m(t)j(is)g(giv)m(en)h(b)m(y) 1463 4862 y Fx(m)1548 4877 y Ft(1)1623 4862 y Fy(\()p Fx(\025)p Fy(\))27 b(=)73 b(lim)1887 4925 y Fr(b)p Ft(!)p FC(+)p Ft(1)2141 4794 y Fx(')p Fy(\()p Fx(b;)17 b(\025)p Fy(\))p 2140 4839 285 4 v 2140 4930 a Fx( )t Fy(\()p Fx(b;)g(\025)p Fy(\))-118 5127 y Fv(De\014nition)36 b(3.2.16)49 b FA(We)37 b(c)-5 b(al)5 b(l)36 b(Weyl's)h Fx(m)p FA(-function,)g Fx(m)31 b Fy(=)g Fx(m)p Fy(\()p Fx(\025)p Fy(\))p FA(,)37 b(for)f(Im)e Fx(\025)d Fs(6)p Fy(=)g(0)36 b FA(asso)-5 b(ciate)g(d)36 b(to)g(the)h(e)-5 b(qua-)-118 5247 y(tion)35 b Fx(H)8 b(f)38 b Fy(=)27 b Fx(\025f)46 b FA(the)34 b(pr)-5 b(evious)35 b(limit.)28 5456 y Fy(The)f(follo)m(wing)c(result)i(giv)m (es)h(the)g(analyticit)m(y)e(of)h(the)h Fx(m)p Fy(-function)-118 5666 y Fv(Theorem)k(3.2.17)h(\([14],)f([17]\))49 b FA(In)27 b(the)i(limit)f(p)-5 b(oint)27 b(c)-5 b(ase,)29 b(the)f(limit)g(p)-5 b(oint)28 b Fx(m)p Fy(\()p Fx(\025)p Fy(\))g FA(is)g(an)g(analytic)g (function)-118 5786 y(of)37 b Fx(\025)f FA(in)h(the)g(upp)-5 b(er)37 b(half)f(plane)g Fy(\005)1173 5801 y FC(+)1270 5786 y FA(and)g(in)h(the)g(lower)f(half)g(plane)h Fy(\005)2535 5801 y Ft(\000)2594 5786 y FA(.)51 b(Im)34 b Fx(m)p Fy(\()p Fx(\025)p Fy(\))e Fx(>)f Fy(0)37 b FA(for)g(Im)d Fx(\025)d(>)h Fy(0)k FA(and)-118 5906 y(if)f Fx(m)p Fy(\()p Fx(\025)p Fy(\))g FA(has)f(zer)-5 b(os)34 b(or)h(p)-5 b(oles)34 b(in)h(the)g(r)-5 b(e)g(al)34 b(axis,)g(the)h(latter)g(ar)-5 b(e)35 b(simpler.)1900 6155 y Fy(46)p eop %%Page: 47 51 47 50 bop 28 219 a Fv(Pro)s(of:)42 b Fy(F)-8 b(rom)28 b(the)h(expression)i(of)e(the)h(radius)f(and)g(the)h(cen)m(ter)h(in)d (terms)i(of)e Fx(')i Fy(and)f Fx( )k Fy(it)c(turns)h(out)f(that)-118 339 y(these)37 b(ob)5 b(jects)37 b(for)e(the)h(circle)f Fx(C)1123 354 y FC(1)1197 339 y Fy(are)h(con)m(tin)m(uous)g(functions)g (of)f Fx(\025)h Fy(whenev)m(er)i(Im)32 b Fx(\025)g(>)h Fy(0.)53 b(Th)m(us,)38 b(since)e Fx(C)3982 354 y Fr(b)-118 459 y Fy(is)h(in)g(the)h(in)m(terior)e(of)h(the)h(disk)g Fx(C)1194 474 y FC(1)1270 459 y Fy(if)f Fx(b)f(>)g Fy(1,)j(it)d(follo)m (ws)h(that)g(if)f Fx(\025)i Fy(is)f(restricted)h(to)f(a)g(compact)h (set)g Fx(K)44 b Fy(in)-118 580 y(\005)-45 595 y FC(+)14 580 y Fy(,)39 b(then)g(the)f(p)s(oin)m(ts)f Fx(m)g Fy(=)f Fx(m)p Fy(\()p Fx(\025;)17 b(b;)g(\014)6 b Fy(\))38 b(on)f Fx(C)1669 595 y Fr(b)1741 580 y Fy(are)h(uniformly)e(b)s(ounded)i(when) h Fx(b)e Fs(!)f(1)p Fy(.)59 b(The)39 b(functions)-118 700 y Fx(m)-33 715 y Fr(b)2 700 y Fy(\()p Fx(\025;)17 b(\014)6 b Fy(\))28 b(=)i Fx(m)p Fy(\()p Fx(\025;)17 b(b;)g(\014)6 b Fy(\),)34 b(as)g(they)g(are)g(meromorphic)e(and)i(b)s (ounded,)h(they)g(are)f(analytic)e(in)h Fx(K)7 b Fy(.)47 b(Hence,)36 b(b)m(y)-118 820 y(Cauc)m(h)m(y's)41 b(theorem)e(they)h (form)e(an)g(equi-con)m(tin)m(uous)i(set)f(on)g Fx(K)7 b Fy(,)41 b(and)e Fx(m)2736 835 y Fr(b)2809 820 y Fy(con)m(v)m(erges)i (uniformly)c(to)i Fx(m)3914 835 y Ft(1)3989 820 y Fy(.)-118 941 y(Being)32 b(the)h(uniform)e(limit)e(of)j(analytic)f(functions,)i Fx(m)1939 956 y Ft(1)2046 941 y Fy(itself)f(is)g(analytic)f(on)h Fx(K)7 b Fy(,)33 b(and)g(hence)h(on)e(\005)3706 956 y FC(+)3765 941 y Fy(.)28 1061 y(Since)k Fx(m)371 1076 y Ft(1)481 1061 y Fy(is)f(in)f(the)i(in)m(terior)e(of)h Fx(C)1400 1076 y Fr(b)1434 1061 y Fy(,)h(it)e(follo)m(ws)g(that)h(Im)d Fx(m)2368 1076 y Ft(1)2476 1061 y Fx(>)g Fy(0)j(on)g(\005)2879 1076 y FC(+)2938 1061 y Fy(.)52 b(This)35 b(also)g(pro)m(v)m(es)i(that) e(if)-118 1182 y Fx(m)-33 1197 y Ft(1)78 1182 y Fy(has)h(zeros)h(or)f (p)s(oles)g(on)g(the)g(real)f(axis,)i(then)g(they)g(are)f(simple)e (\(in)i(the)g(case)h(of)f(p)s(oles\))f(and)h(that)g(the)-118 1302 y(p)s(oles)c(ha)m(v)m(e)i(negativ)m(e)f(residue.)44 b Fh(\003)p Fy(.)28 1422 y(W)-8 b(e)37 b(already)g(kno)m(w)h(that)e(in) h(the)g(cases)h(w)m(e)g(are)f(in)m(terested)h(in,)f(that)f(is,)i(in)e (quasi-p)s(erio)s(dic)f(p)s(oten)m(tials,)-118 1543 y(the)e(imaginary)d (p)s(oin)m(ts)j(Im)f Fx(\025)c Fs(6)p Fy(=)g(0)33 b(are)f(alw)m(a)m(ys) i(in)e(the)h(resolv)m(en)m(t)h(set.)45 b(It)33 b(ma)m(y)g(happ)s(en)g (\(and)g(it)f(fact)h(in)f(our)-118 1663 y(examples)d(it)f(will)f(happ)s (en\))j(that)f(there)g(are)h(op)s(en)f(subsets)i(in)e(the)g(real)f (line)g(whic)m(h)i(b)s(elong)e(to)h(the)h(resolv)m(en)m(t)-118 1783 y(set)k(\(quite)f(t)m(ypically)f(the)i(sp)s(ectrum)g(will)d(b)s(e) j(a)f(Can)m(tor)g(subset)i(of)e(the)h(real)e(line\).)45 b(If)33 b(this)g(is)g(the)h(case,)g(one)-118 1904 y(w)m(ould)28 b(lik)m(e)g(to)h(kno)m(w)g(to)g(what)g(extend)h(are)e(W)-8 b(eyl's)30 b Fx(m)p Fy(-functions)e(extendable)i(through)e(these)i FA(holes)36 b Fy(or)28 b FA(gaps)-118 2024 y Fy(in)k(the)h(sp)s (ectrum.)44 b(The)33 b(follo)m(wing)d(theorem)i(sho)m(ws)j(some)d(ligh) m(t)f(on)h(this)h(problem)-118 2253 y Fv(Theorem)k(3.2.18)h(\([33]\))48 b FA(If)e Fx(\025)k Fs(2)h Fw(R)5 b FA(,)56 b(then)47 b(the)g(eigenvalue)e(e)-5 b(quation)47 b Fx(H)8 b(f)60 b Fy(=)50 b Fx(\025f)58 b FA(has)46 b(a)h(sub)-5 b(or)g(dinate)-118 2373 y(solution)35 b(at)h Fy(+)p Fs(1)g FA(if,)f(and)g(only)h(if,)f (either)h(the)g(b)-5 b(oundary)35 b(limit)h Fx(m)p Fy(\()p Fx(\025)23 b Fy(+)f Fx(i)p Fy(0\))36 b FA(exists)f(as)h(a)f(\014nite)h (r)-5 b(e)g(al)35 b(numb)-5 b(er)-118 2493 y(\(in)45 b(which)g(c)-5 b(ase)45 b Fx(\037)h FA(is)g(sub)-5 b(or)g(dinate)45 b(at)h Fy(+)p Fs(1)p FA(\))f(or)h(if)g Fy(lim)2072 2508 y Fr(\017)p Ft(!)p FC(0+)2282 2493 y Fs(j)p Fx(m)p Fy(\()p Fx(\025)30 b Fy(+)g Fx(i\017)p Fy(\))p Fs(j)49 b Fy(=)e(+)p Fs(1)f FA(\(in)f(which)g(c)-5 b(ase)45 b Fx( )50 b FA(is)-118 2614 y(sub)-5 b(or)g(dinate)34 b(at)h Fy(+)p Fs(1)p FA(\).)28 2842 y Fy(W)-8 b(e)27 b(will)d(come)i(bac)m(k)i(on)e(this)g(question)h (later)e(on,)j(when)f(referring)f(to)g(the)h(case)g(of)f(Sc)m(hr\177) -49 b(odinger)27 b(equation)-118 2962 y(with)32 b(quasi-p)s(erio)s(dic) f(p)s(oten)m(tial.)28 3083 y(In)40 b(the)h(ab)s(o)m(v)m(e)g(discussion) f(of)f(the)i Fx(m)p Fy(-function,)g(the)f(dep)s(endence)i(of)e Fx(m)g Fy(on)g(the)g(endp)s(oin)m(t)g Fx(a)g Fy(and)g(the)-118 3203 y(b)s(oundary)28 b(condition)e Fx(\013)i Fy(w)m(as)h(not)e (considered.)43 b(F)-8 b(rom)26 b(no)m(w)i(on)f(w)m(e)i(will)c (restrict)j(the)g(notation)e Fx(m)3556 3218 y Fr(\013)3606 3203 y Fy(\()p Fx(\025)p Fy(\))h(to)g(the)-118 3324 y(case)36 b Fx(a)d Fy(=)g(0)i(and)h(the)g(notation)e Fx(m)p Fy(\()p Fx(\025)p Fy(\))i(to)f(the)h(case)h Fx(\013)c Fy(=)g Fx(\031)t(=)p Fy(2.)52 b(In)36 b(this)f(case)h(w)m(e)h(ha)m(v)m(e)g (that)e(the)h(b)s(oundary)-118 3444 y(conditions)c(b)s(ecome)1346 3564 y Fx(')p Fy(\()p Fx(a)p Fy(;)17 b Fx(\025)p Fy(\))27 b(=)h(1)p Fx(;)212 b(')2121 3523 y Ft(0)2144 3564 y Fy(\()p Fx(a)p Fy(;)17 b Fx(\025)p Fy(\))27 b(=)h(0)1330 3739 y Fx( )t Fy(\()p Fx(a)p Fy(;)17 b Fx(\025)p Fy(\))27 b(=)g(0)p Fx(;)212 b( )2110 3698 y Ft(0)2133 3739 y Fy(\()p Fx(a)p Fy(;)17 b Fx(\025)p Fy(\))28 b(=)f(1)p Fx(;)-118 3913 y Fy(and)32 b(therefore)h Fx(')p Fy(\()p Fs(\001)p Fx(;)17 b(\025)p Fy(\))31 b(and)h Fx( )t Fy(\()p Fs(\001)p Fx(;)17 b(\025)p Fy(\))31 b(are)h(the)h(normalized)d(solutions)h(at)g Fx(a)p Fy(.)44 b(Therefore)33 b(if,)e(again,)g(w)m(e)i(denote)-118 4033 y(b)m(y)g Fx(\037)g Fy(a)f Fx(L)258 3997 y FC(2)298 4033 y Fy(-solution)f(at)h(+)p Fs(1)p Fy(,)g(one)h(sees)h(that)1597 4303 y Fx(m)p Fy(\()p Fx(a;)17 b(\025)p Fy(\))28 b(=)2052 4235 y Fx(\037)2113 4199 y Ft(0)2136 4235 y Fy(\()p Fx(a)p Fy(\))p 2052 4280 212 4 v 2064 4371 a Fx(\037)p Fy(\()p Fx(a)p Fy(\))2273 4303 y Fx(:)1466 b Fy(\(3.19\))-118 4573 y(If)38 b(w)m(e)i(w)m(an)m(t)g(to)e(reco)m(v)m(er)i(the)f Fx(m)p Fy(-function)f(for)g(the)h(v)-5 b(alues)39 b(of)f Fx(\013)h Fy(at)f Fx(a)h Fy(w)m(e)h(just)f(ha)m(v)m(e)h(to)e(use)h(the) g(follo)m(wing)-118 4694 y(easy)33 b(relation)1270 4863 y Fx(m)1355 4878 y Fr(\013)1405 4863 y Fy(\()p Fx(a;)17 b(\025)p Fy(\))27 b(=)1775 4796 y(sin)16 b Fx(\013)23 b Fy(+)f Fx(m)p Fy(\()p Fx(a)p Fy(;)17 b Fx(\025)p Fy(\)cos)g Fx(\013)p 1774 4840 844 4 v 1774 4932 a Fy(cos)g Fx(\013)23 b Fs(\000)f Fx(m)p Fy(\()p Fx(a;)17 b(\025)p Fy(\)sin)g Fx(\013)-118 5094 y Fy(Moreo)m(v)m(er,)39 b(w)m(e)f(can)f(c)m(hange)h (the)f(p)s(oten)m(tial)e(p)s(erforming)f(a)j(shift)f Fx(t)f Fs(7!)f Fx(V)22 b Fy(\()p Fx(t)j Fy(+)g Fx(s)p Fy(\),)38 b(for)e(a)g(\014xed)i Fx(s)f Fy(to)f(reco)m(v)m(er)-118 5214 y(the)d(v)-5 b(alue)33 b(of)g(the)g(W)-8 b(eyl's)34 b Fx(m)p Fy(-functions)f(for)f(other)h(v)-5 b(alues)33 b(of)g(the)g(initial)d(b)s(oundary)j(condition)f Fx(a)p Fy(.)45 b(Th)m(us)35 b(if,)-118 5334 y(for)e(Im)f Fx(\025)e Fs(6)p Fy(=)f(0,)34 b(w)m(e)h(denote)f(b)m(y)h Fx(\037)p Fy(\()p Fx(t;)17 b(\025)p Fy(\))34 b(the)g(only)f(solution)f(of)i Fx(H)8 b(\037)29 b Fy(=)h Fx(\025\037)j Fy(whic)m(h)i(is)e(square)i(in) m(tegrable)d(near)-118 5455 y(+)p Fs(1)g Fy(and)h(has)g(v)-5 b(alue)32 b(1)g(at)g(zero,)h(then)g(w)m(e)h(can)f(restate)g(the)g (equation)f(\(3.19\))g(b)m(y)1290 5732 y Fx(\037)p Fy(\()p Fx(t;)17 b(\025)p Fy(\))28 b(=)f(exp)1860 5592 y Fq(\022)1933 5596 y(Z)2033 5623 y Fr(s)1988 5822 y FC(0)2086 5732 y Fx(m)p Fy(\()p Fx(s;)17 b(\025)p Fy(\))p Fx(ds)2491 5592 y Fq(\023)2580 5732 y Fx(:)1900 6155 y Fy(47)p eop %%Page: 48 52 48 51 bop -118 219 a Fy(This)36 b(solution)e(implies)g(also)g(that)i (the)g Fx(m)p Fy(-function)f Fx(m)f Fy(=)f Fx(m)p Fy(\()p Fx(t;)17 b(\025)p Fy(\))36 b(is)f(a)h(solution)e(of)h(the)i(follo)m (wing)c(Ricatti)-118 339 y(equation)1283 515 y Fx(d)p 1266 560 86 4 v 1266 651 a(dt)1362 583 y(m)p Fy(\()p Fx(t;)17 b(\025)p Fy(\))22 b(+)g Fx(m)p Fy(\()p Fx(t;)17 b(\025)p Fy(\))2076 542 y FC(2)2143 583 y Fy(=)27 b Fx(V)22 b Fy(\()p Fx(t)p Fy(\))g Fs(\000)h Fx(\025:)1124 b Fy(\(3.20\))-118 808 y(This)30 b(Ricatti)d(equation)j(will)d(b)s(e)j(of)f(fundamen)m (tal)f(imp)s(ortance)g(when)j(sp)s(eaking)f(ab)s(out)f(the)h (reducibilit)m(y)e(on)-118 928 y(the)33 b(resolv)m(en)m(t)h(set)f(in)f (Sc)m(hr\177)-49 b(odinger)33 b(equation)f(with)g(quasi-p)s(erio)s(dic) f(p)s(oten)m(tial.)28 1049 y(W)-8 b(e)33 b(no)m(w)f(in)m(tro)s(duce)g (a)g(useful)g(ob)5 b(ject)33 b(to)e(study)j(the)e(structure)h(of)f(the) g(op)s(erator)g Fx(H)8 b Fy(:)42 b(Green's)33 b(function.)-118 1169 y(T)-8 b(o)33 b(this)f(end,)h(recall)e(that,)i(asso)s(ciated)f(to) g(the)h(b)s(oundary-v)-5 b(alue)32 b(problem)1755 1369 y Fx(H)8 b(x)28 b Fy(=)f Fx(\025x)1399 1569 y Fy(sin)17 b Fx(\013x)p Fy(\()p Fx(a)p Fy(\))23 b Fs(\000)g Fy(cos)17 b Fx(\013)q(x)2168 1528 y Ft(0)2191 1569 y Fy(\()p Fx(a)p Fy(\))28 b(=)g(0)1398 1735 y(cos)18 b Fx(\014)5 b(x)p Fy(\()p Fx(b)p Fy(\))23 b(+)f(sin)16 b Fx(\014)6 b(x)2151 1694 y Ft(0)2175 1735 y Fy(\()p Fx(b)p Fy(\))28 b(=)f(0)p Fx(;)-118 1900 y Fy(and)34 b(for)g(an)m(y)h Fx(\025)e Fy(in)h(the)g(resolv)m(en)m(t)i(set)e(\(that)g(is)g(for)g(an)m(y)g Fx(\025)g Fy(suc)m(h)i(that)e(the)g(op)s(erator)g Fx(H)d Fs(\000)23 b Fx(\025I)42 b Fy(on)34 b(the)h(space)-118 2034 y Fx(L)-52 1998 y FC(2)-12 2034 y Fy(\([)p Fx(a;)17 b(b)p Fy(]\))34 b(has)h(a)f(b)s(ounded)h(in)m(v)m(erse\),)h(there)e(is) g(a)g(Green)g(function,)h Fx(G)2548 1983 y FC([)p Fr(a;b)p FC(])2548 2062 y Fr(\013;\014)2678 2034 y Fy(\()p Fx(s;)17 b(t)p Fy(;)g Fx(\025)p Fy(\),)34 b(whic)m(h)h(is)e(de\014ned)j(as)e (the)-118 2155 y(in)m(tegral)29 b(k)m(ernel)i(of)f(the)h(resolv)m(en)m (t)h(of)e(the)h(op)s(erator)f Fx(H)1950 2170 y Fr(\025)2023 2155 y Fy(=)e Fx(H)d Fs(\000)19 b Fx(\025I)38 b Fy(in)30 b([)p Fx(a;)17 b(b)p Fy(])31 b(with)f(b)s(oundary)h(phases)h Fx(\013)g Fy(and)-118 2289 y Fx(\014)6 b Fy(.)45 b(More)33 b(precisely)-8 b(,)34 b(the)f(op)s(erator)g Fx(H)1337 2238 y FC([)p Fr(a;b)p FC(])1329 2316 y Fr(\013;\014)1500 2289 y Fy(de\014ned)h(b)m(y)g Fx(H)8 b(f)39 b Fy(=)29 b Fs(\000)p Fx(f)2389 2253 y Ft(00)2454 2289 y Fy(+)22 b Fx(V)g(f)43 b Fy(on)34 b(the)f(space)h(of)f(functions)g Fx(f)44 b Fy(on)-118 2409 y([)p Fx(a;)17 b(b)p Fy(])27 b(satisfying)f(the)h(ab)s(o)m(v)m(e)h(b)s(oundary)f(conditions)f(is)h (essen)m(tially)f(self-adjoin)m(t)f(and)i(the)g(resolv)m(en)m(t)h(op)s (erator)-118 2529 y(of)k(its)g(unique)h(self-adjoin)m(t)e(extension)i (has)g(an)g(in)m(tegral)e(k)m(ernel)i(giv)m(en)g(b)m(y)g(the)g(Green's) h(function)600 2841 y Fx(G)677 2790 y FC([)p Fr(a;b)p FC(])677 2869 y Fr(\013;\014)808 2841 y Fy(\()p Fx(s;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))27 b(=)1240 2637 y Fq(8)1240 2726 y(<)1240 2906 y(:)1376 2751 y Fx( )t Fy(\()p Fx(s;)17 b(\025)p Fy(\))1683 2641 y Fq(\020)1741 2751 y Fx(')p Fy(\()p Fx(t;)g(\025)p Fy(\))22 b(+)g Fx(m)2222 2700 y FC([)p Fr(a;b)p FC(])2222 2779 y Fr(\013;\014)2353 2751 y Fy(\()p Fx(\025)p Fy(\))p Fx( )t Fy(\()p Fx(t;)17 b(\025)p Fy(\))2765 2641 y Fq(\021)2912 2751 y Fy(\()p Fx(s)28 b Fs(\024)g Fx(t)p Fy(\))1370 2931 y Fx( )t Fy(\()p Fx(t;)17 b(\025)p Fy(\))1666 2820 y Fq(\020)1725 2931 y Fx(')p Fy(\()p Fx(s;)g(\025)p Fy(\))22 b(+)g Fx(m)2217 2880 y FC([)p Fr(a;b)p FC(])2217 2958 y Fr(\013;\014)2348 2931 y Fy(\()p Fx(\025)p Fy(\))p Fx( )t Fy(\()p Fx(s;)17 b(\025)p Fy(\))2771 2820 y Fq(\021)2913 2931 y Fy(\()p Fx(s)28 b(>)f(t)p Fy(\))3270 2841 y Fx(;)-118 3184 y Fy(and,)f(if)d(w)m(e)i(are)f(in)f(the)i(limit-p)s(oin)m(t)19 b(case,)27 b(w)m(e)e(ma)m(y)f(let)f Fx(b)i Fy(go)e(to)h(+)p Fs(1)f Fy(so)i(that)e Fx(m)2799 3133 y FC([)p Fr(a;b)p FC(])2799 3212 y Fr(\013;\014)2930 3184 y Fy(\()p Fx(\025)p Fy(\))h(con)m(v)m(erges)i(to)e Fx(m)3711 3199 y Fr(\013)3761 3184 y Fy(\()p Fx(a;)17 b(\025)p Fy(\),)-118 3334 y(irresp)s(ectiv)m (ely)33 b(of)f(the)h(p)s(ossible)f(v)-5 b(alues)32 b(of)g(the)h(angle)f Fx(\014)6 b Fy(.)43 b(Then)34 b Fx(G)2412 3283 y FC([)p Fr(a;b)p FC(])2412 3361 y Fr(\013;\014)2575 3334 y Fy(con)m(v)m(erges)h (to)d(the)h(function)592 3607 y Fx(G)669 3566 y FC([)p Fr(a;)p FC(+)p Ft(1)p FC(\))669 3632 y Fr(\013)903 3607 y Fy(\()p Fx(s;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))27 b(=)1335 3467 y Fq(\032)1457 3546 y Fx( )t Fy(\()p Fx(s;)17 b(\025)p Fy(\))g(\()o Fx(')p Fy(\()p Fx(t;)g(\025)p Fy(\))22 b(+)g Fx(m)2282 3561 y Fr(\013)2331 3546 y Fy(\()p Fx(a;)17 b(\025)p Fy(\))p Fx( )t Fy(\()p Fx(t;)g(\025)p Fy(\)\))88 b(\()p Fx(s)28 b Fs(\024)g Fx(t)p Fy(\))1452 3667 y Fx( )t Fy(\()p Fx(t;)17 b(\025)p Fy(\))g(\()o Fx(')p Fy(\()p Fx(s;)g(\025)p Fy(\))k(+)h Fx(m)2276 3682 y Fr(\013)2326 3667 y Fy(\()p Fx(a;)17 b(\025)p Fy(\))p Fx( )t Fy(\()p Fx(s;)g(\025)p Fy(\)\))83 b(\()p Fx(s)28 b(>)f(t)p Fy(\))-118 3860 y(and)43 b(this)g(is)g(also)f(an)h(in)m(tegral)f(k)m(ernel)i(for)e (the)i(resolv)m(en)m(t)g(op)s(erator)f(de\014ned)h(b)m(y)h Fx(H)50 b Fy(on)43 b(the)h(subspace)h(of)-118 3980 y Fx(L)-52 3944 y FC(2)-12 3980 y Fy(\()p Fx(a;)17 b Fy(+)p Fs(1)p Fy(\))31 b(satisfying)h(the)h(b)s(oundary)g(conditions)f(at)g (the)h(left)f(endp)s(oin)m(t.)28 4101 y(Note)38 b(that)f(in)g(b)s(oth)g (cases)i(\()p Fx(b)e(<)f Fy(+)p Fs(1)h Fy(and)h Fx(b)f Fy(=)f(+)p Fs(1)p Fy(\),)i(Green's)g(function)g(v)-5 b(alue)37 b(at)g(\()p Fx(s;)17 b(t)p Fy(\))37 b(with)g Fx(t)g(>)f(s)-118 4221 y Fy(can)i(b)s(e)g(computed)h(b)m(y)g(m)m (ultiplying)34 b(the)39 b(v)-5 b(alue)37 b(of)h(an)m(y)g(solution)f(of) g(the)i(eigen)m(v)-5 b(alue)37 b(equation)h Fx(H)8 b(f)47 b Fy(=)37 b Fx(\025f)-118 4341 y Fy(whic)m(h)42 b(satis\014es)h(the)f (b)s(oundary)g(condition)e(at)h(the)i(left)d(endp)s(oin)m(t)i(of)f(the) h(in)m(terv)-5 b(al)41 b(b)m(y)h(the)h(v)-5 b(alue)41 b(of)g(an)m(y)-118 4462 y(solution)31 b(satisfying)g(the)h(b)s(oundary) h(condition)e(at)h(the)g(righ)m(t)g(endp)s(oin)m(t)g(and)g(dividing)e (the)j(pro)s(duct)f(b)m(y)h(the)-118 4582 y(W)-8 b(ronskian)42 b(of)f(these)i(t)m(w)m(o)g(solutions.)70 b(In)42 b(this)g(in)m (terpretation,)h(the)g(b)s(oundary)f(condition)e(at)i(in\014nit)m(y)f (is)-118 4702 y(understo)s(o)s(d)d(as)f(the)h(square)g(in)m(tegrabilit) m(y)d(of)i(the)h(function.)57 b(This)38 b(last)e(fact)i(is)e(general)h (and)h(this)f(sp)s(ecial)-118 4823 y(form)27 b(of)h(Green's)h(function) f(can)g(also)g(b)s(e)g(used)i(in)d(the)i(case)g(of)f(the)h(Sc)m(hr\177) -49 b(odinger)28 b(op)s(erator)g(in)f(the)i(whole)f(line)-118 4943 y Fw(R)5 b Fy(.)28 5064 y(W)-8 b(e)28 b(shouldn't)g(get)g(the)g (wrong)g(impression)e(that)i(the)g(sp)s(ectrum)g(of)f(the)h(Sc)m (hr\177)-49 b(odinger)28 b(op)s(erator)f(\(ev)m(en)i(in)-118 5184 y(the)g(case)h(of)e(a)g(quasi-p)s(erio)s(dic)f(p)s(oten)m(tial\))g (do)s(es)i(not)f(dep)s(end)i(on)f(whether)h(w)m(e)f(are)g(in)f(the)h (half-line)d(problem)-118 5304 y(or)37 b(in)g(the)h(whole)g(line)e (problem,)i(as)g(this)f(w)m(on't)h(usually)f(b)s(e)h(the)g(case)h(in)e (our)g(problems.)58 b(T)-8 b(o)38 b(distinguish)-118 5425 y(b)s(et)m(w)m(een)28 b(these)e(problems)f(w)m(e)i(will)c(use)k (the)f(sup)s(erscript)g(+)g(\(resp.)42 b Fs(\000)p Fy(\))26 b(to)f(refer)h(to)f(ob)5 b(jects)27 b(in)e(the)h(con)m(text)h(of)-118 5545 y(the)33 b(Sc)m(hr\177)-49 b(odinger)34 b(op)s(erator)e(on)h(the)h (p)s(ositiv)m(e)e(\(resp.)46 b(negativ)m(e\))34 b(half-line,)c(whereas) 35 b(the)e(notation)f(without)-118 5666 y(sup)s(erscripts)i(will)c (refer)j(to)f(the)h(op)s(erator)f(de\014ned)i(on)f(the)g(whole)f(line.) 28 5786 y(F)-8 b(rom)32 b(the)i(follo)m(wing)d(theorems)i(w)m(e)i(will) c(deduce)k(the)f(relationship)d(b)s(et)m(w)m(een)36 b(the)d(sp)s(ectra) h Fx(\033)3587 5750 y FC(+)3647 5786 y Fy(,)f Fx(\033)3766 5750 y Ft(\000)3859 5786 y Fy(and)-118 5906 y Fx(\033)t Fy(,)g(whic)m(h)g(are)f(subsets)j(of)d(the)h(real)f(line.)1900 6155 y(48)p eop %%Page: 49 53 49 52 bop -118 219 a Fv(Theorem)37 b(3.2.19)h(\([14]\))48 b FA(The)29 b(essential)g(supp)-5 b(ort)30 b(of)f(the)h(various)f(p)-5 b(arts)30 b(of)f(the)h(sp)-5 b(e)g(ctr)g(al)29 b(me)-5 b(asur)g(e)29 b Fx(\033)3796 182 y FC(+)3792 243 y Fr(\013)3885 219 y FA(ar)-5 b(e)-118 339 y(given)34 b(by)-33 523 y(\(i\))49 b(ess-supp)34 b Fx(\033)571 486 y FC(+)567 547 y Fr(\013)658 523 y Fy(=)27 b Fw(R)33 b Fs(\000)23 b(f)p Fx(\025)k Fs(2)h Fw(R)5 b Fy(;)58 b FA(a)35 b(sub)-5 b(or)g(dinate)34 b(solution)g(exists,)h(not)g(satisfying)f(the)h(b.c.)44 b(at)35 b(zer)-5 b(o)p Fs(g)-62 719 y FA(\(ii\))48 b(ess-supp)34 b Fx(\033)571 683 y FC(+)567 744 y Fr(\013;ac)732 719 y Fy(=)28 b Fs(f)o Fx(\025)g Fs(2)g Fw(R)5 b Fy(;)58 b FA(no)34 b(sub)-5 b(or)g(dinate)34 b(solution)h(exists)p Fs(g)-92 916 y FA(\(iii\))48 b(ess-supp)34 b Fx(\033)571 880 y FC(+)567 941 y Fr(\013;s)697 916 y Fy(=)27 b Fs(f)p Fx(\025)h Fs(2)g Fw(R)5 b Fy(;)57 b FA(a)35 b(sub)-5 b(or)g(dinate)34 b(solution)h(exists,)f(satisfying)h(the)f(b.c.)45 b(at)35 b(zer)-5 b(o)o Fs(g)-77 1113 y FA(\(iv\))48 b(ess-supp)34 b Fx(\033)571 1077 y FC(+)567 1138 y Fr(\013;sc)727 1113 y Fy(=)28 b Fs(f)p Fx(\025)f Fs(2)h Fw(R)5 b Fy(;)58 b FA(a)34 b(sub)-5 b(or)g(dinate)35 b(solution)f(exists,)h(satisfying)f (the)h(b.c.)44 b(at)35 b(zer)-5 b(o,)136 1272 y(but)35 b(it)g(is)g(not)g(squar)-5 b(e)34 b(inte)-5 b(gr)g(able)34 b(at)58 b Fy(+)22 b Fs(1g)-48 1468 y FA(\(v\))49 b(ess-supp)34 b Fx(\033)571 1432 y FC(+)567 1493 y Fr(\013;pp)735 1468 y Fy(=)28 b Fs(f)o Fx(\025)g Fs(2)g Fw(R)5 b Fy(;)58 b FA(a)34 b(sub)-5 b(or)g(dinate)35 b(solution)f(exists,)h(satisfying)f (the)h(b.c.)44 b(at)35 b(zer)-5 b(o,)136 1627 y(and)34 b(it)h(is)g(squar)-5 b(e)34 b(inte)-5 b(gr)g(able)34 b(at)58 b Fy(+)22 b Fs(1g)-118 1811 y FA(wher)-5 b(e)34 b(b.c.)44 b(states)35 b(for)g(b)-5 b(oundary)35 b(c)-5 b(ondition.)28 2013 y Fy(F)d(or)22 b(the)i(essen)m(tial)f(supp)s(ort)g (of)g(the)g(sp)s(ectrum)h(of)e(the)i(op)s(erator)e(on)h(the)h(whole)f (line)e(w)m(e)k(ha)m(v)m(e)f(the)f(follo)m(wing)-118 2214 y Fv(Theorem)37 b(3.2.20)h(\([14]\))48 b FA(A)n(n)35 b(essential)f(supp)-5 b(ort)35 b(of)f Fx(\033)2061 2229 y Fr(ac)2169 2214 y FA(is)g(given)g(by)h Fx(S)2714 2229 y FC(+)2796 2214 y Fs([)22 b Fx(S)2944 2229 y Ft(\000)3003 2214 y FA(,)35 b(wher)-5 b(e)331 2411 y Fx(S)391 2426 y Ft(\006)478 2411 y Fy(=)27 b Fs(f)p Fx(\025)h Fs(2)g Fw(R)5 b Fy(;)57 b FA(the)35 b(eigenvalue)f(e)-5 b(quation)35 b(has)f(no)g(sub)-5 b(or)g(dinate)35 b(solution)f(at)57 b Fs(\006)23 b(1g)-118 2608 y FA(and)34 b(an)h(essential)f(supp)-5 b(ort)35 b(for)f(the)h(singular)g(c)-5 b(omp)g(onent)33 b(of)i Fx(\033)k FA(is)34 b(given)g(by)44 2805 y Fx(S)104 2820 y Fr(s)169 2805 y Fy(=)27 b Fs(f)p Fx(\025)h Fs(2)g Fw(R)5 b Fy(;)57 b FA(the)35 b(eigenvalue)f(e)-5 b(quation)34 b(has)h(a)f(solution)h(sub)-5 b(or)g(dinate)34 b(b)-5 b(oth)35 b(at)57 b Fy(+)22 b Fs(1)35 b FA(and)56 b Fs(\000)23 b(1g)-118 3091 y Fg(3.2.4)136 b(Some)53 b(sp)t(ectral)h(theory)f(of)h (1D)f(Sc)l(hr\177)-67 b(odinger)53 b(op)t(erators)h(with)f(quasi-)293 3240 y(p)t(erio)t(dic)45 b(p)t(oten)l(tial)-118 3425 y Fy(No)m(w)33 b(w)m(e)h(turn)e(to)h(our)f(case)i(of)e(in)m(terest)1551 3622 y Fs(\000)p Fx(x)1683 3581 y Ft(0)q(0)1748 3622 y Fy(+)22 b Fx(q)t Fy(\()p Fx(t)p Fy(\))p Fx(x)28 b Fy(=)g Fx(\025x;)1436 b Fy(\(3.21\))-118 3819 y(b)s(eing)40 b Fx(q)k Fy(a)d(quasi-p)s(erio)s(dic)d(function.)67 b(W)-8 b(e)41 b(will)d(review)k(the)f(concepts)h(already)e(seen)i(for)e(this)g (particular)-118 3939 y(setup.)76 b(As)43 b(w)m(e)h(will)d(see,)46 b(the)e(particularities)c(of)i(our)h(equation)g(will)e(result)h(imply)g (stronger)h(prop)s(erties)-118 4059 y(of)34 b(the)g(ob)5 b(jects)35 b(already)f(de\014ned.)50 b(As)34 b(w)m(e)h(will)d(use)j (the)g(prop)s(erties)f(of)f(quasi-p)s(erio)s(dicit)m(y)f(of)i(the)g(p)s (oten)m(tial)-118 4180 y Fx(q)t Fy(\()p Fx(t)p Fy(\))27 b(=)h Fx(Q)p Fy(\()p Fx(!)t(t)9 b Fy(+)g Fx(\036)p Fy(\),)28 b(w)m(e)f(will)d(write)i Fx(q)k Fy(instead)d(of)e Fx(V)48 b Fy(in)26 b(order)g(to)g(stress)i(this)e(quasi-p)s(erio)s(dicit)m(y)-8 b(.)39 b(Moreo)m(v)m(er,)29 b(in)-118 4300 y(the)h(sequel)i(w)m(e)f (will)d(assume)i(that)g Fx(Q)h Fy(is)e(con)m(tin)m(uous)i(and)f(that)g (the)h(frequency)h(v)m(ector)f Fx(!)g Fs(2)d Fw(R)3434 4264 y Fr(d)3511 4300 y Fy(is)i(rationally)-118 4421 y(indep)s(enden)m(t,)k(unless)f(otherwise)g(stated.)-118 4677 y Fv(The)c(eigen)m(v)-6 b(alue)29 b(problem)f(on)h(the)g(whole)g (line)e(revisited.)46 b(Characterization)28 b(of)h(the)g(sp)s(ectrum) -118 4862 y Fy(In)i(this)f(subsection,)i(w)m(e)g(plan)e(to)g(giv)m(e)h (a)f(useful)h(c)m(haracterization)e(of)h(the)i(sp)s(ectrum)f(of)f(the)h (op)s(erator)f Fx(H)38 b Fy(on)-118 4982 y(the)h(whole)f(line.)59 b(It)38 b(should)g(b)s(e)g(regarded)h(as)f(a)g(nearly)g(self-con)m (tained)g(accoun)m(t)h(of)f(those)h(results)f(on)g(the)-118 5102 y(eigen)m(v)-5 b(alue)30 b(problem)g(on)h(the)h(whole)e(line)g (whic)m(h)i(will)c(b)s(e)k(hea)m(vily)e(used)i(in)f(the)g(sequel.)44 b(Summing)29 b(up)i(what)-118 5223 y(w)m(e)j(ha)m(v)m(e)f(seen)h(in)e (previous)h(sections)g(w)m(e)h(ha)m(v)m(e)g(the)f(follo)m(wing)-118 5425 y Fv(Theorem)k(3.2.21)49 b FA(Consider)28 b(the)g(Schr\177)-50 b(odinger)27 b(op)-5 b(er)g(ator)28 b(with)h(quasi-p)-5 b(erio)g(dic)27 b(p)-5 b(otential)28 b Fx(q)t FA(,)i(whose)d(exten-) -118 5545 y(sion)36 b Fx(Q)h FA(to)g Fw(T)389 5509 y Fr(d)470 5545 y FA(is)g(c)-5 b(ontinuous)37 b(and)f(whose)g(fr)-5 b(e)g(quency)37 b(ve)-5 b(ctor)36 b(is)h(irr)-5 b(ational.)50 b(L)-5 b(et)37 b Fx(H)45 b FA(b)-5 b(e)36 b(the)h(c)-5 b(orr)g(esp)g(onding)-118 5666 y(self-adjoint)36 b(extension)g(to)h Fx(L)1008 5629 y FC(2)1048 5666 y Fy(\()p Fw(R)5 b Fy(\))p FA(.)57 b(Then,)37 b(for)g Fx(\025)g FA(in)f(the)h(r)-5 b(esolvent)37 b(set,)g Fx(\032)p Fy(\()p Fx(H)8 b Fy(\))p FA(,)38 b(of)e(this)h(op)-5 b(er)g(ator)37 b(\(and,)g(in)-118 5786 y(p)-5 b(articular,)40 b(this)f(is)f(true)i(for)f(those)f Fx(\025)h FA(with)g(Im)34 b Fx(\025)i Fs(6)p Fy(=)f(0)p FA(\))j(ther)-5 b(e)39 b(exist)g(two)g(line)-5 b(arly)39 b(indep)-5 b(endent)37 b(solutions)-118 5906 y Fx(\037)-57 5921 y FC(+)37 5906 y FA(and)d Fx(\037)287 5921 y Ft(\000)381 5906 y FA(which)g(b)-5 b(elong,)34 b(r)-5 b(esp)g(e)g(ctively,)34 b(to)h Fx(L)1701 5870 y FC(2)1741 5906 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))34 b FA(and)g Fx(L)2375 5870 y FC(2)2415 5906 y Fy(\()p Fs(\0001)p Fx(;)17 b Fy(0\))p FA(.)1900 6155 y Fy(49)p eop %%Page: 50 54 50 53 bop 28 219 a Fv(Pro)s(of:)43 b Fy(F)-8 b(or)31 b(Im)h Fx(\025)27 b Fs(6)p Fy(=)h(0)j(this)g(has)h(b)s(een)g(pro)m(v)m (ed)h(in)e(the)h(previous)g(section,)g(in)e(the)i(discussion)g(of)f(W) -8 b(eyl's)-118 339 y Fx(m)p Fy(-functions.)80 b(No)m(w)45 b(assume)g(that)g Fx(\025)f Fy(is)g(real)g(and)h(b)s(elongs)f(to)h(the) g(resolv)m(en)m(t)h(set.)80 b(By)45 b(de\014nition,)i(the)-118 459 y(op)s(erator)27 b(\()p Fx(H)20 b Fs(\000)13 b Fx(\025I)8 b Fy(\))28 b(de\014ned)i(in)d(some)h(dense)h(subset)h(of)d Fx(L)2082 423 y FC(2)2122 459 y Fy(\()p Fw(R)5 b Fy(\),)35 b(and)28 b(therefore)h(the)f(only)f(solution)g(in)g Fx(L)3828 423 y FC(2)3868 459 y Fy(\()p Fw(R)5 b Fy(\))-118 580 y(of)32 b(the)h(eigen)m(v)-5 b(alue)32 b(problem)f(\()p Fx(H)f Fs(\000)23 b Fx(\025I)8 b Fy(\))p Fx(f)38 b Fy(=)27 b(0)33 b(is)f(zero.)28 700 y(W)-8 b(e)33 b(will)d(use)k(theorem)e (2.2.5)g(in)g(a)g(suitable)g(form)m(ulation)e(for)i(our)g(purp)s(oses.) -118 918 y Fv(Theorem)37 b(3.2.22)h(\([78],)f([45]\))49 b FA(Supp)-5 b(ose)48 b(that)i(the)f(\015ow)g(on)g(the)g(c)-5 b(omp)g(act)48 b(sp)-5 b(ac)g(e)48 b Fw(T)3281 882 y Fr(d)3375 918 y FA(is)g(minimal)g(\(in)-118 1038 y(p)-5 b(articular)46 b(this)g(is)h(true)f(is)h(the)f(fr)-5 b(e)g(quency)46 b(ve)-5 b(ctor)46 b Fx(!)k FA(is)c(r)-5 b(ational)5 b(ly)46 b(indep)-5 b(endent)45 b(in)h(the)g(quasi-p)-5 b(erio)g(dic)-118 1159 y(c)g(ase\))34 b(and)g(that)i(the)e(function)h Fx(q)k FA(is)34 b(c)-5 b(ontinuous)35 b(and)f(b)-5 b(ounde)g(d.)44 b(Then)34 b(the)h(system)1086 1298 y Fq(\022)1213 1377 y Fx(x)p Fy(\()p Fx(t)p Fy(\))1201 1498 y Fx(x)1256 1461 y Ft(0)1280 1498 y Fy(\()p Fx(t)p Fy(\))1433 1298 y Fq(\023)1506 1319 y Ft(0)1557 1438 y Fy(=)1660 1298 y Fq(\022)1919 1377 y Fy(0)227 b(1)1775 1498 y Fx(q)t Fy(\()p Fx(t)p Fy(\))22 b Fs(\000)h Fx(\025)83 b Fy(0)2285 1298 y Fq(\023)17 b(\022)2502 1377 y Fx(x)p Fy(\()p Fx(t)p Fy(\))2490 1498 y Fx(x)2545 1461 y Ft(0)2569 1498 y Fy(\()p Fx(t)p Fy(\))2721 1298 y Fq(\023)3766 1438 y Fy(\(3.22\))-118 1707 y FA(admits)30 b(an)g(exp)-5 b(onential)29 b(dichotomy)h(if,)h(and)f(only)g(if,)h(al)5 b(l)30 b(nontrivial)g(solutions)g(of)g(the)h(ab)-5 b(ove)29 b(system)i(\(3.22\))-118 1827 y(ar)-5 b(e)35 b(not)f(b)-5 b(ounde)g(d)34 b(in)h Fw(R)5 b FA(.)28 2045 y Fy(As)37 b(a)f(\014rst)g(consequence,)41 b(for)35 b(Im)d Fx(\025)i Fs(6)p Fy(=)g(0,)i(the)h(system)g(satis\014es)g(an)f(exp)s(onen)m(tial) g(dic)m(hotom)m(y)-8 b(,)36 b(b)s(ecause)-118 2166 y(w)m(e)41 b(sa)m(w)h(that)e(there)i(exist)e(t)m(w)m(o)i(linearly)c(indep)s(enden) m(t)k(solutions,)f(one)g(in)f Fx(L)2874 2129 y FC(2)2914 2166 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))39 b(and)i(the)g(other)f (in)-118 2286 y Fx(L)-52 2250 y FC(2)-12 2286 y Fy(\()p Fs(\0001)p Fx(;)17 b Fy(0\).)79 b(No)m(w)46 b(let)e(Im)32 b Fx(\025)48 b Fy(=)g(0)d(in)f(the)h(resolv)m(en)m(t)h(set.)81 b(W)-8 b(e)45 b(w)m(an)m(t)h(to)e(see)i(that)f(the)g(system)h(has)f(an) -118 2406 y(exp)s(onen)m(tial)37 b(dic)m(hotom)m(y)-8 b(,)39 b(and)f(to)f(this)h(end)g(w)m(e)h(m)m(ust)f(pro)m(v)m(e)i(that)d (the)h(only)g(solution)e(whic)m(h)j(is)e(b)s(ounded)-118 2527 y(is)e(the)h(trivial)c(one.)53 b(Let)35 b(\()p Fx( )t(;)17 b( )1065 2491 y Ft(0)1088 2527 y Fy(\))35 b(b)s(e)h(a)f(solution)f(of)h (the)h(system)g(\(3.22\))e(whic)m(h)i(is)f(b)s(ounded)h(b)m(y)h(a)e (constan)m(t)-118 2647 y Fx(C)49 b Fy(in)42 b Fw(R)5 b Fy(.)78 b(If)42 b(this)g(solution)f(is)h(non-zero,)j(then)e(it)e (cannot)i(b)s(elong)e(to)h Fx(L)2710 2611 y FC(2)2750 2647 y Fy(\()p Fw(R)5 b Fy(\),)50 b(b)s(ecause)44 b(of)e(the)h(resolv)m (en)m(t)-118 2768 y(assumption)32 b(for)g Fx(\025)p Fy(.)43 b(Th)m(us,)34 b(the)f(limit)1441 3040 y(lim)1385 3102 y Fr(T)10 b Ft(!)p FC(+)p Ft(1)1649 2905 y Fq(Z)1748 2931 y Fr(T)1704 3130 y Ft(\000)p Fr(T)1830 3040 y Fs(j)p Fx( )t Fy(\()p Fx(t)p Fy(\))p Fs(j)2064 2992 y FC(2)2120 3040 y Fx(dt)28 b Fy(=)f(+)p Fs(1)-118 3311 y Fy(m)m(ust)j(hold.)42 b(No)m(w)31 b(let)f Fx(g)781 3326 y Fr(n)858 3311 y Fy(a)g(function)g (in)g Fx(C)1506 3275 y Ft(1)1499 3335 y Fr(c)1580 3311 y Fy(\()p Fw(R)5 b Fy(\))37 b(with)30 b(supp)s(ort)g(in)g(the)h(in)m (terv)-5 b(al)29 b([)p Fs(\000)p Fx(n)18 b Fs(\000)h Fy(1)p Fx(;)e(n)g Fy(+)h(1])30 b(and)g(v)-5 b(alue)-118 3431 y(1)32 b(on)h([)p Fs(\000)p Fx(n;)17 b(n)p Fy(].)44 b(Assume,)34 b(moreo)m(v)m(er,)f(that)1279 3642 y(sup)1216 3724 y Fr(t)p Ft(2)p Fk(R)p Fr(;n)p Ft(\025)p FC(1)1506 3642 y Fs(fj)p Fx(g)1631 3657 y Fr(n)1677 3642 y Fy(\()p Fx(t)p Fy(\))p Fs(j)p Fx(;)17 b Fs(j)p Fx(g)1939 3601 y Ft(0)1935 3667 y Fr(n)1981 3642 y Fy(\()p Fx(t)p Fy(\))p Fs(j)p Fx(;)g Fs(j)p Fx(g)2243 3601 y Ft(0)n(0)2239 3667 y Fr(n)2285 3642 y Fy(\()p Fx(t)p Fy(\))p Fs(jg)27 b Fx(<)h(C)-118 3918 y Fy(for)k(a)g(constan)m(t)i Fx(C)583 3882 y Ft(0)606 3918 y Fy(.)43 b(Set)1333 4058 y Fx( )1396 4073 y Fr(n)1443 4058 y Fy(\()p Fx(t)p Fy(\))28 b(=)1896 3991 y(1)p 1695 4035 450 4 v 1695 4127 a Fs(k)p Fx(g)1792 4142 y Fr(n)1839 4127 y Fx( )t Fs(k)1956 4143 y Fr(L)2004 4124 y Fp(2)2038 4143 y FC(\()p Fk(R)p FC(\))2155 4058 y Fx(g)2202 4073 y Fr(n)2249 4058 y Fy(\()p Fx(t)p Fy(\))p Fx( )t Fy(\()p Fx(t)p Fy(\))p Fx(;)-118 4297 y Fy(whic)m(h)33 b(de\014nes)h(a)f(sequence)i(of)d(functions)h(in)e Fx(C)1689 4261 y FC(2)1682 4322 y Fr(c)1729 4297 y Fy(\()p Fw(R)5 b Fy(\))38 b(with)32 b Fx(L)2197 4261 y FC(2)2237 4297 y Fy(\()p Fw(R)5 b Fy(\)-norm)37 b(one.)44 b(No)m(w)33 b(it)f(is)g(satis\014ed)h(that)566 4552 y Fs(\000)p Fx( )710 4511 y Ft(0)q(0)706 4577 y Fr(n)754 4552 y Fy(\()p Fx(t)p Fy(\))23 b(+)f(\()o Fx(q)t Fy(\()p Fx(t)p Fy(\))h Fs(\000)f Fx(\025)p Fy(\))17 b Fx( )1478 4567 y Fr(n)1525 4552 y Fy(\()p Fx(t)p Fy(\))28 b(=)1939 4485 y Fs(\000)p Fy(1)p 1777 4529 V 1777 4621 a Fs(k)p Fx(g)1874 4636 y Fr(n)1921 4621 y Fx( )t Fs(k)2038 4637 y Fr(L)2086 4618 y Fp(2)2120 4637 y FC(\()p Fk(R)p FC(\))2254 4552 y Fy(\()p Fx(g)2343 4511 y Ft(0)o(0)2339 4577 y Fr(n)2385 4552 y Fy(\()p Fx(t)p Fy(\))p Fx( )t Fy(\()p Fx(t)p Fy(\))23 b(+)f(2)p Fx(g)2895 4511 y Ft(0)2891 4577 y Fr(n)2937 4552 y Fy(\()p Fx(t)p Fy(\))p Fx( )3115 4511 y Ft(0)3139 4552 y Fy(\()p Fx(t)p Fy(\)\))16 b Fx(;)-118 4822 y Fy(and)33 b(therefore)1135 4966 y Fs(k\000)p Fx( )1329 4925 y Ft(00)1325 4990 y Fr(n)1395 4966 y Fy(+)22 b(\()o Fx(q)k Fs(\000)d Fx(\025)p Fy(\))17 b Fx( )1874 4981 y Fr(n)1921 4966 y Fs(k)1971 4995 y Fr(L)2019 4976 y Fp(2)2053 4995 y FC(\()p Fk(R)p FC(\))2188 4966 y Fs(\024)2483 4898 y Fx(K)p 2303 4943 V 2303 5034 a Fs(k)p Fx(g)2400 5049 y Fr(n)2447 5034 y Fx( )t Fs(k)2564 5051 y Fr(L)2612 5032 y Fp(2)2646 5051 y FC(\()p Fk(R)p FC(\))-118 5200 y Fy(for)32 b(a)g(suitable)g (constan)m(t)h Fx(K)7 b Fy(,)33 b(b)s(ecause)h(b)s(oth)e Fx(g)1661 5164 y Ft(0)1657 5225 y Fr(n)1737 5200 y Fy(and)g Fx(g)1977 5164 y Ft(00)1973 5225 y Fr(n)2052 5200 y Fy(are)h(zero)g(on) f(\()p Fs(\000)p Fx(n;)17 b(n)p Fy(\).)44 b(This)33 b(w)m(ould)g(mean)f (that)1270 5411 y(lim)1218 5471 y Fr(n)p Ft(!)p FC(+)p Ft(1)1474 5411 y Fs(k\000)p Fx( )1668 5370 y Ft(00)1664 5436 y Fr(n)1734 5411 y Fy(+)22 b(\()p Fx(q)j Fs(\000)e Fx(\025)p Fy(\))17 b Fx( )2213 5426 y Fr(n)2260 5411 y Fs(k)2310 5440 y Fr(L)2358 5421 y Fp(2)2392 5440 y FC(\()p Fk(R)p FC(\))2527 5411 y Fy(=)28 b(0)-118 5666 y(for)36 b(a)g(sequence)j(of)d(elemen)m(ts)g(in)g Fx(C)1240 5629 y FC(2)1233 5690 y Fr(c)1279 5666 y Fy(\()p Fw(R)5 b Fy(\))42 b(with)36 b Fx(L)1755 5629 y FC(2)1795 5666 y Fy(-norm)f(one.)55 b(This)37 b(is,)g(using)f(W)-8 b(eyl's)37 b(criterion)e(3.2.1,)i(w)m(e)-118 5786 y(obtain)26 b(a)h(con)m (tradiction)f(with)g(the)i(fact)f(that)f Fx(H)35 b Fy(is)27 b(a)f(self-adjoin)m(t)g(op)s(erator)g(with)h(domain)e Fs(D)s Fy(\()p Fx(H)8 b Fy(\))27 b(=)g Fx(C)3793 5750 y Ft(1)3786 5811 y Fr(c)3868 5786 y Fy(\()p Fw(R)5 b Fy(\))-118 5906 y(and)39 b Fx(\025)f Fs(2)g Fw(R)50 b Fy(is)38 b(in)g(the)h(resolv)m(en)m(t)h(set)g(of)e(this)g(op)s(erator.) 62 b(This)39 b(altogether)f(giv)m(es)h(the)g(desired)g(conclusion)1900 6155 y(50)p eop %%Page: 51 55 51 54 bop -118 219 a Fy(that)30 b Fx( )35 b Fy(m)m(ust)30 b(necessarily)h(b)s(e)g(zero.)43 b(This)31 b(implies)d(that)j(the)f (system)i(has)f(an)f(exp)s(onen)m(tial)g(dic)m(hotom)m(y)g(and,)-118 339 y(b)s(ecause)41 b(of)e(the)g(v)m(olume)g(preserv)-5 b(ation)40 b(the)f(in)m(v)-5 b(arian)m(t)38 b(bundles)i Fx(V)2455 354 y FC(+)2554 339 y Fy(and)f Fx(V)2807 354 y Ft(\000)2905 339 y Fy(ha)m(v)m(e)i(b)s(oth)e(dimension)f(one.)-118 459 y(Therefore)44 b(the)g(corresp)s(onding)g(solutions)e(are)i (linearly)d(indep)s(enden)m(t.)78 b(These)45 b(solutions)d(are)i (clearly)e(in)-118 580 y Fx(L)-52 544 y FC(2)-12 580 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))30 b(and)j Fx(L)619 544 y FC(2)658 580 y Fy(\()p Fs(\0001)p Fx(;)17 b Fy(0\))32 b(resp)s(ectiv)m(ely)-8 b(,)33 b(b)s(ecause)g(of)f(the)g(exp)s(onen)m (tial)f(b)s(ounds)i(around)f(+)p Fs(1)f Fy(and)i Fs(\0001)p Fy(.)-118 700 y Fh(\003)28 820 y Fy(Note)h(that)f(w)m(e)h(ha)m(v)m(e)g (th)m(us)h(pro)m(v)m(ed)f(the)g(core)f(p)s(oin)m(ts)g(of)g(the)h(follo) m(wing)c(imp)s(ortan)m(t)h(result,)j(whic)m(h)g(is)e(the)-118 941 y(c)m(haracterization)g(of)g(the)h(sp)s(ectrum)g(for)f(the)h(whole) f(line)f(op)s(erator)h(that)h(w)m(e)g(will)d(use:)-118 1144 y Fv(Corollary)36 b(3.2.23)49 b FA(Assume)29 b(that)g(the)f(p)-5 b(otential)29 b Fx(V)50 b FA(is)29 b(of)f(the)h(typ)-5 b(e)29 b Fx(V)21 b Fy(\()p Fx(t)p Fy(\))28 b(=)g Fx(Q)p Fy(\()p Fx(\034)11 b Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\)\))p FA(,)30 b(wher)-5 b(e)28 b Fx(Q)g Fy(:)g Fx(Y)48 b Fs(!)28 b Fw(R)-118 1265 y FA(is)h(c)-5 b(ontinuous,)31 b Fx(Y)51 b FA(is)29 b(a)g(c)-5 b(omp)g(act)29 b(sp)-5 b(ac)g(e)29 b(and)g Fx(\034)11 b Fy(\()p Fs(\001)p Fy(;)17 b Fs(\001)p Fy(\))27 b(:)h Fw(R)16 b Fs(\002)11 b Fx(Y)55 b Fs(!)27 b Fx(Y)51 b FA(is)29 b(a)h(minimal)e(\015ow)i(\(in)f(p)-5 b(articular)29 b(this)h(is)-118 1385 y(true)c(for)f(quasi-p)-5 b(erio)g(dic)24 b(e)-5 b(quations\).)41 b(Then)25 b(a)g(value)g Fx(\025)g FA(is)h(in)f(the)g(r)-5 b(esolvent)25 b(set)g(of)h(the)f (Schr\177)-50 b(odinger)24 b(op)-5 b(er)g(ator)-118 1505 y Fx(H)50 b FA(on)42 b(the)h(whole)e(r)-5 b(e)g(al)42 b(line)g(if,)j(and)c(only)i(if,)h(the)e(two-dimensional)f(system)h (\(3.22\))g(has)g(an)g(exp)-5 b(onential)-118 1626 y(dichotomy.)28 1829 y Fv(Pro)s(of:)52 b Fy(W)-8 b(e)37 b(ha)m(v)m(e)h(already)e(seen)i (in)e(the)h(previous)g(pro)s(of)f(that)h(if)e Fx(\025)i Fy(w)m(as)g(in)f(the)h(resolv)m(en)m(t)h(set,)h(either)-118 1950 y(real)34 b(or)g(complex,)g(then)i(the)f(system)g(had)g(an)f(exp)s (onen)m(tial)g(dic)m(hotom)m(y)-8 b(.)49 b(No)m(w)35 b(w)m(e)h(m)m(ust)e(see)i(the)f(con)m(v)m(erse,)-118 2070 y(namely)-8 b(,)34 b(that)g(if)f(system)i(\(3.22\))f(has)h(an)f (exp)s(onen)m(tial)g(dic)m(hotom)m(y)g(for)g Fx(\025)c Fs(2)h Fw(C)20 b Fy(,)41 b(then)35 b Fx(\025)f Fy(is)g(in)g(the)h (resolv)m(en)m(t)-118 2190 y(set)29 b(of)e(the)i(op)s(erator)e Fx(H)36 b Fy(on)28 b Fx(L)1002 2154 y FC(2)1042 2190 y Fy(.)42 b(If)28 b(Im)k Fx(\025)27 b Fs(6)p Fy(=)h(0)g(then)h(there)f (is)g(nothing)f(to)h(pro)m(v)m(e,)j(as)d(these)h(p)s(oin)m(ts)f(are)g (alw)m(a)m(ys)-118 2311 y(in)h(the)h(resolv)m(en)m(t)h(set.)43 b(If)30 b Fx(\025)g Fy(is)f(real)g(and)h(the)g(system)h(has)f(an)f(exp) s(onen)m(tial)g(dic)m(hotom)m(y)-8 b(,)30 b(w)m(e)h(are)f(going)e(to)i (use)-118 2431 y(the)g(splitting)d(giv)m(en)j(b)m(y)g(the)g(dic)m (hotom)m(y)f(to)g(giv)m(e)h(an)f(in)m(tegral)f(k)m(ernel)i(\(a)f (Green's)i(function\))d(for)h(the)h(in)m(v)m(erse)-118 2552 y(of)40 b(the)i(op)s(erator.)67 b(Let)41 b Fx(\037)877 2567 y FC(+)977 2552 y Fy(and)g Fx(\037)1236 2567 y Ft(\000)1336 2552 y Fy(b)s(e)g(linearly)e(indep)s(enden)m(t)j(solutions)e(of)h(the)g (Sc)m(hr\177)-49 b(odinger)41 b(equation)-118 2672 y(whic)m(h)i(are)f (in)f Fx(L)532 2636 y FC(2)572 2672 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))41 b(and)h Fx(L)1223 2636 y FC(2)1263 2672 y Fy(\()p Fs(\0001)p Fx(;)17 b Fy(0\))41 b(resp)s(ectiv)m(ely)-8 b(.)73 b(F)-8 b(or)42 b(simplicit)m(y)d(let)j(us)g(tak)m(e)h(the)g(W)-8 b(ronskian)-118 2792 y(equal)32 b(to)h(one.)43 b(Let)1093 2965 y Fx(G)p Fy(\()p Fx(s;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))27 b(=)1602 2825 y Fq(\032)1719 2904 y Fx(\037)1780 2919 y Ft(\000)1839 2904 y Fy(\()p Fx(s)p Fy(\))p Fx(\037)2022 2919 y FC(+)2081 2904 y Fy(\()p Fx(t)p Fy(\))115 b(for)g Fx(s)28 b Fs(\024)g Fx(t)1719 3025 y(\037)1780 3040 y Ft(\000)1839 3025 y Fy(\()p Fx(t)p Fy(\))p Fx(\037)2011 3040 y FC(+)2070 3025 y Fy(\()p Fx(s)p Fy(\))115 b(for)h Fx(s)28 b(>)f(t)-118 3183 y Fy(W)-8 b(e)33 b(ha)m(v)m(e)h(that)1200 3303 y Fs(j)p Fx(G)p Fy(\()p Fx(s;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))p Fs(j)26 b(\024)i Fx(C)7 b Fy(\()p Fx(\025)p Fy(\))17 b(exp\()p Fx(\013)q Fy(\()p Fx(\025)p Fy(\))p Fs(j)p Fx(t)22 b Fs(\000)g Fx(s)p Fs(j)p Fy(\))p Fx(;)-118 3468 y Fy(b)s(ecause)38 b(the)e(exp)s(onen)m(tial)g(dic)m(hotom)m(y)g(imp)s (oses)g(a)g(certain)g(rate)g(of)g(decrease)i(of)e Fx(\037)3092 3483 y FC(+)3187 3468 y Fy(and)h Fx(\037)3442 3483 y Ft(\000)3501 3468 y Fy(.)55 b(Hence,)39 b(w)m(e)-118 3589 y(ha)m(v)m(e)34 b(that)e(the)h(op)s(erator)1327 3752 y(\()p Fx(T)14 b( )t Fy(\)\()p Fx(t)p Fy(\))27 b(=)1783 3616 y Fq(Z)1838 3842 y Fk(R)1907 3752 y Fx(G)p Fy(\()p Fx(s;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))p Fx( )t Fy(\()p Fx(s)p Fy(\))p Fx(ds)-118 3974 y Fy(is)32 b(a)g(b)s(ounded)i(linear)d (map)g(from)h Fx(L)1244 3938 y FC(2)1283 3974 y Fy(\()p Fw(R)5 b Fy(\))39 b(to)32 b(itself)f(\(see,)j(for)e(instance,)h([74]\)) f(and)h(satis\014es)1584 4172 y(\()p Fx(H)c Fs(\000)23 b Fx(\025I)8 b Fy(\))p Fx(T)14 b( )31 b Fy(=)d Fx( )-118 4370 y Fy(for)k(all)e Fx( )i Fs(2)c Fx(C)432 4334 y Ft(1)425 4395 y Fr(c)507 4370 y Fy(\()p Fw(R)5 b Fy(\),)38 b(and)33 b(th)m(us)h(for)e(all)e Fx( )i Fs(2)c Fx(L)1658 4334 y FC(2)1698 4370 y Fy(\()p Fw(R)5 b Fy(\).)49 b Fh(\003)28 4491 y Fy(In)29 b(the)h(case)g(of)e(quasi-p)s(erio)s(dic)f(p)s(oten)m (tials)g(with)i(con)m(tin)m(uous)h(extensions)g(to)e Fw(T)3038 4454 y Fr(d)3111 4491 y Fy(w)m(e)i(ha)m(v)m(e)g(the)f(follo)m (wing)-118 4694 y Fv(Corollary)36 b(3.2.24)49 b FA(The)f(sp)-5 b(e)g(ctrum)48 b(of)g(the)g(Schr\177)-50 b(odinger)47 b(op)-5 b(er)g(ator)48 b(on)g Fx(L)2824 4658 y FC(2)2864 4694 y Fy(\()p Fw(R)5 b Fy(\))54 b FA(with)49 b(p)-5 b(otential)48 b Fx(V)21 b Fy(\()p Fx(t)p Fy(\))53 b(=)-118 4814 y Fx(Q)p Fy(\()p Fx(!)t(t)23 b Fy(+)f Fx(\036)p Fy(\))p FA(,)36 b(wher)-5 b(e)35 b Fx(Q)30 b Fy(:)f Fw(T)882 4778 y Fr(d)956 4814 y Fs(!)g Fw(R)47 b FA(is)35 b(c)-5 b(ontinuous)36 b(and)f Fx(!)40 b FA(is)35 b(r)-5 b(ational)5 b(ly)36 b(indep)-5 b(endent,)34 b(do)-5 b(es)36 b(not)f(dep)-5 b(end)35 b(on)-118 4935 y(the)g(initial)f(phase)g Fx(\036)h FA(on)f(the)h(torus)g Fw(T)1306 4899 y Fr(d)1350 4935 y FA(.)28 5138 y Fv(Pro)s(of:)42 b Fy(If)29 b Fx(!)k Fy(is)c(rationally)e(indep)s(enden)m(t,)32 b(the)e(exp)s(onen)m(tial)e (dic)m(hotom)m(y)i(extends)h(all)d(o)m(v)m(er)i Fw(T)3582 5102 y Fr(d)3656 5138 y Fy(b)m(y)g(2.2.5,)-118 5259 y(and)j(b)m(y)g (the)g(previous)g(c)m(haracterization)f(of)g(the)h(sp)s(ectrum,)g(the)g (result)f(follo)m(ws.)43 b Fh(\003)-118 5462 y Fv(Remark)37 b(3.2.25)49 b FA(Her)-5 b(e)36 b(we)g(have)g(use)-5 b(d)36 b(that)h Fx(Q)f FA(is)g(c)-5 b(ontinuous,)37 b(although)f(c)-5 b(or)g(ol)5 b(laries)35 b(3.2.23)g(and)h(3.2.24)-118 5582 y(ar)-5 b(e)35 b(also)f(true)h(if)g(the)g(\015ow)g(de\014ne)-5 b(d)33 b(by)i(e)-5 b(quations)35 b(\(3.22\))f(is)g(only)h(c)-5 b(ontinuous)35 b(on)f Fw(R)3106 5546 y FC(2)3174 5582 y Fs(\002)22 b Fw(T)3336 5546 y Fr(d)3380 5582 y FA(.)28 5786 y Fy(With)g(these)i(useful)e(prop)s(erties)h(of)f(the)g(sp)s 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Ft(0)2088 777 y FC(+)2147 752 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))p 2027 799 332 4 v 2027 891 a Fx(\037)2088 906 y FC(+)2147 891 y Fy(\()p Fx(t)p Fy(;)g Fx(\025)p Fy(\))-118 1047 y(where)38 b(w)m(e)g(write)e(the)i(underscript)f(+)g(for)f(the)i (eigen)m(v)-5 b(alue)36 b(problem)g(on)g(the)i(p)s(ositiv)m(e)e(line,)g (that)h(is,)h(when)-118 1167 y Fx(\037)-57 1182 y FC(+)33 1167 y Fy(is)30 b(square)h(in)m(tegrable)f(at)g(+)p Fs(1)g Fy(and)h(satis\014es)g(the)g(b)s(oundary)g(conditions)f(at)g(zero.)44 b(The)31 b(reason)g(for)f(suc)m(h)-118 1288 y(a)40 b(distinction)e(is)i (that)g(one)g(ma)m(y)g(\(and)g(should\))g(also)f(consider)h(the)h (problem)e(on)h(the)g(negativ)m(e)h(half-line,)-118 1408 y(whic)m(h)33 b(will)c(b)s(e)k(denoted)g(b)m(y)g(the)f(underscript)h Fs(\000)p Fy(,)g(and)f(the)h(corresp)s(onding)f(W)-8 b(eyl's)32 b Fx(m)p Fy(-functions,)h(giv)m(en)f(b)m(y)1516 1681 y Fx(m)1601 1696 y Ft(\000)1660 1681 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))27 b(=)2013 1610 y 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Fy(\))1779 4618 y FC(2)1846 4660 y Fy(=)28 b Fs(\000)p Fx(\025)22 b Fy(+)g Fx(Q)p Fy(\()p Fx(\022)s Fy(\))p Fx(;)212 b(\022)2692 4618 y Ft(0)2744 4660 y Fy(=)27 b Fx(!)t(:)827 b Fy(\(3.23\))28 4911 y(By)31 b(a)f(theorem)h(of)e(Sc)m(harf)i(\([79],)g([65]\),)g(in)e (the)i(quasi-p)s(erio)s(dic)d(case,)k(the)f(functions)g Fx(m)3339 4926 y Ft(\006)3398 4911 y Fy(,)g(for)f(Im)i Fx(\025)27 b Fs(6)p Fy(=)h(0,)-118 5031 y(are)34 b(also)f(quasi-p)s (erio)s(dic)f(with)i(the)g(same)g(frequency)i(v)m(ector)f Fx(!)t Fy(.)47 b(That)34 b(is,)g(there)h(exist)f(unique)h(con)m(tin)m (uous)-118 5151 y(functions)d Fx(M)396 5166 y Ft(\006)456 5151 y Fy(\()p Fs(\001)p Fx(;)17 b(\025)p Fy(\))27 b(:)g Fw(T)805 5115 y Fr(d)877 5151 y Fs(!)g Fw(P)1063 5115 y FC(1)1105 5151 y Fy(\()p Fw(C)20 b Fy(\))38 b(suc)m(h)c(that)1337 5371 y Fx(m)1422 5386 y Ft(\006)1482 5371 y Fy(\()p Fx(t;)17 b(\025;)g(\036)p Fy(\))26 b(=)i Fx(M)2020 5386 y Ft(\006)2079 5371 y Fy(\()p Fx(!)t(t)22 b Fy(+)g Fx(\036)p Fy(;)17 b Fx(\025)p Fy(\))p Fx(;)-118 5591 y Fy(where)34 b Fx(\036)e Fy(is)g(the)h(initial)c(phase)k(for)f Fx(\022)s Fy(.)44 b(Moreo)m(v)m(er,)34 b(for)e(Im)g Fx(\025)c Fs(6)p Fy(=)f(0,)33 b(these)h(extensions)f(are)g(giv)m(en)g(b)m(y)1444 5861 y Fx(M)1538 5876 y Ft(\006)1598 5861 y Fy(\()p Fx(\036)p Fy(;)17 b Fx(\025)p Fy(\))27 b(=)1973 5793 y Fx(\037)2034 5808 y Ft(\006)2093 5757 y(0)2117 5793 y Fy(\(0;)17 b Fx(\025;)g(\036)p Fy(\))p 1973 5838 471 4 v 1985 5929 a Fx(\037)2046 5944 y Ft(\006)2105 5929 y Fy(\(0;)g Fx(\025;)g(\036)p Fy(\))1900 6155 y(52)p eop %%Page: 53 57 53 56 bop -118 219 a Fy(where,)34 b(again,)d(w)m(e)j(wrote)f Fx(\037)p Fy(\()p Fs(\001)p Fy(;)17 b Fs(\001)p Fx(;)g(\036)p Fy(\))31 b(to)h(stress)i(the)f(dep)s(endence)i(on)d(the)h(initial)c (phase)34 b(on)e(the)h(torus)g Fw(T)3813 182 y Fr(d)3857 219 y Fy(.)28 339 y(The)39 b(men)m(tioned)f(theorem)g(b)m(y)h(Sc)m (harf)f(just)g(states)h(the)g(con)m(tin)m(uit)m(y)f(of)f(the)i (functions)f Fx(M)3474 354 y Ft(\006)3571 339 y Fy(de\014ned)i(on)-118 459 y(the)34 b(torus.)49 b(Nev)m(ertheless,)36 b(it)d(is)h(also)f(true) h(that)g(these)i(functions)e(ha)m(v)m(e)h(alw)m(a)m(ys)g(directional)c (deriv)-5 b(ativ)m(es)34 b(in)-118 580 y(the)f(direction)e(of)h Fx(!)t Fy(,)g(so)h(that)g(w)m(e)g(can)g(write)f(do)m(wn)i(the)f (Ricatti)d(equation)j(\(3.23\))e(for)h(these)i(extensions)865 796 y Fx(D)946 811 y Fr(!)997 796 y Fx(M)1091 811 y Ft(\006)1150 796 y Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))22 b(+)g Fx(M)1589 811 y Ft(\006)1648 796 y Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))1873 755 y FC(2)1940 796 y Fy(=)28 b Fs(\000)p Fx(\025)22 b Fy(+)g Fx(Q)p Fy(\()p Fx(\022)s Fy(\))p Fx(;)212 b(\022)2786 755 y Ft(0)2837 796 y Fy(=)28 b Fx(!)t(:)733 b Fy(\(3.24\))28 1012 y(In)34 b(the)g(case)g(when)h(the)f(extension)g Fx(Q)g Fy(of)f(the)h(quasi-p)s (erio)s(dic)d(function)i Fx(q)k Fy(is)c(smo)s(oth,)g(this)g(smo)s (othness)-118 1132 y(will)d(imply)h(a)h(greater)h(regularit)m(y)e(of)h (the)h(extensions)h Fx(M)2039 1147 y Ft(\006)2131 1132 y Fy(as)f(it)e(is)h(stated)i(b)m(y)f(the)g(follo)m(wing)d(theorem:)-118 1356 y Fv(Theorem)37 b(3.2.26)h(\([65]\))48 b FA(L)-5 b(et)39 b Fs(F)k Fy(=)33 b Fs(D)1467 1320 y Fr(\013)1517 1356 y Fy(\()p Fx(!)t Fy(\))k FA(\(wher)-5 b(e)37 b(,)i(fr)-5 b(om)38 b(now)f(on,)i Fs(D)2777 1320 y Fr(\013)2826 1356 y Fy(\()p Fx(!)t Fy(\))e FA(wil)5 b(l)38 b(stand)g(for)g(the)g(set)g (of)-118 1476 y(quasi-p)-5 b(erio)g(dic)37 b(functions)i(with)f(fr)-5 b(e)g(quency)39 b Fx(!)f Fs(2)e Fw(R)1846 1440 y Fr(d)1931 1476 y FA(whose)i(extension)g(to)h(the)g Fx(d)p FA(-dimensional)d (torus)j(is)g(of)-118 1597 y(class)34 b Fx(C)193 1560 y Fr(\013)277 1597 y FA(\))h(for)g Fx(\013)28 b Fy(=)g Fx(r)m(;)17 b Fs(1)p Fx(;)g(a)p FA(.)44 b(If)34 b Fx(q)e Fs(2)c(F)44 b FA(and)34 b(if)h Fx(\025)g FA(is)f(in)h(the)g(r)-5 b(esolvent)34 b(set)h Fx(\032)p Fy(\()p Fx(\025)p Fy(\))g FA(then)g(the)g(functions)1629 1813 y Fx(m)28 b Fy(=)f Fx(m)1930 1828 y Ft(\006)2017 1813 y Fs(2)h(F)2183 1828 y Fr(P)2242 1813 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5921 y FC(2)2091 5906 y Fx(M)10 b Fy(\))p Fx(X)36 b Fy(=)27 b Fx(G)1900 6155 y Fy(53)p eop %%Page: 54 58 54 57 bop -118 219 a Fy(on)32 b Fx(C)94 182 y FC(0)165 219 y Fy(has)h(a)f(b)s(ounded)h(in)m(v)m(erse)g(if)e(the)i(conditions)e (of)h(the)g(lemma)e(are)i(satis\014ed.)44 b(The)33 b(b)s(ound)g(comes)f (from)-118 339 y(an)k(application)d(of)j(the)h(theorem)e(on)h(a)m(v)m (erages)i(for)d(con)m(tin)m(uous)i(mappings)e(of)h(the)g(torus,)i(and)e (the)g(b)s(ound)-118 459 y(for)e Fs(j)p Fy(Re)f([)p Fx(A)309 474 y FC(1)370 459 y Fy(+)22 b Fx(A)541 474 y FC(2)581 459 y Fx(M)10 b Fy(])p Fs(j)35 b Fy(giv)m(es)h(a)e(b)s(ound)h(for)f (the)i(in)m(v)m(erse)g(of)e(the)i(ab)s(o)m(v)m(e)f(linear)f(mapping,)g (whic)m(h)h(completes)-118 580 y(the)e(pro)s(of)f(of)g(the)h(lemma.)41 b Fh(\003)28 700 y Fy(The)48 b(second)g(step)g(is)f(to)f(use)i(the)g (previous)f(result)g(to)g(deduce)h(the)g(regularit)m(y)d(of)i(the)g (solutions)f(of)-118 820 y(equation)32 b(\(3.24\).)-118 1049 y Fv(Lemma)37 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Fy(\()p Fw(T)933 3059 y Fr(d)977 3095 y Fy(\).)57 b(T)-8 b(o)37 b(sho)m(w)h(analyticit)m (y)-8 b(,)37 b(the)g(Cauc)m(h)m(y-Riemann)g(equations)h(are)f(c)m(hec)m (k)m(ed)j(b)m(y)-118 3215 y(the)33 b(same)f(metho)s(d.)43 b Fh(\003)28 3336 y Fy(No)m(w)33 b(w)m(e)h(can)f(\014nish)g(the)g(pro)s (of)e(of)h(theorem)h(3.2.26)28 3456 y(F)-8 b(or)34 b(Im)e Fx(\025)e Fs(6)p Fy(=)g(0,)35 b(the)f(extension)h Fx(M)42 b Fy(=)30 b Fx(M)1596 3471 y Ft(\006)1689 3456 y Fy(of)k Fx(m)d Fy(=)f Fx(m)2109 3471 y Ft(\006)2203 3456 y Fy(is)j(a)h (solution)f(of)h(the)g(Ricatti)f(equation)g(\(3.24\))-118 3576 y(whic)m(h)g(corresp)s(onds)h(to)e(the)h(c)m(hoice)1287 3796 y Fx(A)1360 3811 y FC(0)1427 3796 y Fy(=)28 b Fx(Q)22 b Fs(\000)h Fx(\025;)114 b(A)2001 3811 y FC(1)2068 3796 y Fy(=)28 b(0)97 b Fx(A)2391 3811 y FC(2)2458 3796 y Fy(=)28 b(2)-118 4016 y(in)42 b(equation)g(\(3.25\).)72 b(Assume)44 b(Im)32 b Fx(\025)44 b(>)g Fy(0)e(and)h(the)g(case)g(Im)32 b Fx(\025)45 b(<)f Fy(0)e(follo)m(ws)f(similarly)-8 b(.)70 b(Then,)46 b(b)m(y)d(the)-118 4137 y(ergo)s(dicit)m(y)31 b(of)i(the)g(\015o)m(w)1080 4257 y(Re)g([)p Fx(A)1328 4272 y FC(1)1390 4257 y Fy(+)22 b Fx(A)1561 4272 y FC(2)1600 4257 y Fx(M)10 b Fy(])29 b(=)e(2Re)33 b([)p Fx(M)10 b Fy(])29 b(=)e(Re)33 b([)p Fx(m)p Fy(])28 b Fx(<)g Fy(0)-118 4432 y(b)s(ecause)j Fx(m)p Fy(\()p Fx(t;)17 b(\025)p Fy(\))29 b(is)h(a)f(Herglotz)g(function)g(and)g(therefore)i(has)f (negativ)m(e)f(real)g(part)g(in)g(the)h(upp)s(er)g(plane)g(and)-118 4552 y(hence)f(negativ)m(e)e(a)m(v)m(erage.)43 b(This)27 b(completes)g(the)h(pro)s(of)f(when)h(Im)k Fx(\025)c Fs(6)p Fy(=)f(0,)h(as)g(lemma)d(3.2.28)i(can)g(b)s(e)h(applied.)28 4672 y(F)-8 b(or)32 b(Im)g Fx(\025)c Fy(=)f(0)32 b(and)h Fx(\025)f Fy(in)g(the)h(resolv)m(en)m(t)h(set,)f(w)m(e)h(consider,)f (as)g(b)s(efore,)1699 4940 y(~)-67 b Fx(m)28 b Fy(=)2015 4873 y Fx(\037)2076 4837 y Ft(0)p 1908 4917 299 4 v 1908 5008 a Fx(\037)22 b Fy(+)g Fx(i\037)2183 4980 y Ft(0)-118 5205 y Fy(for)32 b(whic)m(h)h(one)g(c)m(hec)m(ks)i(that)e(it)e (satis\014es)i(the)g(Ricatti)e(equation)1294 5477 y(~)-67 b Fx(m)1361 5436 y Ft(0)1406 5477 y Fy(+)1504 5336 y Fq(\022)1578 5477 y Fx(a)1629 5492 y FC(0)1690 5477 y Fy(+)22 b Fx(a)1839 5492 y FC(1)1897 5477 y Fy(~)-67 b Fx(m)23 b Fy(+)2095 5409 y(1)p 2095 5454 49 4 v 2095 5545 a(2)2153 5477 y Fx(a)2204 5492 y FC(2)2262 5477 y Fy(~)-67 b Fx(m)2329 5436 y FC(2)2369 5336 y Fq(\023)2470 5477 y Fy(=)27 b(0)-118 5748 y(with)897 5869 y Fx(a)948 5884 y FC(0)1016 5869 y Fy(=)g Fx(\025)22 b Fs(\000)h Fx(q)t(;)114 b(a)1537 5884 y FC(1)1604 5869 y Fy(=)28 b(2)p Fx(i)p Fy(\()p Fx(q)e Fs(\000)c Fx(\025)p Fy(\))p Fx(;)114 b(a)2283 5884 y FC(2)2351 5869 y Fy(=)27 b(2\()p Fx(q)f Fs(\000)d Fx(\025)p Fy(\))f(+)g(2)p Fx(:)1900 6155 y Fy(54)p eop %%Page: 55 59 55 58 bop -118 219 a Fy(Moreo)m(v)m(er,)51 b(~)-67 b Fx(m)28 b Fs(2)g(D)623 182 y FC(0)662 219 y Fy(\()p Fx(!)t Fy(\),)i(for)g(the)h(same)g(argumen)m(ts)g(\(due)g(to)g(Sc)m(harf)7 b(\))31 b(as)g(for)f(Im)i Fx(\025)c Fs(6)p Fy(=)f(0.)43 b(Hence)50 b(~)-67 b Fx(m)31 b Fy(extends)-118 339 y(to)h(a)g(function) 501 314 y(~)464 339 y Fx(M)39 b Fs(2)28 b Fx(C)768 303 y FC(0)761 364 y Fr(!)811 339 y Fy(\()p Fw(T)912 303 y Fr(d)956 339 y Fy(\))k(whic)m(h)i(satis\014es)f(the)g(equation)1146 580 y Fx(D)1227 595 y Fr(!)1313 555 y Fy(~)1277 580 y Fx(M)g Fy(+)1502 440 y Fq(\022)1576 580 y Fx(A)1649 595 y FC(0)1710 580 y Fy(+)22 b Fx(A)1881 595 y FC(1)1957 555 y Fy(~)1921 580 y Fx(M)33 b Fy(+)2156 513 y(1)p 2156 557 49 4 v 2156 649 a(2)2215 580 y Fx(A)2288 595 y FC(2)2363 555 y Fy(~)2327 580 y Fx(M)2431 539 y FC(2)2471 440 y Fq(\023)2572 580 y Fy(=)28 b(0)p Fx(;)-118 829 y Fy(b)s(eing)34 b Fx(A)220 844 y Fr(j)291 829 y Fy(the)h(extensions)h(to)f Fw(T)1118 793 y Fr(d)1196 829 y Fy(of)g Fx(a)1361 844 y Fr(j)1397 829 y Fy(,)h(for)e Fx(j)j Fy(=)31 b(1)p Fx(;)17 b Fy(2)p Fx(;)g Fy(3)34 b(resp)s(ectiv)m(ely)-8 b(.)51 b(In)35 b(order)g(to)f(pro)m(v)m(e)i(the)f(theorem)g(it)-118 950 y(su\016ces)g(to)d(sho)m(w)h(that)g([)p Fx(A)886 965 y FC(1)948 950 y Fy(+)22 b Fx(A)1119 965 y FC(2)1194 925 y Fy(~)1158 950 y Fx(M)11 b Fy(])28 b(=)f([)p Fx(a)1499 965 y FC(1)1561 950 y Fy(+)22 b Fx(a)1710 965 y FC(2)1768 950 y Fy(~)-67 b Fx(m)p Fy(])33 b(has)g(non-zero)f(real)g(part.)43 b(T)-8 b(o)33 b(this)f(end,)i(notice)e(that)1443 1189 y Fx(a)1494 1204 y FC(1)1556 1189 y Fy(+)22 b Fx(a)1705 1204 y FC(2)1763 1189 y Fy(~)-68 b Fx(m)28 b Fy(=)g(2)2020 1122 y(\()p Fx(\037)22 b Fy(+)g Fx(i\037)2333 1085 y Ft(0)2356 1122 y Fy(\))2394 1085 y Ft(0)p 2020 1166 399 4 v 2069 1257 a Fx(\037)g Fy(+)g Fx(i\037)2344 1229 y Ft(0)2428 1189 y Fx(;)-118 1418 y Fy(and)33 b(therefore,)g(b)m(y)g(the) g(ergo)s(dicit)m(y)-8 b(,)1026 1664 y([)p Fx(a)1104 1679 y FC(1)1166 1664 y Fy(+)22 b Fx(a)1315 1679 y FC(2)1373 1664 y Fy(~)-67 b Fx(m)p Fy(])28 b(=)56 b(lim)1599 1726 y Fr(T)10 b Ft(!1)1828 1597 y Fy(2)p 1817 1641 71 4 v 1817 1733 a Fx(T)1915 1529 y Fq(Z)2014 1555 y Fr(T)1970 1754 y FC(0)2096 1597 y Fy(\()p Fx(\037)p Fy(\()p Fx(s)p Fy(\))22 b(+)g Fx(i\037)2531 1561 y Ft(0)2554 1597 y Fy(\()p Fx(s)p Fy(\)\))2714 1561 y Ft(0)p 2096 1641 642 4 v 2145 1733 a Fx(\037)p Fy(\()p Fx(s)p Fy(\))g(+)g Fx(i\037)2542 1704 y Ft(0)2566 1733 y Fy(\()p Fx(s)p Fy(\))2747 1664 y Fx(ds:)-118 1904 y Fy(Since)30 b Fx(\037;)17 b(\037)300 1868 y Ft(0)353 1904 y Fy(do)29 b(not)h(v)-5 b(anish)29 b(sim)m(ultaneously)g(\(this)g(is)g(an)h(easy)g(consequence) j(of)c(Sturmian)g(theory\),)i(w)m(e)f(can)-118 2025 y(write)1409 2145 y Fx(\037)p Fy(\()p Fx(s)p Fy(\))22 b(+)g Fx(i\037)1806 2104 y Ft(0)1830 2145 y Fy(\()p Fx(s)p Fy(\))27 b(=)h(exp)q(\()p Fx(f)11 b Fy(\()p Fx(s)p Fy(\)\))-118 2304 y(for)32 b(a)g(suitable)g (function)g Fx(f)38 b Fs(2)28 b Fx(C)1115 2268 y Fr(\013)1165 2304 y Fy(\()p Fw(R)5 b Fy(\).)49 b(Then)34 b(the)f(ab)s(o)m(v)m(e)g (limit)c(can)k(b)s(e)g(expressed)i(as)1078 2556 y(lim)1049 2618 y Fr(T)10 b Ft(!1)1279 2488 y Fy(2)p 1268 2533 71 4 v 1268 2624 a Fx(T)1365 2420 y Fq(Z)1465 2447 y Fr(T)1421 2646 y FC(0)1536 2556 y Fx(f)1595 2515 y Ft(0)1618 2556 y Fy(\()p Fx(s)p Fy(\))p Fx(ds)28 b Fy(=)55 b(lim)1968 2618 y Fr(T)10 b Ft(!1)2187 2488 y Fy(2)17 b(\()o Fx(f)11 b Fy(\()p Fx(T)j Fy(\))21 b Fs(\000)i Fx(f)11 b Fy(\(0\)\))p 2187 2533 652 4 v 2477 2624 a Fx(T)-118 2787 y Fy(so)33 b(that)945 2938 y(Re)g([)p Fx(A)1193 2953 y FC(1)1255 2938 y Fy(+)22 b Fx(A)1426 2953 y FC(2)1501 2912 y Fy(~)1465 2938 y Fx(M)11 b Fy(])28 b(=)83 b(lim)1728 3000 y Fr(T)10 b Ft(!)p FC(+)p Ft(1)1992 2938 y Fy(2)2051 2870 y(ln)15 b Fs(j)p Fx(\037)p Fy(\()p Fx(T)f Fy(\))22 b(+)g Fx(\037)2565 2834 y Ft(0)2588 2870 y Fy(\()p Fx(T)14 b Fy(\))p Fs(j)p 2051 2915 713 4 v 2371 3006 a Fx(T)2800 2938 y(<)28 b Fy(0)-118 3133 y(b)s(ecause)34 b(of)e(the)h(exp)s(onen)m(tial)f(dic)m (hotom)m(y)g(in)g(the)h(resolv)m(en)m(t)h(set.)44 b Fh(\003)p Fy(.)28 3253 y(So,)33 b(for)f(eac)m(h)h Fx(\025)g Fy(the)g(resolv)m(en) m(t)g(set,)h(w)m(e)f(ha)m(v)m(e)h(de\014ned)g(a)f(map)1290 3437 y Fx(M)38 b Fy(:)28 b Fx(\022)j Fs(2)d Fw(T)1710 3396 y Fr(d)1781 3437 y Fs(7!)g Fx(M)10 b Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))28 b Fs(2)g Fw(P)2419 3396 y FC(1)2460 3437 y Fy(\()p Fw(C)20 b Fy(\))-118 3621 y(whic)m(h)38 b(is)e(of)h(the)h(same)f(kind)g(of)g(regularit)m(y)f(as)i Fx(Q)p Fy(,)g(and)g(suc)m(h)g(that)g(W)-8 b(eyl's)37 b Fx(m)h Fy(functions)f(are)g(of)g(the)h(form)-118 3742 y Fx(m)p Fy(\()p Fx(t)p Fy(;)17 b Fx(\025;)g(\036)p Fy(\))30 b(=)g Fx(M)10 b Fy(\()p Fx(!)t(t)23 b Fy(+)g Fx(\036)p Fy(;)17 b Fx(\025)p Fy(\))34 b(for)f(all)f Fx(\036)f Fs(2)f Fw(T)1547 3706 y Fr(d)1625 3742 y Fy(and)k Fx(t)d Fs(2)g Fw(R)5 b Fy(.)54 b(A)34 b(similar)d(study)k(can)g(b)s(e)f(done)h (for)e(the)i(Green's)-118 3862 y(functions)j(in)g(the)h(resolv)m(en)m (t)h(set)f(for)f(the)h(eigen)m(v)-5 b(alue)37 b(problem)h(either)g(in)g (the)h(half)e(or)h(in)g(the)h(whole)f(line.)-118 3983 y(Recall)31 b(that,)h(in)g(our)h(case,)g(these)h(functions)f(could)f(b) s(e)g(written)h(as)1000 4216 y Fx(G)p Fy(\()p Fx(s;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))27 b(=)1619 4149 y Fx(\037)1680 4164 y FC(+)1739 4149 y Fy(\()p Fx(t;)17 b(\025)p Fy(\))p Fx(\037)2012 4164 y FC(+)2071 4149 y Fy(\()p Fx(s;)g(\025)p Fy(\))p 1519 4193 874 4 v 1519 4285 a Fx(W)d Fy(\()p Fx(\037)1724 4300 y FC(+)1783 4285 y Fy(\()p Fs(\001)p Fx(;)j(\025)p Fy(\))p Fx(;)g(\037)2093 4300 y Ft(\000)2151 4285 y Fy(\()p Fs(\001)p Fx(;)g(\025)p Fy(\)\))2403 4216 y Fx(;)105 b Fy(for)32 b Fx(s)27 b Fs(\024)i Fx(t)-118 4424 y Fy(so)760 4575 y Fx(G)p Fy(\()p Fx(t;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))27 b(=)1374 4507 y Fx(\037)1435 4522 y FC(+)1494 4507 y Fy(\()p Fx(t;)17 b(\025)p Fy(\))p Fx(\037)1767 4522 y FC(+)1826 4507 y Fy(\()p Fx(t;)g(\025)p Fy(\))p 1269 4552 V 1269 4643 a Fx(W)d Fy(\()p Fx(\037)1474 4658 y FC(+)1533 4643 y Fy(\()p Fs(\001)p Fx(;)j(\025)p Fy(\))p Fx(;)g(\037)1843 4658 y Ft(\000)1901 4643 y Fy(\()p Fs(\001)p Fx(;)g(\025)p Fy(\)\))2180 4575 y(=)2686 4507 y(1)p 2294 4552 834 4 v 2294 4643 a Fx(m)2379 4658 y Ft(\000)2438 4643 y Fy(\()p Fx(t)p Fy(;)g Fx(\025)p Fy(\))22 b Fs(\000)h Fx(m)2857 4658 y FC(+)2916 4643 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))-118 4784 y(and)45 b(this)f(last)g (expression)i(alw)m(a)m(ys)f(mak)m(es)h(sense,)j(b)s(ecause)d Fx(m)2345 4799 y Ft(\000)2435 4784 y Fs(\000)31 b Fx(m)2628 4799 y FC(+)2732 4784 y Fy(is)44 b(nev)m(er)i(zero)f(in)f(the)i(resolv) m(en)m(t)-118 4905 y(set.)74 b(Indeed,)46 b(if)41 b(Im)32 b Fx(\025)45 b Fs(6)p Fy(=)f(0,)h(this)d(is)g(true,)j(b)s(ecause)f(in)e (this)g(set)h(the)g(imaginary)c(parts)k(of)f Fx(m)3570 4920 y Ft(\000)3672 4905 y Fy(and)g Fx(m)3956 4920 y FC(+)-118 5025 y Fy(ha)m(v)m(e)k(di\013eren)m(t)e(signs)g(and)h(are)f (not)g(zero.)79 b(F)-8 b(or)44 b(real)f Fx(\025)h Fy(in)g(the)h(resolv) m(en)m(t)g(set)g(w)m(e)g(ha)m(v)m(e)h(an)e(exp)s(onen)m(tial)-118 5145 y(dic)m(hotom)m(y)27 b(due)h(to)f(the)h(already)g(men)m(tioned)f (results)h(and)f(w)m(e)i(ha)m(v)m(e)g(that,)f(for)f(some)h Fx(t)p Fy(,)g Fx(m)3257 5160 y Ft(\000)3317 5145 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))27 b(=)h Fx(m)3745 5160 y FC(+)3804 5145 y Fy(\()p Fx(t;)17 b(\025)p Fy(\))-118 5266 y(is)32 b(equiv)-5 b(alen)m(t)32 b(to)h(that)1134 5386 y Fx(\037)1195 5401 y FC(+)1254 5386 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))p Fx(\037)1527 5345 y Ft(0)1527 5411 y(\000)1586 5386 y Fy(\()p Fx(t)p Fy(;)g Fx(\025)p Fy(\))22 b Fs(\000)h Fx(\037)1981 5345 y Ft(0)1981 5411 y FC(+)2040 5386 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))p Fx(\037)2313 5401 y Ft(\000)2372 5386 y Fy(\()p Fx(t)p Fy(;)g Fx(\025)p Fy(\))27 b(=)h(0)-118 5545 y(whic)m(h)35 b(is)f(the)h(W)-8 b(ronskian.)50 b(This)34 b(cannot)h(b)s(e)g(zero,)h (b)s(ecause)g Fx(\037)2304 5560 y FC(+)2397 5545 y Fy(and)f Fx(\037)2650 5560 y Ft(\000)2744 5545 y Fy(are)f(linearly)f(indep)s (enden)m(t)j(solu-)-118 5666 y(tions,)c(since)h(they)g(corresp)s(ond)h (to)e(the)h(t)m(w)m(o)g(bundles)h(that)e(giv)m(e)g(the)h(exp)s(onen)m (tial)f(dic)m(hotom)m(y)-8 b(.)28 5786 y(W)g(e)38 b(\014nish)g(this)g (subsection)h(with)e(a)h(prop)s(osition)e(that)i(relates)f(the)i(exp)s (onen)m(tial)e(dic)m(hotom)m(y)g(and)h(the)-118 5906 y(asso)s(ciated)32 b(in)m(v)-5 b(arian)m(t)32 b(subbundles)i(to)e(the)h (pro)5 b(jectiv)m(e)33 b(co)s(ordinates)f Fx(M)2612 5921 y Ft(\006)2704 5906 y Fy(in)g(the)h(case)h(of)e(Im)g Fx(\025)27 b Fs(6)p Fy(=)h(0)1900 6155 y(55)p eop %%Page: 56 60 56 59 bop -118 219 a Fv(Prop)s(osition)35 b(3.2.29)j(\([45]\))49 b FA(Supp)-5 b(ose)33 b(that)h Fx(\025)27 b Fs(2)h Fw(C)60 b FA(with)34 b(Im)g Fx(\025)28 b Fs(6)p Fy(=)f(0)34 b FA(in)f(e)-5 b(quation)34 b(\(3.21\))f(with)g(irr)-5 b(ational)-118 339 y(fr)g(e)g(quency)35 b(ve)-5 b(ctor)34 b Fx(!)t FA(.)44 b(Write)36 b(as)1382 459 y Fw(C)1448 418 y FC(2)1515 459 y Fs(\002)23 b Fw(T)1678 418 y Fr(d)1750 459 y Fy(=)k Fx(V)1932 418 y FC(+)1991 459 y Fy(\()p Fx(\025)p Fy(\))22 b Fs(\010)g Fx(V)2324 418 y Ft(\000)2383 459 y Fy(\()p Fx(\025)p Fy(\))-118 626 y FA(the)43 b(de)-5 b(c)g(omp)g(osition)41 b(in)i(invariant)f(subbund)5 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1255 y(\(ii\))48 b Fx(N)10 b Fy(\()p Fx(\036)p Fy(;)17 b Fx(\025)p Fy(\))27 b(=)h Fx(M)10 b Fy(\()p Fx(\036)p Fy(;)17 b Fx(\025)p Fy(\))35 b FA(ar)-5 b(e)34 b(the)h(Weyl)g Fx(M)10 b FA(-functions)35 b(in)g(the)g(notation) f(ab)-5 b(ove.)-118 1512 y Fv(The)38 b(sp)s(ectrum)f(on)g(the)h(whole)e (line)g(v.s.)51 b(the)37 b(sp)s(ectrum)g(on)h(the)f(half-line)-118 1697 y Fy(W)-8 b(e)46 b(end)h(the)f(exp)s(osition)f(on)h(the)g(sp)s (ectral)g(prop)s(erties)f(of)h(equation)f(\(3.21\))h(b)m(y)g(exp)s (osing)g(some)g(of)f(the)-118 1817 y(di\013erences)g(b)s(et)m(w)m(een)i (the)d(sp)s(ectrum)h(of)f(the)h(op)s(erator)e Fx(H)52 b Fy(on)44 b(the)h(whole)f(line)f(and)i(the)f(sp)s(ectra)h(of)f(the) -118 1937 y(op)s(erators)32 b Fx(H)402 1901 y Ft(\006)493 1937 y Fy(on)h(the)g(p)s(ositiv)m(e)f(and)g(negativ)m(e)h(half-lines.) -118 2146 y Fv(Theorem)k(3.2.30)h(\([79],)f([43],)h([45]\))49 b FA(L)-5 b(et)28 b(the)g(\015ow)f(on)g Fy(\012)i FA(b)-5 b(e)27 b(minimal)f(\(this)i(is)f(true)i(for)e(e)-5 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y(\015o)m(w)h(is)f(giv)m(en)h(b)m(y)g(the)g (equation)1319 551 y Fx(')1383 510 y Ft(0)1434 551 y Fy(=)27 b(cos)1668 510 y FC(2)1724 551 y Fx(')22 b Fs(\000)h Fy(\()p Fx(q)t Fy(\()p Fx(t)p Fy(\))f Fs(\000)g Fx(\025)p Fy(\))17 b(sin)2459 510 y FC(2)2515 551 y Fx(')1187 b Fy(\(3.27\))-118 762 y(where,)36 b(b)m(y)f(the)g(argumen)m(ts)f(with)g (whic)m(h)h(w)m(e)h(b)s(egan)e(the)h(c)m(hapter,)g(the)g(argumen)m(t)f Fx(')g Fy(is)g(tak)m(en)i(mo)s(dulus)d Fx(\031)t Fy(.)-118 883 y(W)-8 b(e)33 b(will)e(denote)i(b)m(y)h Fx(F)14 b Fy(\()p Fx(';)j(\036)p Fy(\))32 b(the)h(left)f(hand)h(side)g(of)g(this) f(equation.)44 b(This)34 b(equation)e(giv)m(es)h(raise)g(to)f(a)h (\015o)m(w)-118 1003 y(on)42 b(the)g(space)h Fx(B)48 b Fy(=)43 b Fw(P)774 967 y FC(1)816 1003 y Fy(\()p Fw(R)5 b Fy(\))34 b Fs(\002)29 b Fy(\012)42 b(\(homeomorphic)e(to)h Fw(T)2104 967 y Fr(d)p FC(+1)2280 1003 y Fy(b)s(ecause)i(\012)h(=)f Fw(T)2946 967 y Fr(d)2990 1003 y Fy(\),)h(whic)m(h)f(w)m(e)f(will)e (denote)-118 1124 y(b)m(y)k(\010\()p Fx(t)p Fy(;)17 b Fx(\036)273 1139 y FC(0)313 1124 y Fx(;)g(')421 1139 y FC(0)460 1124 y Fy(\))46 b(=)g(\()p Fx(')p Fy(\()p Fx(t)p Fy(;)17 b Fx(')949 1139 y FC(0)988 1124 y Fx(;)g(\036)1090 1139 y FC(0)1128 1124 y Fy(\))p Fx(;)g(\036)1268 1139 y FC(0)1337 1124 y Fy(+)29 b Fx(!)t(t)p Fy(\),)46 b(where)e Fx(')p Fy(\()p Fx(t)p Fy(;)17 b Fx(')2190 1139 y FC(0)2229 1124 y Fx(;)g(\036)2331 1139 y FC(0)2370 1124 y Fy(\))43 b(is)g(the)h(solution)e(of)h(\(3.27\))f(with)h(initial)-118 1244 y(conditions)32 b Fx(')27 b Fy(=)h Fx(')608 1259 y FC(0)647 1244 y Fy(,)33 b Fx(\036)27 b Fy(=)h Fx(\036)954 1259 y FC(0)1025 1244 y Fy(for)k Fx(t)c Fy(=)g(0.)28 1364 y(The)g(rotation)e(n)m(um)m(b)s(er)h(is)g(in)m(tro)s(duced)g(to)g (study)h(the)f FA(aver)-5 b(age)34 b Fy(v)-5 b(ariation)24 b(of)j(the)g(phase)h Fx(')f Fy(o)m(v)m(er)h(the)g(time.)-118 1485 y(It)33 b(is)f(therefore)h(sensible)g(to)f(de\014ne)i(it)d(as)i (the)g(follo)m(wing)d(limit)834 1686 y Fx(')p Fy(\()p Fx(t)p Fy(;)17 b Fx(')1079 1701 y FC(0)1118 1686 y Fx(;)g(\036)1220 1701 y FC(0)1259 1686 y Fy(\))22 b Fs(\000)h Fx(')p Fy(\(0;)17 b Fx(')1678 1701 y FC(0)1716 1686 y Fx(;)g(\036)1818 1701 y FC(0)1857 1686 y Fy(\))p 834 1731 1062 4 v 1347 1822 a Fx(t)1933 1754 y Fy(=)2046 1686 y(1)p 2046 1731 49 4 v 2053 1822 a Fx(t)2122 1618 y Fq(Z)2221 1645 y Fr(t)2177 1844 y FC(0)2267 1754 y Fx(F)31 b Fy(\(\010\()p Fx(s)p Fy(;)17 b Fx(')2661 1769 y FC(0)2700 1754 y Fx(;)g(\036)2802 1769 y FC(0)2841 1754 y Fy(\)\))f Fx(ds;)709 b Fy(\(3.28\))-118 2013 y(when)35 b(the)g(time)e Fx(t)h Fy(go)s(es)h(to)f(in\014nit)m(y)f (and)i Fx(')f Fy(is)g(considered)h(in)e(the)i(lift.)46 b(There)36 b(are)e(man)m(y)h(things)e(to)h(pro)m(v)m(e.)-118 2133 y(W)-8 b(e)29 b(will)e(\014rst)i(sho)m(w)h(that)e(these)i(time)e (a)m(v)m(erages)i(con)m(v)m(erge)h(for)d FA(al)5 b(l)39 b Fy(\()p Fx(')2504 2148 y FC(0)2543 2133 y Fx(;)17 b(\036)2645 2148 y FC(0)2684 2133 y Fy(\))27 b Fs(2)h Fx(B)34 b Fy(and)29 b(that)g(the)g(con)m(v)m(ergence)-118 2253 y(is)c(uniform)f(in)h Fx(B)5 b Fy(.)42 b(Note)26 b(that)f(this)h(is)f(a)g(remark)-5 b(able)25 b(fact,)i(b)s(ecause)g(the)f(con)m(v)m(ergence)j(is)c(for)g (all)f(the)i(elemen)m(ts)-118 2374 y(of)32 b Fx(B)38 b Fy(instead)32 b(of)g(a)h(set)g(of)f(total)f(measure)i(as)g(it)e(is)h (t)m(ypical)g(in)g(the)h(con)m(text)h(of)e(ergo)s(dic)g(ob)5 b(jects.)28 2494 y(There)42 b(are)f(some)g(easy)h(remarks)f(on)g(the)g (ab)s(o)m(v)m(e)h(limit)37 b(that)k(can)g(b)s(e)g(made.)69 b(First,)42 b(note)f(that)f(if)g(the)-118 2615 y(limit)27 b(\(3.28\))j(con)m(v)m(erges)j(for)e(some)f(\()p Fx(')1320 2630 y FC(0)1360 2615 y Fx(;)17 b(\036)1462 2630 y FC(0)1501 2615 y Fy(\))27 b Fs(2)h Fx(B)5 b Fy(,)32 b(then)f(it)f(m)m(ust)h(also) f(con)m(v)m(erge)i(for)f(another)g(\()p Fx(')3550 2630 y FC(1)3589 2615 y Fx(;)17 b(\036)3691 2630 y FC(0)3730 2615 y Fy(\),)31 b(with)-118 2735 y(0)i Fx(<)h(')138 2750 y FC(1)201 2735 y Fs(\000)25 b Fx(')367 2750 y FC(2)440 2735 y Fx(<)34 b(\031)40 b Fy(and)c(the)g(limit)d(is)i(indep)s(enden)m (t)i(of)f Fx(')p Fy(.)54 b(Indeed,)38 b(let)e Fx(')2720 2750 y Fr(j)2756 2735 y Fy(\()p Fx(t)p Fy(\))e(=)f Fx(')p Fy(\()p Fx(t)p Fy(;)17 b Fx(')3255 2750 y Fr(j)3291 2735 y Fx(;)g(\036)3393 2750 y FC(0)3432 2735 y Fy(\))36 b(for)f Fx(j)40 b Fy(=)33 b(0)p Fx(;)17 b Fy(1.)-118 2855 y(Then)36 b(w)m(e)h(ha)m(v)m(e)f(that)f(0)d Fx(<)g(')980 2870 y FC(1)1020 2855 y Fy(\()p Fx(t)p Fy(\))24 b Fs(\000)g Fx(')1320 2870 y FC(2)1360 2855 y Fy(\()p Fx(t)p Fy(\))32 b Fx(<)g(\031)39 b Fy(for)c(all)e Fx(t)i Fy(b)s(ecause)i(otherwise)f (the)g(theorem)f(on)g(uniqueness)-118 2976 y(of)d(solutions)g(of)g (equation)g(\(3.27\))g(w)m(ould)g(b)s(e)h(con)m(tradicted.)44 b(Hence)1673 3187 y Fx(')1737 3202 y FC(1)1777 3187 y Fy(\()p Fx(t)p Fy(\))22 b Fs(\000)h Fx(')2074 3202 y FC(0)2113 3187 y Fy(\()p Fx(t)p Fy(\))-118 3399 y(is)32 b(b)s(ounded)h(and)g(the)g(claim)d(follo)m(ws.)28 3520 y(W)-8 b(e)37 b(no)m(w)g(w)m(an)m(t)h(to)e(state)h(the)g(ab)s(o)m(v)m 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Fx(T)1490 4145 y Fq(Z)1589 4171 y Fr(T)1545 4370 y FC(0)1661 4280 y Fx(F)k Fy(\(\010\()p Fx(t)p Fy(;)j Fx(';)g(\036)p Fy(\)\))p Fx(dt)27 b Fy(=)g Fx(F)2498 4239 y Ft(\003)2537 4280 y Fy(\()p Fx(';)17 b(\036)p Fy(\))-118 4545 y(for)23 b Fx(\027)6 b Fy(-almost)22 b(ev)m(ery)j(\()p Fx(';)17 b(\036)p Fy(\))27 b Fs(2)h Fx(B)5 b Fy(,)25 b(where)g(the)f(in)m (tegration)e(is)h(tak)m(en)h(with)g(resp)s(ect)g(to)g(the)f(Leb)s (esgue)i(measure.)-118 4665 y(More)31 b(precisely)-8 b(,)32 b(there)g(is)f(a)f(set)i Fx(B)1198 4680 y FC(0)1265 4665 y Fs(\032)d Fx(B)5 b Fy(,)31 b(with)g Fx(\027)6 b Fy(\()p Fx(B)25 b Fs(\000)19 b Fx(B)2090 4680 y FC(0)2130 4665 y Fy(\))27 b(=)h(0)j(suc)m(h)h(that)f(the)h(con)m(v)m(ergence)h (holds)e(for)g(all)-118 4785 y(\()p Fx(';)17 b(\036)p Fy(\))27 b Fs(2)h Fx(B)319 4800 y FC(0)358 4785 y Fy(.)44 b(Moreo)m(v)m(er)34 b(w)m(e)g(ha)m(v)m(e)f(that)g Fx(F)1514 4749 y Ft(\003)1585 4785 y Fy(satis\014es)1549 4910 y Fq(Z)1604 5136 y Fr(B)1682 5046 y Fx(F)1759 5004 y Ft(\003)1798 5046 y Fx(d\027)h Fy(=)2034 4910 y Fq(Z)2089 5136 y Fr(B)2167 5046 y Fx(F)14 b(d\027)-118 5304 y Fy(and)33 b Fx(F)149 5268 y Ft(\003)220 5304 y Fy(is)f(in)m(v)-5 b(arian)m(t)31 b(under)j(the)f(\015o)m(w)g(\010)1445 5319 y Fr(t)1475 5304 y Fy(.)28 5425 y(W)-8 b(e)35 b(ha)m(v)m(e)g(already)f(sho)m(wn)i (that)e(the)g(con)m(v)m(ergence)j(for)d(one)g Fx(')2386 5440 y FC(0)2460 5425 y Fy(implies)e(the)i(con)m(v)m(ergence)j(of)d (\(3.28\))f(for)-118 5545 y FA(al)5 b(l)48 b Fx(')36 b Fs(2)h Fw(T)301 5509 y FC(1)344 5545 y Fy(.)60 b(F)-8 b(rom)36 b(this)i(w)m(e)h(conclude)f(that)g Fx(B)1734 5560 y FC(0)1811 5545 y Fy(can)g(b)s(e)g(c)m(hosen)i(to)d(b)s(e)h(of)g (the)g(form)f Fx(B)3313 5560 y FC(0)3389 5545 y Fy(=)g Fw(P)3561 5509 y FC(1)3602 5545 y Fy(\()p Fw(R)5 b Fy(\))32 b Fs(\002)26 b Fy(\012)3949 5560 y FC(0)3989 5545 y Fy(,)-118 5666 y(where)40 b(\012)240 5681 y FC(0)317 5666 y Fs(\032)f Fy(\012,)h(and)f Fx(F)843 5629 y Ft(\003)920 5666 y Fy(is)f(indep)s (enden)m(t)i(of)e Fx(')p Fy(.)61 b(Therefore,)41 b(w)m(e)f(can)f (consider)f Fx(F)3125 5629 y Ft(\003)3203 5666 y Fy(as)h(a)f(function)g (on)g(\012)-118 5786 y(\(de\014ned)d Fx(\027)6 b Fy(-almost)31 b(ev)m(erywhere\))37 b(whic)m(h)d(is)f(in)m(v)-5 b(arian)m(t)32 b(under)i(the)g(irrational)c(\015o)m(w)k(giv)m(en)g(b)m(y)g Fx(!)t Fy(.)45 b(Since)34 b(this)-118 5906 y(\015o)m(w)e(is)f(uniquely) h(ergo)s(dic)f(and)h(preserv)m(es)i(only)d(the)h(usual)g(Haar)f (measure)h Fx(\026)f Fy(on)h Fw(T)3062 5870 y Fr(d)3105 5906 y Fy(,)g(whic)m(h)g(is)g(ergo)s(dic,)f(w)m(e)1900 6155 y(57)p eop %%Page: 58 62 58 61 bop -118 219 a Fy(conclude)38 b(that)g Fx(F)582 182 y Ft(\003)659 219 y Fy(agrees)g(with)g(a)f(constan)m(t,)j(sa)m(y)f Fx(\013)q Fy(,)g(on)f(a)g(set)g Fx(B)2460 234 y FC(1)2536 219 y Fy(=)f Fw(T)2712 182 y FC(1)2780 219 y Fs(\002)27 b Fy(\012)2954 234 y FC(1)2994 219 y Fy(,)39 b(where)g Fx(\026)p Fy(\(\012)26 b Fs(\000)g Fy(\012)3713 234 y FC(1)3753 219 y Fy(\))36 b(=)h(0.)-118 339 y(Being)32 b(the)h(measure)g(normalized,)d(it)i(m)m(ust)h(agree)f(with)1684 586 y Fx(\013)c Fy(=)1878 450 y Fq(Z)1933 676 y Fr(B)2010 586 y Fx(F)14 b(d\027)q(:)1552 b Fy(\(3.29\))-118 836 y(As)33 b(this)f(argumen)m(t)g(w)m(orks)i(for)e(an)m(y)i(in)m(v)-5 b(arian)m(t)31 b(measure)i Fx(\027)39 b Fy(of)32 b(the)h(\015o)m(w)g (\010)2691 851 y Fr(t)2754 836 y Fy(on)f Fx(B)5 b Fy(,)33 b(w)m(e)h(conclude)f(that)1563 947 y Fq(Z)1618 1173 y Fr(B)1696 1083 y Fy(\()p Fx(F)j Fs(\000)22 b Fx(\013)q Fy(\))16 b Fx(d\027)34 b Fy(=)28 b(0)-118 1334 y(holds)36 b(for)g(an)m(y)h(in)m(v)-5 b(arian)m(t)36 b(measure)g Fx(\027)43 b Fy(on)37 b Fx(B)5 b Fy(.)55 b(F)-8 b(rom)35 b(all)g(these)j(preliminaries,)c(the)j(existence)h(of)e(the)h(limit) -118 1454 y(in)30 b(\(3.28\))f(will)f(follo)m(w)h(from)g(the)i(follo)m (wing)d(lemma,)h(whic)m(h)i(uses)h(the)f(metho)s(ds)f(of)g(Krylo)m(v)g (and)h(Bogoljub)s(o)m(v)-118 1575 y(\([6]\))-118 1784 y Fv(Lemma)37 b(3.3.1)h(\([47]\))48 b FA(L)-5 b(et)35 b Fx(G)g FA(b)-5 b(e)35 b(a)f(c)-5 b(ontinuous)35 b(function)g(on)f Fx(B)40 b FA(such)35 b(that)1701 1901 y Fq(Z)1757 2127 y Fr(B)1834 2037 y Fx(Gd\027)f Fy(=)27 b(0)-118 2287 y FA(for)35 b(any)f(invariant)h(me)-5 b(asur)g(e)34 b(for)h(the)f (\015ow)h Fx(F)49 b FA(on)34 b Fx(B)5 b FA(.)45 b(Then)1138 2486 y Fy(1)p 1055 2531 215 4 v 1055 2622 a Fx(b)22 b Fs(\000)h Fx(a)1296 2418 y Fq(Z)1395 2445 y Fr(b)1351 2644 y(a)1446 2554 y Fx(G)p Fy(\(\010)1631 2569 y Fr(t)1661 2554 y Fy(\()p Fx(\014)6 b Fy(\)\))p Fx(dt)27 b Fs(!)h Fy(0)p Fx(;)106 b FA(as)35 b Fx(b)22 b Fs(\000)h Fx(a)28 b Fs(!)f(1)-118 2810 y FA(for)35 b(al)5 b(l)34 b Fx(\014)f Fy(=)28 b(\()p Fx(';)17 b(\036)p Fy(\))34 b FA(and)g(the)h(c)-5 b(onver)g(genc)g(e)33 b(is)i(uniform.)28 3020 y Fv(Pro)s(of:)60 b Fy(Since)41 b(the)g(space)h(of)f(con)m(tin)m(uous)g(functions)g(on)g Fx(B)5 b Fy(,)43 b Fx(C)7 b Fy(\()p Fx(B)e Fy(\),)43 b(with)d(the)i(uniform)d(top)s(ology)-8 b(,)41 b(is)-118 3140 y(separable,)c(w)m(e)f(can)g(\014nd)h(a)e(dense)j(linear)c (subspace)j Fx(D)i Fy(generated)d(b)m(y)h(a)f(coun)m(table)f(set)i(of)e (functions.)53 b(W)-8 b(e)-118 3261 y(assume)35 b(that)f(the)h (statemen)m(t)h(is)e(false)g(for)g(some)g(function)g Fx(G)e Fs(2)f Fx(C)7 b Fy(\()p Fx(B)e Fy(\).)50 b(W)-8 b(e)35 b(ma)m(y)f(then)h(c)m(ho)s(ose)h Fx(D)h Fy(so)e(that)-118 3381 y Fx(G)28 b Fs(2)g Fx(D)35 b Fy(and)e(select)g(sequences)i Fx(b)1136 3396 y Fr(j)1173 3381 y Fx(;)17 b(a)1268 3396 y Fr(j)1305 3381 y Fx(;)g(\014)1404 3396 y Fr(j)1473 3381 y Fy(suc)m(h)34 b(that)e Fx(b)1945 3396 y Fr(j)2004 3381 y Fs(\000)23 b Fx(a)2155 3396 y Fr(j)2219 3381 y Fs(!)k(1)33 b Fy(and)1145 3656 y(lim)1126 3717 y Fr(j)t Ft(!1)1445 3589 y Fy(1)p 1326 3633 288 4 v 1326 3724 a Fx(b)1367 3739 y Fr(j)1426 3724 y Fs(\000)23 b Fx(a)1577 3739 y Fr(j)1640 3520 y Fq(Z)1740 3547 y Fr(b)1770 3557 y Fj(j)1695 3746 y Fr(a)1732 3756 y Fj(j)1823 3656 y Fx(G)17 b Fy(\(\010)2025 3671 y Fr(t)2055 3656 y Fy(\()p Fx(\014)2148 3671 y Fr(j)2184 3656 y Fy(\)\))g Fx(dt)27 b Fs(!)h Fx(\016)j Fs(6)p Fy(=)d(0)p Fx(:)-118 3927 y Fy(W)-8 b(e)46 b(ma)m(y)f(also)g(assume)h(that)f Fx(\014)1125 3942 y Fr(j)1211 3927 y Fs(!)50 b Fx(\014)6 b Fy(.)82 b(Using)45 b(the)h(Can)m(tor)g(diagonal)d(pro)s(cess,)50 b(and)45 b(using)h(that)f Fx(D)j Fy(is)-118 4047 y(coun)m(table,)33 b(w)m(e)g(can)g(pic)m(k)g(a)f(subsequence,)37 b(whic)m(h)c(w)m(e)g (call)e Fx(a)2191 4062 y Fr(j)2228 4047 y Fx(;)17 b(b)2313 4062 y Fr(j)2350 4047 y Fx(;)g(\014)2449 4062 y Fr(j)2517 4047 y Fy(again,)32 b(suc)m(h)i(that)1549 4255 y(1)p 1430 4299 V 1430 4390 a Fx(b)1471 4405 y Fr(j)1530 4390 y Fs(\000)22 b Fx(a)1680 4405 y Fr(j)1744 4186 y Fq(Z)1843 4213 y Fr(b)1873 4223 y Fj(j)1799 4412 y Fr(a)1836 4422 y Fj(j)1927 4322 y Fx(H)i Fy(\(\010)2140 4337 y Fr(t)2170 4322 y Fy(\()p Fx(\014)2263 4337 y Fr(j)2300 4322 y Fy(\)\))16 b Fx(dt)-118 4598 y Fy(con)m(v)m(erges)43 b(for)e(all)e Fx(H)50 b Fs(2)43 b Fx(D)s Fy(.)69 b(This)41 b(limit)d(de\014nes)43 b(a)e(linear)e(functional)h Fx(l)45 b Fy(=)d Fx(l)r Fy(\()p Fx(H)8 b Fy(\),)43 b Fx(H)50 b Fs(2)42 b Fx(D)s Fy(,)i(and)d(since)g Fx(l)-118 4719 y Fy(is)h(b)s(ounded)h(\(with)f(norm)f(1\))h(it)g (extends)i(uniquely)f(to)f(a)g(b)s(ounded)h(linear)e(functional)f(on)j Fx(C)7 b Fy(\()p Fx(B)e Fy(\).)73 b(Since)-118 4839 y Fx(b)-77 4854 y Fr(j)-10 4839 y Fs(\000)31 b Fx(a)149 4854 y Fr(j)233 4839 y Fs(!)47 b(1)d Fy(one)h(v)m(eri\014es)g(that)f Fx(l)j Fy(is)c(in)m(v)-5 b(arian)m(t.)78 b(By)44 b(the)h(Riesz)f (Represen)m(tation)h(Theorem)g(\(see,)j(for)-118 4959 y(instance,)33 b([73]\))f Fx(l)j Fy(de\014nes)f(an)f(in)m(v)-5 b(arian)m(t)31 b(measure)i Fx(\027)39 b Fy(on)32 b Fx(B)5 b Fy(.)44 b(But)32 b(b)m(y)i(our)e(assumption,)1455 5076 y Fq(Z)1510 5302 y Fr(B)1587 5212 y Fx(Gd\027)i Fy(=)28 b Fx(l)r Fy(\()p Fx(G)p Fy(\))f(=)h Fx(\016)k Fs(6)p Fy(=)27 b(0)-118 5462 y(whic)m(h)33 b(is)f(a)g(con)m(tradiction.)43 b Fh(\003)28 5583 y Fy(No)m(w)d(w)m(e)h(apply)e(the)h(ab)s(o)m(v)m(e)g (lemma)e(to)h Fx(G)g Fy(=)g Fx(F)i Fs(\000)27 b Fx(\013)40 b Fy(for)f(whic)m(h)h(the)g(h)m(yp)s(othesis)h(ha)m(v)m(e)g(already)e (b)s(een)-118 5703 y(v)m(eri\014ed.)44 b(Hence)1455 5799 y(1)p 1444 5843 71 4 v 1444 5935 a Fx(T)1541 5731 y Fq(Z)1640 5757 y Fr(T)1596 5956 y FC(0)1712 5866 y Fy(\()p Fx(F)14 b Fy(\(\010)1935 5881 y Fr(t)1965 5866 y Fy(\()p Fx(\014)6 b Fy(\)\))21 b Fs(\000)i Fx(\013)q Fy(\))16 b Fx(dt)1900 6155 y Fy(58)p eop %%Page: 59 63 59 62 bop -118 219 a Fy(con)m(v)m(erges)32 b(to)f(zero)f(and)h(the)g (con)m(v)m(ergence)i(is)c(uniform.)41 b(Th)m(us)32 b(the)f(expression)h (\(3.28\))e(con)m(v)m(erges)i(to)e(a)g(limit)-118 339 y Fx(\013)j Fy(whic)m(h)g(is)f(indep)s(enden)m(t)i(of)e Fx(\014)h Fy(=)28 b(\()p Fx(')1312 354 y FC(0)1351 339 y Fx(;)17 b(\036)1453 354 y FC(0)1492 339 y Fy(\).)44 b(Moreo)m(v)m(er,)34 b(one)f(has)g(that)1268 500 y Fx(')p Fy(\()p Fx(b)p Fy(\))22 b Fs(\000)h Fx(')p Fy(\()p Fx(a)p Fy(\))p 1268 545 494 4 v 1407 636 a Fx(b)g Fs(\000)g Fx(a)1799 568 y Fs(!)28 b Fx(\013)33 b Fy(for)f Fx(b)22 b Fs(\000)h Fx(a)28 b Fs(!)f(1)-118 781 y Fy(for)i(an)m(y)h(solution)d (of)i(\(3.27\))g(and)g(the)h(existence)g(of)f(the)h(rotation)d(n)m(um)m (b)s(er)j(has)g(therefore)f(b)s(een)h(established.)-118 901 y(W)-8 b(e)33 b(no)m(w)g(w)m(an)m(t)h(to)e(see)h(more)f(prop)s (erties)h(of)f(this)g(newly)h(de\014ned)h(ob)5 b(ject.)28 1021 y(Consider)32 b(the)g(rotation)e(n)m(um)m(b)s(er)h Fx(\013)e Fy(=)e Fx(\013)q Fy(\()p Fx(\025)p Fy(\))k(as)h(a)f(function) g(of)g Fx(\025)c Fs(2)h Fw(R)43 b Fy(and)31 b(let)g(us)h(pro)m(v)m(e)h (its)e(con)m(tin)m(uit)m(y)-8 b(.)-118 1142 y(Assume)40 b(for)f(con)m(tradiction)g(that)g Fx(\013)h Fy(is)g(not)f(con)m(tin)m (uous)h(at)g(a)f(p)s(oin)m(t)g Fx(\025)2642 1157 y FC(0)2721 1142 y Fs(2)h Fw(R)50 b Fy(and)40 b(let)f Fx(\025)3340 1157 y Fr(j)3416 1142 y Fy(b)s(e)h(a)f(sequence)-118 1262 y(con)m(v)m(erging)32 b(to)e Fx(\025)h Fy(with)g Fx(\013)q Fy(\()p Fx(\025)950 1277 y Fr(j)986 1262 y Fy(\))d Fs(!)f Fx(\013)1242 1226 y Ft(\003)1309 1262 y Fs(6)p Fy(=)h Fx(\013)q Fy(\()p Fx(\025)1571 1277 y FC(0)1610 1262 y Fy(\).)42 b(Let)32 b Fx(F)1954 1277 y Fr(j)2018 1262 y Fy(=)c Fx(F)14 b Fy(\()p Fx(';)j(\036)p Fy(;)g Fx(\025)2504 1277 y Fr(j)2539 1262 y Fy(\))27 b(=)h(cos)2839 1226 y FC(2)2895 1262 y Fx(')19 b Fs(\000)h Fy(\()p Fx(Q)p Fy(\()p Fx(\036)p Fy(\))f Fs(\000)h Fx(\025)3497 1277 y Fr(j)3533 1262 y Fy(\))d(sin)3707 1222 y FC(2)3764 1262 y Fx(')31 b Fy(and)-118 1383 y(\010)-48 1335 y Fr(j)-48 1405 y(t)21 1383 y Fy(b)s(e)h(the)g(corresp)s(onding)g(\015o)m(w)g(for) f Fx(\025)d Fy(=)f Fx(\025)1541 1398 y Fr(j)1578 1383 y Fy(.)43 b(If)32 b Fx(\027)1793 1398 y Fr(j)1861 1383 y Fy(is)g(an)m(y)g(in)m(v)-5 b(arian)m(t)31 b(measure)h(for)f(this)g (\015o)m(w)i(w)m(e)g(ha)m(v)m(e)g(that)1570 1608 y Fx(\013)q Fy(\()p Fx(\025)1728 1623 y Fr(j)1764 1608 y Fy(\))27 b(=)1933 1472 y Fq(Z)1988 1698 y Fr(B)2066 1608 y Fx(F)2129 1623 y Fr(j)2165 1608 y Fx(d\027)2264 1623 y Fr(j)2301 1608 y Fx(:)-118 1829 y Fy(W)-8 b(e)31 b(ma)m(y)f(supp)s(ose,)i(due)g (to)e(the)h(con)m(v)m(ergence)i Fx(F)1715 1844 y Fr(j)1779 1829 y Fs(!)28 b Fx(F)14 b Fy(,)30 b(that)h Fx(\027)2299 1844 y Fr(j)2363 1829 y Fs(!)c Fx(\027)37 b Fy(in)30 b(the)h(w)m(eak)h(top)s(ology)d(of)h(measures,)-118 1949 y(so)j(that)1505 1957 y Fq(Z)1560 2183 y Fr(B)1638 2093 y Fx(F)1701 2108 y FC(0)1740 2093 y Fx(d\027)1839 2108 y Fr(j)1903 2093 y Fs(!)2031 1957 y Fq(Z)2086 2183 y Fr(B)2163 2093 y Fx(F)2226 2108 y FC(0)2266 2093 y Fx(d\027)q(;)-118 2304 y Fy(and)g(due)g(to)f(all)f(this)h Fx(\027)39 b Fy(is)32 b(in)m(v)-5 b(arian)m(t)31 b(under)i(the)g(\015o)m(w)h (induced)f(b)m(y)g Fx(F)2504 2319 y FC(0)2544 2304 y Fy(,)f(\010)2673 2268 y FC(0)2673 2329 y Fr(t)2713 2304 y Fy(.)44 b(Finally)-8 b(,)30 b(since)1394 2478 y Fs(j)p Fx(F)1485 2493 y Fr(j)1543 2478 y Fs(\000)23 b Fx(F)1706 2493 y FC(0)1746 2478 y Fs(j)k(\024)h(j)p Fx(\025)1991 2493 y Fr(j)2049 2478 y Fs(\000)23 b Fx(\025)2206 2493 y FC(0)2245 2478 y Fs(j)28 b(!)f Fy(0)p Fx(;)-118 2652 y Fy(w)m(e)34 b(conclude)f(that)582 2849 y Fx(\013)q Fy(\()p Fx(\025)740 2864 y Fr(j)776 2849 y Fy(\))27 b(=)945 2714 y Fq(Z)1000 2939 y Fr(B)1077 2849 y Fx(F)1140 2864 y Fr(j)1177 2849 y Fx(d\027)1276 2864 y Fr(j)1340 2849 y Fy(=)h Fx(O)s Fy(\()p Fs(j)p Fx(\025)1645 2864 y Fr(j)1702 2849 y Fs(\000)23 b Fx(\025)1859 2864 y FC(0)1898 2849 y Fs(j)p Fy(\))f(+)2084 2714 y Fq(Z)2139 2939 y Fr(B)2217 2849 y Fx(F)2280 2864 y FC(0)2319 2849 y Fx(d\027)2418 2864 y Fr(j)2482 2849 y Fs(!)2609 2714 y Fq(Z)2665 2939 y Fr(B)2742 2849 y Fx(F)2805 2864 y FC(0)2845 2849 y Fx(d\027)34 b Fy(=)27 b Fx(\013)q Fy(\()p Fx(\025)3239 2864 y FC(0)3278 2849 y Fy(\))-118 3070 y(whic)m(h)33 b(is)f(a)g(con)m(tradiction)g(with)g(the)h(assumptions.)43 b(W)-8 b(e)33 b(therefore)g(obtain)-118 3246 y Fv(Theorem)k(3.3.2)h (\([47]\))48 b FA(F)-7 b(or)34 b(r)-5 b(e)g(al)34 b Fx(\025)h FA(the)g(expr)-5 b(ession)34 b(\(3.28\))g(or,)g(for)h(short,)1706 3407 y Fx(')p Fy(\()p Fx(t)p Fy(\))22 b Fs(\000)h Fx(')p Fy(\(0\))p 1706 3451 486 4 v 1931 3543 a Fx(t)-118 3679 y FA(c)-5 b(onver)g(ges)36 b(when)g Fx(t)31 b Fs(!)g Fy(+)p Fs(1)p FA(,)37 b(uniformly)g(with)g(r)-5 b(esp)g(e)g(ct)36 b(to)h(initial)g(c)-5 b(onditions)35 b Fy(\()p Fx(')2967 3694 y FC(0)3007 3679 y Fx(;)17 b(\036)3109 3694 y FC(0)3148 3679 y Fy(\))31 b Fs(2)h Fx(B)5 b FA(,)37 b(to)g(a)g(function)-118 3799 y Fx(\013)f Fy(=)f Fx(\013)q Fy(\()p Fx(\025)p Fy(\))j FA(which)g(is)h(indep)-5 b(endent)38 b(of)g Fy(\()p Fx(')1477 3814 y FC(0)1517 3799 y Fx(;)17 b(\036)1619 3814 y FC(0)1657 3799 y Fy(\))p FA(,)40 b(but)g(c)-5 b(ontinuously)39 b(dep)-5 b(endent)38 b(on)g Fx(\025)p FA(.)57 b(Mor)-5 b(e)g(over,)40 b Fx(\013)q Fy(\()p Fx(\025)p Fy(\))e FA(is)-118 3920 y(monotone)c(incr)-5 b(e)g(asing,)33 b(e)-5 b(qual)35 b(to)g(zer)-5 b(o)34 b(for)h Fx(\025)27 b Fs(\024)i Fx(\025)1794 3884 y Ft(\003)1868 3920 y FA(for)35 b(some)f Fx(\025)2330 3884 y Ft(\003)2404 3920 y FA(and)g Fx(\013)q Fy(\()p Fx(\025)p Fy(\))27 b Fs(!)h Fy(+)p Fs(1)34 b FA(for)h Fx(\025)27 b Fs(!)g(1)p FA(.)-118 4095 y Fv(Remark)37 b(3.3.3)49 b FA(In)29 b(some)g(c)-5 b(ases)29 b(the)g(r)-5 b(otation)30 b(numb)-5 b(er)29 b(for)h(the)g(discr)-5 b(ete)29 b(analo)-5 b(g)28 b(of)i(Schr\177)-50 b(odinger)28 b(e)-5 b(quation)-118 4216 y(with)38 b(quasi-p)-5 b(erio)g(dic)38 b(p)-5 b(otential)38 b(has)g(b)-5 b(e)g(en)38 b(shown)g(to)h(have)e(str)-5 b(onger)39 b(r)-5 b(e)g(gularity)39 b(pr)-5 b(op)g(erties)38 b(on)g(the)h(ener)-5 b(gy)-118 4336 y(\(of)41 b(H\177)-50 b(older)41 b(typ)-5 b(e)41 b(in)g(the)h(c)-5 b(ase)40 b(of)h(one)g(fr)-5 b(e)g(quency)41 b(and)g(a)g(c)-5 b(ertain)41 b(mo)-5 b(dulus)41 b(of)g(c)-5 b(ontinuity)41 b(in)g(the)g(c)-5 b(ase)41 b(of)-118 4457 y(sever)-5 b(al)34 b(b)-5 b(asic)34 b(fr)-5 b(e)g(quencies,)34 b(se)-5 b(e)34 b([34])h(and)f(the)h(r)-5 b(efer)g(enc)g(es)34 b(ther)-5 b(ein\).)28 4632 y Fy(There)30 b(only)f(remain)e(few)i(items) f(of)h(the)g(ab)s(o)m(v)m(e)h(theorem)e(to)h(b)s(e)g(pro)m(v)m(ed.)44 b(As)29 b(already)f(men)m(tioned,)i(a)e(zero)-118 4752 y(of)j(a)h(solution)f Fx(x)p Fy(\()p Fx(t)p Fy(\))h(of)g(equation)f (\(3.21\))h(corresp)s(onds)h(to)f(a)f(v)-5 b(alue)32 b Fx(t)2450 4767 y FC(0)2521 4752 y Fy(for)g(whic)m(h)g Fx(')p Fy(\()p Fx(t)3085 4767 y FC(0)3125 4752 y Fy(\))c(=)f(0)32 b(mo)s(dulus)f Fx(\031)36 b Fy(and)-118 4873 y(b)m(y)h(\(3.27\))e(one)h (has)g(that)g Fx(')944 4837 y Ft(0)967 4873 y Fy(\()p Fx(t)1040 4888 y FC(0)1080 4873 y Fy(\))d(=)g(0.)53 b(This)36 b(sho)m(ws)i(that)e Fx(')f Fy(increases)i(at)f(suc)m(h)h(a)f(zero)g (and)g(therefore)g(one)-118 4993 y(has)f(one)g(zero)g(p)s(er)f (increase)i(of)e Fx(')g Fy(b)m(y)i Fx(\031)t Fy(.)49 b(Therefore,)36 b(if)e Fx(N)10 b Fy(\()p Fx(T)k Fy(;)j Fx(\025;)g(x)p Fy(\))34 b(is)g(the)i(n)m(um)m(b)s(er)f(of)f(zero)s(es)h (in)f(the)h(time)-118 5114 y(in)m(terv)-5 b(al)31 b([0)p Fx(;)17 b(T)d Fy(])32 b(of)g(a)g(solution)g Fx(x)p Fy(\()p Fx(t)p Fy(\),)h(one)g(has)f(that)1439 5342 y(lim)1411 5404 y Fr(T)10 b Ft(!1)1629 5275 y Fx(\031)t(N)g Fy(\()p Fx(T)k Fy(;)j Fx(\025;)g(x)p Fy(\))p 1629 5319 494 4 v 1841 5411 a Fx(T)2161 5342 y Fy(=)27 b Fx(\013)q Fy(\()p Fx(\025)p Fy(\))p Fx(:)28 5545 y Fy(With)47 b(this)f(iden)m(tit)m(y)-8 b(,)51 b(the)c(remaining)e(items)h(of)g(the)i(ab)s(o)m(v)m(e)f(theorem) g(are)g(an)g(easy)h(consequence)i(of)-118 5666 y(Sturmian)34 b(theory)-8 b(.)54 b(No)m(w)36 b(w)m(e)h(pro)m(v)m(e)g(an)f(imp)s (ortan)m(t)e(result)i(whic)m(h)g(rev)m(eals)h(the)f(imp)s(ortance)e(of) i(the)g(ob)5 b(ject)-118 5786 y(already)32 b(de\014ned.)46 b(W)-8 b(e)34 b(will)c(call)i FA(sp)-5 b(e)g(ctr)g(al)34 b(gap)39 b Fy(to)32 b(an)m(y)i(connected)h(comp)s(onen)m(t)e(of)f(the)h (resolv)m(en)m(t)h(set)g(in)e(the)-118 5906 y(real)g(line.)1900 6155 y(59)p eop %%Page: 60 64 60 63 bop -118 219 a Fv(Theorem)37 b(3.3.4)h(\(Gap)f(lab)s(elling,)f ([47]\))49 b FA(If)34 b Fx(q)k FA(is)c(quasi-p)-5 b(erio)g(dic)34 b(with)g(fr)-5 b(e)g(quency)34 b Fx(!)k FA(and)c Fx(I)43 b FA(is)34 b(an)g(op)-5 b(en)-118 339 y(interval)34 b(in)h(a)g(sp)-5 b(e)g(ctr)g(al)34 b(gap,)g(then)h(ther)-5 b(e)35 b(exist)f(inte)-5 b(gers)35 b Fv(k)28 b Fs(2)g Fw(Z)2299 303 y Fr(d)2371 339 y FA(such)35 b(that)1653 581 y Fx(\013)q Fy(\()p Fx(\025)p Fy(\))27 b(=)1990 514 y Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)p 1990 558 245 4 v 2088 649 a Fy(2)-118 799 y FA(for)35 b(al)5 b(l)34 b Fx(\025)28 b Fs(2)g Fx(I)8 b FA(.)28 996 y Fv(Pro)s(of:)39 b Fy(W)-8 b(e)24 b(recall)e(that,)j(in) d(the)i(resolv)m(en)m(t)h(set)f(and)f(in)g(the)h(previous)f(notations,) i Fx(\037)3150 1011 y Ft(\006)3209 996 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))23 b(\(the)h(solutions)-118 1116 y(square)31 b(in)m(tegrable)f(in)f Fs(\0061)p Fy(\))i(and)f Fx(G)p Fy(\()p Fx(t;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))30 b(\(the)g(Green's)i(function)e(for)f Fx(s)f Fy(=)f Fx(t)p Fy(\))k(are)f(w)m(ell)g(de\014ned)i(and)e(b)s(oth)1255 1358 y Fx(G)p Fy(\()p Fx(t;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))194 b(and)2197 1291 y Fx(d)p 2180 1336 86 4 v 2180 1427 a(dt)2275 1358 y(G)p Fy(\()p Fx(t;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))-118 1577 y(are)37 b(quasi-p)s(erio)s(dic)e(functions)i (with)f(frequency)j(v)m(ector)f Fx(!)t Fy(.)56 b(The)38 b(result)f(for)g Fx(G)g Fy(w)m(as)h(obtained)e(at)h(the)g(end)-118 1697 y(of)d(section)h(3.2.4,)g(whereas)h(the)f(result)g(for)f(its)g (deriv)-5 b(ativ)m(e)35 b(uses)h(similar)31 b(tec)m(hniques)37 b(and)e(can)g(b)s(e)g(found)f(in)-118 1817 y([79].)43 b(No)m(w)33 b(w)m(e)h(normalize)c Fx(\037)975 1832 y FC(+)1067 1817 y Fy(and)j Fx(\037)1318 1832 y Ft(\000)1409 1817 y Fy(so)g(that)f(their)g(W)-8 b(ronskian)33 b(equals)g(one.)44 b(W)-8 b(e)33 b(then)g(ha)m(v)m(e)1368 2010 y Fx(G)p Fy(\()p Fx(t;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))27 b(=)g Fx(\037)1927 2025 y FC(+)1987 2010 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))p Fx(\037)2260 2025 y Ft(\000)2318 2010 y Fy(\()p Fx(t;)g(\025)p Fy(\))-118 2203 y(and)937 2280 y Fx(d)p 919 2325 V 919 2416 a(dt)1015 2347 y(G)p Fy(\()p Fx(t;)g(t)p Fy(;)g Fx(\025)p Fy(\))27 b(=)h Fx(\037)1575 2362 y FC(+)1634 2347 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))p Fx(\037)1907 2306 y Ft(0)1907 2372 y(\000)1966 2347 y Fy(\()p Fx(t)p Fy(;)g Fx(\025)p Fy(\))22 b(+)g Fx(\037)2359 2306 y Ft(0)2359 2372 y FC(+)2418 2347 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))p Fx(\037)2691 2362 y Ft(\000)2750 2347 y Fy(\()p Fx(t)p Fy(;)g Fx(\025)p Fy(\))p Fx(;)-118 2541 y Fy(so)33 b(it)e(is)h(clear)g(that)h(at)f(a)g (zero)h(of)f Fx(G)p Fy(\()p Fx(t;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))32 b(either)h Fx(\037)1897 2556 y FC(+)1988 2541 y Fy(or)g Fx(\037)2169 2556 y Ft(\000)2260 2541 y Fy(will)e(v)-5 b(anish.)43 b(Hence,)34 b(at)e(suc)m(h)i(zero)1139 2716 y Fx(d)p 1121 2760 V 1121 2851 a(dt)1217 2783 y(G)p Fy(\()p Fx(t;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))27 b(=)h Fs(\006)1810 2702 y Fq(\000)1856 2783 y Fx(\037)1917 2798 y FC(+)1976 2783 y Fx(\037)2037 2742 y Ft(0)2037 2808 y(\000)2118 2783 y Fy(+)22 b Fx(\037)2277 2742 y Ft(0)2277 2808 y FC(+)2336 2783 y Fx(\037)2397 2798 y Ft(\000)2456 2702 y Fq(\001)2530 2783 y Fy(=)27 b Fs(\006)p Fy(1)p Fx(;)-118 3007 y Fy(the)37 b(sign)g(dep)s(ending)g(on)g(whether)h Fx(\037)1312 3022 y FC(+)1408 3007 y Fy(or)f Fx(\037)1593 3022 y Ft(\000)1689 3007 y Fy(is)f(zero.)57 b(An)m(yw)m(a)m(y)-8 b(,)41 b Fx(G)p Fy(\()p Fx(t;)17 b(t)p Fy(;)g Fx(\025)p Fy(\),)37 b(and)g(similarly)c(its)k(extension)-118 3127 y(\000\()p Fs(\001)p Fx(;)17 b(\025)p Fy(\))32 b(to)g Fw(T)362 3091 y Fr(d)406 3127 y Fy(,)g(has)h(only)f(simple)f(zero)s (es.)45 b(No)m(w)33 b(w)m(e)g(use)h(the)f(follo)m(wing)d(result,)-118 3324 y Fv(Lemma)37 b(3.3.5)h(\([47],)f([79],[31)q(]\))48 b FA(L)-5 b(et)34 b Fx(f)11 b Fy(\()p Fx(t)p Fy(\))32 b FA(b)-5 b(e)33 b(a)f(function)h(such)f(that)h(b)-5 b(oth)33 b Fx(f)44 b FA(and)32 b Fx(f)3215 3288 y Ft(0)3271 3324 y FA(ar)-5 b(e)32 b(quasi-p)-5 b(erio)g(dic)-118 3444 y(with)34 b(the)f(same)g(fr)-5 b(e)g(quency)34 b(ve)-5 b(ctor)34 b Fx(!)t FA(.)43 b(Assume)34 b(that)g Fx(f)45 b FA(has)33 b(only)h(simple)f(zer)-5 b(o)g(es.)43 b(Then)33 b(the)h(numb)-5 b(er)34 b Fx(N)10 b Fy(\()p Fx(T)k Fy(\))-118 3565 y FA(of)34 b(zer)-5 b(o)g(es)35 b(of)f Fx(f)11 b Fy(\()p Fx(t)p Fy(\))35 b FA(in)f Fy([0)p Fx(;)17 b(T)d Fy(])35 b FA(satis\014es)f(the)h(identity)1528 3812 y Fy(lim)1500 3874 y Fr(T)10 b Ft(!1)1718 3745 y Fx(\031)t(N)g Fy(\()p Fx(T)k Fy(\))p 1718 3790 295 4 v 1830 3881 a Fx(T)2050 3812 y Fy(=)27 b Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)-118 4047 y FA(for)35 b(a)f(suitable)h Fv(k)28 b Fs(2)g Fw(Z)726 4011 y Fr(d)764 4047 y FA(.)-118 4244 y Fy(to)k(deduce)i(that)f(the)g(limit)1467 4395 y(lim)1452 4454 y Fr(t)p Ft(!1)1645 4327 y Fx(\031)t(N)1782 4342 y FC(2)1821 4327 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))p 1645 4372 389 4 v 1821 4463 a Fx(t)2071 4395 y Fy(=)27 b Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)o Fx(;)-118 4597 y Fy(for)28 b(some)h Fv(k)f Fs(2)g Fw(Z)518 4561 y Fr(d)556 4597 y Fy(,)i(if)e Fx(N)777 4612 y FC(2)816 4597 y Fy(\()p Fx(T)14 b Fy(;)j Fx(\025)p Fy(\))28 b(is)h(the)g(n)m(um) m(b)s(er)g(of)g(zero)s(es)h(of)e Fx(G)p Fy(\()p Fs(\001)p Fx(;)17 b Fs(\001)p Fy(;)g Fx(\025)p Fy(\))28 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b(is)h(con)m(tin)m(uous)g(it)e(follo)m(ws)h(that)g Fx(\013)q Fy(\()p Fx(\025)p Fy(\))g(is)g(constan)m(t)i(in)d Fx(I)8 b Fy(.)p Fh(\003)28 5567 y Fy(Once)34 b(a)e(frequency)i(v)m (ector)g Fx(!)i Fy(is)c(\014xed,)i(w)m(e)g(will)c(refer)j(to)f(the)h FA(mo)-5 b(dule)34 b(of)h(half-r)-5 b(esonanc)g(es)38 b Fy(as)33 b(the)g(set)1359 5812 y Fs(M)p Fy(\()p Fx(!)t Fy(\))26 b(=)1750 5671 y Fq(\032)1834 5744 y Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)p 1834 5789 245 4 v 1932 5880 a Fy(2)2089 5812 y(;)g Fv(k)28 b Fs(2)g Fw(Z)2383 5771 y Fr(d)2421 5671 y Fq(\033)2512 5812 y Fx(:)1900 6155 y Fy(60)p eop %%Page: 61 65 61 64 bop -118 219 a Fv(Remark)37 b(3.3.6)h(\(Geometric)d(in)m (terpretation)g(of)j(the)f(rotation)f(n)m(um)m(b)s(er\))49 b FA(This)26 b(the)-5 b(or)g(em)27 b(gives)g(an)-118 339 y(interpr)-5 b(etation)41 b(of)f(the)h(r)-5 b(otation)41 b(numb)-5 b(er)41 b(when)f Fx(\025)h FA(is)f(in)h(the)g(interior)g(of)f (a)h(sp)-5 b(e)g(ctr)g(al)41 b(gap.)62 b(We)41 b(have)f(just)-118 459 y(se)-5 b(en)37 b(that)h(in)f(this)g(situation)h(the)f(r)-5 b(otation)38 b(numb)-5 b(er)37 b(is)g(in)g Fs(M)p Fy(\()p Fx(!)t Fy(\))p FA(,)g(the)g(mo)-5 b(dule)37 b(of)g(half-fr)-5 b(e)g(quencies.)51 b(That)-118 580 y(is,)34 b(ther)-5 b(e)35 b(exist)g(inte)-5 b(gers)34 b Fv(k)28 b Fy(=)g(\()p Fx(k)1128 595 y FC(1)1167 580 y Fx(;)17 b(:)g(:)g(:)f(;)h(k)1437 595 y Fr(d)1476 580 y Fy(\))28 b Fs(2)g Fw(Z)1705 544 y Fr(d)1778 580 y FA(such)35 b(that)1640 819 y Fx(\013)q Fy(\()p Fx(\025)p Fy(\))27 b(=)1976 751 y Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)p 1976 796 245 4 v 2074 887 a Fy(2)2231 819 y Fx(:)-118 1029 y FA(We)41 b(ar)-5 b(e)40 b(going)g(to)h(give)f(a) h(me)-5 b(aning)39 b(to)i(these)g(inte)-5 b(gers.)62 b(We)41 b(have)f(alr)-5 b(e)g(ady)40 b(se)-5 b(en,)41 b(in)g(se)-5 b(ction)40 b(3.2.4,)h(that)-118 1149 y(the)33 b(pr)-5 b(oje)g(ctivization)31 b(of)i(the)g(solution)f(in)h Fx(L)1541 1113 y FC(2)1581 1149 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))31 b FA(is)i(quasi-p)-5 b(erio)g(dic)31 b(with)i(fr)-5 b(e)g(quency)32 b Fx(!)t FA(.)44 b(In)32 b(p)-5 b(articular,)-118 1269 y(the)32 b(evolution)f(of)h(this)f(angle) g Fx(')p Fy(\()p Fx(t)p Fy(\))h FA(is)g(also)f(a)h(quasi-p)-5 b(erio)g(dic)30 b(function,)i(but)h(p)-5 b(ossibly)31 b(with)g(fr)-5 b(e)g(quency)32 b(ve)-5 b(ctor)-118 1390 y Fx(!)t(=)p Fy(2)34 b FA(inste)-5 b(ad)34 b(of)g Fx(!)t FA(.)44 b(This)35 b(implies)e(that)j(ther)-5 b(e)35 b(exists)f(a)h(c)-5 b(ontinuous)34 b(function)1708 1574 y Fy(\010)28 b(:)g Fw(T)1924 1533 y Fr(d)1996 1574 y Fs(!)f Fw(T)-118 1758 y FA(such)35 b(that)1597 1911 y Fx(')p Fy(\()p Fx(t)p Fy(\))28 b(=)f(\010)1990 1770 y Fq(\022)2074 1843 y Fx(!)t(t)p 2074 1888 100 4 v 2099 1979 a Fy(2)2183 1770 y Fq(\023)2273 1911 y Fx(:)-118 2135 y FA(Then,)k(for)g(e)-5 b(ach)30 b(fundamental)g(gener)-5 b(ator,)32 b(given)e(by)h(taking)g(al)5 b(l)31 b(c)-5 b(omp)g(onents)29 b(in)i Fw(T)3042 2099 y Fr(d)3117 2135 y FA(c)-5 b(onstant,)32 b(exc)-5 b(ept)30 b(fr)-5 b(om)-118 2255 y(one,)45 b(say)f Fx(\022)333 2270 y Fr(j)370 2255 y FA(,)h(which)e(turns)h(ar)-5 b(ound)43 b Fw(T)1391 2219 y FC(1)1434 2255 y FA(,)j(we)d(c)-5 b(an)43 b(me)-5 b(asur)g(e)43 b(its)h(winding)e(by)i Fy(\010)p FA(,)i(by)e(me)-5 b(ans)42 b(of)i(a)f(suitable)-118 2376 y(lifting,)34 b(and)h(asso)-5 b(ciate)33 b(natur)-5 b(al)5 b(ly)36 b(to)f(it)g(an)f(inte)-5 b(ger)35 b Fx(i)1915 2391 y Fr(j)1987 2376 y FA(which)f(turns)h(to)g(b)-5 b(e)34 b Fx(k)2803 2391 y Fr(j)2840 2376 y FA(.)28 2563 y Fy(W)-8 b(e)46 b(ha)m(v)m(e)i(seen)f(that)e(the)i(rotation)d(n)m(um)m (b)s(er)i(is)g(a)f(con)m(tin)m(uous)i(function)e(of)h Fx(\025)p Fy(,)j(whic)m(h)d(is)f(monotone)-118 2683 y(increasing)35 b(and)h(is)f(constan)m(t)h(at)g(the)g(in)m(terior)e(of)h(the)h(sp)s (ectral)g(gaps.)53 b(In)36 b(principle)e(it)g(could)i(happ)s(en)g(that) -118 2804 y(there)28 b(existed)g(other)f(in)m(terv)-5 b(als)26 b(of)h(constancy)-8 b(,)29 b(apart)e(from)f(the)h(sp)s(ectral) g(gaps.)42 b(It)27 b(will)e(b)s(e)i(sho)m(wn)h(in)f(section)-118 2924 y(3.3.3)k(that)g(this)h(is)f(not)g(the)i(case,)f(i.e.)43 b(that)32 b(the)g(in)m(terv)-5 b(als)30 b(of)i(constancy)h(of)e Fx(\013)q Fy(\()p Fx(\025)p Fy(\))g(are)h(exactly)g(the)g(sp)s(ectral) -118 3044 y(gaps.)47 b(As)34 b(w)m(e)h(will)c(need)k(some)e(more)g (theory)i(on)e(the)h(extension)h(of)e(the)h(rotation)e(n)m(um)m(b)s(er) i(for)g(complex)f Fx(\025)p Fy(,)-118 3165 y(whic)m(h)g(will)d(b)s(e)j (giv)m(en)g(in)f(section)g(3.3.2,)g(w)m(e)i(go)e(on)h(with)f(other)g (prop)s(erties)h(of)f(the)h(rotation)e(n)m(um)m(b)s(er.)-118 3352 y Fv(Remark)37 b(3.3.7)h(\(Hill's)c(equation)j(with)g(quasi-p)s (erio)s(dic)f(forcing,)h([13],)h([12]\))49 b FA(The)24 b(the)-5 b(or)g(em)24 b(on)h(gap)-118 3473 y(lab)-5 b(elling)33 b(state)-5 b(d)34 b(ab)-5 b(ove)33 b(\(to)-5 b(gether)34 b(with)g(its)43 b Fy(con)m(v)m(erse)p FA(,)36 b(which)d(wil)5 b(l)34 b(b)-5 b(e)34 b(given)f(in)h(se)-5 b(ction)33 b(3.3.3\))h(c)-5 b(an)33 b(b)-5 b(e)34 b(use)-5 b(d)-118 3593 y(to)35 b(de\014ne)f(the)h(analo)-5 b(g)34 b(of)g(r)-5 b(esonanc)g(e)34 b(tongues)g(for)h(Hil)5 b(l's)35 b(e)-5 b(quation)34 b(with)h(p)-5 b(erio)g(dic)34 b(for)-5 b(cing,)34 b(namely)1469 3777 y Fx(x)1524 3736 y Ft(00)1589 3777 y Fy(+)22 b(\()p Fx(a)g Fy(+)g Fx(bq)t Fy(\()p Fx(t)p Fy(\)\))17 b Fx(x)28 b Fy(=)g(0)p Fx(;)1353 b Fy(\(3.30\))-118 3962 y FA(with)43 b Fy(\()p Fx(a;)17 b(b)p Fy(\))45 b Fs(2)f Fw(R)535 3925 y FC(2)624 3962 y FA(and)f Fx(q)48 b FA(a)43 b(quasi-p)-5 b(erio)g(dic)42 b(function)i(with)f(fr)-5 b(e)g(quency)44 b Fx(!)t FA(.)70 b(Inde)-5 b(e)g(d,)45 b(when)d Fx(d)i Fy(=)g(1)f FA(in)h(the)-118 4082 y(ab)-5 b(ove)32 b(e)-5 b(quation,)32 b(the)h(instability)g(zones)e(\(c)-5 b(al)5 b(le)-5 b(d)42 b Fy(resonance)32 b(tongues)p FA(,)i(b)-5 b(e)g(c)g(ause)32 b(of)g(their)h(shap)-5 b(e)31 b(and)h(b)-5 b(e)g(c)g(ause)-118 4202 y(this)34 b(instability)g(c)-5 b(omes)33 b(fr)-5 b(om)34 b(p)-5 b(ar)g(ametric)33 b(r)-5 b(esonanc)g(e\))33 b(ar)-5 b(e)34 b(identi\014e)-5 b(d)33 b(by)h(the)g(c)-5 b(ondition)33 b Fs(j)p FA(tr)-5 b(ac)g(e)p Fy(\()p Fs(P)3647 4217 y Fr(a;b)3739 4202 y Fy(\))p Fs(j)27 b(\025)h Fy(2)p FA(,)-118 4323 y(wher)-5 b(e)35 b Fs(P)227 4338 y Fr(a;b)355 4323 y FA(is)g(the)h(time-p)-5 b(erio)g(d)35 b(or)g(Poinc)-5 b(ar)n(\023)-47 b(e)34 b(matrix)i(for)f(the)h (\014rst-or)-5 b(der)36 b(system)f(asso)-5 b(ciate)g(d)35 b(to)h(e)-5 b(quation)-118 4443 y(\(3.30\).)47 b(As)36 b(al)5 b(l)35 b(tongues)h(ar)-5 b(e)36 b(sep)-5 b(ar)g(ate)g(d)35 b(by)h(the)g(sp)-5 b(e)g(ctrum,)36 b(which)f(in)h(this)f(p)-5 b(erio)g(dic)35 b(c)-5 b(ase)36 b(c)-5 b(onsists)35 b(of)g(non-)-118 4563 y(void)g(intervals,)h(we)f(c)-5 b(an)35 b(e)-5 b(asily)36 b(lab)-5 b(el)35 b(tongues)h(fr)-5 b(om)35 b(left)h(to)g(right.)48 b(However,)36 b(in)f(the)h(quasi-p)-5 b(erio)g(dic)35 b(c)-5 b(ase,)-118 4684 y(the)46 b(sp)-5 b(e)g(ctrum)46 b(is)g(quite)g(usual)5 b(ly)47 b(a)f(Cantor)f(set)h(\(this)g(wil)5 b(l)46 b(b)-5 b(e)46 b(se)-5 b(en)45 b(in)h(se)-5 b(ction)45 b(3.3.5\),)j(so)e(determining)-118 4804 y(instability)33 b(zones)f(is)h(not)g(so)g(e)-5 b(asy.)44 b(However,)32 b(we)h(c)-5 b(an)33 b(de\014ne)f(r)-5 b(esonanc)g(e)32 b(tongues)g(in)h(the)g Fy(\()p Fx(a;)17 b(b)p Fy(\))p FA(-p)-5 b(ar)g(ameter)-118 4925 y(plane)34 b(as)g(those)h(p)-5 b(oints)34 b Fy(\()p Fx(a;)17 b(b)p Fy(\))28 b Fs(2)g Fw(R)1203 4888 y FC(2)1284 4925 y FA(such)34 b(that)i(the)f(r)-5 b(otation)34 b(numb)-5 b(er)35 b(is)f(of)h(the)g(form)1826 5096 y Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)p 1826 5141 245 4 v 1924 5232 a Fy(2)-118 5387 y FA(for)32 b(a)f Fv(k)d Fs(2)g Fw(Z)366 5350 y Fr(d)404 5387 y FA(.)44 b(A)n(n)31 b(e)-5 b(asy)32 b(c)-5 b(omputation)31 b(shows)g(that)h(these)g (tongues)f(emanate)g(fr)-5 b(om)32 b(the)g(p)-5 b(oints)31 b(in)g(the)h Fx(b)c Fy(=)g(0)-118 5507 y FA(axis)34 b(of)h(the)g(form) 1610 5696 y Fx(a)28 b Fy(=)1793 5555 y Fq(\022)1876 5628 y Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)p 1876 5673 V 1974 5764 a Fy(2)2131 5555 y Fq(\023)2204 5577 y FC(2)2260 5696 y Fx(:)-118 5906 y FA(This)34 b(c)-5 b(an)34 b(b)-5 b(e)35 b(se)-5 b(en)34 b(in)h(\014gur)-5 b(e)34 b(3.1.)45 b(In)34 b(this)h(c)-5 b(ontext)34 b(c)-5 b(ol)5 b(lapse)-5 b(d)34 b(gaps)g(ar)-5 b(e)35 b(c)-5 b(al)5 b(le)-5 b(d)44 b Fy(instabilit)m(y)30 b(p)s(o)s(c)m(k)m(ets)p FA(.)1900 6155 y Fy(61)p eop %%Page: 62 66 62 65 bop 709 1855 a currentpoint currentpoint translate 0.82678 0.82678 scale neg exch neg exch translate 709 1855 a @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 3600 @rwi @setspecial %%BeginDocument: gap_label.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: gap_label.eps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Sat Jun 9 17:54:56 2001 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 1890 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (\(0,0\)) Cshow 4318 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (\(0,1\)) Cshow 2818 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (\(1,-1\)) Cshow 5599 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (\(2,-2\)) Cshow 2245 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (\(-1,2\)) Cshow 6527 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (\(-1,3\)) Cshow 3307 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (\(-2,4\)) Cshow 1.000 UL LTb 238 280 M 6724 0 V 0 4592 V -6724 0 V 238 280 L 1.000 UL LT0 238 4413 Pnt 568 3862 Pnt 670 3586 Pnt 785 3448 Pnt 950 3035 Pnt 950 4505 Pnt 1013 2806 Pnt 1153 2622 Pnt 1191 3954 Pnt 1204 3954 Pnt 1217 2484 Pnt 1229 2346 Pnt 1280 2484 Pnt 1306 2163 Pnt 1306 2300 Pnt 1318 2163 Pnt 1331 2117 Pnt 1344 2392 Pnt 1369 2163 Pnt 1382 2117 Pnt 1382 2300 Pnt 1395 3540 Pnt 1433 3494 Pnt 1458 1841 Pnt 1458 2117 Pnt 1458 2163 Pnt 1458 3357 Pnt 1471 1933 Pnt 1471 2117 Pnt 1484 1749 Pnt 1484 1887 Pnt 1484 2117 Pnt 1496 1704 Pnt 1496 1749 Pnt 1496 3357 Pnt 1522 1658 Pnt 1522 1795 Pnt 1522 1979 Pnt 1547 1612 Pnt 1560 1704 Pnt 1560 1933 Pnt 1573 1520 Pnt 1573 1933 Pnt 1585 1520 Pnt 1585 1841 Pnt 1598 3081 Pnt 1611 1566 Pnt 1623 1382 Pnt 1623 1428 Pnt 1623 1749 Pnt 1623 1795 Pnt 1636 1382 Pnt 1636 1520 Pnt 1636 2989 Pnt 1649 1290 Pnt 1649 1474 Pnt 1649 1704 Pnt 1649 2897 Pnt 1662 1428 Pnt 1662 1474 Pnt 1662 1658 Pnt 1662 1749 Pnt 1662 2852 Pnt 1662 2989 Pnt 1674 1428 Pnt 1687 1198 Pnt 1687 1244 Pnt 1687 1612 Pnt 1687 1658 Pnt 1687 2943 Pnt 1700 1152 Pnt 1700 1198 Pnt 1700 1336 Pnt 1700 1382 Pnt 1700 1566 Pnt 1712 1152 Pnt 1712 1290 Pnt 1712 1336 Pnt 1725 1244 Pnt 1725 1290 Pnt 1725 1520 Pnt 1725 1566 Pnt 1725 1612 Pnt 1738 1015 Pnt 1738 1244 Pnt 1738 1474 Pnt 1751 969 Pnt 1751 1061 Pnt 1751 1244 Pnt 1751 1520 Pnt 1751 1566 Pnt 1751 2714 Pnt 1763 923 Pnt 1763 1015 Pnt 1763 1152 Pnt 1763 1198 Pnt 1763 1428 Pnt 1763 1474 Pnt 1776 877 Pnt 1776 969 Pnt 1776 1107 Pnt 1776 1152 Pnt 1776 1382 Pnt 1789 831 Pnt 1789 923 Pnt 1789 1061 Pnt 1789 1107 Pnt 1789 1336 Pnt 1789 1428 Pnt 1789 1474 Pnt 1789 2622 Pnt 1801 877 Pnt 1801 1015 Pnt 1801 1061 Pnt 1801 1107 Pnt 1801 1336 Pnt 1801 1382 Pnt 1801 2668 Pnt 1814 739 Pnt 1814 831 Pnt 1814 969 Pnt 1814 1015 Pnt 1814 1061 Pnt 1814 1290 Pnt 1814 1428 Pnt 1827 647 Pnt 1827 693 Pnt 1827 739 Pnt 1827 785 Pnt 1827 923 Pnt 1827 969 Pnt 1827 1015 Pnt 1827 1244 Pnt 1827 1336 Pnt 1827 1382 Pnt 1827 2530 Pnt 1840 601 Pnt 1840 647 Pnt 1840 693 Pnt 1840 739 Pnt 1840 877 Pnt 1840 923 Pnt 1840 969 Pnt 1840 1198 Pnt 1840 1290 Pnt 1840 2576 Pnt 1852 556 Pnt 1852 601 Pnt 1852 647 Pnt 1852 693 Pnt 1852 831 Pnt 1852 877 Pnt 1852 923 Pnt 1852 1152 Pnt 1852 1198 Pnt 1852 1336 Pnt 1852 2346 Pnt 1865 464 Pnt 1865 510 Pnt 1865 601 Pnt 1865 647 Pnt 1865 739 Pnt 1865 785 Pnt 1865 831 Pnt 1865 877 Pnt 1865 1107 Pnt 1865 1152 Pnt 1865 1244 Pnt 1865 1290 Pnt 1878 372 Pnt 1878 418 Pnt 1878 464 Pnt 1878 510 Pnt 1878 556 Pnt 1878 601 Pnt 1878 693 Pnt 1878 739 Pnt 1878 785 Pnt 1878 831 Pnt 1878 877 Pnt 1878 1061 Pnt 1878 1107 Pnt 1878 1198 Pnt 1878 1290 Pnt 1878 2255 Pnt 1878 2392 Pnt 1878 2484 Pnt 1890 280 Pnt 1890 326 Pnt 1890 372 Pnt 1890 418 Pnt 1890 464 Pnt 1890 510 Pnt 1890 556 Pnt 1890 647 Pnt 1890 693 Pnt 1890 739 Pnt 1890 785 Pnt 1890 831 Pnt 1890 1015 Pnt 1890 1061 Pnt 1890 1152 Pnt 1890 1244 Pnt 1890 2255 Pnt 1890 4872 Pnt 1903 280 Pnt 1903 326 Pnt 1903 372 Pnt 1903 418 Pnt 1903 464 Pnt 1903 556 Pnt 1903 601 Pnt 1903 647 Pnt 1903 693 Pnt 1903 739 Pnt 1903 785 Pnt 1903 969 Pnt 1903 1015 Pnt 1903 1061 Pnt 1903 1107 Pnt 1903 1152 Pnt 1903 1198 Pnt 1903 2438 Pnt 1916 280 Pnt 1916 326 Pnt 1916 372 Pnt 1916 418 Pnt 1916 464 Pnt 1916 510 Pnt 1916 556 Pnt 1916 601 Pnt 1916 647 Pnt 1916 693 Pnt 1916 739 Pnt 1916 923 Pnt 1916 969 Pnt 1916 1015 Pnt 1916 1107 Pnt 1916 1198 Pnt 1916 2300 Pnt 1916 2392 Pnt 1929 280 Pnt 1929 326 Pnt 1929 418 Pnt 1929 464 Pnt 1929 510 Pnt 1929 556 Pnt 1929 601 Pnt 1929 647 Pnt 1929 693 Pnt 1929 877 Pnt 1929 923 Pnt 1929 969 Pnt 1929 1061 Pnt 1929 1152 Pnt 1929 2117 Pnt 1929 2255 Pnt 1941 280 Pnt 1941 326 Pnt 1941 372 Pnt 1941 418 Pnt 1941 464 Pnt 1941 510 Pnt 1941 556 Pnt 1941 601 Pnt 1941 647 Pnt 1941 831 Pnt 1941 877 Pnt 1941 923 Pnt 1941 1015 Pnt 1941 1061 Pnt 1941 1107 Pnt 1941 2346 Pnt 1954 280 Pnt 1954 326 Pnt 1954 372 Pnt 1954 418 Pnt 1954 464 Pnt 1954 510 Pnt 1954 556 Pnt 1954 601 Pnt 1954 785 Pnt 1954 831 Pnt 1954 877 Pnt 1954 969 Pnt 1954 1015 Pnt 1954 1061 Pnt 1954 1107 Pnt 1954 2025 Pnt 1954 2071 Pnt 1954 2300 Pnt 1967 280 Pnt 1967 326 Pnt 1967 372 Pnt 1967 418 Pnt 1967 464 Pnt 1967 510 Pnt 1967 556 Pnt 1967 739 Pnt 1967 785 Pnt 1967 831 Pnt 1967 923 Pnt 1967 969 Pnt 1967 1061 Pnt 1967 1979 Pnt 1967 2025 Pnt 1967 2163 Pnt 1979 280 Pnt 1979 326 Pnt 1979 372 Pnt 1979 418 Pnt 1979 464 Pnt 1979 510 Pnt 1979 647 Pnt 1979 693 Pnt 1979 739 Pnt 1979 785 Pnt 1979 877 Pnt 1979 923 Pnt 1979 1015 Pnt 1979 1933 Pnt 1979 1979 Pnt 1979 2117 Pnt 1979 2255 Pnt 1992 280 Pnt 1992 326 Pnt 1992 372 Pnt 1992 418 Pnt 1992 601 Pnt 1992 647 Pnt 1992 693 Pnt 1992 739 Pnt 1992 831 Pnt 1992 877 Pnt 1992 923 Pnt 1992 969 Pnt 1992 1015 Pnt 1992 1887 Pnt 1992 2209 Pnt 2005 280 Pnt 2005 326 Pnt 2005 372 Pnt 2005 510 Pnt 2005 556 Pnt 2005 601 Pnt 2005 647 Pnt 2005 693 Pnt 2005 785 Pnt 2005 831 Pnt 2005 877 Pnt 2005 923 Pnt 2005 969 Pnt 2005 1841 Pnt 2005 2209 Pnt 2018 280 Pnt 2018 326 Pnt 2018 372 Pnt 2018 418 Pnt 2018 464 Pnt 2018 510 Pnt 2018 556 Pnt 2018 601 Pnt 2018 647 Pnt 2018 739 Pnt 2018 785 Pnt 2018 831 Pnt 2018 877 Pnt 2018 923 Pnt 2018 1795 Pnt 2018 2025 Pnt 2018 2163 Pnt 2030 280 Pnt 2030 326 Pnt 2030 372 Pnt 2030 418 Pnt 2030 464 Pnt 2030 510 Pnt 2030 556 Pnt 2030 601 Pnt 2030 647 Pnt 2030 693 Pnt 2030 739 Pnt 2030 785 Pnt 2030 877 Pnt 2030 923 Pnt 2030 1749 Pnt 2030 1979 Pnt 2030 2117 Pnt 2043 280 Pnt 2043 326 Pnt 2043 372 Pnt 2043 418 Pnt 2043 464 Pnt 2043 510 Pnt 2043 556 Pnt 2043 601 Pnt 2043 647 Pnt 2043 693 Pnt 2043 739 Pnt 2043 831 Pnt 2043 877 Pnt 2043 1704 Pnt 2043 1795 Pnt 2043 1933 Pnt 2043 2071 Pnt 2043 2117 Pnt 2056 280 Pnt 2056 326 Pnt 2056 372 Pnt 2056 418 Pnt 2056 464 Pnt 2056 510 Pnt 2056 556 Pnt 2056 601 Pnt 2056 647 Pnt 2056 693 Pnt 2056 785 Pnt 2056 831 Pnt 2056 1658 Pnt 2056 1749 Pnt 2056 2071 Pnt 2068 280 Pnt 2068 326 Pnt 2068 372 Pnt 2068 418 Pnt 2068 464 Pnt 2068 510 Pnt 2068 556 Pnt 2068 601 Pnt 2068 647 Pnt 2068 739 Pnt 2068 785 Pnt 2068 831 Pnt 2068 1612 Pnt 2068 1704 Pnt 2068 1887 Pnt 2068 4459 Pnt 2081 280 Pnt 2081 326 Pnt 2081 372 Pnt 2081 418 Pnt 2081 464 Pnt 2081 510 Pnt 2081 556 Pnt 2081 601 Pnt 2081 647 Pnt 2081 693 Pnt 2081 739 Pnt 2081 785 Pnt 2081 1566 Pnt 2081 1658 Pnt 2081 1841 Pnt 2081 1979 Pnt 2081 2025 Pnt 2094 280 Pnt 2094 326 Pnt 2094 372 Pnt 2094 418 Pnt 2094 464 Pnt 2094 510 Pnt 2094 556 Pnt 2094 601 Pnt 2094 647 Pnt 2094 693 Pnt 2094 739 Pnt 2094 1520 Pnt 2094 1612 Pnt 2094 1795 Pnt 2106 280 Pnt 2106 326 Pnt 2106 372 Pnt 2106 418 Pnt 2106 464 Pnt 2106 510 Pnt 2106 556 Pnt 2106 601 Pnt 2106 647 Pnt 2106 693 Pnt 2106 1566 Pnt 2106 1749 Pnt 2119 280 Pnt 2119 326 Pnt 2119 372 Pnt 2119 418 Pnt 2119 464 Pnt 2119 510 Pnt 2119 556 Pnt 2119 601 Pnt 2119 647 Pnt 2119 1382 Pnt 2119 1520 Pnt 2119 1704 Pnt 2119 1749 Pnt 2119 1887 Pnt 2119 1933 Pnt 2132 280 Pnt 2132 326 Pnt 2132 372 Pnt 2132 418 Pnt 2132 464 Pnt 2132 510 Pnt 2132 556 Pnt 2132 601 Pnt 2132 647 Pnt 2132 1336 Pnt 2132 1382 Pnt 2132 1474 Pnt 2132 1658 Pnt 2132 1704 Pnt 2132 1841 Pnt 2145 280 Pnt 2145 326 Pnt 2145 372 Pnt 2145 418 Pnt 2145 464 Pnt 2145 510 Pnt 2145 556 Pnt 2145 601 Pnt 2145 1244 Pnt 2145 1290 Pnt 2145 1336 Pnt 2145 1428 Pnt 2145 1658 Pnt 2145 4551 Pnt 2157 280 Pnt 2157 326 Pnt 2157 372 Pnt 2157 418 Pnt 2157 464 Pnt 2157 510 Pnt 2157 556 Pnt 2157 1198 Pnt 2157 1244 Pnt 2157 1290 Pnt 2157 1336 Pnt 2157 1382 Pnt 2157 1612 Pnt 2157 1795 Pnt 2157 1841 Pnt 2170 280 Pnt 2170 326 Pnt 2170 372 Pnt 2170 418 Pnt 2170 464 Pnt 2170 510 Pnt 2170 1107 Pnt 2170 1152 Pnt 2170 1198 Pnt 2170 1244 Pnt 2170 1290 Pnt 2170 1336 Pnt 2170 1566 Pnt 2170 1749 Pnt 2183 280 Pnt 2183 326 Pnt 2183 372 Pnt 2183 418 Pnt 2183 464 Pnt 2183 1061 Pnt 2183 1107 Pnt 2183 1152 Pnt 2183 1244 Pnt 2183 1290 Pnt 2183 1520 Pnt 2183 1566 Pnt 2183 1704 Pnt 2183 1749 Pnt 2195 280 Pnt 2195 326 Pnt 2195 372 Pnt 2195 418 Pnt 2195 969 Pnt 2195 1015 Pnt 2195 1061 Pnt 2195 1107 Pnt 2195 1198 Pnt 2195 1244 Pnt 2195 1474 Pnt 2195 1520 Pnt 2208 280 Pnt 2208 326 Pnt 2208 372 Pnt 2208 418 Pnt 2208 831 Pnt 2208 877 Pnt 2208 923 Pnt 2208 969 Pnt 2208 1015 Pnt 2208 1061 Pnt 2208 1107 Pnt 2208 1152 Pnt 2208 1198 Pnt 2208 1428 Pnt 2208 1474 Pnt 2208 1658 Pnt 2208 1704 Pnt 2208 4413 Pnt 2221 280 Pnt 2221 326 Pnt 2221 372 Pnt 2221 739 Pnt 2221 785 Pnt 2221 831 Pnt 2221 877 Pnt 2221 923 Pnt 2221 969 Pnt 2221 1015 Pnt 2221 1061 Pnt 2221 1107 Pnt 2221 1152 Pnt 2221 1382 Pnt 2221 1428 Pnt 2221 1612 Pnt 2221 1658 Pnt 2234 280 Pnt 2234 326 Pnt 2234 556 Pnt 2234 601 Pnt 2234 647 Pnt 2234 693 Pnt 2234 739 Pnt 2234 785 Pnt 2234 831 Pnt 2234 877 Pnt 2234 923 Pnt 2234 1015 Pnt 2234 1061 Pnt 2234 1107 Pnt 2234 1336 Pnt 2234 1382 Pnt 2234 1566 Pnt 2246 280 Pnt 2246 326 Pnt 2246 372 Pnt 2246 418 Pnt 2246 464 Pnt 2246 510 Pnt 2246 556 Pnt 2246 601 Pnt 2246 647 Pnt 2246 693 Pnt 2246 739 Pnt 2246 785 Pnt 2246 831 Pnt 2246 877 Pnt 2246 923 Pnt 2246 969 Pnt 2246 1015 Pnt 2246 1061 Pnt 2246 1290 Pnt 2246 1336 Pnt 2246 1382 Pnt 2246 1520 Pnt 2246 1612 Pnt 2259 280 Pnt 2259 326 Pnt 2259 372 Pnt 2259 418 Pnt 2259 464 Pnt 2259 510 Pnt 2259 556 Pnt 2259 601 Pnt 2259 647 Pnt 2259 693 Pnt 2259 739 Pnt 2259 785 Pnt 2259 831 Pnt 2259 877 Pnt 2259 923 Pnt 2259 969 Pnt 2259 1015 Pnt 2259 1244 Pnt 2259 1290 Pnt 2259 1336 Pnt 2259 1520 Pnt 2272 280 Pnt 2272 326 Pnt 2272 372 Pnt 2272 418 Pnt 2272 464 Pnt 2272 510 Pnt 2272 556 Pnt 2272 601 Pnt 2272 647 Pnt 2272 693 Pnt 2272 739 Pnt 2272 785 Pnt 2272 831 Pnt 2272 877 Pnt 2272 923 Pnt 2272 969 Pnt 2272 1198 Pnt 2272 1244 Pnt 2272 1290 Pnt 2272 1474 Pnt 2272 1520 Pnt 2284 280 Pnt 2284 326 Pnt 2284 372 Pnt 2284 418 Pnt 2284 464 Pnt 2284 510 Pnt 2284 556 Pnt 2284 601 Pnt 2284 647 Pnt 2284 693 Pnt 2284 739 Pnt 2284 785 Pnt 2284 831 Pnt 2284 877 Pnt 2284 923 Pnt 2284 1152 Pnt 2284 1198 Pnt 2284 1244 Pnt 2284 1428 Pnt 2284 1520 Pnt 2297 280 Pnt 2297 326 Pnt 2297 372 Pnt 2297 418 Pnt 2297 464 Pnt 2297 510 Pnt 2297 556 Pnt 2297 601 Pnt 2297 647 Pnt 2297 693 Pnt 2297 739 Pnt 2297 785 Pnt 2297 831 Pnt 2297 877 Pnt 2297 1061 Pnt 2297 1107 Pnt 2297 1152 Pnt 2297 1198 Pnt 2297 1382 Pnt 2297 1428 Pnt 2297 3908 Pnt 2310 280 Pnt 2310 326 Pnt 2310 372 Pnt 2310 418 Pnt 2310 464 Pnt 2310 510 Pnt 2310 556 Pnt 2310 601 Pnt 2310 647 Pnt 2310 693 Pnt 2310 739 Pnt 2310 785 Pnt 2310 831 Pnt 2310 1015 Pnt 2310 1061 Pnt 2310 1107 Pnt 2310 1152 Pnt 2310 1336 Pnt 2310 1382 Pnt 2310 1428 Pnt 2310 4413 Pnt 2323 280 Pnt 2323 326 Pnt 2323 372 Pnt 2323 418 Pnt 2323 464 Pnt 2323 510 Pnt 2323 556 Pnt 2323 601 Pnt 2323 647 Pnt 2323 693 Pnt 2323 739 Pnt 2323 785 Pnt 2323 969 Pnt 2323 1015 Pnt 2323 1061 Pnt 2323 1107 Pnt 2323 1152 Pnt 2323 1336 Pnt 2323 1382 Pnt 2323 1428 Pnt 2335 280 Pnt 2335 326 Pnt 2335 372 Pnt 2335 418 Pnt 2335 464 Pnt 2335 510 Pnt 2335 556 Pnt 2335 601 Pnt 2335 647 Pnt 2335 693 Pnt 2335 739 Pnt 2335 923 Pnt 2335 969 Pnt 2335 1015 Pnt 2335 1061 Pnt 2335 1107 Pnt 2335 1290 Pnt 2335 1382 Pnt 2348 280 Pnt 2348 326 Pnt 2348 372 Pnt 2348 418 Pnt 2348 464 Pnt 2348 510 Pnt 2348 556 Pnt 2348 601 Pnt 2348 647 Pnt 2348 693 Pnt 2348 831 Pnt 2348 877 Pnt 2348 923 Pnt 2348 969 Pnt 2348 1015 Pnt 2348 1061 Pnt 2348 1244 Pnt 2348 1290 Pnt 2348 1336 Pnt 2361 280 Pnt 2361 326 Pnt 2361 372 Pnt 2361 418 Pnt 2361 464 Pnt 2361 510 Pnt 2361 556 Pnt 2361 601 Pnt 2361 785 Pnt 2361 831 Pnt 2361 877 Pnt 2361 923 Pnt 2361 969 Pnt 2361 1015 Pnt 2361 1198 Pnt 2361 1244 Pnt 2361 1290 Pnt 2361 1336 Pnt 2373 280 Pnt 2373 326 Pnt 2373 372 Pnt 2373 418 Pnt 2373 464 Pnt 2373 510 Pnt 2373 556 Pnt 2373 739 Pnt 2373 785 Pnt 2373 831 Pnt 2373 877 Pnt 2373 923 Pnt 2373 969 Pnt 2373 1152 Pnt 2373 1198 Pnt 2373 1244 Pnt 2373 1290 Pnt 2386 280 Pnt 2386 326 Pnt 2386 372 Pnt 2386 418 Pnt 2386 464 Pnt 2386 510 Pnt 2386 647 Pnt 2386 693 Pnt 2386 739 Pnt 2386 785 Pnt 2386 831 Pnt 2386 877 Pnt 2386 923 Pnt 2386 1107 Pnt 2386 1152 Pnt 2386 3678 Pnt 2386 4045 Pnt 2399 280 Pnt 2399 326 Pnt 2399 372 Pnt 2399 418 Pnt 2399 464 Pnt 2399 556 Pnt 2399 601 Pnt 2399 647 Pnt 2399 693 Pnt 2399 739 Pnt 2399 785 Pnt 2399 831 Pnt 2399 877 Pnt 2399 923 Pnt 2399 1061 Pnt 2399 1107 Pnt 2399 1152 Pnt 2399 1198 Pnt 2399 1244 Pnt 2399 3678 Pnt 2412 280 Pnt 2412 326 Pnt 2412 372 Pnt 2412 418 Pnt 2412 464 Pnt 2412 510 Pnt 2412 556 Pnt 2412 601 Pnt 2412 647 Pnt 2412 693 Pnt 2412 739 Pnt 2412 785 Pnt 2412 831 Pnt 2412 877 Pnt 2412 1015 Pnt 2412 1061 Pnt 2412 1107 Pnt 2412 1152 Pnt 2412 1198 Pnt 2412 3678 Pnt 2424 280 Pnt 2424 326 Pnt 2424 372 Pnt 2424 418 Pnt 2424 464 Pnt 2424 510 Pnt 2424 556 Pnt 2424 601 Pnt 2424 647 Pnt 2424 693 Pnt 2424 739 Pnt 2424 785 Pnt 2424 831 Pnt 2424 969 Pnt 2424 1015 Pnt 2424 1061 Pnt 2424 1107 Pnt 2424 1152 Pnt 2424 4183 Pnt 2437 280 Pnt 2437 326 Pnt 2437 372 Pnt 2437 418 Pnt 2437 464 Pnt 2437 510 Pnt 2437 556 Pnt 2437 601 Pnt 2437 647 Pnt 2437 693 Pnt 2437 739 Pnt 2437 785 Pnt 2437 923 Pnt 2437 969 Pnt 2437 1015 Pnt 2437 1107 Pnt 2437 1152 Pnt 2437 3540 Pnt 2450 280 Pnt 2450 326 Pnt 2450 372 Pnt 2450 418 Pnt 2450 464 Pnt 2450 510 Pnt 2450 556 Pnt 2450 601 Pnt 2450 647 Pnt 2450 693 Pnt 2450 739 Pnt 2450 877 Pnt 2450 923 Pnt 2450 969 Pnt 2450 1015 Pnt 2450 1061 Pnt 2450 1107 Pnt 2450 3540 Pnt 2450 3586 Pnt 2450 3908 Pnt 2462 280 Pnt 2462 326 Pnt 2462 372 Pnt 2462 418 Pnt 2462 464 Pnt 2462 510 Pnt 2462 556 Pnt 2462 601 Pnt 2462 647 Pnt 2462 693 Pnt 2462 831 Pnt 2462 877 Pnt 2462 923 Pnt 2462 969 Pnt 2462 1015 Pnt 2462 1061 Pnt 2475 280 Pnt 2475 326 Pnt 2475 372 Pnt 2475 418 Pnt 2475 464 Pnt 2475 510 Pnt 2475 556 Pnt 2475 601 Pnt 2475 647 Pnt 2475 785 Pnt 2475 831 Pnt 2475 877 Pnt 2475 923 Pnt 2475 969 Pnt 2475 1015 Pnt 2475 1061 Pnt 2488 280 Pnt 2488 326 Pnt 2488 372 Pnt 2488 418 Pnt 2488 464 Pnt 2488 510 Pnt 2488 556 Pnt 2488 601 Pnt 2488 739 Pnt 2488 785 Pnt 2488 831 Pnt 2488 877 Pnt 2488 923 Pnt 2488 969 Pnt 2488 1015 Pnt 2488 3403 Pnt 2488 3816 Pnt 2501 280 Pnt 2501 326 Pnt 2501 372 Pnt 2501 418 Pnt 2501 464 Pnt 2501 510 Pnt 2501 556 Pnt 2501 647 Pnt 2501 693 Pnt 2501 739 Pnt 2501 785 Pnt 2501 831 Pnt 2501 877 Pnt 2501 923 Pnt 2501 969 Pnt 2501 3403 Pnt 2501 3448 Pnt 2513 280 Pnt 2513 326 Pnt 2513 372 Pnt 2513 418 Pnt 2513 464 Pnt 2513 510 Pnt 2513 601 Pnt 2513 647 Pnt 2513 693 Pnt 2513 739 Pnt 2513 785 Pnt 2513 831 Pnt 2513 877 Pnt 2513 923 Pnt 2513 969 Pnt 2513 3770 Pnt 2526 280 Pnt 2526 326 Pnt 2526 372 Pnt 2526 418 Pnt 2526 464 Pnt 2526 556 Pnt 2526 601 Pnt 2526 647 Pnt 2526 693 Pnt 2526 739 Pnt 2526 785 Pnt 2526 831 Pnt 2526 877 Pnt 2526 923 Pnt 2539 280 Pnt 2539 326 Pnt 2539 372 Pnt 2539 418 Pnt 2539 464 Pnt 2539 510 Pnt 2539 556 Pnt 2539 601 Pnt 2539 647 Pnt 2539 693 Pnt 2539 739 Pnt 2539 785 Pnt 2539 831 Pnt 2539 877 Pnt 2539 3954 Pnt 2539 4000 Pnt 2551 280 Pnt 2551 326 Pnt 2551 372 Pnt 2551 418 Pnt 2551 464 Pnt 2551 510 Pnt 2551 556 Pnt 2551 601 Pnt 2551 647 Pnt 2551 693 Pnt 2551 739 Pnt 2551 785 Pnt 2551 831 Pnt 2551 877 Pnt 2551 3265 Pnt 2564 280 Pnt 2564 326 Pnt 2564 372 Pnt 2564 418 Pnt 2564 464 Pnt 2564 510 Pnt 2564 556 Pnt 2564 601 Pnt 2564 647 Pnt 2564 693 Pnt 2564 739 Pnt 2564 785 Pnt 2564 831 Pnt 2564 3632 Pnt 2577 280 Pnt 2577 326 Pnt 2577 372 Pnt 2577 418 Pnt 2577 464 Pnt 2577 510 Pnt 2577 556 Pnt 2577 601 Pnt 2577 647 Pnt 2577 693 Pnt 2577 739 Pnt 2577 785 Pnt 2589 280 Pnt 2589 326 Pnt 2589 372 Pnt 2589 418 Pnt 2589 464 Pnt 2589 510 Pnt 2589 556 Pnt 2589 601 Pnt 2589 647 Pnt 2589 693 Pnt 2589 739 Pnt 2589 785 Pnt 2602 280 Pnt 2602 326 Pnt 2602 372 Pnt 2602 418 Pnt 2602 464 Pnt 2602 510 Pnt 2602 556 Pnt 2602 601 Pnt 2602 647 Pnt 2602 693 Pnt 2602 739 Pnt 2602 3173 Pnt 2615 280 Pnt 2615 326 Pnt 2615 372 Pnt 2615 418 Pnt 2615 464 Pnt 2615 510 Pnt 2615 556 Pnt 2615 601 Pnt 2615 647 Pnt 2615 693 Pnt 2615 3081 Pnt 2615 3540 Pnt 2628 280 Pnt 2628 326 Pnt 2628 372 Pnt 2628 418 Pnt 2628 464 Pnt 2628 510 Pnt 2628 556 Pnt 2628 601 Pnt 2628 647 Pnt 2628 693 Pnt 2628 3035 Pnt 2640 280 Pnt 2640 326 Pnt 2640 372 Pnt 2640 418 Pnt 2640 464 Pnt 2640 510 Pnt 2640 556 Pnt 2640 601 Pnt 2640 647 Pnt 2640 2943 Pnt 2640 3081 Pnt 2653 280 Pnt 2653 326 Pnt 2653 372 Pnt 2653 418 Pnt 2653 464 Pnt 2653 510 Pnt 2653 556 Pnt 2653 601 Pnt 2653 647 Pnt 2653 2897 Pnt 2653 3035 Pnt 2666 280 Pnt 2666 326 Pnt 2666 372 Pnt 2666 418 Pnt 2666 464 Pnt 2666 510 Pnt 2666 556 Pnt 2666 601 Pnt 2666 2852 Pnt 2666 2989 Pnt 2678 280 Pnt 2678 326 Pnt 2678 372 Pnt 2678 418 Pnt 2678 464 Pnt 2678 510 Pnt 2678 556 Pnt 2678 2806 Pnt 2691 280 Pnt 2691 326 Pnt 2691 372 Pnt 2691 418 Pnt 2691 464 Pnt 2691 510 Pnt 2691 556 Pnt 2691 2760 Pnt 2691 3678 Pnt 2704 280 Pnt 2704 326 Pnt 2704 372 Pnt 2704 418 Pnt 2704 464 Pnt 2704 510 Pnt 2704 2714 Pnt 2704 2897 Pnt 2717 280 Pnt 2717 326 Pnt 2717 372 Pnt 2717 418 Pnt 2717 464 Pnt 2717 2668 Pnt 2717 2760 Pnt 2717 2852 Pnt 2717 3311 Pnt 2729 280 Pnt 2729 326 Pnt 2729 372 Pnt 2729 418 Pnt 2729 464 Pnt 2729 2622 Pnt 2729 2714 Pnt 2729 2806 Pnt 2742 280 Pnt 2742 326 Pnt 2742 372 Pnt 2742 418 Pnt 2742 2576 Pnt 2742 2668 Pnt 2755 280 Pnt 2755 326 Pnt 2755 372 Pnt 2755 418 Pnt 2755 2530 Pnt 2755 2622 Pnt 2755 2714 Pnt 2755 3219 Pnt 2767 280 Pnt 2767 326 Pnt 2767 372 Pnt 2767 2484 Pnt 2767 2576 Pnt 2767 2668 Pnt 2780 280 Pnt 2780 326 Pnt 2780 2530 Pnt 2793 280 Pnt 2793 326 Pnt 2793 2346 Pnt 2793 2484 Pnt 2793 2622 Pnt 2793 3081 Pnt 2793 3127 Pnt 2806 280 Pnt 2806 2300 Pnt 2806 2438 Pnt 2806 2576 Pnt 2806 3448 Pnt 2818 280 Pnt 2818 2255 Pnt 2818 2392 Pnt 2818 2530 Pnt 2818 3448 Pnt 2831 280 Pnt 2831 326 Pnt 2831 2163 Pnt 2831 2300 Pnt 2831 2484 Pnt 2831 2989 Pnt 2831 3035 Pnt 2831 3357 Pnt 2844 280 Pnt 2844 326 Pnt 2844 372 Pnt 2844 418 Pnt 2844 2071 Pnt 2844 2117 Pnt 2844 2255 Pnt 2844 2438 Pnt 2856 280 Pnt 2856 326 Pnt 2856 372 Pnt 2856 418 Pnt 2856 464 Pnt 2856 2025 Pnt 2856 2071 Pnt 2856 2209 Pnt 2856 2392 Pnt 2869 280 Pnt 2869 326 Pnt 2869 372 Pnt 2869 418 Pnt 2869 464 Pnt 2869 510 Pnt 2869 556 Pnt 2869 1933 Pnt 2869 1979 Pnt 2869 2117 Pnt 2869 2163 Pnt 2869 2346 Pnt 2869 2943 Pnt 2869 3265 Pnt 2882 280 Pnt 2882 326 Pnt 2882 372 Pnt 2882 418 Pnt 2882 464 Pnt 2882 510 Pnt 2882 556 Pnt 2882 601 Pnt 2882 647 Pnt 2882 1841 Pnt 2882 1887 Pnt 2882 2071 Pnt 2882 2117 Pnt 2882 2255 Pnt 2882 2300 Pnt 2882 2852 Pnt 2895 280 Pnt 2895 326 Pnt 2895 372 Pnt 2895 418 Pnt 2895 464 Pnt 2895 510 Pnt 2895 556 Pnt 2895 601 Pnt 2895 647 Pnt 2895 693 Pnt 2895 739 Pnt 2895 785 Pnt 2895 1704 Pnt 2895 1749 Pnt 2895 1795 Pnt 2895 1841 Pnt 2895 1979 Pnt 2895 2025 Pnt 2895 2209 Pnt 2895 2255 Pnt 2895 2852 Pnt 2895 3265 Pnt 2907 280 Pnt 2907 326 Pnt 2907 372 Pnt 2907 418 Pnt 2907 464 Pnt 2907 510 Pnt 2907 556 Pnt 2907 601 Pnt 2907 647 Pnt 2907 693 Pnt 2907 739 Pnt 2907 785 Pnt 2907 831 Pnt 2907 877 Pnt 2907 923 Pnt 2907 969 Pnt 2907 1520 Pnt 2907 1566 Pnt 2907 1612 Pnt 2907 1658 Pnt 2907 1704 Pnt 2907 1749 Pnt 2907 1933 Pnt 2907 1979 Pnt 2907 2163 Pnt 2907 2209 Pnt 2907 2852 Pnt 2920 280 Pnt 2920 326 Pnt 2920 372 Pnt 2920 418 Pnt 2920 464 Pnt 2920 510 Pnt 2920 556 Pnt 2920 601 Pnt 2920 647 Pnt 2920 693 Pnt 2920 739 Pnt 2920 785 Pnt 2920 831 Pnt 2920 877 Pnt 2920 923 Pnt 2920 969 Pnt 2920 1015 Pnt 2920 1061 Pnt 2920 1107 Pnt 2920 1152 Pnt 2920 1198 Pnt 2920 1244 Pnt 2920 1290 Pnt 2920 1336 Pnt 2920 1382 Pnt 2920 1428 Pnt 2920 1474 Pnt 2920 1520 Pnt 2920 1566 Pnt 2920 1612 Pnt 2920 1841 Pnt 2920 1887 Pnt 2920 1933 Pnt 2920 2117 Pnt 2920 2163 Pnt 2933 280 Pnt 2933 326 Pnt 2933 372 Pnt 2933 418 Pnt 2933 464 Pnt 2933 510 Pnt 2933 556 Pnt 2933 601 Pnt 2933 647 Pnt 2933 693 Pnt 2933 739 Pnt 2933 785 Pnt 2933 831 Pnt 2933 877 Pnt 2933 923 Pnt 2933 969 Pnt 2933 1015 Pnt 2933 1061 Pnt 2933 1107 Pnt 2933 1152 Pnt 2933 1198 Pnt 2933 1244 Pnt 2933 1290 Pnt 2933 1336 Pnt 2933 1382 Pnt 2933 1428 Pnt 2933 1474 Pnt 2933 1520 Pnt 2933 1749 Pnt 2933 1795 Pnt 2933 1841 Pnt 2933 2071 Pnt 2933 2117 Pnt 2933 2714 Pnt 2933 3127 Pnt 2933 3219 Pnt 2945 280 Pnt 2945 326 Pnt 2945 372 Pnt 2945 418 Pnt 2945 464 Pnt 2945 510 Pnt 2945 556 Pnt 2945 601 Pnt 2945 647 Pnt 2945 693 Pnt 2945 739 Pnt 2945 785 Pnt 2945 831 Pnt 2945 877 Pnt 2945 923 Pnt 2945 969 Pnt 2945 1015 Pnt 2945 1061 Pnt 2945 1107 Pnt 2945 1152 Pnt 2945 1198 Pnt 2945 1244 Pnt 2945 1290 Pnt 2945 1336 Pnt 2945 1612 Pnt 2945 1658 Pnt 2945 1704 Pnt 2945 1749 Pnt 2945 1795 Pnt 2945 1979 Pnt 2945 2025 Pnt 2945 2071 Pnt 2945 2668 Pnt 2945 2714 Pnt 2945 2760 Pnt 2958 280 Pnt 2958 326 Pnt 2958 372 Pnt 2958 418 Pnt 2958 464 Pnt 2958 510 Pnt 2958 556 Pnt 2958 601 Pnt 2958 647 Pnt 2958 693 Pnt 2958 739 Pnt 2958 785 Pnt 2958 831 Pnt 2958 877 Pnt 2958 923 Pnt 2958 969 Pnt 2958 1015 Pnt 2958 1061 Pnt 2958 1107 Pnt 2958 1474 Pnt 2958 1520 Pnt 2958 1566 Pnt 2958 1612 Pnt 2958 1658 Pnt 2958 1704 Pnt 2958 1933 Pnt 2958 1979 Pnt 2958 2025 Pnt 2958 2714 Pnt 2958 3173 Pnt 2971 280 Pnt 2971 326 Pnt 2971 372 Pnt 2971 418 Pnt 2971 464 Pnt 2971 510 Pnt 2971 556 Pnt 2971 601 Pnt 2971 647 Pnt 2971 693 Pnt 2971 739 Pnt 2971 785 Pnt 2971 831 Pnt 2971 877 Pnt 2971 923 Pnt 2971 1152 Pnt 2971 1198 Pnt 2971 1244 Pnt 2971 1290 Pnt 2971 1336 Pnt 2971 1382 Pnt 2971 1428 Pnt 2971 1474 Pnt 2971 1520 Pnt 2971 1566 Pnt 2971 1612 Pnt 2971 1841 Pnt 2971 1887 Pnt 2971 1933 Pnt 2971 1979 Pnt 2984 280 Pnt 2984 326 Pnt 2984 372 Pnt 2984 418 Pnt 2984 464 Pnt 2984 510 Pnt 2984 556 Pnt 2984 601 Pnt 2984 647 Pnt 2984 693 Pnt 2984 739 Pnt 2984 785 Pnt 2984 831 Pnt 2984 877 Pnt 2984 923 Pnt 2984 969 Pnt 2984 1015 Pnt 2984 1061 Pnt 2984 1107 Pnt 2984 1152 Pnt 2984 1198 Pnt 2984 1244 Pnt 2984 1290 Pnt 2984 1336 Pnt 2984 1382 Pnt 2984 1428 Pnt 2984 1474 Pnt 2984 1520 Pnt 2984 1795 Pnt 2984 1841 Pnt 2984 1887 Pnt 2984 1933 Pnt 2984 2576 Pnt 2984 3081 Pnt 2996 280 Pnt 2996 326 Pnt 2996 372 Pnt 2996 418 Pnt 2996 464 Pnt 2996 510 Pnt 2996 556 Pnt 2996 601 Pnt 2996 647 Pnt 2996 693 Pnt 2996 739 Pnt 2996 785 Pnt 2996 831 Pnt 2996 877 Pnt 2996 923 Pnt 2996 969 Pnt 2996 1015 Pnt 2996 1061 Pnt 2996 1107 Pnt 2996 1152 Pnt 2996 1198 Pnt 2996 1244 Pnt 2996 1290 Pnt 2996 1336 Pnt 2996 1382 Pnt 2996 1428 Pnt 2996 1704 Pnt 2996 1749 Pnt 2996 1795 Pnt 2996 1841 Pnt 2996 2530 Pnt 2996 2576 Pnt 2996 2622 Pnt 2996 2989 Pnt 3009 280 Pnt 3009 326 Pnt 3009 372 Pnt 3009 418 Pnt 3009 464 Pnt 3009 510 Pnt 3009 556 Pnt 3009 601 Pnt 3009 647 Pnt 3009 693 Pnt 3009 739 Pnt 3009 785 Pnt 3009 831 Pnt 3009 877 Pnt 3009 923 Pnt 3009 969 Pnt 3009 1015 Pnt 3009 1061 Pnt 3009 1107 Pnt 3009 1152 Pnt 3009 1198 Pnt 3009 1244 Pnt 3009 1290 Pnt 3009 1336 Pnt 3009 1612 Pnt 3009 1658 Pnt 3009 1704 Pnt 3009 1749 Pnt 3009 1795 Pnt 3009 2484 Pnt 3009 2576 Pnt 3022 280 Pnt 3022 326 Pnt 3022 372 Pnt 3022 418 Pnt 3022 464 Pnt 3022 510 Pnt 3022 556 Pnt 3022 601 Pnt 3022 647 Pnt 3022 693 Pnt 3022 739 Pnt 3022 785 Pnt 3022 831 Pnt 3022 877 Pnt 3022 923 Pnt 3022 969 Pnt 3022 1015 Pnt 3022 1061 Pnt 3022 1107 Pnt 3022 1152 Pnt 3022 1198 Pnt 3022 1520 Pnt 3022 1566 Pnt 3022 1612 Pnt 3022 1658 Pnt 3022 1704 Pnt 3022 1749 Pnt 3022 2438 Pnt 3022 2943 Pnt 3022 3035 Pnt 3034 280 Pnt 3034 326 Pnt 3034 372 Pnt 3034 418 Pnt 3034 464 Pnt 3034 510 Pnt 3034 556 Pnt 3034 601 Pnt 3034 647 Pnt 3034 693 Pnt 3034 739 Pnt 3034 785 Pnt 3034 831 Pnt 3034 877 Pnt 3034 923 Pnt 3034 969 Pnt 3034 1015 Pnt 3034 1061 Pnt 3034 1107 Pnt 3034 1428 Pnt 3034 1474 Pnt 3034 1520 Pnt 3034 1566 Pnt 3034 1612 Pnt 3034 1658 Pnt 3034 1704 Pnt 3034 2530 Pnt 3034 2897 Pnt 3047 280 Pnt 3047 326 Pnt 3047 372 Pnt 3047 418 Pnt 3047 464 Pnt 3047 510 Pnt 3047 556 Pnt 3047 601 Pnt 3047 647 Pnt 3047 693 Pnt 3047 739 Pnt 3047 785 Pnt 3047 831 Pnt 3047 877 Pnt 3047 923 Pnt 3047 969 Pnt 3047 1290 Pnt 3047 1336 Pnt 3047 1382 Pnt 3047 1428 Pnt 3047 1474 Pnt 3047 1520 Pnt 3047 1566 Pnt 3047 1612 Pnt 3047 1658 Pnt 3047 2392 Pnt 3047 2438 Pnt 3047 2484 Pnt 3047 2943 Pnt 3047 2989 Pnt 3060 280 Pnt 3060 326 Pnt 3060 372 Pnt 3060 418 Pnt 3060 464 Pnt 3060 510 Pnt 3060 556 Pnt 3060 601 Pnt 3060 647 Pnt 3060 693 Pnt 3060 739 Pnt 3060 785 Pnt 3060 831 Pnt 3060 1152 Pnt 3060 1198 Pnt 3060 1244 Pnt 3060 1290 Pnt 3060 1336 Pnt 3060 1382 Pnt 3060 1428 Pnt 3060 1474 Pnt 3060 1520 Pnt 3060 1566 Pnt 3060 2346 Pnt 3060 2392 Pnt 3060 2438 Pnt 3060 2852 Pnt 3073 280 Pnt 3073 326 Pnt 3073 372 Pnt 3073 418 Pnt 3073 464 Pnt 3073 510 Pnt 3073 556 Pnt 3073 601 Pnt 3073 647 Pnt 3073 693 Pnt 3073 923 Pnt 3073 969 Pnt 3073 1015 Pnt 3073 1061 Pnt 3073 1107 Pnt 3073 1152 Pnt 3073 1198 Pnt 3073 1244 Pnt 3073 1290 Pnt 3073 1336 Pnt 3073 1382 Pnt 3073 1428 Pnt 3073 1474 Pnt 3073 1520 Pnt 3073 2300 Pnt 3073 2346 Pnt 3073 2897 Pnt 3085 280 Pnt 3085 326 Pnt 3085 372 Pnt 3085 418 Pnt 3085 464 Pnt 3085 510 Pnt 3085 556 Pnt 3085 601 Pnt 3085 647 Pnt 3085 693 Pnt 3085 739 Pnt 3085 785 Pnt 3085 831 Pnt 3085 877 Pnt 3085 923 Pnt 3085 969 Pnt 3085 1015 Pnt 3085 1061 Pnt 3085 1107 Pnt 3085 1152 Pnt 3085 1198 Pnt 3085 1244 Pnt 3085 1290 Pnt 3085 1336 Pnt 3085 1382 Pnt 3085 1428 Pnt 3085 1474 Pnt 3085 2255 Pnt 3085 2392 Pnt 3085 2806 Pnt 3098 280 Pnt 3098 326 Pnt 3098 372 Pnt 3098 418 Pnt 3098 464 Pnt 3098 510 Pnt 3098 556 Pnt 3098 601 Pnt 3098 647 Pnt 3098 693 Pnt 3098 739 Pnt 3098 785 Pnt 3098 831 Pnt 3098 877 Pnt 3098 923 Pnt 3098 969 Pnt 3098 1015 Pnt 3098 1061 Pnt 3098 1107 Pnt 3098 1152 Pnt 3098 1198 Pnt 3098 1244 Pnt 3098 1290 Pnt 3098 1336 Pnt 3098 1382 Pnt 3098 2209 Pnt 3098 2346 Pnt 3098 2760 Pnt 3111 280 Pnt 3111 326 Pnt 3111 372 Pnt 3111 418 Pnt 3111 464 Pnt 3111 510 Pnt 3111 556 Pnt 3111 601 Pnt 3111 647 Pnt 3111 693 Pnt 3111 739 Pnt 3111 785 Pnt 3111 831 Pnt 3111 877 Pnt 3111 923 Pnt 3111 969 Pnt 3111 1015 Pnt 3111 1061 Pnt 3111 1107 Pnt 3111 1152 Pnt 3111 1198 Pnt 3111 1244 Pnt 3111 1290 Pnt 3111 1336 Pnt 3111 2163 Pnt 3111 2300 Pnt 3111 2806 Pnt 3111 2852 Pnt 3123 280 Pnt 3123 326 Pnt 3123 372 Pnt 3123 418 Pnt 3123 464 Pnt 3123 510 Pnt 3123 556 Pnt 3123 601 Pnt 3123 647 Pnt 3123 693 Pnt 3123 739 Pnt 3123 785 Pnt 3123 831 Pnt 3123 877 Pnt 3123 923 Pnt 3123 969 Pnt 3123 1015 Pnt 3123 1061 Pnt 3123 1107 Pnt 3123 1152 Pnt 3123 1198 Pnt 3123 1244 Pnt 3123 2117 Pnt 3123 2209 Pnt 3123 2255 Pnt 3123 2714 Pnt 3136 280 Pnt 3136 326 Pnt 3136 372 Pnt 3136 418 Pnt 3136 464 Pnt 3136 510 Pnt 3136 556 Pnt 3136 601 Pnt 3136 647 Pnt 3136 693 Pnt 3136 739 Pnt 3136 785 Pnt 3136 831 Pnt 3136 877 Pnt 3136 923 Pnt 3136 969 Pnt 3136 1015 Pnt 3136 1061 Pnt 3136 1107 Pnt 3136 1152 Pnt 3136 1198 Pnt 3136 2025 Pnt 3136 2071 Pnt 3136 2163 Pnt 3136 2209 Pnt 3136 2255 Pnt 3136 2668 Pnt 3136 2760 Pnt 3136 2806 Pnt 3149 280 Pnt 3149 326 Pnt 3149 372 Pnt 3149 418 Pnt 3149 464 Pnt 3149 510 Pnt 3149 556 Pnt 3149 601 Pnt 3149 647 Pnt 3149 693 Pnt 3149 739 Pnt 3149 785 Pnt 3149 831 Pnt 3149 877 Pnt 3149 923 Pnt 3149 969 Pnt 3149 1015 Pnt 3149 1061 Pnt 3149 1107 Pnt 3149 1979 Pnt 3149 2025 Pnt 3149 2117 Pnt 3149 2209 Pnt 3161 280 Pnt 3161 326 Pnt 3161 372 Pnt 3161 418 Pnt 3161 464 Pnt 3161 510 Pnt 3161 556 Pnt 3161 601 Pnt 3161 647 Pnt 3161 693 Pnt 3161 739 Pnt 3161 785 Pnt 3161 831 Pnt 3161 877 Pnt 3161 923 Pnt 3161 969 Pnt 3161 1015 Pnt 3161 1061 Pnt 3161 1933 Pnt 3161 1979 Pnt 3161 2071 Pnt 3161 2163 Pnt 3161 2622 Pnt 3161 2760 Pnt 3174 280 Pnt 3174 326 Pnt 3174 372 Pnt 3174 418 Pnt 3174 464 Pnt 3174 510 Pnt 3174 556 Pnt 3174 601 Pnt 3174 647 Pnt 3174 693 Pnt 3174 739 Pnt 3174 785 Pnt 3174 831 Pnt 3174 877 Pnt 3174 923 Pnt 3174 969 Pnt 3174 1887 Pnt 3174 1933 Pnt 3174 2025 Pnt 3174 2117 Pnt 3174 2576 Pnt 3174 2668 Pnt 3187 280 Pnt 3187 326 Pnt 3187 372 Pnt 3187 418 Pnt 3187 464 Pnt 3187 510 Pnt 3187 556 Pnt 3187 601 Pnt 3187 647 Pnt 3187 693 Pnt 3187 739 Pnt 3187 785 Pnt 3187 831 Pnt 3187 877 Pnt 3187 923 Pnt 3187 1795 Pnt 3187 1841 Pnt 3187 1887 Pnt 3187 1979 Pnt 3187 2071 Pnt 3187 2576 Pnt 3200 280 Pnt 3200 326 Pnt 3200 372 Pnt 3200 418 Pnt 3200 464 Pnt 3200 510 Pnt 3200 556 Pnt 3200 601 Pnt 3200 647 Pnt 3200 693 Pnt 3200 739 Pnt 3200 785 Pnt 3200 831 Pnt 3200 1749 Pnt 3200 1795 Pnt 3200 1841 Pnt 3200 1933 Pnt 3200 2025 Pnt 3200 2071 Pnt 3200 2530 Pnt 3200 2622 Pnt 3200 2668 Pnt 3212 280 Pnt 3212 326 Pnt 3212 372 Pnt 3212 418 Pnt 3212 464 Pnt 3212 510 Pnt 3212 556 Pnt 3212 601 Pnt 3212 647 Pnt 3212 693 Pnt 3212 739 Pnt 3212 785 Pnt 3212 1658 Pnt 3212 1704 Pnt 3212 1749 Pnt 3212 1795 Pnt 3212 1887 Pnt 3212 1979 Pnt 3212 2025 Pnt 3212 2484 Pnt 3225 280 Pnt 3225 326 Pnt 3225 372 Pnt 3225 418 Pnt 3225 464 Pnt 3225 510 Pnt 3225 556 Pnt 3225 601 Pnt 3225 647 Pnt 3225 693 Pnt 3225 1612 Pnt 3225 1658 Pnt 3225 1704 Pnt 3225 1749 Pnt 3225 1841 Pnt 3225 1933 Pnt 3225 1979 Pnt 3225 2438 Pnt 3225 2484 Pnt 3225 2622 Pnt 3238 280 Pnt 3238 326 Pnt 3238 372 Pnt 3238 418 Pnt 3238 464 Pnt 3238 510 Pnt 3238 556 Pnt 3238 601 Pnt 3238 1520 Pnt 3238 1566 Pnt 3238 1612 Pnt 3238 1658 Pnt 3238 1704 Pnt 3238 1749 Pnt 3238 1795 Pnt 3238 1887 Pnt 3238 1933 Pnt 3238 2438 Pnt 3238 2530 Pnt 3250 280 Pnt 3250 326 Pnt 3250 372 Pnt 3250 418 Pnt 3250 464 Pnt 3250 510 Pnt 3250 556 Pnt 3250 1428 Pnt 3250 1474 Pnt 3250 1520 Pnt 3250 1566 Pnt 3250 1612 Pnt 3250 1704 Pnt 3250 1749 Pnt 3250 1841 Pnt 3250 1887 Pnt 3250 2392 Pnt 3250 2576 Pnt 3263 280 Pnt 3263 326 Pnt 3263 372 Pnt 3263 418 Pnt 3263 464 Pnt 3263 510 Pnt 3263 1290 Pnt 3263 1336 Pnt 3263 1382 Pnt 3263 1428 Pnt 3263 1474 Pnt 3263 1520 Pnt 3263 1566 Pnt 3263 1658 Pnt 3263 1704 Pnt 3263 1795 Pnt 3263 1841 Pnt 3263 2346 Pnt 3263 2392 Pnt 3263 2484 Pnt 3276 280 Pnt 3276 326 Pnt 3276 372 Pnt 3276 418 Pnt 3276 1107 Pnt 3276 1152 Pnt 3276 1198 Pnt 3276 1244 Pnt 3276 1290 Pnt 3276 1336 Pnt 3276 1382 Pnt 3276 1428 Pnt 3276 1474 Pnt 3276 1520 Pnt 3276 1612 Pnt 3276 1658 Pnt 3276 1749 Pnt 3276 1795 Pnt 3276 1841 Pnt 3276 2346 Pnt 3276 2438 Pnt 3289 280 Pnt 3289 326 Pnt 3289 372 Pnt 3289 739 Pnt 3289 785 Pnt 3289 831 Pnt 3289 877 Pnt 3289 923 Pnt 3289 969 Pnt 3289 1015 Pnt 3289 1061 Pnt 3289 1107 Pnt 3289 1152 Pnt 3289 1198 Pnt 3289 1244 Pnt 3289 1290 Pnt 3289 1336 Pnt 3289 1382 Pnt 3289 1428 Pnt 3289 1474 Pnt 3289 1520 Pnt 3289 1566 Pnt 3289 1612 Pnt 3289 1704 Pnt 3289 1749 Pnt 3289 1795 Pnt 3289 2300 Pnt 3289 2484 Pnt 3301 280 Pnt 3301 326 Pnt 3301 372 Pnt 3301 418 Pnt 3301 464 Pnt 3301 510 Pnt 3301 556 Pnt 3301 601 Pnt 3301 647 Pnt 3301 693 Pnt 3301 739 Pnt 3301 785 Pnt 3301 831 Pnt 3301 877 Pnt 3301 923 Pnt 3301 969 Pnt 3301 1015 Pnt 3301 1061 Pnt 3301 1107 Pnt 3301 1152 Pnt 3301 1198 Pnt 3301 1244 Pnt 3301 1290 Pnt 3301 1336 Pnt 3301 1382 Pnt 3301 1428 Pnt 3301 1474 Pnt 3301 1520 Pnt 3301 1566 Pnt 3301 1658 Pnt 3301 1704 Pnt 3301 1749 Pnt 3301 2255 Pnt 3301 2300 Pnt 3301 2392 Pnt 3314 280 Pnt 3314 326 Pnt 3314 372 Pnt 3314 418 Pnt 3314 464 Pnt 3314 510 Pnt 3314 556 Pnt 3314 601 Pnt 3314 647 Pnt 3314 693 Pnt 3314 739 Pnt 3314 785 Pnt 3314 831 Pnt 3314 877 Pnt 3314 923 Pnt 3314 969 Pnt 3314 1015 Pnt 3314 1061 Pnt 3314 1107 Pnt 3314 1152 Pnt 3314 1198 Pnt 3314 1244 Pnt 3314 1290 Pnt 3314 1336 Pnt 3314 1382 Pnt 3314 1428 Pnt 3314 1474 Pnt 3314 1520 Pnt 3314 1566 Pnt 3314 1612 Pnt 3314 1658 Pnt 3314 1704 Pnt 3314 2209 Pnt 3314 2255 Pnt 3314 2438 Pnt 3327 280 Pnt 3327 326 Pnt 3327 372 Pnt 3327 418 Pnt 3327 464 Pnt 3327 510 Pnt 3327 556 Pnt 3327 601 Pnt 3327 647 Pnt 3327 693 Pnt 3327 739 Pnt 3327 785 Pnt 3327 831 Pnt 3327 877 Pnt 3327 923 Pnt 3327 969 Pnt 3327 1015 Pnt 3327 1061 Pnt 3327 1107 Pnt 3327 1152 Pnt 3327 1198 Pnt 3327 1244 Pnt 3327 1290 Pnt 3327 1336 Pnt 3327 1382 Pnt 3327 1428 Pnt 3327 1474 Pnt 3327 1520 Pnt 3327 1566 Pnt 3327 1612 Pnt 3327 1658 Pnt 3327 2209 Pnt 3327 2346 Pnt 3339 280 Pnt 3339 326 Pnt 3339 372 Pnt 3339 418 Pnt 3339 464 Pnt 3339 510 Pnt 3339 556 Pnt 3339 601 Pnt 3339 647 Pnt 3339 693 Pnt 3339 739 Pnt 3339 785 Pnt 3339 831 Pnt 3339 877 Pnt 3339 923 Pnt 3339 969 Pnt 3339 1015 Pnt 3339 1061 Pnt 3339 1107 Pnt 3339 1152 Pnt 3339 1198 Pnt 3339 1244 Pnt 3339 1290 Pnt 3339 1336 Pnt 3339 1382 Pnt 3339 1474 Pnt 3339 1520 Pnt 3339 1566 Pnt 3339 1612 Pnt 3339 2163 Pnt 3339 2209 Pnt 3339 2300 Pnt 3339 2392 Pnt 3352 280 Pnt 3352 326 Pnt 3352 372 Pnt 3352 418 Pnt 3352 464 Pnt 3352 510 Pnt 3352 556 Pnt 3352 601 Pnt 3352 647 Pnt 3352 693 Pnt 3352 739 Pnt 3352 785 Pnt 3352 831 Pnt 3352 877 Pnt 3352 923 Pnt 3352 969 Pnt 3352 1015 Pnt 3352 1061 Pnt 3352 1107 Pnt 3352 1152 Pnt 3352 1198 Pnt 3352 1244 Pnt 3352 1290 Pnt 3352 1336 Pnt 3352 1382 Pnt 3352 1428 Pnt 3352 1474 Pnt 3352 1520 Pnt 3352 1566 Pnt 3352 2117 Pnt 3352 2163 Pnt 3352 2346 Pnt 3365 280 Pnt 3365 326 Pnt 3365 372 Pnt 3365 418 Pnt 3365 464 Pnt 3365 510 Pnt 3365 556 Pnt 3365 601 Pnt 3365 647 Pnt 3365 693 Pnt 3365 739 Pnt 3365 785 Pnt 3365 831 Pnt 3365 877 Pnt 3365 923 Pnt 3365 969 Pnt 3365 1015 Pnt 3365 1061 Pnt 3365 1107 Pnt 3365 1152 Pnt 3365 1198 Pnt 3365 1244 Pnt 3365 1290 Pnt 3365 1336 Pnt 3365 1382 Pnt 3365 1428 Pnt 3365 1474 Pnt 3365 1520 Pnt 3365 2071 Pnt 3365 2117 Pnt 3365 2255 Pnt 3378 280 Pnt 3378 326 Pnt 3378 372 Pnt 3378 418 Pnt 3378 464 Pnt 3378 510 Pnt 3378 556 Pnt 3378 601 Pnt 3378 647 Pnt 3378 693 Pnt 3378 739 Pnt 3378 785 Pnt 3378 831 Pnt 3378 877 Pnt 3378 923 Pnt 3378 969 Pnt 3378 1015 Pnt 3378 1061 Pnt 3378 1107 Pnt 3378 1152 Pnt 3378 1198 Pnt 3378 1244 Pnt 3378 1290 Pnt 3378 1336 Pnt 3378 1382 Pnt 3378 1428 Pnt 3378 1474 Pnt 3378 2025 Pnt 3378 2071 Pnt 3378 2117 Pnt 3378 2209 Pnt 3378 2300 Pnt 3390 280 Pnt 3390 326 Pnt 3390 372 Pnt 3390 418 Pnt 3390 464 Pnt 3390 510 Pnt 3390 556 Pnt 3390 601 Pnt 3390 647 Pnt 3390 693 Pnt 3390 739 Pnt 3390 785 Pnt 3390 831 Pnt 3390 877 Pnt 3390 923 Pnt 3390 969 Pnt 3390 1015 Pnt 3390 1061 Pnt 3390 1107 Pnt 3390 1152 Pnt 3390 1198 Pnt 3390 1244 Pnt 3390 1290 Pnt 3390 1336 Pnt 3390 1382 Pnt 3390 1428 Pnt 3390 2025 Pnt 3390 2071 Pnt 3390 2209 Pnt 3403 280 Pnt 3403 326 Pnt 3403 372 Pnt 3403 418 Pnt 3403 464 Pnt 3403 510 Pnt 3403 556 Pnt 3403 601 Pnt 3403 647 Pnt 3403 693 Pnt 3403 739 Pnt 3403 785 Pnt 3403 831 Pnt 3403 877 Pnt 3403 923 Pnt 3403 969 Pnt 3403 1015 Pnt 3403 1061 Pnt 3403 1107 Pnt 3403 1152 Pnt 3403 1198 Pnt 3403 1244 Pnt 3403 1290 Pnt 3403 1336 Pnt 3403 1382 Pnt 3403 1979 Pnt 3403 2025 Pnt 3403 2163 Pnt 3403 2255 Pnt 3416 280 Pnt 3416 326 Pnt 3416 372 Pnt 3416 418 Pnt 3416 464 Pnt 3416 510 Pnt 3416 556 Pnt 3416 601 Pnt 3416 647 Pnt 3416 693 Pnt 3416 739 Pnt 3416 785 Pnt 3416 831 Pnt 3416 877 Pnt 3416 923 Pnt 3416 969 Pnt 3416 1015 Pnt 3416 1061 Pnt 3416 1107 Pnt 3416 1152 Pnt 3416 1198 Pnt 3416 1244 Pnt 3416 1290 Pnt 3416 1336 Pnt 3416 1933 Pnt 3416 1979 Pnt 3416 2117 Pnt 3416 2209 Pnt 3428 280 Pnt 3428 326 Pnt 3428 372 Pnt 3428 418 Pnt 3428 464 Pnt 3428 510 Pnt 3428 556 Pnt 3428 601 Pnt 3428 647 Pnt 3428 693 Pnt 3428 739 Pnt 3428 785 Pnt 3428 831 Pnt 3428 877 Pnt 3428 923 Pnt 3428 969 Pnt 3428 1015 Pnt 3428 1061 Pnt 3428 1107 Pnt 3428 1152 Pnt 3428 1198 Pnt 3428 1244 Pnt 3428 1290 Pnt 3428 1887 Pnt 3428 1933 Pnt 3428 1979 Pnt 3428 2117 Pnt 3428 2209 Pnt 3441 280 Pnt 3441 326 Pnt 3441 372 Pnt 3441 418 Pnt 3441 464 Pnt 3441 510 Pnt 3441 556 Pnt 3441 601 Pnt 3441 647 Pnt 3441 693 Pnt 3441 739 Pnt 3441 785 Pnt 3441 831 Pnt 3441 877 Pnt 3441 923 Pnt 3441 969 Pnt 3441 1015 Pnt 3441 1061 Pnt 3441 1107 Pnt 3441 1152 Pnt 3441 1198 Pnt 3441 1244 Pnt 3441 1841 Pnt 3441 1887 Pnt 3441 1933 Pnt 3441 2071 Pnt 3441 2163 Pnt 3454 280 Pnt 3454 326 Pnt 3454 372 Pnt 3454 418 Pnt 3454 464 Pnt 3454 510 Pnt 3454 556 Pnt 3454 601 Pnt 3454 647 Pnt 3454 693 Pnt 3454 739 Pnt 3454 785 Pnt 3454 831 Pnt 3454 877 Pnt 3454 923 Pnt 3454 969 Pnt 3454 1015 Pnt 3454 1061 Pnt 3454 1107 Pnt 3454 1152 Pnt 3454 1198 Pnt 3454 1795 Pnt 3454 1841 Pnt 3454 1887 Pnt 3454 2025 Pnt 3454 2071 Pnt 3454 2117 Pnt 3467 280 Pnt 3467 326 Pnt 3467 372 Pnt 3467 418 Pnt 3467 464 Pnt 3467 510 Pnt 3467 556 Pnt 3467 601 Pnt 3467 647 Pnt 3467 693 Pnt 3467 739 Pnt 3467 785 Pnt 3467 831 Pnt 3467 877 Pnt 3467 923 Pnt 3467 969 Pnt 3467 1015 Pnt 3467 1061 Pnt 3467 1107 Pnt 3467 1152 Pnt 3467 1749 Pnt 3467 1795 Pnt 3467 1841 Pnt 3467 1887 Pnt 3467 2025 Pnt 3467 2117 Pnt 3479 280 Pnt 3479 326 Pnt 3479 372 Pnt 3479 418 Pnt 3479 464 Pnt 3479 510 Pnt 3479 556 Pnt 3479 601 Pnt 3479 647 Pnt 3479 693 Pnt 3479 739 Pnt 3479 785 Pnt 3479 831 Pnt 3479 877 Pnt 3479 923 Pnt 3479 969 Pnt 3479 1015 Pnt 3479 1061 Pnt 3479 1107 Pnt 3479 1704 Pnt 3479 1749 Pnt 3479 1795 Pnt 3479 1841 Pnt 3479 1979 Pnt 3479 2071 Pnt 3492 280 Pnt 3492 326 Pnt 3492 372 Pnt 3492 418 Pnt 3492 464 Pnt 3492 510 Pnt 3492 556 Pnt 3492 601 Pnt 3492 647 Pnt 3492 693 Pnt 3492 739 Pnt 3492 785 Pnt 3492 831 Pnt 3492 877 Pnt 3492 923 Pnt 3492 969 Pnt 3492 1015 Pnt 3492 1061 Pnt 3492 1658 Pnt 3492 1704 Pnt 3492 1749 Pnt 3492 1795 Pnt 3492 1933 Pnt 3492 1979 Pnt 3492 2071 Pnt 3505 280 Pnt 3505 326 Pnt 3505 372 Pnt 3505 418 Pnt 3505 464 Pnt 3505 510 Pnt 3505 556 Pnt 3505 601 Pnt 3505 647 Pnt 3505 693 Pnt 3505 739 Pnt 3505 785 Pnt 3505 831 Pnt 3505 877 Pnt 3505 923 Pnt 3505 969 Pnt 3505 1015 Pnt 3505 1612 Pnt 3505 1658 Pnt 3505 1704 Pnt 3505 1749 Pnt 3505 1887 Pnt 3505 1933 Pnt 3505 2025 Pnt 3517 280 Pnt 3517 326 Pnt 3517 372 Pnt 3517 418 Pnt 3517 464 Pnt 3517 510 Pnt 3517 556 Pnt 3517 601 Pnt 3517 647 Pnt 3517 693 Pnt 3517 739 Pnt 3517 785 Pnt 3517 831 Pnt 3517 877 Pnt 3517 923 Pnt 3517 969 Pnt 3517 1566 Pnt 3517 1612 Pnt 3517 1658 Pnt 3517 1704 Pnt 3517 1749 Pnt 3517 1887 Pnt 3517 1979 Pnt 3530 280 Pnt 3530 326 Pnt 3530 372 Pnt 3530 418 Pnt 3530 464 Pnt 3530 510 Pnt 3530 556 Pnt 3530 601 Pnt 3530 647 Pnt 3530 693 Pnt 3530 739 Pnt 3530 785 Pnt 3530 831 Pnt 3530 877 Pnt 3530 1520 Pnt 3530 1566 Pnt 3530 1612 Pnt 3530 1658 Pnt 3530 1704 Pnt 3530 1841 Pnt 3530 1887 Pnt 3530 1979 Pnt 3543 280 Pnt 3543 326 Pnt 3543 372 Pnt 3543 418 Pnt 3543 464 Pnt 3543 510 Pnt 3543 556 Pnt 3543 601 Pnt 3543 647 Pnt 3543 693 Pnt 3543 739 Pnt 3543 785 Pnt 3543 831 Pnt 3543 1428 Pnt 3543 1474 Pnt 3543 1520 Pnt 3543 1566 Pnt 3543 1612 Pnt 3543 1658 Pnt 3543 1795 Pnt 3543 1841 Pnt 3543 1933 Pnt 3556 280 Pnt 3556 326 Pnt 3556 372 Pnt 3556 418 Pnt 3556 464 Pnt 3556 510 Pnt 3556 556 Pnt 3556 601 Pnt 3556 647 Pnt 3556 693 Pnt 3556 739 Pnt 3556 785 Pnt 3556 1382 Pnt 3556 1428 Pnt 3556 1474 Pnt 3556 1520 Pnt 3556 1566 Pnt 3556 1612 Pnt 3556 1749 Pnt 3556 1795 Pnt 3556 1887 Pnt 3556 1933 Pnt 3568 280 Pnt 3568 326 Pnt 3568 372 Pnt 3568 418 Pnt 3568 464 Pnt 3568 510 Pnt 3568 556 Pnt 3568 601 Pnt 3568 647 Pnt 3568 693 Pnt 3568 739 Pnt 3568 1290 Pnt 3568 1336 Pnt 3568 1382 Pnt 3568 1428 Pnt 3568 1474 Pnt 3568 1520 Pnt 3568 1566 Pnt 3568 1749 Pnt 3568 1795 Pnt 3568 1887 Pnt 3581 280 Pnt 3581 326 Pnt 3581 372 Pnt 3581 418 Pnt 3581 464 Pnt 3581 510 Pnt 3581 556 Pnt 3581 601 Pnt 3581 647 Pnt 3581 693 Pnt 3581 1244 Pnt 3581 1290 Pnt 3581 1336 Pnt 3581 1382 Pnt 3581 1428 Pnt 3581 1474 Pnt 3581 1520 Pnt 3581 1566 Pnt 3581 1704 Pnt 3581 1749 Pnt 3581 1841 Pnt 3594 280 Pnt 3594 326 Pnt 3594 372 Pnt 3594 418 Pnt 3594 464 Pnt 3594 510 Pnt 3594 556 Pnt 3594 601 Pnt 3594 1152 Pnt 3594 1198 Pnt 3594 1244 Pnt 3594 1290 Pnt 3594 1336 Pnt 3594 1382 Pnt 3594 1428 Pnt 3594 1474 Pnt 3594 1520 Pnt 3594 1658 Pnt 3594 1704 Pnt 3594 1749 Pnt 3594 1841 Pnt 3606 280 Pnt 3606 326 Pnt 3606 372 Pnt 3606 418 Pnt 3606 464 Pnt 3606 510 Pnt 3606 556 Pnt 3606 1061 Pnt 3606 1107 Pnt 3606 1152 Pnt 3606 1198 Pnt 3606 1244 Pnt 3606 1290 Pnt 3606 1336 Pnt 3606 1382 Pnt 3606 1428 Pnt 3606 1474 Pnt 3606 1612 Pnt 3606 1658 Pnt 3606 1704 Pnt 3606 1795 Pnt 3619 280 Pnt 3619 326 Pnt 3619 372 Pnt 3619 418 Pnt 3619 464 Pnt 3619 510 Pnt 3619 969 Pnt 3619 1015 Pnt 3619 1061 Pnt 3619 1107 Pnt 3619 1152 Pnt 3619 1198 Pnt 3619 1244 Pnt 3619 1290 Pnt 3619 1336 Pnt 3619 1382 Pnt 3619 1428 Pnt 3619 1612 Pnt 3619 1658 Pnt 3619 1749 Pnt 3619 1795 Pnt 3632 280 Pnt 3632 326 Pnt 3632 372 Pnt 3632 418 Pnt 3632 785 Pnt 3632 831 Pnt 3632 877 Pnt 3632 923 Pnt 3632 969 Pnt 3632 1015 Pnt 3632 1061 Pnt 3632 1107 Pnt 3632 1152 Pnt 3632 1198 Pnt 3632 1244 Pnt 3632 1290 Pnt 3632 1336 Pnt 3632 1382 Pnt 3632 1566 Pnt 3632 1612 Pnt 3632 1658 Pnt 3632 1749 Pnt 3644 280 Pnt 3644 326 Pnt 3644 372 Pnt 3644 601 Pnt 3644 647 Pnt 3644 693 Pnt 3644 739 Pnt 3644 785 Pnt 3644 831 Pnt 3644 877 Pnt 3644 923 Pnt 3644 969 Pnt 3644 1015 Pnt 3644 1061 Pnt 3644 1107 Pnt 3644 1152 Pnt 3644 1198 Pnt 3644 1244 Pnt 3644 1290 Pnt 3644 1336 Pnt 3644 1520 Pnt 3644 1566 Pnt 3644 1612 Pnt 3644 1704 Pnt 3644 1749 Pnt 3657 280 Pnt 3657 326 Pnt 3657 372 Pnt 3657 418 Pnt 3657 464 Pnt 3657 510 Pnt 3657 556 Pnt 3657 601 Pnt 3657 647 Pnt 3657 693 Pnt 3657 739 Pnt 3657 785 Pnt 3657 831 Pnt 3657 877 Pnt 3657 923 Pnt 3657 969 Pnt 3657 1015 Pnt 3657 1061 Pnt 3657 1107 Pnt 3657 1152 Pnt 3657 1198 Pnt 3657 1244 Pnt 3657 1290 Pnt 3657 1336 Pnt 3657 1474 Pnt 3657 1520 Pnt 3657 1566 Pnt 3657 1658 Pnt 3657 1704 Pnt 3670 280 Pnt 3670 326 Pnt 3670 372 Pnt 3670 418 Pnt 3670 464 Pnt 3670 510 Pnt 3670 556 Pnt 3670 601 Pnt 3670 647 Pnt 3670 693 Pnt 3670 739 Pnt 3670 785 Pnt 3670 831 Pnt 3670 877 Pnt 3670 923 Pnt 3670 969 Pnt 3670 1015 Pnt 3670 1061 Pnt 3670 1107 Pnt 3670 1152 Pnt 3670 1198 Pnt 3670 1244 Pnt 3670 1290 Pnt 3670 1428 Pnt 3670 1474 Pnt 3670 1520 Pnt 3670 1566 Pnt 3670 1658 Pnt 3683 280 Pnt 3683 326 Pnt 3683 372 Pnt 3683 418 Pnt 3683 464 Pnt 3683 510 Pnt 3683 556 Pnt 3683 601 Pnt 3683 647 Pnt 3683 693 Pnt 3683 739 Pnt 3683 785 Pnt 3683 831 Pnt 3683 877 Pnt 3683 923 Pnt 3683 969 Pnt 3683 1015 Pnt 3683 1061 Pnt 3683 1107 Pnt 3683 1152 Pnt 3683 1198 Pnt 3683 1244 Pnt 3683 1382 Pnt 3683 1428 Pnt 3683 1474 Pnt 3683 1520 Pnt 3683 1612 Pnt 3683 1658 Pnt 3695 280 Pnt 3695 326 Pnt 3695 372 Pnt 3695 418 Pnt 3695 464 Pnt 3695 510 Pnt 3695 556 Pnt 3695 601 Pnt 3695 647 Pnt 3695 693 Pnt 3695 739 Pnt 3695 785 Pnt 3695 831 Pnt 3695 877 Pnt 3695 923 Pnt 3695 969 Pnt 3695 1015 Pnt 3695 1061 Pnt 3695 1107 Pnt 3695 1152 Pnt 3695 1198 Pnt 3695 1336 Pnt 3695 1382 Pnt 3695 1428 Pnt 3695 1474 Pnt 3695 1566 Pnt 3695 1612 Pnt 3708 280 Pnt 3708 326 Pnt 3708 372 Pnt 3708 418 Pnt 3708 464 Pnt 3708 510 Pnt 3708 556 Pnt 3708 601 Pnt 3708 647 Pnt 3708 693 Pnt 3708 739 Pnt 3708 785 Pnt 3708 831 Pnt 3708 877 Pnt 3708 923 Pnt 3708 969 Pnt 3708 1015 Pnt 3708 1061 Pnt 3708 1107 Pnt 3708 1152 Pnt 3708 1290 Pnt 3708 1336 Pnt 3708 1382 Pnt 3708 1428 Pnt 3708 1474 Pnt 3708 1520 Pnt 3708 1566 Pnt 3708 1612 Pnt 3721 280 Pnt 3721 326 Pnt 3721 372 Pnt 3721 418 Pnt 3721 464 Pnt 3721 510 Pnt 3721 556 Pnt 3721 601 Pnt 3721 647 Pnt 3721 693 Pnt 3721 739 Pnt 3721 785 Pnt 3721 831 Pnt 3721 877 Pnt 3721 923 Pnt 3721 969 Pnt 3721 1015 Pnt 3721 1061 Pnt 3721 1107 Pnt 3721 1244 Pnt 3721 1290 Pnt 3721 1336 Pnt 3721 1382 Pnt 3721 1428 Pnt 3721 1520 Pnt 3721 1566 Pnt 3733 280 Pnt 3733 326 Pnt 3733 372 Pnt 3733 418 Pnt 3733 464 Pnt 3733 510 Pnt 3733 556 Pnt 3733 601 Pnt 3733 647 Pnt 3733 693 Pnt 3733 739 Pnt 3733 785 Pnt 3733 831 Pnt 3733 877 Pnt 3733 923 Pnt 3733 969 Pnt 3733 1015 Pnt 3733 1061 Pnt 3733 1198 Pnt 3733 1244 Pnt 3733 1290 Pnt 3733 1336 Pnt 3733 1382 Pnt 3733 1474 Pnt 3733 1520 Pnt 3746 280 Pnt 3746 326 Pnt 3746 372 Pnt 3746 418 Pnt 3746 464 Pnt 3746 510 Pnt 3746 556 Pnt 3746 601 Pnt 3746 647 Pnt 3746 693 Pnt 3746 739 Pnt 3746 785 Pnt 3746 831 Pnt 3746 877 Pnt 3746 923 Pnt 3746 969 Pnt 3746 1015 Pnt 3746 1152 Pnt 3746 1198 Pnt 3746 1244 Pnt 3746 1290 Pnt 3746 1336 Pnt 3746 1382 Pnt 3746 1428 Pnt 3746 1474 Pnt 3746 1520 Pnt 3759 280 Pnt 3759 326 Pnt 3759 372 Pnt 3759 418 Pnt 3759 464 Pnt 3759 510 Pnt 3759 556 Pnt 3759 601 Pnt 3759 647 Pnt 3759 693 Pnt 3759 739 Pnt 3759 785 Pnt 3759 831 Pnt 3759 877 Pnt 3759 923 Pnt 3759 969 Pnt 3759 1107 Pnt 3759 1152 Pnt 3759 1198 Pnt 3759 1244 Pnt 3759 1290 Pnt 3759 1336 Pnt 3759 1428 Pnt 3759 1474 Pnt 3772 280 Pnt 3772 326 Pnt 3772 372 Pnt 3772 418 Pnt 3772 464 Pnt 3772 510 Pnt 3772 556 Pnt 3772 601 Pnt 3772 647 Pnt 3772 693 Pnt 3772 739 Pnt 3772 785 Pnt 3772 831 Pnt 3772 877 Pnt 3772 923 Pnt 3772 1061 Pnt 3772 1107 Pnt 3772 1152 Pnt 3772 1198 Pnt 3772 1244 Pnt 3772 1290 Pnt 3772 1382 Pnt 3772 1428 Pnt 3772 1474 Pnt 3784 280 Pnt 3784 326 Pnt 3784 372 Pnt 3784 418 Pnt 3784 464 Pnt 3784 510 Pnt 3784 556 Pnt 3784 601 Pnt 3784 647 Pnt 3784 693 Pnt 3784 739 Pnt 3784 785 Pnt 3784 831 Pnt 3784 877 Pnt 3784 969 Pnt 3784 1015 Pnt 3784 1061 Pnt 3784 1107 Pnt 3784 1152 Pnt 3784 1198 Pnt 3784 1244 Pnt 3784 1336 Pnt 3784 1382 Pnt 3784 1428 Pnt 3797 280 Pnt 3797 326 Pnt 3797 372 Pnt 3797 418 Pnt 3797 464 Pnt 3797 510 Pnt 3797 556 Pnt 3797 601 Pnt 3797 647 Pnt 3797 693 Pnt 3797 739 Pnt 3797 785 Pnt 3797 831 Pnt 3797 923 Pnt 3797 969 Pnt 3797 1015 Pnt 3797 1061 Pnt 3797 1107 Pnt 3797 1152 Pnt 3797 1198 Pnt 3797 1244 Pnt 3797 1290 Pnt 3797 1336 Pnt 3797 1382 Pnt 3810 280 Pnt 3810 326 Pnt 3810 372 Pnt 3810 418 Pnt 3810 464 Pnt 3810 510 Pnt 3810 556 Pnt 3810 601 Pnt 3810 647 Pnt 3810 693 Pnt 3810 739 Pnt 3810 831 Pnt 3810 877 Pnt 3810 923 Pnt 3810 969 Pnt 3810 1015 Pnt 3810 1061 Pnt 3810 1107 Pnt 3810 1152 Pnt 3810 1198 Pnt 3810 1290 Pnt 3810 1336 Pnt 3810 1382 Pnt 3822 280 Pnt 3822 326 Pnt 3822 372 Pnt 3822 418 Pnt 3822 464 Pnt 3822 510 Pnt 3822 556 Pnt 3822 601 Pnt 3822 647 Pnt 3822 693 Pnt 3822 785 Pnt 3822 831 Pnt 3822 877 Pnt 3822 923 Pnt 3822 969 Pnt 3822 1015 Pnt 3822 1061 Pnt 3822 1107 Pnt 3822 1152 Pnt 3822 1244 Pnt 3822 1290 Pnt 3822 1336 Pnt 3835 280 Pnt 3835 326 Pnt 3835 372 Pnt 3835 418 Pnt 3835 464 Pnt 3835 510 Pnt 3835 556 Pnt 3835 601 Pnt 3835 647 Pnt 3835 693 Pnt 3835 739 Pnt 3835 785 Pnt 3835 831 Pnt 3835 877 Pnt 3835 923 Pnt 3835 969 Pnt 3835 1015 Pnt 3835 1061 Pnt 3835 1107 Pnt 3835 1152 Pnt 3835 1198 Pnt 3835 1244 Pnt 3835 1290 Pnt 3835 1336 Pnt 3848 280 Pnt 3848 326 Pnt 3848 372 Pnt 3848 418 Pnt 3848 464 Pnt 3848 510 Pnt 3848 556 Pnt 3848 601 Pnt 3848 647 Pnt 3848 693 Pnt 3848 739 Pnt 3848 785 Pnt 3848 831 Pnt 3848 877 Pnt 3848 923 Pnt 3848 969 Pnt 3848 1015 Pnt 3848 1061 Pnt 3848 1107 Pnt 3848 1152 Pnt 3848 1198 Pnt 3848 1244 Pnt 3848 1290 Pnt 3861 280 Pnt 3861 326 Pnt 3861 372 Pnt 3861 418 Pnt 3861 464 Pnt 3861 510 Pnt 3861 556 Pnt 3861 601 Pnt 3861 647 Pnt 3861 693 Pnt 3861 739 Pnt 3861 785 Pnt 3861 831 Pnt 3861 877 Pnt 3861 923 Pnt 3861 969 Pnt 3861 1015 Pnt 3861 1061 Pnt 3861 1152 Pnt 3861 1198 Pnt 3861 1244 Pnt 3873 280 Pnt 3873 326 Pnt 3873 372 Pnt 3873 418 Pnt 3873 464 Pnt 3873 510 Pnt 3873 556 Pnt 3873 601 Pnt 3873 647 Pnt 3873 693 Pnt 3873 739 Pnt 3873 785 Pnt 3873 831 Pnt 3873 877 Pnt 3873 923 Pnt 3873 969 Pnt 3873 1015 Pnt 3873 1107 Pnt 3873 1152 Pnt 3873 1198 Pnt 3873 1244 Pnt 3886 280 Pnt 3886 326 Pnt 3886 372 Pnt 3886 418 Pnt 3886 464 Pnt 3886 510 Pnt 3886 556 Pnt 3886 601 Pnt 3886 647 Pnt 3886 693 Pnt 3886 739 Pnt 3886 785 Pnt 3886 831 Pnt 3886 877 Pnt 3886 923 Pnt 3886 969 Pnt 3886 1015 Pnt 3886 1061 Pnt 3886 1107 Pnt 3886 1152 Pnt 3886 1198 Pnt 3899 280 Pnt 3899 326 Pnt 3899 372 Pnt 3899 418 Pnt 3899 464 Pnt 3899 510 Pnt 3899 556 Pnt 3899 601 Pnt 3899 647 Pnt 3899 693 Pnt 3899 739 Pnt 3899 785 Pnt 3899 831 Pnt 3899 877 Pnt 3899 923 Pnt 3899 969 Pnt 3899 1015 Pnt 3899 1061 Pnt 3899 1107 Pnt 3899 1152 Pnt 3899 1198 Pnt 3911 280 Pnt 3911 326 Pnt 3911 372 Pnt 3911 418 Pnt 3911 464 Pnt 3911 510 Pnt 3911 556 Pnt 3911 601 Pnt 3911 647 Pnt 3911 693 Pnt 3911 739 Pnt 3911 785 Pnt 3911 831 Pnt 3911 877 Pnt 3911 923 Pnt 3911 969 Pnt 3911 1015 Pnt 3911 1061 Pnt 3911 1107 Pnt 3911 1152 Pnt 3924 280 Pnt 3924 326 Pnt 3924 372 Pnt 3924 418 Pnt 3924 464 Pnt 3924 510 Pnt 3924 556 Pnt 3924 601 Pnt 3924 647 Pnt 3924 693 Pnt 3924 739 Pnt 3924 785 Pnt 3924 831 Pnt 3924 877 Pnt 3924 923 Pnt 3924 969 Pnt 3924 1015 Pnt 3924 1061 Pnt 3924 1107 Pnt 3937 280 Pnt 3937 326 Pnt 3937 372 Pnt 3937 418 Pnt 3937 464 Pnt 3937 510 Pnt 3937 556 Pnt 3937 601 Pnt 3937 647 Pnt 3937 693 Pnt 3937 739 Pnt 3937 785 Pnt 3937 831 Pnt 3937 877 Pnt 3937 923 Pnt 3937 969 Pnt 3937 1015 Pnt 3937 1061 Pnt 3937 1107 Pnt 3950 280 Pnt 3950 326 Pnt 3950 372 Pnt 3950 418 Pnt 3950 464 Pnt 3950 510 Pnt 3950 556 Pnt 3950 601 Pnt 3950 647 Pnt 3950 693 Pnt 3950 739 Pnt 3950 785 Pnt 3950 831 Pnt 3950 877 Pnt 3950 923 Pnt 3950 969 Pnt 3950 1015 Pnt 3950 1061 Pnt 3962 280 Pnt 3962 326 Pnt 3962 372 Pnt 3962 418 Pnt 3962 464 Pnt 3962 510 Pnt 3962 556 Pnt 3962 601 Pnt 3962 647 Pnt 3962 693 Pnt 3962 739 Pnt 3962 785 Pnt 3962 831 Pnt 3962 877 Pnt 3962 923 Pnt 3962 969 Pnt 3962 1015 Pnt 3962 1061 Pnt 3975 280 Pnt 3975 326 Pnt 3975 372 Pnt 3975 418 Pnt 3975 464 Pnt 3975 510 Pnt 3975 556 Pnt 3975 601 Pnt 3975 647 Pnt 3975 693 Pnt 3975 739 Pnt 3975 785 Pnt 3975 831 Pnt 3975 877 Pnt 3975 923 Pnt 3975 969 Pnt 3975 1015 Pnt 3988 280 Pnt 3988 326 Pnt 3988 372 Pnt 3988 418 Pnt 3988 464 Pnt 3988 510 Pnt 3988 556 Pnt 3988 601 Pnt 3988 647 Pnt 3988 693 Pnt 3988 739 Pnt 3988 785 Pnt 3988 831 Pnt 3988 877 Pnt 3988 923 Pnt 3988 969 Pnt 4000 280 Pnt 4000 326 Pnt 4000 372 Pnt 4000 418 Pnt 4000 464 Pnt 4000 510 Pnt 4000 556 Pnt 4000 601 Pnt 4000 647 Pnt 4000 693 Pnt 4000 739 Pnt 4000 785 Pnt 4000 831 Pnt 4000 877 Pnt 4000 923 Pnt 4000 969 Pnt 4013 280 Pnt 4013 326 Pnt 4013 372 Pnt 4013 418 Pnt 4013 464 Pnt 4013 510 Pnt 4013 556 Pnt 4013 601 Pnt 4013 647 Pnt 4013 693 Pnt 4013 739 Pnt 4013 785 Pnt 4013 831 Pnt 4013 877 Pnt 4013 923 Pnt 4026 280 Pnt 4026 326 Pnt 4026 372 Pnt 4026 418 Pnt 4026 464 Pnt 4026 510 Pnt 4026 556 Pnt 4026 601 Pnt 4026 647 Pnt 4026 693 Pnt 4026 739 Pnt 4026 785 Pnt 4026 831 Pnt 4026 877 Pnt 4039 280 Pnt 4039 326 Pnt 4039 372 Pnt 4039 418 Pnt 4039 464 Pnt 4039 510 Pnt 4039 556 Pnt 4039 601 Pnt 4039 647 Pnt 4039 693 Pnt 4039 739 Pnt 4039 785 Pnt 4039 831 Pnt 4039 877 Pnt 4051 280 Pnt 4051 326 Pnt 4051 372 Pnt 4051 418 Pnt 4051 464 Pnt 4051 510 Pnt 4051 556 Pnt 4051 601 Pnt 4051 647 Pnt 4051 693 Pnt 4051 739 Pnt 4051 785 Pnt 4051 831 Pnt 4064 280 Pnt 4064 326 Pnt 4064 372 Pnt 4064 418 Pnt 4064 464 Pnt 4064 510 Pnt 4064 556 Pnt 4064 601 Pnt 4064 647 Pnt 4064 693 Pnt 4064 739 Pnt 4064 785 Pnt 4064 831 Pnt 4077 280 Pnt 4077 326 Pnt 4077 372 Pnt 4077 418 Pnt 4077 464 Pnt 4077 510 Pnt 4077 556 Pnt 4077 601 Pnt 4077 647 Pnt 4077 693 Pnt 4077 739 Pnt 4077 785 Pnt 4089 280 Pnt 4089 326 Pnt 4089 372 Pnt 4089 418 Pnt 4089 464 Pnt 4089 510 Pnt 4089 556 Pnt 4089 601 Pnt 4089 647 Pnt 4089 693 Pnt 4089 739 Pnt 4102 280 Pnt 4102 326 Pnt 4102 372 Pnt 4102 418 Pnt 4102 464 Pnt 4102 510 Pnt 4102 556 Pnt 4102 601 Pnt 4102 647 Pnt 4102 693 Pnt 4102 739 Pnt 4115 280 Pnt 4115 326 Pnt 4115 372 Pnt 4115 418 Pnt 4115 464 Pnt 4115 510 Pnt 4115 556 Pnt 4115 601 Pnt 4115 647 Pnt 4115 693 Pnt 4127 280 Pnt 4127 326 Pnt 4127 372 Pnt 4127 418 Pnt 4127 464 Pnt 4127 510 Pnt 4127 556 Pnt 4127 601 Pnt 4127 647 Pnt 4127 693 Pnt 4140 280 Pnt 4140 326 Pnt 4140 372 Pnt 4140 418 Pnt 4140 464 Pnt 4140 510 Pnt 4140 556 Pnt 4140 601 Pnt 4140 647 Pnt 4153 280 Pnt 4153 326 Pnt 4153 372 Pnt 4153 418 Pnt 4153 464 Pnt 4153 510 Pnt 4153 556 Pnt 4153 601 Pnt 4166 280 Pnt 4166 326 Pnt 4166 372 Pnt 4166 418 Pnt 4166 464 Pnt 4166 510 Pnt 4166 556 Pnt 4166 601 Pnt 4178 280 Pnt 4178 326 Pnt 4178 372 Pnt 4178 418 Pnt 4178 464 Pnt 4178 510 Pnt 4178 556 Pnt 4191 280 Pnt 4191 326 Pnt 4191 372 Pnt 4191 418 Pnt 4191 464 Pnt 4191 510 Pnt 4191 556 Pnt 4204 280 Pnt 4204 326 Pnt 4204 372 Pnt 4204 418 Pnt 4204 464 Pnt 4204 510 Pnt 4216 280 Pnt 4216 326 Pnt 4216 372 Pnt 4216 418 Pnt 4216 464 Pnt 4229 280 Pnt 4229 326 Pnt 4229 372 Pnt 4229 418 Pnt 4229 464 Pnt 4242 280 Pnt 4242 326 Pnt 4242 372 Pnt 4242 418 Pnt 4255 280 Pnt 4255 326 Pnt 4255 372 Pnt 4255 418 Pnt 4267 280 Pnt 4267 326 Pnt 4267 372 Pnt 4280 280 Pnt 4280 326 Pnt 4293 280 Pnt 4293 326 Pnt 4305 280 Pnt 4318 280 Pnt 4331 280 Pnt 4331 326 Pnt 4344 280 Pnt 4344 326 Pnt 4344 372 Pnt 4356 280 Pnt 4356 326 Pnt 4356 372 Pnt 4356 418 Pnt 4356 464 Pnt 4369 280 Pnt 4369 326 Pnt 4369 372 Pnt 4369 418 Pnt 4369 464 Pnt 4369 510 Pnt 4382 280 Pnt 4382 326 Pnt 4382 372 Pnt 4382 418 Pnt 4382 464 Pnt 4382 510 Pnt 4382 556 Pnt 4394 280 Pnt 4394 326 Pnt 4394 372 Pnt 4394 418 Pnt 4394 464 Pnt 4394 510 Pnt 4394 556 Pnt 4394 601 Pnt 4407 280 Pnt 4407 326 Pnt 4407 372 Pnt 4407 418 Pnt 4407 464 Pnt 4407 510 Pnt 4407 556 Pnt 4407 601 Pnt 4407 647 Pnt 4420 280 Pnt 4420 326 Pnt 4420 372 Pnt 4420 418 Pnt 4420 464 Pnt 4420 510 Pnt 4420 556 Pnt 4420 601 Pnt 4420 647 Pnt 4420 693 Pnt 4433 280 Pnt 4433 326 Pnt 4433 372 Pnt 4433 418 Pnt 4433 464 Pnt 4433 510 Pnt 4433 556 Pnt 4433 601 Pnt 4433 647 Pnt 4433 693 Pnt 4433 739 Pnt 4445 280 Pnt 4445 326 Pnt 4445 372 Pnt 4445 418 Pnt 4445 464 Pnt 4445 510 Pnt 4445 556 Pnt 4445 601 Pnt 4445 647 Pnt 4445 693 Pnt 4445 739 Pnt 4458 280 Pnt 4458 326 Pnt 4458 372 Pnt 4458 418 Pnt 4458 464 Pnt 4458 510 Pnt 4458 556 Pnt 4458 601 Pnt 4458 647 Pnt 4458 693 Pnt 4458 739 Pnt 4458 785 Pnt 4471 280 Pnt 4471 326 Pnt 4471 372 Pnt 4471 418 Pnt 4471 464 Pnt 4471 510 Pnt 4471 556 Pnt 4471 601 Pnt 4471 647 Pnt 4471 693 Pnt 4471 739 Pnt 4471 785 Pnt 4471 831 Pnt 4483 280 Pnt 4483 326 Pnt 4483 372 Pnt 4483 418 Pnt 4483 464 Pnt 4483 510 Pnt 4483 556 Pnt 4483 601 Pnt 4483 647 Pnt 4483 693 Pnt 4483 739 Pnt 4483 785 Pnt 4483 831 Pnt 4483 877 Pnt 4496 280 Pnt 4496 326 Pnt 4496 372 Pnt 4496 418 Pnt 4496 464 Pnt 4496 510 Pnt 4496 556 Pnt 4496 601 Pnt 4496 647 Pnt 4496 693 Pnt 4496 739 Pnt 4496 785 Pnt 4496 831 Pnt 4496 877 Pnt 4496 923 Pnt 4509 280 Pnt 4509 326 Pnt 4509 372 Pnt 4509 418 Pnt 4509 464 Pnt 4509 510 Pnt 4509 556 Pnt 4509 601 Pnt 4509 647 Pnt 4509 693 Pnt 4509 739 Pnt 4509 785 Pnt 4509 831 Pnt 4509 877 Pnt 4509 923 Pnt 4509 969 Pnt 4522 280 Pnt 4522 326 Pnt 4522 372 Pnt 4522 418 Pnt 4522 464 Pnt 4522 510 Pnt 4522 556 Pnt 4522 601 Pnt 4522 647 Pnt 4522 693 Pnt 4522 739 Pnt 4522 785 Pnt 4522 831 Pnt 4522 877 Pnt 4522 923 Pnt 4522 969 Pnt 4534 280 Pnt 4534 326 Pnt 4534 372 Pnt 4534 418 Pnt 4534 464 Pnt 4534 510 Pnt 4534 556 Pnt 4534 601 Pnt 4534 647 Pnt 4534 693 Pnt 4534 739 Pnt 4534 785 Pnt 4534 831 Pnt 4534 877 Pnt 4534 923 Pnt 4534 969 Pnt 4534 1015 Pnt 4547 280 Pnt 4547 326 Pnt 4547 372 Pnt 4547 418 Pnt 4547 464 Pnt 4547 510 Pnt 4547 556 Pnt 4547 601 Pnt 4547 647 Pnt 4547 693 Pnt 4547 739 Pnt 4547 785 Pnt 4547 831 Pnt 4547 877 Pnt 4547 923 Pnt 4547 969 Pnt 4547 1015 Pnt 4547 1061 Pnt 4560 280 Pnt 4560 326 Pnt 4560 372 Pnt 4560 418 Pnt 4560 464 Pnt 4560 510 Pnt 4560 556 Pnt 4560 601 Pnt 4560 647 Pnt 4560 693 Pnt 4560 739 Pnt 4560 785 Pnt 4560 831 Pnt 4560 877 Pnt 4560 923 Pnt 4560 969 Pnt 4560 1015 Pnt 4560 1061 Pnt 4560 1107 Pnt 4572 280 Pnt 4572 326 Pnt 4572 372 Pnt 4572 418 Pnt 4572 464 Pnt 4572 510 Pnt 4572 556 Pnt 4572 601 Pnt 4572 647 Pnt 4572 693 Pnt 4572 739 Pnt 4572 785 Pnt 4572 831 Pnt 4572 877 Pnt 4572 923 Pnt 4572 969 Pnt 4572 1015 Pnt 4572 1061 Pnt 4572 1107 Pnt 4585 280 Pnt 4585 326 Pnt 4585 372 Pnt 4585 418 Pnt 4585 464 Pnt 4585 510 Pnt 4585 556 Pnt 4585 601 Pnt 4585 647 Pnt 4585 693 Pnt 4585 739 Pnt 4585 785 Pnt 4585 831 Pnt 4585 877 Pnt 4585 923 Pnt 4585 969 Pnt 4585 1015 Pnt 4585 1061 Pnt 4585 1107 Pnt 4585 1152 Pnt 4598 280 Pnt 4598 326 Pnt 4598 372 Pnt 4598 418 Pnt 4598 464 Pnt 4598 510 Pnt 4598 556 Pnt 4598 601 Pnt 4598 647 Pnt 4598 693 Pnt 4598 739 Pnt 4598 785 Pnt 4598 831 Pnt 4598 877 Pnt 4598 923 Pnt 4598 969 Pnt 4598 1015 Pnt 4598 1061 Pnt 4598 1107 Pnt 4598 1152 Pnt 4598 1198 Pnt 4611 280 Pnt 4611 326 Pnt 4611 372 Pnt 4611 418 Pnt 4611 464 Pnt 4611 510 Pnt 4611 556 Pnt 4611 601 Pnt 4611 647 Pnt 4611 693 Pnt 4611 739 Pnt 4611 785 Pnt 4611 831 Pnt 4611 877 Pnt 4611 923 Pnt 4611 969 Pnt 4611 1015 Pnt 4611 1061 Pnt 4611 1107 Pnt 4611 1152 Pnt 4611 1198 Pnt 4611 1244 Pnt 4623 280 Pnt 4623 326 Pnt 4623 372 Pnt 4623 418 Pnt 4623 464 Pnt 4623 510 Pnt 4623 556 Pnt 4623 601 Pnt 4623 647 Pnt 4623 693 Pnt 4623 739 Pnt 4623 785 Pnt 4623 831 Pnt 4623 877 Pnt 4623 923 Pnt 4623 969 Pnt 4623 1015 Pnt 4623 1061 Pnt 4623 1107 Pnt 4623 1152 Pnt 4623 1198 Pnt 4623 1244 Pnt 4636 280 Pnt 4636 326 Pnt 4636 372 Pnt 4636 418 Pnt 4636 464 Pnt 4636 510 Pnt 4636 556 Pnt 4636 601 Pnt 4636 647 Pnt 4636 693 Pnt 4636 739 Pnt 4636 785 Pnt 4636 831 Pnt 4636 877 Pnt 4636 923 Pnt 4636 969 Pnt 4636 1015 Pnt 4636 1061 Pnt 4636 1107 Pnt 4636 1152 Pnt 4636 1198 Pnt 4636 1244 Pnt 4636 1290 Pnt 4649 280 Pnt 4649 326 Pnt 4649 372 Pnt 4649 418 Pnt 4649 464 Pnt 4649 510 Pnt 4649 556 Pnt 4649 601 Pnt 4649 647 Pnt 4649 693 Pnt 4649 739 Pnt 4649 785 Pnt 4649 831 Pnt 4649 877 Pnt 4649 923 Pnt 4649 969 Pnt 4649 1015 Pnt 4649 1061 Pnt 4649 1107 Pnt 4649 1152 Pnt 4649 1198 Pnt 4649 1244 Pnt 4649 1290 Pnt 4649 1336 Pnt 4661 280 Pnt 4661 326 Pnt 4661 372 Pnt 4661 418 Pnt 4661 464 Pnt 4661 510 Pnt 4661 556 Pnt 4661 601 Pnt 4661 647 Pnt 4661 693 Pnt 4661 739 Pnt 4661 785 Pnt 4661 831 Pnt 4661 877 Pnt 4661 923 Pnt 4661 969 Pnt 4661 1015 Pnt 4661 1061 Pnt 4661 1107 Pnt 4661 1152 Pnt 4661 1198 Pnt 4661 1244 Pnt 4661 1290 Pnt 4661 1336 Pnt 4674 280 Pnt 4674 326 Pnt 4674 372 Pnt 4674 418 Pnt 4674 464 Pnt 4674 510 Pnt 4674 556 Pnt 4674 601 Pnt 4674 647 Pnt 4674 693 Pnt 4674 739 Pnt 4674 785 Pnt 4674 831 Pnt 4674 877 Pnt 4674 923 Pnt 4674 969 Pnt 4674 1015 Pnt 4674 1061 Pnt 4674 1107 Pnt 4674 1152 Pnt 4674 1198 Pnt 4674 1244 Pnt 4674 1290 Pnt 4674 1336 Pnt 4674 1382 Pnt 4687 280 Pnt 4687 326 Pnt 4687 372 Pnt 4687 418 Pnt 4687 464 Pnt 4687 510 Pnt 4687 556 Pnt 4687 601 Pnt 4687 647 Pnt 4687 693 Pnt 4687 739 Pnt 4687 785 Pnt 4687 831 Pnt 4687 877 Pnt 4687 923 Pnt 4687 969 Pnt 4687 1015 Pnt 4687 1061 Pnt 4687 1107 Pnt 4687 1152 Pnt 4687 1198 Pnt 4687 1244 Pnt 4687 1290 Pnt 4687 1336 Pnt 4687 1382 Pnt 4687 1428 Pnt 4699 280 Pnt 4699 326 Pnt 4699 372 Pnt 4699 418 Pnt 4699 464 Pnt 4699 510 Pnt 4699 556 Pnt 4699 601 Pnt 4699 647 Pnt 4699 693 Pnt 4699 739 Pnt 4699 785 Pnt 4699 831 Pnt 4699 877 Pnt 4699 923 Pnt 4699 969 Pnt 4699 1015 Pnt 4699 1061 Pnt 4699 1107 Pnt 4699 1152 Pnt 4699 1198 Pnt 4699 1244 Pnt 4699 1290 Pnt 4699 1336 Pnt 4699 1382 Pnt 4699 1428 Pnt 4712 280 Pnt 4712 326 Pnt 4712 372 Pnt 4712 418 Pnt 4712 464 Pnt 4712 510 Pnt 4712 556 Pnt 4712 601 Pnt 4712 647 Pnt 4712 693 Pnt 4712 739 Pnt 4712 785 Pnt 4712 831 Pnt 4712 877 Pnt 4712 923 Pnt 4712 969 Pnt 4712 1015 Pnt 4712 1061 Pnt 4712 1107 Pnt 4712 1152 Pnt 4712 1198 Pnt 4712 1244 Pnt 4712 1290 Pnt 4712 1336 Pnt 4712 1382 Pnt 4712 1428 Pnt 4712 1474 Pnt 4725 280 Pnt 4725 326 Pnt 4725 372 Pnt 4725 418 Pnt 4725 464 Pnt 4725 510 Pnt 4725 556 Pnt 4725 601 Pnt 4725 647 Pnt 4725 693 Pnt 4725 739 Pnt 4725 785 Pnt 4725 831 Pnt 4725 877 Pnt 4725 923 Pnt 4725 969 Pnt 4725 1015 Pnt 4725 1061 Pnt 4725 1107 Pnt 4725 1152 Pnt 4725 1198 Pnt 4725 1244 Pnt 4725 1290 Pnt 4725 1336 Pnt 4725 1382 Pnt 4725 1428 Pnt 4725 1474 Pnt 4725 1520 Pnt 4738 280 Pnt 4738 326 Pnt 4738 372 Pnt 4738 418 Pnt 4738 464 Pnt 4738 510 Pnt 4738 556 Pnt 4738 601 Pnt 4738 647 Pnt 4738 693 Pnt 4738 739 Pnt 4738 785 Pnt 4738 831 Pnt 4738 877 Pnt 4738 923 Pnt 4738 969 Pnt 4738 1015 Pnt 4738 1061 Pnt 4738 1107 Pnt 4738 1152 Pnt 4738 1198 Pnt 4738 1244 Pnt 4738 1290 Pnt 4738 1336 Pnt 4738 1382 Pnt 4738 1428 Pnt 4738 1474 Pnt 4738 1520 Pnt 4750 280 Pnt 4750 326 Pnt 4750 372 Pnt 4750 418 Pnt 4750 464 Pnt 4750 510 Pnt 4750 556 Pnt 4750 601 Pnt 4750 647 Pnt 4750 693 Pnt 4750 739 Pnt 4750 785 Pnt 4750 831 Pnt 4750 877 Pnt 4750 923 Pnt 4750 969 Pnt 4750 1015 Pnt 4750 1061 Pnt 4750 1107 Pnt 4750 1152 Pnt 4750 1198 Pnt 4750 1244 Pnt 4750 1290 Pnt 4750 1336 Pnt 4750 1382 Pnt 4750 1428 Pnt 4750 1474 Pnt 4750 1520 Pnt 4750 1566 Pnt 4763 280 Pnt 4763 326 Pnt 4763 372 Pnt 4763 418 Pnt 4763 464 Pnt 4763 510 Pnt 4763 556 Pnt 4763 601 Pnt 4763 647 Pnt 4763 923 Pnt 4763 969 Pnt 4763 1015 Pnt 4763 1061 Pnt 4763 1107 Pnt 4763 1152 Pnt 4763 1198 Pnt 4763 1244 Pnt 4763 1290 Pnt 4763 1336 Pnt 4763 1382 Pnt 4763 1428 Pnt 4763 1474 Pnt 4763 1520 Pnt 4763 1566 Pnt 4763 1612 Pnt 4776 280 Pnt 4776 326 Pnt 4776 372 Pnt 4776 418 Pnt 4776 464 Pnt 4776 510 Pnt 4776 556 Pnt 4776 601 Pnt 4776 647 Pnt 4776 693 Pnt 4776 739 Pnt 4776 785 Pnt 4776 1061 Pnt 4776 1107 Pnt 4776 1152 Pnt 4776 1198 Pnt 4776 1244 Pnt 4776 1290 Pnt 4776 1336 Pnt 4776 1382 Pnt 4776 1428 Pnt 4776 1474 Pnt 4776 1520 Pnt 4776 1566 Pnt 4776 1612 Pnt 4788 280 Pnt 4788 326 Pnt 4788 372 Pnt 4788 418 Pnt 4788 464 Pnt 4788 510 Pnt 4788 556 Pnt 4788 601 Pnt 4788 647 Pnt 4788 693 Pnt 4788 739 Pnt 4788 785 Pnt 4788 831 Pnt 4788 877 Pnt 4788 1152 Pnt 4788 1198 Pnt 4788 1244 Pnt 4788 1290 Pnt 4788 1336 Pnt 4788 1382 Pnt 4788 1428 Pnt 4788 1474 Pnt 4788 1520 Pnt 4788 1566 Pnt 4788 1612 Pnt 4788 1658 Pnt 4801 280 Pnt 4801 326 Pnt 4801 372 Pnt 4801 418 Pnt 4801 464 Pnt 4801 510 Pnt 4801 556 Pnt 4801 601 Pnt 4801 647 Pnt 4801 693 Pnt 4801 739 Pnt 4801 785 Pnt 4801 831 Pnt 4801 877 Pnt 4801 923 Pnt 4801 969 Pnt 4801 1198 Pnt 4801 1244 Pnt 4801 1290 Pnt 4801 1336 Pnt 4801 1382 Pnt 4801 1428 Pnt 4801 1474 Pnt 4801 1520 Pnt 4801 1566 Pnt 4801 1612 Pnt 4801 1658 Pnt 4801 1704 Pnt 4814 280 Pnt 4814 326 Pnt 4814 372 Pnt 4814 418 Pnt 4814 464 Pnt 4814 510 Pnt 4814 556 Pnt 4814 601 Pnt 4814 647 Pnt 4814 693 Pnt 4814 739 Pnt 4814 785 Pnt 4814 831 Pnt 4814 877 Pnt 4814 923 Pnt 4814 969 Pnt 4814 1015 Pnt 4814 1061 Pnt 4814 1290 Pnt 4814 1336 Pnt 4814 1382 Pnt 4814 1428 Pnt 4814 1474 Pnt 4814 1520 Pnt 4814 1566 Pnt 4814 1612 Pnt 4814 1658 Pnt 4814 1704 Pnt 4827 280 Pnt 4827 326 Pnt 4827 372 Pnt 4827 418 Pnt 4827 464 Pnt 4827 510 Pnt 4827 556 Pnt 4827 601 Pnt 4827 647 Pnt 4827 693 Pnt 4827 739 Pnt 4827 785 Pnt 4827 831 Pnt 4827 877 Pnt 4827 923 Pnt 4827 969 Pnt 4827 1015 Pnt 4827 1061 Pnt 4827 1107 Pnt 4827 1336 Pnt 4827 1382 Pnt 4827 1428 Pnt 4827 1474 Pnt 4827 1520 Pnt 4827 1566 Pnt 4827 1612 Pnt 4827 1658 Pnt 4827 1704 Pnt 4827 1749 Pnt 4839 280 Pnt 4839 326 Pnt 4839 372 Pnt 4839 418 Pnt 4839 464 Pnt 4839 510 Pnt 4839 556 Pnt 4839 601 Pnt 4839 647 Pnt 4839 693 Pnt 4839 739 Pnt 4839 785 Pnt 4839 831 Pnt 4839 877 Pnt 4839 923 Pnt 4839 969 Pnt 4839 1015 Pnt 4839 1061 Pnt 4839 1107 Pnt 4839 1152 Pnt 4839 1382 Pnt 4839 1428 Pnt 4839 1474 Pnt 4839 1520 Pnt 4839 1566 Pnt 4839 1612 Pnt 4839 1658 Pnt 4839 1704 Pnt 4839 1749 Pnt 4839 1795 Pnt 4852 280 Pnt 4852 326 Pnt 4852 372 Pnt 4852 418 Pnt 4852 464 Pnt 4852 510 Pnt 4852 556 Pnt 4852 601 Pnt 4852 647 Pnt 4852 693 Pnt 4852 739 Pnt 4852 785 Pnt 4852 831 Pnt 4852 877 Pnt 4852 923 Pnt 4852 969 Pnt 4852 1015 Pnt 4852 1061 Pnt 4852 1107 Pnt 4852 1152 Pnt 4852 1198 Pnt 4852 1428 Pnt 4852 1474 Pnt 4852 1520 Pnt 4852 1566 Pnt 4852 1612 Pnt 4852 1658 Pnt 4852 1704 Pnt 4852 1749 Pnt 4852 1795 Pnt 4865 280 Pnt 4865 326 Pnt 4865 372 Pnt 4865 418 Pnt 4865 464 Pnt 4865 510 Pnt 4865 556 Pnt 4865 601 Pnt 4865 647 Pnt 4865 693 Pnt 4865 739 Pnt 4865 785 Pnt 4865 831 Pnt 4865 877 Pnt 4865 923 Pnt 4865 969 Pnt 4865 1015 Pnt 4865 1061 Pnt 4865 1107 Pnt 4865 1152 Pnt 4865 1198 Pnt 4865 1244 Pnt 4865 1474 Pnt 4865 1520 Pnt 4865 1566 Pnt 4865 1612 Pnt 4865 1658 Pnt 4865 1704 Pnt 4865 1749 Pnt 4865 1795 Pnt 4865 1841 Pnt 4877 280 Pnt 4877 326 Pnt 4877 372 Pnt 4877 418 Pnt 4877 464 Pnt 4877 510 Pnt 4877 556 Pnt 4877 601 Pnt 4877 647 Pnt 4877 693 Pnt 4877 739 Pnt 4877 785 Pnt 4877 831 Pnt 4877 877 Pnt 4877 923 Pnt 4877 969 Pnt 4877 1015 Pnt 4877 1061 Pnt 4877 1107 Pnt 4877 1152 Pnt 4877 1198 Pnt 4877 1244 Pnt 4877 1290 Pnt 4877 1520 Pnt 4877 1566 Pnt 4877 1612 Pnt 4877 1658 Pnt 4877 1704 Pnt 4877 1749 Pnt 4877 1795 Pnt 4877 1841 Pnt 4890 280 Pnt 4890 326 Pnt 4890 372 Pnt 4890 418 Pnt 4890 464 Pnt 4890 510 Pnt 4890 556 Pnt 4890 601 Pnt 4890 647 Pnt 4890 693 Pnt 4890 739 Pnt 4890 785 Pnt 4890 831 Pnt 4890 877 Pnt 4890 923 Pnt 4890 969 Pnt 4890 1015 Pnt 4890 1061 Pnt 4890 1107 Pnt 4890 1152 Pnt 4890 1198 Pnt 4890 1244 Pnt 4890 1290 Pnt 4890 1336 Pnt 4890 1566 Pnt 4890 1612 Pnt 4890 1658 Pnt 4890 1704 Pnt 4890 1749 Pnt 4890 1795 Pnt 4890 1841 Pnt 4890 1887 Pnt 4903 280 Pnt 4903 326 Pnt 4903 372 Pnt 4903 418 Pnt 4903 464 Pnt 4903 510 Pnt 4903 556 Pnt 4903 601 Pnt 4903 647 Pnt 4903 693 Pnt 4903 739 Pnt 4903 785 Pnt 4903 831 Pnt 4903 877 Pnt 4903 923 Pnt 4903 969 Pnt 4903 1015 Pnt 4903 1061 Pnt 4903 1107 Pnt 4903 1152 Pnt 4903 1198 Pnt 4903 1244 Pnt 4903 1290 Pnt 4903 1336 Pnt 4903 1382 Pnt 4903 1612 Pnt 4903 1658 Pnt 4903 1704 Pnt 4903 1749 Pnt 4903 1795 Pnt 4903 1841 Pnt 4903 1887 Pnt 4903 1933 Pnt 4916 280 Pnt 4916 326 Pnt 4916 372 Pnt 4916 418 Pnt 4916 464 Pnt 4916 510 Pnt 4916 556 Pnt 4916 601 Pnt 4916 647 Pnt 4916 693 Pnt 4916 739 Pnt 4916 785 Pnt 4916 831 Pnt 4916 877 Pnt 4916 923 Pnt 4916 969 Pnt 4916 1015 Pnt 4916 1061 Pnt 4916 1107 Pnt 4916 1152 Pnt 4916 1198 Pnt 4916 1244 Pnt 4916 1290 Pnt 4916 1336 Pnt 4916 1382 Pnt 4916 1428 Pnt 4916 1658 Pnt 4916 1704 Pnt 4916 1749 Pnt 4916 1795 Pnt 4916 1841 Pnt 4916 1887 Pnt 4916 1933 Pnt 4928 280 Pnt 4928 326 Pnt 4928 372 Pnt 4928 418 Pnt 4928 464 Pnt 4928 510 Pnt 4928 556 Pnt 4928 601 Pnt 4928 647 Pnt 4928 693 Pnt 4928 739 Pnt 4928 785 Pnt 4928 831 Pnt 4928 877 Pnt 4928 923 Pnt 4928 969 Pnt 4928 1015 Pnt 4928 1061 Pnt 4928 1107 Pnt 4928 1152 Pnt 4928 1198 Pnt 4928 1244 Pnt 4928 1290 Pnt 4928 1336 Pnt 4928 1382 Pnt 4928 1428 Pnt 4928 1474 Pnt 4928 1704 Pnt 4928 1749 Pnt 4928 1795 Pnt 4928 1841 Pnt 4928 1887 Pnt 4928 1933 Pnt 4928 1979 Pnt 4941 280 Pnt 4941 326 Pnt 4941 372 Pnt 4941 418 Pnt 4941 464 Pnt 4941 510 Pnt 4941 556 Pnt 4941 601 Pnt 4941 647 Pnt 4941 693 Pnt 4941 739 Pnt 4941 785 Pnt 4941 831 Pnt 4941 877 Pnt 4941 923 Pnt 4941 969 Pnt 4941 1015 Pnt 4941 1061 Pnt 4941 1107 Pnt 4941 1152 Pnt 4941 1198 Pnt 4941 1244 Pnt 4941 1290 Pnt 4941 1336 Pnt 4941 1382 Pnt 4941 1428 Pnt 4941 1474 Pnt 4941 1520 Pnt 4941 1749 Pnt 4941 1795 Pnt 4941 1841 Pnt 4941 1887 Pnt 4941 1933 Pnt 4941 1979 Pnt 4954 280 Pnt 4954 326 Pnt 4954 372 Pnt 4954 418 Pnt 4954 464 Pnt 4954 510 Pnt 4954 556 Pnt 4954 601 Pnt 4954 647 Pnt 4954 693 Pnt 4954 739 Pnt 4954 785 Pnt 4954 831 Pnt 4954 877 Pnt 4954 923 Pnt 4954 969 Pnt 4954 1015 Pnt 4954 1061 Pnt 4954 1107 Pnt 4954 1152 Pnt 4954 1198 Pnt 4954 1244 Pnt 4954 1290 Pnt 4954 1336 Pnt 4954 1382 Pnt 4954 1428 Pnt 4954 1474 Pnt 4954 1520 Pnt 4954 1566 Pnt 4954 1749 Pnt 4954 1795 Pnt 4954 1841 Pnt 4954 1887 Pnt 4954 1979 Pnt 4954 2025 Pnt 4966 280 Pnt 4966 326 Pnt 4966 372 Pnt 4966 418 Pnt 4966 464 Pnt 4966 510 Pnt 4966 556 Pnt 4966 601 Pnt 4966 647 Pnt 4966 693 Pnt 4966 739 Pnt 4966 785 Pnt 4966 831 Pnt 4966 877 Pnt 4966 923 Pnt 4966 969 Pnt 4966 1015 Pnt 4966 1061 Pnt 4966 1107 Pnt 4966 1152 Pnt 4966 1198 Pnt 4966 1244 Pnt 4966 1290 Pnt 4966 1336 Pnt 4966 1382 Pnt 4966 1428 Pnt 4966 1474 Pnt 4966 1520 Pnt 4966 1566 Pnt 4966 1612 Pnt 4966 1795 Pnt 4966 1841 Pnt 4966 1887 Pnt 4966 1933 Pnt 4966 1979 Pnt 4966 2025 Pnt 4966 2071 Pnt 4979 280 Pnt 4979 326 Pnt 4979 372 Pnt 4979 418 Pnt 4979 464 Pnt 4979 510 Pnt 4979 556 Pnt 4979 601 Pnt 4979 647 Pnt 4979 693 Pnt 4979 739 Pnt 4979 785 Pnt 4979 831 Pnt 4979 877 Pnt 4979 923 Pnt 4979 969 Pnt 4979 1015 Pnt 4979 1061 Pnt 4979 1107 Pnt 4979 1152 Pnt 4979 1198 Pnt 4979 1244 Pnt 4979 1290 Pnt 4979 1336 Pnt 4979 1382 Pnt 4979 1428 Pnt 4979 1474 Pnt 4979 1520 Pnt 4979 1566 Pnt 4979 1612 Pnt 4979 1658 Pnt 4979 1841 Pnt 4979 1887 Pnt 4979 1933 Pnt 4979 1979 Pnt 4979 2025 Pnt 4979 2071 Pnt 4992 280 Pnt 4992 326 Pnt 4992 372 Pnt 4992 418 Pnt 4992 464 Pnt 4992 510 Pnt 4992 556 Pnt 4992 601 Pnt 4992 647 Pnt 4992 693 Pnt 4992 739 Pnt 4992 785 Pnt 4992 831 Pnt 4992 877 Pnt 4992 923 Pnt 4992 969 Pnt 4992 1015 Pnt 4992 1061 Pnt 4992 1107 Pnt 4992 1152 Pnt 4992 1198 Pnt 4992 1244 Pnt 4992 1290 Pnt 4992 1336 Pnt 4992 1382 Pnt 4992 1428 Pnt 4992 1474 Pnt 4992 1520 Pnt 4992 1566 Pnt 4992 1612 Pnt 4992 1658 Pnt 4992 1887 Pnt 4992 1933 Pnt 4992 1979 Pnt 4992 2071 Pnt 4992 2117 Pnt 5005 280 Pnt 5005 326 Pnt 5005 372 Pnt 5005 418 Pnt 5005 464 Pnt 5005 510 Pnt 5005 556 Pnt 5005 601 Pnt 5005 647 Pnt 5005 693 Pnt 5005 739 Pnt 5005 785 Pnt 5005 831 Pnt 5005 877 Pnt 5005 923 Pnt 5005 969 Pnt 5005 1015 Pnt 5005 1061 Pnt 5005 1107 Pnt 5005 1152 Pnt 5005 1198 Pnt 5005 1244 Pnt 5005 1290 Pnt 5005 1336 Pnt 5005 1382 Pnt 5005 1428 Pnt 5005 1474 Pnt 5005 1520 Pnt 5005 1566 Pnt 5005 1612 Pnt 5005 1658 Pnt 5005 1704 Pnt 5005 1887 Pnt 5005 1933 Pnt 5005 1979 Pnt 5005 2025 Pnt 5005 2071 Pnt 5005 2117 Pnt 5017 280 Pnt 5017 326 Pnt 5017 372 Pnt 5017 418 Pnt 5017 464 Pnt 5017 510 Pnt 5017 556 Pnt 5017 601 Pnt 5017 647 Pnt 5017 693 Pnt 5017 739 Pnt 5017 785 Pnt 5017 831 Pnt 5017 877 Pnt 5017 923 Pnt 5017 969 Pnt 5017 1015 Pnt 5017 1061 Pnt 5017 1107 Pnt 5017 1152 Pnt 5017 1198 Pnt 5017 1244 Pnt 5017 1290 Pnt 5017 1336 Pnt 5017 1382 Pnt 5017 1428 Pnt 5017 1474 Pnt 5017 1520 Pnt 5017 1566 Pnt 5017 1612 Pnt 5017 1658 Pnt 5017 1704 Pnt 5017 1749 Pnt 5017 1933 Pnt 5017 1979 Pnt 5017 2025 Pnt 5017 2071 Pnt 5017 2117 Pnt 5017 2163 Pnt 5030 280 Pnt 5030 326 Pnt 5030 372 Pnt 5030 418 Pnt 5030 464 Pnt 5030 510 Pnt 5030 556 Pnt 5030 601 Pnt 5030 647 Pnt 5030 693 Pnt 5030 739 Pnt 5030 785 Pnt 5030 831 Pnt 5030 877 Pnt 5030 923 Pnt 5030 969 Pnt 5030 1015 Pnt 5030 1061 Pnt 5030 1107 Pnt 5030 1152 Pnt 5030 1198 Pnt 5030 1244 Pnt 5030 1290 Pnt 5030 1336 Pnt 5030 1382 Pnt 5030 1428 Pnt 5030 1474 Pnt 5030 1520 Pnt 5030 1566 Pnt 5030 1612 Pnt 5030 1658 Pnt 5030 1704 Pnt 5030 1749 Pnt 5030 1795 Pnt 5030 1979 Pnt 5030 2025 Pnt 5030 2071 Pnt 5030 2117 Pnt 5030 2163 Pnt 5030 2209 Pnt 5043 280 Pnt 5043 326 Pnt 5043 372 Pnt 5043 418 Pnt 5043 464 Pnt 5043 510 Pnt 5043 556 Pnt 5043 1152 Pnt 5043 1198 Pnt 5043 1244 Pnt 5043 1290 Pnt 5043 1336 Pnt 5043 1382 Pnt 5043 1428 Pnt 5043 1474 Pnt 5043 1520 Pnt 5043 1566 Pnt 5043 1612 Pnt 5043 1658 Pnt 5043 1704 Pnt 5043 1749 Pnt 5043 1795 Pnt 5043 2025 Pnt 5043 2071 Pnt 5043 2117 Pnt 5043 2163 Pnt 5043 2209 Pnt 5055 280 Pnt 5055 326 Pnt 5055 372 Pnt 5055 418 Pnt 5055 464 Pnt 5055 510 Pnt 5055 556 Pnt 5055 601 Pnt 5055 647 Pnt 5055 693 Pnt 5055 739 Pnt 5055 785 Pnt 5055 1290 Pnt 5055 1336 Pnt 5055 1382 Pnt 5055 1428 Pnt 5055 1474 Pnt 5055 1520 Pnt 5055 1566 Pnt 5055 1612 Pnt 5055 1658 Pnt 5055 1704 Pnt 5055 1749 Pnt 5055 1795 Pnt 5055 1841 Pnt 5055 2025 Pnt 5055 2071 Pnt 5055 2117 Pnt 5055 2209 Pnt 5055 2255 Pnt 5068 280 Pnt 5068 326 Pnt 5068 372 Pnt 5068 418 Pnt 5068 464 Pnt 5068 510 Pnt 5068 556 Pnt 5068 601 Pnt 5068 647 Pnt 5068 693 Pnt 5068 739 Pnt 5068 785 Pnt 5068 831 Pnt 5068 877 Pnt 5068 923 Pnt 5068 1382 Pnt 5068 1428 Pnt 5068 1474 Pnt 5068 1520 Pnt 5068 1566 Pnt 5068 1612 Pnt 5068 1658 Pnt 5068 1704 Pnt 5068 1749 Pnt 5068 1795 Pnt 5068 1841 Pnt 5068 1887 Pnt 5068 2071 Pnt 5068 2117 Pnt 5068 2163 Pnt 5068 2209 Pnt 5068 2255 Pnt 5081 280 Pnt 5081 326 Pnt 5081 372 Pnt 5081 418 Pnt 5081 464 Pnt 5081 510 Pnt 5081 556 Pnt 5081 601 Pnt 5081 647 Pnt 5081 693 Pnt 5081 739 Pnt 5081 785 Pnt 5081 831 Pnt 5081 877 Pnt 5081 923 Pnt 5081 969 Pnt 5081 1015 Pnt 5081 1061 Pnt 5081 1428 Pnt 5081 1474 Pnt 5081 1520 Pnt 5081 1566 Pnt 5081 1612 Pnt 5081 1658 Pnt 5081 1704 Pnt 5081 1749 Pnt 5081 1795 Pnt 5081 1841 Pnt 5081 1887 Pnt 5081 1933 Pnt 5081 2117 Pnt 5081 2163 Pnt 5081 2209 Pnt 5081 2255 Pnt 5081 2300 Pnt 5094 280 Pnt 5094 326 Pnt 5094 372 Pnt 5094 418 Pnt 5094 464 Pnt 5094 510 Pnt 5094 556 Pnt 5094 601 Pnt 5094 647 Pnt 5094 693 Pnt 5094 739 Pnt 5094 785 Pnt 5094 831 Pnt 5094 877 Pnt 5094 923 Pnt 5094 969 Pnt 5094 1015 Pnt 5094 1061 Pnt 5094 1107 Pnt 5094 1520 Pnt 5094 1566 Pnt 5094 1612 Pnt 5094 1658 Pnt 5094 1704 Pnt 5094 1749 Pnt 5094 1795 Pnt 5094 1841 Pnt 5094 1887 Pnt 5094 1933 Pnt 5094 2117 Pnt 5094 2163 Pnt 5094 2209 Pnt 5094 2300 Pnt 5106 280 Pnt 5106 326 Pnt 5106 372 Pnt 5106 418 Pnt 5106 464 Pnt 5106 510 Pnt 5106 556 Pnt 5106 601 Pnt 5106 647 Pnt 5106 693 Pnt 5106 739 Pnt 5106 785 Pnt 5106 831 Pnt 5106 877 Pnt 5106 923 Pnt 5106 969 Pnt 5106 1015 Pnt 5106 1061 Pnt 5106 1107 Pnt 5106 1152 Pnt 5106 1198 Pnt 5106 1566 Pnt 5106 1612 Pnt 5106 1658 Pnt 5106 1704 Pnt 5106 1749 Pnt 5106 1795 Pnt 5106 1841 Pnt 5106 1887 Pnt 5106 1933 Pnt 5106 1979 Pnt 5106 2163 Pnt 5106 2209 Pnt 5106 2255 Pnt 5106 2300 Pnt 5106 2346 Pnt 5119 280 Pnt 5119 326 Pnt 5119 372 Pnt 5119 418 Pnt 5119 464 Pnt 5119 510 Pnt 5119 556 Pnt 5119 601 Pnt 5119 647 Pnt 5119 693 Pnt 5119 739 Pnt 5119 785 Pnt 5119 831 Pnt 5119 877 Pnt 5119 923 Pnt 5119 969 Pnt 5119 1015 Pnt 5119 1061 Pnt 5119 1107 Pnt 5119 1152 Pnt 5119 1198 Pnt 5119 1244 Pnt 5119 1612 Pnt 5119 1658 Pnt 5119 1704 Pnt 5119 1749 Pnt 5119 1795 Pnt 5119 1841 Pnt 5119 1887 Pnt 5119 1933 Pnt 5119 1979 Pnt 5119 2025 Pnt 5119 2209 Pnt 5119 2255 Pnt 5119 2346 Pnt 5119 2392 Pnt 5132 280 Pnt 5132 326 Pnt 5132 372 Pnt 5132 418 Pnt 5132 464 Pnt 5132 510 Pnt 5132 556 Pnt 5132 601 Pnt 5132 647 Pnt 5132 693 Pnt 5132 739 Pnt 5132 785 Pnt 5132 831 Pnt 5132 877 Pnt 5132 923 Pnt 5132 969 Pnt 5132 1015 Pnt 5132 1061 Pnt 5132 1107 Pnt 5132 1152 Pnt 5132 1198 Pnt 5132 1244 Pnt 5132 1290 Pnt 5132 1336 Pnt 5132 1704 Pnt 5132 1749 Pnt 5132 1795 Pnt 5132 1841 Pnt 5132 1887 Pnt 5132 1933 Pnt 5132 1979 Pnt 5132 2025 Pnt 5132 2209 Pnt 5132 2255 Pnt 5132 2300 Pnt 5132 2346 Pnt 5132 2392 Pnt 5144 280 Pnt 5144 326 Pnt 5144 372 Pnt 5144 418 Pnt 5144 464 Pnt 5144 510 Pnt 5144 556 Pnt 5144 601 Pnt 5144 647 Pnt 5144 693 Pnt 5144 739 Pnt 5144 785 Pnt 5144 831 Pnt 5144 877 Pnt 5144 923 Pnt 5144 969 Pnt 5144 1015 Pnt 5144 1061 Pnt 5144 1107 Pnt 5144 1152 Pnt 5144 1198 Pnt 5144 1244 Pnt 5144 1290 Pnt 5144 1336 Pnt 5144 1382 Pnt 5144 1749 Pnt 5144 1795 Pnt 5144 1841 Pnt 5144 1887 Pnt 5144 1933 Pnt 5144 1979 Pnt 5144 2025 Pnt 5144 2071 Pnt 5144 2255 Pnt 5144 2300 Pnt 5144 2346 Pnt 5144 2392 Pnt 5144 2438 Pnt 5157 280 Pnt 5157 326 Pnt 5157 372 Pnt 5157 418 Pnt 5157 464 Pnt 5157 510 Pnt 5157 556 Pnt 5157 601 Pnt 5157 647 Pnt 5157 693 Pnt 5157 739 Pnt 5157 785 Pnt 5157 831 Pnt 5157 877 Pnt 5157 923 Pnt 5157 969 Pnt 5157 1015 Pnt 5157 1061 Pnt 5157 1107 Pnt 5157 1152 Pnt 5157 1198 Pnt 5157 1244 Pnt 5157 1290 Pnt 5157 1336 Pnt 5157 1382 Pnt 5157 1428 Pnt 5157 1795 Pnt 5157 1841 Pnt 5157 1887 Pnt 5157 1933 Pnt 5157 1979 Pnt 5157 2025 Pnt 5157 2071 Pnt 5157 2117 Pnt 5157 2300 Pnt 5157 2346 Pnt 5157 2392 Pnt 5157 2438 Pnt 5170 280 Pnt 5170 326 Pnt 5170 372 Pnt 5170 418 Pnt 5170 464 Pnt 5170 510 Pnt 5170 556 Pnt 5170 601 Pnt 5170 647 Pnt 5170 693 Pnt 5170 739 Pnt 5170 785 Pnt 5170 831 Pnt 5170 877 Pnt 5170 923 Pnt 5170 969 Pnt 5170 1015 Pnt 5170 1061 Pnt 5170 1107 Pnt 5170 1152 Pnt 5170 1198 Pnt 5170 1244 Pnt 5170 1290 Pnt 5170 1336 Pnt 5170 1382 Pnt 5170 1428 Pnt 5170 1474 Pnt 5170 1841 Pnt 5170 1887 Pnt 5170 1933 Pnt 5170 1979 Pnt 5170 2025 Pnt 5170 2071 Pnt 5170 2117 Pnt 5170 2300 Pnt 5170 2346 Pnt 5170 2392 Pnt 5170 2438 Pnt 5170 2484 Pnt 5182 280 Pnt 5182 326 Pnt 5182 372 Pnt 5182 418 Pnt 5182 464 Pnt 5182 510 Pnt 5182 556 Pnt 5182 601 Pnt 5182 647 Pnt 5182 693 Pnt 5182 739 Pnt 5182 785 Pnt 5182 831 Pnt 5182 877 Pnt 5182 923 Pnt 5182 969 Pnt 5182 1015 Pnt 5182 1061 Pnt 5182 1107 Pnt 5182 1152 Pnt 5182 1198 Pnt 5182 1244 Pnt 5182 1290 Pnt 5182 1336 Pnt 5182 1382 Pnt 5182 1428 Pnt 5182 1474 Pnt 5182 1520 Pnt 5182 1887 Pnt 5182 1933 Pnt 5182 1979 Pnt 5182 2025 Pnt 5182 2071 Pnt 5182 2117 Pnt 5182 2163 Pnt 5182 2346 Pnt 5182 2392 Pnt 5182 2484 Pnt 5195 280 Pnt 5195 326 Pnt 5195 372 Pnt 5195 418 Pnt 5195 464 Pnt 5195 510 Pnt 5195 556 Pnt 5195 601 Pnt 5195 647 Pnt 5195 693 Pnt 5195 739 Pnt 5195 785 Pnt 5195 831 Pnt 5195 877 Pnt 5195 923 Pnt 5195 969 Pnt 5195 1015 Pnt 5195 1061 Pnt 5195 1107 Pnt 5195 1152 Pnt 5195 1198 Pnt 5195 1244 Pnt 5195 1290 Pnt 5195 1336 Pnt 5195 1382 Pnt 5195 1428 Pnt 5195 1474 Pnt 5195 1520 Pnt 5195 1566 Pnt 5195 1933 Pnt 5195 1979 Pnt 5195 2025 Pnt 5195 2071 Pnt 5195 2117 Pnt 5195 2163 Pnt 5195 2209 Pnt 5195 2392 Pnt 5195 2438 Pnt 5195 2484 Pnt 5195 2530 Pnt 5208 280 Pnt 5208 326 Pnt 5208 372 Pnt 5208 418 Pnt 5208 464 Pnt 5208 510 Pnt 5208 556 Pnt 5208 601 Pnt 5208 647 Pnt 5208 693 Pnt 5208 739 Pnt 5208 785 Pnt 5208 831 Pnt 5208 877 Pnt 5208 923 Pnt 5208 969 Pnt 5208 1015 Pnt 5208 1061 Pnt 5208 1107 Pnt 5208 1152 Pnt 5208 1198 Pnt 5208 1244 Pnt 5208 1290 Pnt 5208 1336 Pnt 5208 1382 Pnt 5208 1428 Pnt 5208 1474 Pnt 5208 1520 Pnt 5208 1566 Pnt 5208 1612 Pnt 5208 1979 Pnt 5208 2025 Pnt 5208 2071 Pnt 5208 2117 Pnt 5208 2163 Pnt 5208 2209 Pnt 5208 2392 Pnt 5208 2438 Pnt 5208 2484 Pnt 5208 2530 Pnt 5221 280 Pnt 5221 326 Pnt 5221 372 Pnt 5221 418 Pnt 5221 464 Pnt 5221 510 Pnt 5221 556 Pnt 5221 601 Pnt 5221 647 Pnt 5221 693 Pnt 5221 739 Pnt 5221 785 Pnt 5221 831 Pnt 5221 877 Pnt 5221 923 Pnt 5221 969 Pnt 5221 1015 Pnt 5221 1061 Pnt 5221 1107 Pnt 5221 1152 Pnt 5221 1198 Pnt 5221 1244 Pnt 5221 1290 Pnt 5221 1336 Pnt 5221 1382 Pnt 5221 1428 Pnt 5221 1474 Pnt 5221 1520 Pnt 5221 1566 Pnt 5221 1612 Pnt 5221 1658 Pnt 5221 1979 Pnt 5221 2025 Pnt 5221 2071 Pnt 5221 2117 Pnt 5221 2163 Pnt 5221 2209 Pnt 5221 2255 Pnt 5221 2438 Pnt 5221 2484 Pnt 5221 2530 Pnt 5221 2576 Pnt 5233 280 Pnt 5233 326 Pnt 5233 372 Pnt 5233 418 Pnt 5233 464 Pnt 5233 510 Pnt 5233 556 Pnt 5233 601 Pnt 5233 647 Pnt 5233 693 Pnt 5233 739 Pnt 5233 785 Pnt 5233 831 Pnt 5233 877 Pnt 5233 923 Pnt 5233 969 Pnt 5233 1015 Pnt 5233 1061 Pnt 5233 1107 Pnt 5233 1152 Pnt 5233 1198 Pnt 5233 1244 Pnt 5233 1290 Pnt 5233 1336 Pnt 5233 1382 Pnt 5233 1428 Pnt 5233 1474 Pnt 5233 1520 Pnt 5233 1566 Pnt 5233 1612 Pnt 5233 1658 Pnt 5233 1704 Pnt 5233 2025 Pnt 5233 2071 Pnt 5233 2117 Pnt 5233 2163 Pnt 5233 2209 Pnt 5233 2255 Pnt 5233 2300 Pnt 5233 2438 Pnt 5233 2484 Pnt 5233 2530 Pnt 5233 2576 Pnt 5233 2622 Pnt 5246 280 Pnt 5246 326 Pnt 5246 372 Pnt 5246 418 Pnt 5246 464 Pnt 5246 510 Pnt 5246 556 Pnt 5246 601 Pnt 5246 647 Pnt 5246 693 Pnt 5246 739 Pnt 5246 785 Pnt 5246 831 Pnt 5246 877 Pnt 5246 923 Pnt 5246 969 Pnt 5246 1015 Pnt 5246 1061 Pnt 5246 1107 Pnt 5246 1152 Pnt 5246 1198 Pnt 5246 1244 Pnt 5246 1290 Pnt 5246 1336 Pnt 5246 1382 Pnt 5246 1428 Pnt 5246 1474 Pnt 5246 1520 Pnt 5246 1566 Pnt 5246 1612 Pnt 5246 1658 Pnt 5246 1704 Pnt 5246 1749 Pnt 5246 2071 Pnt 5246 2117 Pnt 5246 2163 Pnt 5246 2209 Pnt 5246 2255 Pnt 5246 2300 Pnt 5246 2484 Pnt 5246 2530 Pnt 5246 2622 Pnt 5259 280 Pnt 5259 326 Pnt 5259 372 Pnt 5259 418 Pnt 5259 464 Pnt 5259 510 Pnt 5259 556 Pnt 5259 601 Pnt 5259 647 Pnt 5259 693 Pnt 5259 739 Pnt 5259 785 Pnt 5259 831 Pnt 5259 877 Pnt 5259 923 Pnt 5259 969 Pnt 5259 1015 Pnt 5259 1061 Pnt 5259 1107 Pnt 5259 1152 Pnt 5259 1198 Pnt 5259 1244 Pnt 5259 1290 Pnt 5259 1336 Pnt 5259 1382 Pnt 5259 1428 Pnt 5259 1474 Pnt 5259 1520 Pnt 5259 1566 Pnt 5259 1612 Pnt 5259 1658 Pnt 5259 1704 Pnt 5259 1749 Pnt 5259 1795 Pnt 5259 2117 Pnt 5259 2163 Pnt 5259 2209 Pnt 5259 2255 Pnt 5259 2300 Pnt 5259 2346 Pnt 5259 2530 Pnt 5259 2576 Pnt 5259 2622 Pnt 5259 2668 Pnt 5271 280 Pnt 5271 326 Pnt 5271 372 Pnt 5271 418 Pnt 5271 464 Pnt 5271 510 Pnt 5271 556 Pnt 5271 601 Pnt 5271 647 Pnt 5271 693 Pnt 5271 739 Pnt 5271 785 Pnt 5271 831 Pnt 5271 877 Pnt 5271 923 Pnt 5271 969 Pnt 5271 1015 Pnt 5271 1061 Pnt 5271 1107 Pnt 5271 1152 Pnt 5271 1198 Pnt 5271 1244 Pnt 5271 1290 Pnt 5271 1336 Pnt 5271 1382 Pnt 5271 1428 Pnt 5271 1474 Pnt 5271 1520 Pnt 5271 1566 Pnt 5271 1612 Pnt 5271 1658 Pnt 5271 1704 Pnt 5271 1749 Pnt 5271 1795 Pnt 5271 1841 Pnt 5271 2163 Pnt 5271 2209 Pnt 5271 2255 Pnt 5271 2300 Pnt 5271 2346 Pnt 5271 2392 Pnt 5271 2530 Pnt 5271 2576 Pnt 5271 2668 Pnt 5284 280 Pnt 5284 326 Pnt 5284 372 Pnt 5284 418 Pnt 5284 464 Pnt 5284 510 Pnt 5284 556 Pnt 5284 601 Pnt 5284 647 Pnt 5284 693 Pnt 5284 739 Pnt 5284 785 Pnt 5284 831 Pnt 5284 877 Pnt 5284 923 Pnt 5284 969 Pnt 5284 1015 Pnt 5284 1061 Pnt 5284 1107 Pnt 5284 1152 Pnt 5284 1198 Pnt 5284 1244 Pnt 5284 1290 Pnt 5284 1336 Pnt 5284 1382 Pnt 5284 1428 Pnt 5284 1474 Pnt 5284 1520 Pnt 5284 1566 Pnt 5284 1612 Pnt 5284 1658 Pnt 5284 1704 Pnt 5284 1749 Pnt 5284 1795 Pnt 5284 1841 Pnt 5284 1887 Pnt 5284 2163 Pnt 5284 2209 Pnt 5284 2255 Pnt 5284 2300 Pnt 5284 2346 Pnt 5284 2392 Pnt 5284 2576 Pnt 5284 2622 Pnt 5284 2668 Pnt 5284 2714 Pnt 5297 280 Pnt 5297 326 Pnt 5297 372 Pnt 5297 418 Pnt 5297 464 Pnt 5297 510 Pnt 5297 556 Pnt 5297 601 Pnt 5297 647 Pnt 5297 693 Pnt 5297 739 Pnt 5297 785 Pnt 5297 831 Pnt 5297 877 Pnt 5297 923 Pnt 5297 969 Pnt 5297 1015 Pnt 5297 1061 Pnt 5297 1107 Pnt 5297 1152 Pnt 5297 1198 Pnt 5297 1244 Pnt 5297 1290 Pnt 5297 1336 Pnt 5297 1382 Pnt 5297 1428 Pnt 5297 1474 Pnt 5297 1520 Pnt 5297 1566 Pnt 5297 1612 Pnt 5297 1658 Pnt 5297 1704 Pnt 5297 1749 Pnt 5297 1795 Pnt 5297 1841 Pnt 5297 1887 Pnt 5297 1933 Pnt 5297 2209 Pnt 5297 2255 Pnt 5297 2300 Pnt 5297 2346 Pnt 5297 2392 Pnt 5297 2438 Pnt 5297 2622 Pnt 5297 2668 Pnt 5297 2714 Pnt 5310 280 Pnt 5310 326 Pnt 5310 372 Pnt 5310 418 Pnt 5310 464 Pnt 5310 510 Pnt 5310 556 Pnt 5310 601 Pnt 5310 647 Pnt 5310 693 Pnt 5310 739 Pnt 5310 785 Pnt 5310 831 Pnt 5310 877 Pnt 5310 923 Pnt 5310 969 Pnt 5310 1015 Pnt 5310 1061 Pnt 5310 1107 Pnt 5310 1152 Pnt 5310 1198 Pnt 5310 1244 Pnt 5310 1290 Pnt 5310 1336 Pnt 5310 1382 Pnt 5310 1428 Pnt 5310 1474 Pnt 5310 1520 Pnt 5310 1566 Pnt 5310 1612 Pnt 5310 1658 Pnt 5310 1704 Pnt 5310 1749 Pnt 5310 1795 Pnt 5310 1841 Pnt 5310 1887 Pnt 5310 1933 Pnt 5310 2255 Pnt 5310 2300 Pnt 5310 2346 Pnt 5310 2392 Pnt 5310 2438 Pnt 5310 2622 Pnt 5310 2668 Pnt 5310 2714 Pnt 5310 2760 Pnt 5322 280 Pnt 5322 326 Pnt 5322 372 Pnt 5322 418 Pnt 5322 464 Pnt 5322 510 Pnt 5322 556 Pnt 5322 601 Pnt 5322 647 Pnt 5322 693 Pnt 5322 739 Pnt 5322 785 Pnt 5322 831 Pnt 5322 877 Pnt 5322 923 Pnt 5322 969 Pnt 5322 1015 Pnt 5322 1061 Pnt 5322 1107 Pnt 5322 1152 Pnt 5322 1198 Pnt 5322 1244 Pnt 5322 1290 Pnt 5322 1336 Pnt 5322 1382 Pnt 5322 1428 Pnt 5322 1474 Pnt 5322 1520 Pnt 5322 1566 Pnt 5322 1612 Pnt 5322 1658 Pnt 5322 1704 Pnt 5322 1749 Pnt 5322 1795 Pnt 5322 1841 Pnt 5322 1887 Pnt 5322 1933 Pnt 5322 1979 Pnt 5322 2300 Pnt 5322 2346 Pnt 5322 2392 Pnt 5322 2438 Pnt 5322 2484 Pnt 5322 2668 Pnt 5322 2714 Pnt 5322 2760 Pnt 5335 280 Pnt 5335 326 Pnt 5335 372 Pnt 5335 418 Pnt 5335 464 Pnt 5335 510 Pnt 5335 556 Pnt 5335 601 Pnt 5335 647 Pnt 5335 693 Pnt 5335 739 Pnt 5335 785 Pnt 5335 831 Pnt 5335 877 Pnt 5335 923 Pnt 5335 969 Pnt 5335 1015 Pnt 5335 1061 Pnt 5335 1107 Pnt 5335 1152 Pnt 5335 1198 Pnt 5335 1244 Pnt 5335 1290 Pnt 5335 1336 Pnt 5335 1382 Pnt 5335 1428 Pnt 5335 1474 Pnt 5335 1520 Pnt 5335 1566 Pnt 5335 1612 Pnt 5335 1658 Pnt 5335 1704 Pnt 5335 1749 Pnt 5335 1795 Pnt 5335 1841 Pnt 5335 1887 Pnt 5335 1933 Pnt 5335 1979 Pnt 5335 2025 Pnt 5335 2300 Pnt 5335 2346 Pnt 5335 2392 Pnt 5335 2438 Pnt 5335 2484 Pnt 5335 2530 Pnt 5335 2668 Pnt 5335 2714 Pnt 5335 2806 Pnt 5348 280 Pnt 5348 326 Pnt 5348 372 Pnt 5348 418 Pnt 5348 464 Pnt 5348 510 Pnt 5348 556 Pnt 5348 601 Pnt 5348 647 Pnt 5348 693 Pnt 5348 739 Pnt 5348 785 Pnt 5348 831 Pnt 5348 877 Pnt 5348 923 Pnt 5348 969 Pnt 5348 1015 Pnt 5348 1061 Pnt 5348 1107 Pnt 5348 1152 Pnt 5348 1198 Pnt 5348 1244 Pnt 5348 1290 Pnt 5348 1336 Pnt 5348 1382 Pnt 5348 1428 Pnt 5348 1474 Pnt 5348 1520 Pnt 5348 1566 Pnt 5348 1612 Pnt 5348 1658 Pnt 5348 1704 Pnt 5348 1749 Pnt 5348 1795 Pnt 5348 1841 Pnt 5348 1887 Pnt 5348 1933 Pnt 5348 1979 Pnt 5348 2025 Pnt 5348 2071 Pnt 5348 2346 Pnt 5348 2392 Pnt 5348 2438 Pnt 5348 2484 Pnt 5348 2530 Pnt 5348 2714 Pnt 5348 2760 Pnt 5348 2806 Pnt 5348 2852 Pnt 5360 280 Pnt 5360 326 Pnt 5360 372 Pnt 5360 418 Pnt 5360 464 Pnt 5360 510 Pnt 5360 556 Pnt 5360 601 Pnt 5360 647 Pnt 5360 693 Pnt 5360 739 Pnt 5360 785 Pnt 5360 831 Pnt 5360 877 Pnt 5360 923 Pnt 5360 969 Pnt 5360 1015 Pnt 5360 1061 Pnt 5360 1107 Pnt 5360 1152 Pnt 5360 1198 Pnt 5360 1244 Pnt 5360 1290 Pnt 5360 1336 Pnt 5360 1382 Pnt 5360 1428 Pnt 5360 1474 Pnt 5360 1520 Pnt 5360 1566 Pnt 5360 1612 Pnt 5360 1658 Pnt 5360 1704 Pnt 5360 1749 Pnt 5360 1795 Pnt 5360 1841 Pnt 5360 1887 Pnt 5360 1933 Pnt 5360 1979 Pnt 5360 2025 Pnt 5360 2071 Pnt 5360 2117 Pnt 5360 2392 Pnt 5360 2438 Pnt 5360 2484 Pnt 5360 2530 Pnt 5360 2576 Pnt 5360 2714 Pnt 5360 2760 Pnt 5360 2852 Pnt 5373 280 Pnt 5373 326 Pnt 5373 372 Pnt 5373 418 Pnt 5373 464 Pnt 5373 510 Pnt 5373 556 Pnt 5373 601 Pnt 5373 647 Pnt 5373 693 Pnt 5373 739 Pnt 5373 785 Pnt 5373 831 Pnt 5373 877 Pnt 5373 923 Pnt 5373 969 Pnt 5373 1015 Pnt 5373 1061 Pnt 5373 1107 Pnt 5373 1152 Pnt 5373 1198 Pnt 5373 1244 Pnt 5373 1290 Pnt 5373 1336 Pnt 5373 1382 Pnt 5373 1428 Pnt 5373 1474 Pnt 5373 1520 Pnt 5373 1566 Pnt 5373 1612 Pnt 5373 1658 Pnt 5373 1704 Pnt 5373 1749 Pnt 5373 1795 Pnt 5373 1841 Pnt 5373 1887 Pnt 5373 1933 Pnt 5373 1979 Pnt 5373 2025 Pnt 5373 2071 Pnt 5373 2117 Pnt 5373 2438 Pnt 5373 2484 Pnt 5373 2530 Pnt 5373 2576 Pnt 5373 2760 Pnt 5373 2806 Pnt 5373 2852 Pnt 5373 2897 Pnt 5386 280 Pnt 5386 326 Pnt 5386 372 Pnt 5386 418 Pnt 5386 464 Pnt 5386 510 Pnt 5386 556 Pnt 5386 601 Pnt 5386 647 Pnt 5386 693 Pnt 5386 739 Pnt 5386 785 Pnt 5386 831 Pnt 5386 877 Pnt 5386 923 Pnt 5386 969 Pnt 5386 1015 Pnt 5386 1061 Pnt 5386 1107 Pnt 5386 1152 Pnt 5386 1198 Pnt 5386 1244 Pnt 5386 1290 Pnt 5386 1336 Pnt 5386 1382 Pnt 5386 1428 Pnt 5386 1474 Pnt 5386 1520 Pnt 5386 1566 Pnt 5386 1612 Pnt 5386 1658 Pnt 5386 1704 Pnt 5386 1749 Pnt 5386 1795 Pnt 5386 1841 Pnt 5386 1887 Pnt 5386 1933 Pnt 5386 1979 Pnt 5386 2025 Pnt 5386 2071 Pnt 5386 2117 Pnt 5386 2163 Pnt 5386 2438 Pnt 5386 2484 Pnt 5386 2530 Pnt 5386 2576 Pnt 5386 2622 Pnt 5386 2806 Pnt 5386 2897 Pnt 5399 280 Pnt 5399 326 Pnt 5399 372 Pnt 5399 418 Pnt 5399 464 Pnt 5399 510 Pnt 5399 556 Pnt 5399 601 Pnt 5399 647 Pnt 5399 693 Pnt 5399 739 Pnt 5399 785 Pnt 5399 831 Pnt 5399 877 Pnt 5399 923 Pnt 5399 969 Pnt 5399 1015 Pnt 5399 1061 Pnt 5399 1107 Pnt 5399 1152 Pnt 5399 1198 Pnt 5399 1244 Pnt 5399 1290 Pnt 5399 1336 Pnt 5399 1382 Pnt 5399 1428 Pnt 5399 1474 Pnt 5399 1520 Pnt 5399 1566 Pnt 5399 1612 Pnt 5399 1658 Pnt 5399 1704 Pnt 5399 1749 Pnt 5399 1795 Pnt 5399 1841 Pnt 5399 1887 Pnt 5399 1933 Pnt 5399 1979 Pnt 5399 2025 Pnt 5399 2071 Pnt 5399 2117 Pnt 5399 2163 Pnt 5399 2209 Pnt 5399 2484 Pnt 5399 2530 Pnt 5399 2576 Pnt 5399 2622 Pnt 5399 2668 Pnt 5399 2806 Pnt 5399 2852 Pnt 5399 2897 Pnt 5399 2943 Pnt 5411 280 Pnt 5411 326 Pnt 5411 372 Pnt 5411 418 Pnt 5411 464 Pnt 5411 510 Pnt 5411 556 Pnt 5411 601 Pnt 5411 647 Pnt 5411 693 Pnt 5411 739 Pnt 5411 785 Pnt 5411 831 Pnt 5411 877 Pnt 5411 923 Pnt 5411 969 Pnt 5411 1015 Pnt 5411 1061 Pnt 5411 1107 Pnt 5411 1152 Pnt 5411 1198 Pnt 5411 1244 Pnt 5411 1290 Pnt 5411 1336 Pnt 5411 1382 Pnt 5411 1428 Pnt 5411 1474 Pnt 5411 1520 Pnt 5411 1566 Pnt 5411 1612 Pnt 5411 1658 Pnt 5411 1704 Pnt 5411 1749 Pnt 5411 1795 Pnt 5411 1841 Pnt 5411 1887 Pnt 5411 1933 Pnt 5411 1979 Pnt 5411 2025 Pnt 5411 2071 Pnt 5411 2117 Pnt 5411 2163 Pnt 5411 2209 Pnt 5411 2255 Pnt 5411 2530 Pnt 5411 2576 Pnt 5411 2622 Pnt 5411 2668 Pnt 5411 2852 Pnt 5411 2897 Pnt 5424 280 Pnt 5424 326 Pnt 5424 372 Pnt 5424 418 Pnt 5424 464 Pnt 5424 510 Pnt 5424 556 Pnt 5424 601 Pnt 5424 647 Pnt 5424 693 Pnt 5424 739 Pnt 5424 785 Pnt 5424 831 Pnt 5424 877 Pnt 5424 923 Pnt 5424 969 Pnt 5424 1015 Pnt 5424 1061 Pnt 5424 1107 Pnt 5424 1152 Pnt 5424 1198 Pnt 5424 1244 Pnt 5424 1290 Pnt 5424 1336 Pnt 5424 1382 Pnt 5424 1428 Pnt 5424 1474 Pnt 5424 1520 Pnt 5424 1566 Pnt 5424 1612 Pnt 5424 1658 Pnt 5424 1704 Pnt 5424 1749 Pnt 5424 1795 Pnt 5424 1841 Pnt 5424 1887 Pnt 5424 1933 Pnt 5424 1979 Pnt 5424 2025 Pnt 5424 2071 Pnt 5424 2117 Pnt 5424 2163 Pnt 5424 2209 Pnt 5424 2255 Pnt 5424 2530 Pnt 5424 2576 Pnt 5424 2622 Pnt 5424 2668 Pnt 5424 2714 Pnt 5424 2852 Pnt 5424 2897 Pnt 5424 2943 Pnt 5424 2989 Pnt 5437 280 Pnt 5437 326 Pnt 5437 372 Pnt 5437 418 Pnt 5437 464 Pnt 5437 510 Pnt 5437 556 Pnt 5437 601 Pnt 5437 647 Pnt 5437 693 Pnt 5437 739 Pnt 5437 785 Pnt 5437 831 Pnt 5437 877 Pnt 5437 923 Pnt 5437 969 Pnt 5437 1015 Pnt 5437 1061 Pnt 5437 1107 Pnt 5437 1152 Pnt 5437 1198 Pnt 5437 1244 Pnt 5437 1290 Pnt 5437 1336 Pnt 5437 1382 Pnt 5437 1428 Pnt 5437 1474 Pnt 5437 1520 Pnt 5437 1566 Pnt 5437 1612 Pnt 5437 1658 Pnt 5437 1704 Pnt 5437 1749 Pnt 5437 1795 Pnt 5437 1841 Pnt 5437 1887 Pnt 5437 1933 Pnt 5437 1979 Pnt 5437 2025 Pnt 5437 2071 Pnt 5437 2117 Pnt 5437 2163 Pnt 5437 2209 Pnt 5437 2255 Pnt 5437 2300 Pnt 5437 2576 Pnt 5437 2622 Pnt 5437 2668 Pnt 5437 2714 Pnt 5437 2897 Pnt 5437 2943 Pnt 5437 2989 Pnt 5449 280 Pnt 5449 326 Pnt 5449 372 Pnt 5449 418 Pnt 5449 464 Pnt 5449 510 Pnt 5449 556 Pnt 5449 601 Pnt 5449 647 Pnt 5449 693 Pnt 5449 739 Pnt 5449 785 Pnt 5449 831 Pnt 5449 877 Pnt 5449 923 Pnt 5449 969 Pnt 5449 1015 Pnt 5449 1061 Pnt 5449 1107 Pnt 5449 1152 Pnt 5449 1198 Pnt 5449 1244 Pnt 5449 1290 Pnt 5449 1336 Pnt 5449 1382 Pnt 5449 1428 Pnt 5449 1474 Pnt 5449 1520 Pnt 5449 1566 Pnt 5449 1612 Pnt 5449 1658 Pnt 5449 1704 Pnt 5449 1749 Pnt 5449 1795 Pnt 5449 1841 Pnt 5449 1887 Pnt 5449 1933 Pnt 5449 1979 Pnt 5449 2025 Pnt 5449 2071 Pnt 5449 2117 Pnt 5449 2163 Pnt 5449 2209 Pnt 5449 2255 Pnt 5449 2300 Pnt 5449 2346 Pnt 5449 2622 Pnt 5449 2668 Pnt 5449 2714 Pnt 5449 2760 Pnt 5449 2897 Pnt 5449 2943 Pnt 5449 2989 Pnt 5449 3035 Pnt 5462 280 Pnt 5462 326 Pnt 5462 372 Pnt 5462 418 Pnt 5462 464 Pnt 5462 510 Pnt 5462 556 Pnt 5462 601 Pnt 5462 647 Pnt 5462 693 Pnt 5462 739 Pnt 5462 785 Pnt 5462 831 Pnt 5462 877 Pnt 5462 923 Pnt 5462 969 Pnt 5462 1015 Pnt 5462 1061 Pnt 5462 1107 Pnt 5462 1152 Pnt 5462 1198 Pnt 5462 1244 Pnt 5462 1290 Pnt 5462 1336 Pnt 5462 1382 Pnt 5462 1428 Pnt 5462 1474 Pnt 5462 1520 Pnt 5462 1566 Pnt 5462 1612 Pnt 5462 1658 Pnt 5462 1704 Pnt 5462 1749 Pnt 5462 1795 Pnt 5462 1841 Pnt 5462 1887 Pnt 5462 1933 Pnt 5462 1979 Pnt 5462 2025 Pnt 5462 2071 Pnt 5462 2117 Pnt 5462 2163 Pnt 5462 2209 Pnt 5462 2255 Pnt 5462 2300 Pnt 5462 2346 Pnt 5462 2622 Pnt 5462 2668 Pnt 5462 2714 Pnt 5462 2760 Pnt 5462 2943 Pnt 5462 2989 Pnt 5462 3035 Pnt 5475 280 Pnt 5475 326 Pnt 5475 372 Pnt 5475 418 Pnt 5475 464 Pnt 5475 510 Pnt 5475 556 Pnt 5475 601 Pnt 5475 647 Pnt 5475 693 Pnt 5475 739 Pnt 5475 785 Pnt 5475 831 Pnt 5475 877 Pnt 5475 923 Pnt 5475 969 Pnt 5475 1015 Pnt 5475 1061 Pnt 5475 1107 Pnt 5475 1152 Pnt 5475 1198 Pnt 5475 1244 Pnt 5475 1290 Pnt 5475 1336 Pnt 5475 1382 Pnt 5475 1428 Pnt 5475 1474 Pnt 5475 1520 Pnt 5475 1566 Pnt 5475 1612 Pnt 5475 1658 Pnt 5475 1704 Pnt 5475 1749 Pnt 5475 1795 Pnt 5475 1841 Pnt 5475 1887 Pnt 5475 1933 Pnt 5475 1979 Pnt 5475 2025 Pnt 5475 2071 Pnt 5475 2117 Pnt 5475 2163 Pnt 5475 2209 Pnt 5475 2255 Pnt 5475 2300 Pnt 5475 2346 Pnt 5475 2392 Pnt 5475 2668 Pnt 5475 2714 Pnt 5475 2760 Pnt 5475 2806 Pnt 5475 2989 Pnt 5475 3081 Pnt 5488 280 Pnt 5488 326 Pnt 5488 372 Pnt 5488 418 Pnt 5488 464 Pnt 5488 510 Pnt 5488 556 Pnt 5488 601 Pnt 5488 647 Pnt 5488 693 Pnt 5488 739 Pnt 5488 785 Pnt 5488 831 Pnt 5488 877 Pnt 5488 923 Pnt 5488 969 Pnt 5488 1015 Pnt 5488 1061 Pnt 5488 1107 Pnt 5488 1152 Pnt 5488 1198 Pnt 5488 1244 Pnt 5488 1290 Pnt 5488 1336 Pnt 5488 1382 Pnt 5488 1428 Pnt 5488 1474 Pnt 5488 1520 Pnt 5488 1566 Pnt 5488 1612 Pnt 5488 1658 Pnt 5488 1704 Pnt 5488 1749 Pnt 5488 1795 Pnt 5488 1841 Pnt 5488 1887 Pnt 5488 1933 Pnt 5488 1979 Pnt 5488 2025 Pnt 5488 2071 Pnt 5488 2117 Pnt 5488 2163 Pnt 5488 2209 Pnt 5488 2255 Pnt 5488 2300 Pnt 5488 2346 Pnt 5488 2392 Pnt 5488 2438 Pnt 5488 2668 Pnt 5488 2714 Pnt 5488 2760 Pnt 5488 2806 Pnt 5488 2852 Pnt 5488 2989 Pnt 5488 3035 Pnt 5488 3081 Pnt 5500 280 Pnt 5500 326 Pnt 5500 372 Pnt 5500 418 Pnt 5500 464 Pnt 5500 510 Pnt 5500 556 Pnt 5500 601 Pnt 5500 647 Pnt 5500 693 Pnt 5500 739 Pnt 5500 785 Pnt 5500 831 Pnt 5500 877 Pnt 5500 1474 Pnt 5500 1520 Pnt 5500 1566 Pnt 5500 1612 Pnt 5500 1658 Pnt 5500 1704 Pnt 5500 1749 Pnt 5500 1795 Pnt 5500 1841 Pnt 5500 1887 Pnt 5500 1933 Pnt 5500 1979 Pnt 5500 2025 Pnt 5500 2071 Pnt 5500 2117 Pnt 5500 2163 Pnt 5500 2209 Pnt 5500 2255 Pnt 5500 2300 Pnt 5500 2346 Pnt 5500 2392 Pnt 5500 2438 Pnt 5500 2714 Pnt 5500 2760 Pnt 5500 2806 Pnt 5500 2852 Pnt 5500 3035 Pnt 5500 3127 Pnt 5513 280 Pnt 5513 326 Pnt 5513 372 Pnt 5513 418 Pnt 5513 464 Pnt 5513 510 Pnt 5513 556 Pnt 5513 601 Pnt 5513 647 Pnt 5513 693 Pnt 5513 739 Pnt 5513 1566 Pnt 5513 1612 Pnt 5513 1658 Pnt 5513 1704 Pnt 5513 1749 Pnt 5513 1795 Pnt 5513 1841 Pnt 5513 1887 Pnt 5513 1933 Pnt 5513 1979 Pnt 5513 2025 Pnt 5513 2071 Pnt 5513 2117 Pnt 5513 2163 Pnt 5513 2209 Pnt 5513 2255 Pnt 5513 2300 Pnt 5513 2346 Pnt 5513 2392 Pnt 5513 2438 Pnt 5513 2484 Pnt 5513 2760 Pnt 5513 2806 Pnt 5513 2852 Pnt 5513 2897 Pnt 5513 3035 Pnt 5513 3081 Pnt 5513 3127 Pnt 5526 280 Pnt 5526 326 Pnt 5526 372 Pnt 5526 418 Pnt 5526 464 Pnt 5526 510 Pnt 5526 556 Pnt 5526 601 Pnt 5526 647 Pnt 5526 1658 Pnt 5526 1704 Pnt 5526 1749 Pnt 5526 1795 Pnt 5526 1841 Pnt 5526 1887 Pnt 5526 1933 Pnt 5526 1979 Pnt 5526 2025 Pnt 5526 2071 Pnt 5526 2117 Pnt 5526 2163 Pnt 5526 2209 Pnt 5526 2255 Pnt 5526 2300 Pnt 5526 2346 Pnt 5526 2392 Pnt 5526 2438 Pnt 5526 2484 Pnt 5526 2530 Pnt 5526 2760 Pnt 5526 2806 Pnt 5526 2852 Pnt 5526 2897 Pnt 5526 3081 Pnt 5526 3173 Pnt 5538 280 Pnt 5538 326 Pnt 5538 372 Pnt 5538 418 Pnt 5538 464 Pnt 5538 510 Pnt 5538 556 Pnt 5538 601 Pnt 5538 1749 Pnt 5538 1795 Pnt 5538 1841 Pnt 5538 1887 Pnt 5538 1933 Pnt 5538 1979 Pnt 5538 2025 Pnt 5538 2071 Pnt 5538 2117 Pnt 5538 2163 Pnt 5538 2209 Pnt 5538 2255 Pnt 5538 2300 Pnt 5538 2346 Pnt 5538 2392 Pnt 5538 2438 Pnt 5538 2484 Pnt 5538 2530 Pnt 5538 2806 Pnt 5538 2852 Pnt 5538 2897 Pnt 5538 2943 Pnt 5538 3081 Pnt 5538 3127 Pnt 5538 3173 Pnt 5538 3219 Pnt 5551 280 Pnt 5551 326 Pnt 5551 372 Pnt 5551 418 Pnt 5551 464 Pnt 5551 510 Pnt 5551 1841 Pnt 5551 1887 Pnt 5551 1933 Pnt 5551 1979 Pnt 5551 2025 Pnt 5551 2071 Pnt 5551 2117 Pnt 5551 2163 Pnt 5551 2209 Pnt 5551 2255 Pnt 5551 2300 Pnt 5551 2346 Pnt 5551 2392 Pnt 5551 2438 Pnt 5551 2484 Pnt 5551 2530 Pnt 5551 2576 Pnt 5551 2852 Pnt 5551 2897 Pnt 5551 2943 Pnt 5551 3127 Pnt 5551 3173 Pnt 5551 3219 Pnt 5564 280 Pnt 5564 326 Pnt 5564 372 Pnt 5564 418 Pnt 5564 464 Pnt 5564 1887 Pnt 5564 1933 Pnt 5564 1979 Pnt 5564 2025 Pnt 5564 2071 Pnt 5564 2117 Pnt 5564 2163 Pnt 5564 2209 Pnt 5564 2255 Pnt 5564 2300 Pnt 5564 2346 Pnt 5564 2392 Pnt 5564 2438 Pnt 5564 2484 Pnt 5564 2530 Pnt 5564 2576 Pnt 5564 2622 Pnt 5564 2852 Pnt 5564 2897 Pnt 5564 2943 Pnt 5564 2989 Pnt 5564 3173 Pnt 5564 3219 Pnt 5564 3265 Pnt 5577 280 Pnt 5577 326 Pnt 5577 372 Pnt 5577 1933 Pnt 5577 1979 Pnt 5577 2025 Pnt 5577 2071 Pnt 5577 2117 Pnt 5577 2163 Pnt 5577 2209 Pnt 5577 2255 Pnt 5577 2300 Pnt 5577 2346 Pnt 5577 2392 Pnt 5577 2438 Pnt 5577 2484 Pnt 5577 2530 Pnt 5577 2576 Pnt 5577 2622 Pnt 5577 2897 Pnt 5577 2943 Pnt 5577 2989 Pnt 5577 3035 Pnt 5577 3173 Pnt 5577 3219 Pnt 5577 3265 Pnt 5589 280 Pnt 5589 326 Pnt 5589 2025 Pnt 5589 2071 Pnt 5589 2117 Pnt 5589 2163 Pnt 5589 2209 Pnt 5589 2255 Pnt 5589 2300 Pnt 5589 2346 Pnt 5589 2392 Pnt 5589 2438 Pnt 5589 2484 Pnt 5589 2530 Pnt 5589 2576 Pnt 5589 2622 Pnt 5589 2668 Pnt 5589 2897 Pnt 5589 2943 Pnt 5589 2989 Pnt 5589 3035 Pnt 5589 3219 Pnt 5589 3265 Pnt 5589 3311 Pnt 5602 280 Pnt 5602 326 Pnt 5602 372 Pnt 5602 418 Pnt 5602 464 Pnt 5602 510 Pnt 5602 2071 Pnt 5602 2117 Pnt 5602 2163 Pnt 5602 2209 Pnt 5602 2255 Pnt 5602 2300 Pnt 5602 2346 Pnt 5602 2392 Pnt 5602 2438 Pnt 5602 2530 Pnt 5602 2576 Pnt 5602 2622 Pnt 5602 2668 Pnt 5602 2943 Pnt 5602 2989 Pnt 5602 3035 Pnt 5602 3081 Pnt 5602 3219 Pnt 5602 3265 Pnt 5615 280 Pnt 5615 326 Pnt 5615 372 Pnt 5615 418 Pnt 5615 464 Pnt 5615 510 Pnt 5615 556 Pnt 5615 601 Pnt 5615 647 Pnt 5615 693 Pnt 5615 2117 Pnt 5615 2163 Pnt 5615 2209 Pnt 5615 2255 Pnt 5615 2300 Pnt 5615 2346 Pnt 5615 2392 Pnt 5615 2438 Pnt 5615 2484 Pnt 5615 2530 Pnt 5615 2576 Pnt 5615 2622 Pnt 5615 2668 Pnt 5615 2714 Pnt 5615 2989 Pnt 5615 3035 Pnt 5615 3081 Pnt 5615 3265 Pnt 5615 3311 Pnt 5615 3357 Pnt 5627 280 Pnt 5627 326 Pnt 5627 372 Pnt 5627 418 Pnt 5627 464 Pnt 5627 510 Pnt 5627 556 Pnt 5627 601 Pnt 5627 647 Pnt 5627 693 Pnt 5627 739 Pnt 5627 785 Pnt 5627 831 Pnt 5627 2163 Pnt 5627 2209 Pnt 5627 2255 Pnt 5627 2300 Pnt 5627 2346 Pnt 5627 2392 Pnt 5627 2438 Pnt 5627 2484 Pnt 5627 2530 Pnt 5627 2576 Pnt 5627 2622 Pnt 5627 2668 Pnt 5627 2714 Pnt 5627 2760 Pnt 5627 2989 Pnt 5627 3035 Pnt 5627 3081 Pnt 5627 3127 Pnt 5627 3265 Pnt 5627 3311 Pnt 5627 3357 Pnt 5640 280 Pnt 5640 326 Pnt 5640 372 Pnt 5640 418 Pnt 5640 464 Pnt 5640 510 Pnt 5640 556 Pnt 5640 601 Pnt 5640 647 Pnt 5640 693 Pnt 5640 739 Pnt 5640 785 Pnt 5640 831 Pnt 5640 877 Pnt 5640 923 Pnt 5640 2209 Pnt 5640 2255 Pnt 5640 2300 Pnt 5640 2346 Pnt 5640 2392 Pnt 5640 2438 Pnt 5640 2484 Pnt 5640 2530 Pnt 5640 2622 Pnt 5640 2668 Pnt 5640 2714 Pnt 5640 2760 Pnt 5640 3035 Pnt 5640 3081 Pnt 5640 3127 Pnt 5640 3311 Pnt 5640 3403 Pnt 5653 280 Pnt 5653 326 Pnt 5653 372 Pnt 5653 418 Pnt 5653 464 Pnt 5653 510 Pnt 5653 556 Pnt 5653 601 Pnt 5653 647 Pnt 5653 693 Pnt 5653 739 Pnt 5653 785 Pnt 5653 831 Pnt 5653 877 Pnt 5653 923 Pnt 5653 969 Pnt 5653 1015 Pnt 5653 2255 Pnt 5653 2300 Pnt 5653 2346 Pnt 5653 2392 Pnt 5653 2438 Pnt 5653 2484 Pnt 5653 2530 Pnt 5653 2576 Pnt 5653 2622 Pnt 5653 2668 Pnt 5653 2714 Pnt 5653 2760 Pnt 5653 2806 Pnt 5653 3035 Pnt 5653 3081 Pnt 5653 3127 Pnt 5653 3173 Pnt 5653 3311 Pnt 5653 3357 Pnt 5653 3403 Pnt 5666 280 Pnt 5666 326 Pnt 5666 372 Pnt 5666 418 Pnt 5666 464 Pnt 5666 510 Pnt 5666 556 Pnt 5666 601 Pnt 5666 647 Pnt 5666 693 Pnt 5666 739 Pnt 5666 785 Pnt 5666 831 Pnt 5666 877 Pnt 5666 923 Pnt 5666 969 Pnt 5666 1015 Pnt 5666 1061 Pnt 5666 2300 Pnt 5666 2346 Pnt 5666 2392 Pnt 5666 2438 Pnt 5666 2484 Pnt 5666 2530 Pnt 5666 2576 Pnt 5666 2622 Pnt 5666 2668 Pnt 5666 2714 Pnt 5666 2760 Pnt 5666 2806 Pnt 5666 3081 Pnt 5666 3127 Pnt 5666 3173 Pnt 5666 3357 Pnt 5666 3448 Pnt 5678 280 Pnt 5678 326 Pnt 5678 372 Pnt 5678 418 Pnt 5678 464 Pnt 5678 510 Pnt 5678 556 Pnt 5678 601 Pnt 5678 647 Pnt 5678 693 Pnt 5678 739 Pnt 5678 785 Pnt 5678 831 Pnt 5678 877 Pnt 5678 923 Pnt 5678 969 Pnt 5678 1015 Pnt 5678 1061 Pnt 5678 1107 Pnt 5678 1152 Pnt 5678 2346 Pnt 5678 2392 Pnt 5678 2438 Pnt 5678 2484 Pnt 5678 2530 Pnt 5678 2576 Pnt 5678 2622 Pnt 5678 2714 Pnt 5678 2760 Pnt 5678 2806 Pnt 5678 2852 Pnt 5678 3081 Pnt 5678 3127 Pnt 5678 3173 Pnt 5678 3219 Pnt 5678 3357 Pnt 5678 3403 Pnt 5678 3448 Pnt 5691 280 Pnt 5691 326 Pnt 5691 372 Pnt 5691 418 Pnt 5691 464 Pnt 5691 510 Pnt 5691 556 Pnt 5691 601 Pnt 5691 647 Pnt 5691 693 Pnt 5691 739 Pnt 5691 785 Pnt 5691 831 Pnt 5691 877 Pnt 5691 923 Pnt 5691 969 Pnt 5691 1015 Pnt 5691 1061 Pnt 5691 1107 Pnt 5691 1152 Pnt 5691 1198 Pnt 5691 2392 Pnt 5691 2438 Pnt 5691 2484 Pnt 5691 2530 Pnt 5691 2576 Pnt 5691 2622 Pnt 5691 2668 Pnt 5691 2714 Pnt 5691 2760 Pnt 5691 2806 Pnt 5691 2852 Pnt 5691 2897 Pnt 5691 3127 Pnt 5691 3173 Pnt 5691 3219 Pnt 5691 3403 Pnt 5691 3494 Pnt 5704 280 Pnt 5704 326 Pnt 5704 372 Pnt 5704 418 Pnt 5704 464 Pnt 5704 510 Pnt 5704 556 Pnt 5704 601 Pnt 5704 647 Pnt 5704 693 Pnt 5704 739 Pnt 5704 785 Pnt 5704 831 Pnt 5704 877 Pnt 5704 923 Pnt 5704 969 Pnt 5704 1015 Pnt 5704 1061 Pnt 5704 1107 Pnt 5704 1152 Pnt 5704 1198 Pnt 5704 1244 Pnt 5704 1290 Pnt 5704 2392 Pnt 5704 2438 Pnt 5704 2484 Pnt 5704 2530 Pnt 5704 2576 Pnt 5704 2622 Pnt 5704 2668 Pnt 5704 2714 Pnt 5704 2760 Pnt 5704 2806 Pnt 5704 2852 Pnt 5704 2897 Pnt 5704 3173 Pnt 5704 3219 Pnt 5704 3265 Pnt 5704 3403 Pnt 5704 3448 Pnt 5704 3494 Pnt 5716 280 Pnt 5716 326 Pnt 5716 372 Pnt 5716 418 Pnt 5716 464 Pnt 5716 510 Pnt 5716 556 Pnt 5716 601 Pnt 5716 647 Pnt 5716 693 Pnt 5716 739 Pnt 5716 785 Pnt 5716 831 Pnt 5716 877 Pnt 5716 923 Pnt 5716 969 Pnt 5716 1015 Pnt 5716 1061 Pnt 5716 1107 Pnt 5716 1152 Pnt 5716 1198 Pnt 5716 1244 Pnt 5716 1290 Pnt 5716 1336 Pnt 5716 2438 Pnt 5716 2484 Pnt 5716 2530 Pnt 5716 2576 Pnt 5716 2622 Pnt 5716 2668 Pnt 5716 2714 Pnt 5716 2806 Pnt 5716 2852 Pnt 5716 2897 Pnt 5716 2943 Pnt 5716 3173 Pnt 5716 3219 Pnt 5716 3265 Pnt 5716 3311 Pnt 5716 3448 Pnt 5716 3540 Pnt 5729 280 Pnt 5729 326 Pnt 5729 372 Pnt 5729 418 Pnt 5729 464 Pnt 5729 510 Pnt 5729 556 Pnt 5729 601 Pnt 5729 647 Pnt 5729 693 Pnt 5729 739 Pnt 5729 785 Pnt 5729 831 Pnt 5729 877 Pnt 5729 923 Pnt 5729 969 Pnt 5729 1015 Pnt 5729 1061 Pnt 5729 1107 Pnt 5729 1152 Pnt 5729 1198 Pnt 5729 1244 Pnt 5729 1290 Pnt 5729 1336 Pnt 5729 1382 Pnt 5729 2484 Pnt 5729 2530 Pnt 5729 2576 Pnt 5729 2622 Pnt 5729 2668 Pnt 5729 2714 Pnt 5729 2760 Pnt 5729 2806 Pnt 5729 2852 Pnt 5729 2897 Pnt 5729 2943 Pnt 5729 3219 Pnt 5729 3265 Pnt 5729 3311 Pnt 5729 3494 Pnt 5729 3540 Pnt 5742 280 Pnt 5742 326 Pnt 5742 372 Pnt 5742 418 Pnt 5742 464 Pnt 5742 510 Pnt 5742 556 Pnt 5742 601 Pnt 5742 647 Pnt 5742 693 Pnt 5742 739 Pnt 5742 785 Pnt 5742 831 Pnt 5742 877 Pnt 5742 923 Pnt 5742 969 Pnt 5742 1015 Pnt 5742 1061 Pnt 5742 1107 Pnt 5742 1152 Pnt 5742 1198 Pnt 5742 1244 Pnt 5742 1290 Pnt 5742 1336 Pnt 5742 1382 Pnt 5742 1428 Pnt 5742 2530 Pnt 5742 2576 Pnt 5742 2622 Pnt 5742 2668 Pnt 5742 2714 Pnt 5742 2760 Pnt 5742 2806 Pnt 5742 2852 Pnt 5742 2897 Pnt 5742 2943 Pnt 5742 2989 Pnt 5742 3219 Pnt 5742 3265 Pnt 5742 3311 Pnt 5742 3357 Pnt 5742 3494 Pnt 5742 3586 Pnt 5754 280 Pnt 5754 326 Pnt 5754 372 Pnt 5754 418 Pnt 5754 464 Pnt 5754 510 Pnt 5754 556 Pnt 5754 601 Pnt 5754 647 Pnt 5754 693 Pnt 5754 739 Pnt 5754 785 Pnt 5754 831 Pnt 5754 877 Pnt 5754 923 Pnt 5754 969 Pnt 5754 1015 Pnt 5754 1061 Pnt 5754 1107 Pnt 5754 1152 Pnt 5754 1198 Pnt 5754 1244 Pnt 5754 1290 Pnt 5754 1336 Pnt 5754 1382 Pnt 5754 1428 Pnt 5754 1474 Pnt 5754 2576 Pnt 5754 2622 Pnt 5754 2668 Pnt 5754 2714 Pnt 5754 2760 Pnt 5754 2806 Pnt 5754 2897 Pnt 5754 2943 Pnt 5754 2989 Pnt 5754 3035 Pnt 5754 3265 Pnt 5754 3311 Pnt 5754 3357 Pnt 5754 3540 Pnt 5754 3586 Pnt 5767 280 Pnt 5767 326 Pnt 5767 372 Pnt 5767 418 Pnt 5767 464 Pnt 5767 510 Pnt 5767 556 Pnt 5767 601 Pnt 5767 647 Pnt 5767 693 Pnt 5767 739 Pnt 5767 785 Pnt 5767 831 Pnt 5767 877 Pnt 5767 923 Pnt 5767 969 Pnt 5767 1015 Pnt 5767 1061 Pnt 5767 1107 Pnt 5767 1152 Pnt 5767 1198 Pnt 5767 1244 Pnt 5767 1290 Pnt 5767 1336 Pnt 5767 1382 Pnt 5767 1428 Pnt 5767 1474 Pnt 5767 1520 Pnt 5767 2576 Pnt 5767 2622 Pnt 5767 2668 Pnt 5767 2714 Pnt 5767 2760 Pnt 5767 2806 Pnt 5767 2852 Pnt 5767 2897 Pnt 5767 2943 Pnt 5767 2989 Pnt 5767 3035 Pnt 5767 3265 Pnt 5767 3311 Pnt 5767 3357 Pnt 5767 3403 Pnt 5767 3540 Pnt 5767 3632 Pnt 5780 280 Pnt 5780 326 Pnt 5780 372 Pnt 5780 418 Pnt 5780 464 Pnt 5780 510 Pnt 5780 556 Pnt 5780 601 Pnt 5780 647 Pnt 5780 693 Pnt 5780 739 Pnt 5780 785 Pnt 5780 831 Pnt 5780 877 Pnt 5780 923 Pnt 5780 969 Pnt 5780 1015 Pnt 5780 1061 Pnt 5780 1107 Pnt 5780 1152 Pnt 5780 1198 Pnt 5780 1244 Pnt 5780 1290 Pnt 5780 1336 Pnt 5780 1382 Pnt 5780 1428 Pnt 5780 1474 Pnt 5780 1520 Pnt 5780 1566 Pnt 5780 2622 Pnt 5780 2668 Pnt 5780 2714 Pnt 5780 2760 Pnt 5780 2806 Pnt 5780 2852 Pnt 5780 2897 Pnt 5780 2943 Pnt 5780 2989 Pnt 5780 3035 Pnt 5780 3081 Pnt 5780 3311 Pnt 5780 3357 Pnt 5780 3403 Pnt 5780 3586 Pnt 5780 3632 Pnt 5793 280 Pnt 5793 326 Pnt 5793 372 Pnt 5793 418 Pnt 5793 464 Pnt 5793 510 Pnt 5793 556 Pnt 5793 601 Pnt 5793 647 Pnt 5793 693 Pnt 5793 739 Pnt 5793 785 Pnt 5793 831 Pnt 5793 877 Pnt 5793 923 Pnt 5793 969 Pnt 5793 1015 Pnt 5793 1061 Pnt 5793 1107 Pnt 5793 1152 Pnt 5793 1198 Pnt 5793 1244 Pnt 5793 1290 Pnt 5793 1336 Pnt 5793 1382 Pnt 5793 1428 Pnt 5793 1474 Pnt 5793 1520 Pnt 5793 1566 Pnt 5793 1612 Pnt 5793 2668 Pnt 5793 2714 Pnt 5793 2760 Pnt 5793 2806 Pnt 5793 2852 Pnt 5793 2897 Pnt 5793 2943 Pnt 5793 2989 Pnt 5793 3035 Pnt 5793 3081 Pnt 5793 3357 Pnt 5793 3403 Pnt 5793 3448 Pnt 5793 3586 Pnt 5793 3632 Pnt 5793 3678 Pnt 5805 280 Pnt 5805 326 Pnt 5805 372 Pnt 5805 418 Pnt 5805 464 Pnt 5805 510 Pnt 5805 556 Pnt 5805 601 Pnt 5805 647 Pnt 5805 693 Pnt 5805 739 Pnt 5805 785 Pnt 5805 831 Pnt 5805 877 Pnt 5805 923 Pnt 5805 969 Pnt 5805 1015 Pnt 5805 1061 Pnt 5805 1107 Pnt 5805 1152 Pnt 5805 1198 Pnt 5805 1244 Pnt 5805 1290 Pnt 5805 1336 Pnt 5805 1382 Pnt 5805 1428 Pnt 5805 1474 Pnt 5805 1520 Pnt 5805 1566 Pnt 5805 1612 Pnt 5805 1658 Pnt 5805 2714 Pnt 5805 2760 Pnt 5805 2806 Pnt 5805 2852 Pnt 5805 2897 Pnt 5805 2943 Pnt 5805 2989 Pnt 5805 3035 Pnt 5805 3081 Pnt 5805 3127 Pnt 5805 3357 Pnt 5805 3403 Pnt 5805 3448 Pnt 5805 3632 Pnt 5805 3678 Pnt 5818 280 Pnt 5818 326 Pnt 5818 372 Pnt 5818 418 Pnt 5818 464 Pnt 5818 510 Pnt 5818 556 Pnt 5818 601 Pnt 5818 647 Pnt 5818 693 Pnt 5818 739 Pnt 5818 785 Pnt 5818 831 Pnt 5818 877 Pnt 5818 923 Pnt 5818 969 Pnt 5818 1015 Pnt 5818 1061 Pnt 5818 1107 Pnt 5818 1152 Pnt 5818 1198 Pnt 5818 1244 Pnt 5818 1290 Pnt 5818 1336 Pnt 5818 1382 Pnt 5818 1428 Pnt 5818 1474 Pnt 5818 1520 Pnt 5818 1566 Pnt 5818 1612 Pnt 5818 1658 Pnt 5818 1704 Pnt 5818 2714 Pnt 5818 2760 Pnt 5818 2806 Pnt 5818 2852 Pnt 5818 2897 Pnt 5818 2943 Pnt 5818 3035 Pnt 5818 3081 Pnt 5818 3127 Pnt 5818 3403 Pnt 5818 3448 Pnt 5818 3494 Pnt 5818 3632 Pnt 5818 3678 Pnt 5818 3724 Pnt 5831 280 Pnt 5831 326 Pnt 5831 372 Pnt 5831 418 Pnt 5831 464 Pnt 5831 510 Pnt 5831 556 Pnt 5831 601 Pnt 5831 647 Pnt 5831 693 Pnt 5831 739 Pnt 5831 785 Pnt 5831 831 Pnt 5831 877 Pnt 5831 923 Pnt 5831 969 Pnt 5831 1015 Pnt 5831 1061 Pnt 5831 1107 Pnt 5831 1152 Pnt 5831 1198 Pnt 5831 1244 Pnt 5831 1290 Pnt 5831 1336 Pnt 5831 1382 Pnt 5831 1428 Pnt 5831 1474 Pnt 5831 1520 Pnt 5831 1566 Pnt 5831 1612 Pnt 5831 1658 Pnt 5831 1704 Pnt 5831 1749 Pnt 5831 2760 Pnt 5831 2806 Pnt 5831 2852 Pnt 5831 2897 Pnt 5831 2943 Pnt 5831 2989 Pnt 5831 3035 Pnt 5831 3081 Pnt 5831 3127 Pnt 5831 3173 Pnt 5831 3403 Pnt 5831 3448 Pnt 5831 3494 Pnt 5831 3678 Pnt 5831 3724 Pnt 5843 280 Pnt 5843 326 Pnt 5843 372 Pnt 5843 418 Pnt 5843 464 Pnt 5843 510 Pnt 5843 556 Pnt 5843 601 Pnt 5843 647 Pnt 5843 693 Pnt 5843 739 Pnt 5843 785 Pnt 5843 831 Pnt 5843 877 Pnt 5843 923 Pnt 5843 969 Pnt 5843 1015 Pnt 5843 1061 Pnt 5843 1107 Pnt 5843 1152 Pnt 5843 1198 Pnt 5843 1244 Pnt 5843 1290 Pnt 5843 1336 Pnt 5843 1382 Pnt 5843 1428 Pnt 5843 1474 Pnt 5843 1520 Pnt 5843 1566 Pnt 5843 1612 Pnt 5843 1658 Pnt 5843 1704 Pnt 5843 1749 Pnt 5843 1795 Pnt 5843 2806 Pnt 5843 2852 Pnt 5843 2897 Pnt 5843 2943 Pnt 5843 2989 Pnt 5843 3035 Pnt 5843 3081 Pnt 5843 3127 Pnt 5843 3173 Pnt 5843 3219 Pnt 5843 3448 Pnt 5843 3494 Pnt 5843 3540 Pnt 5843 3678 Pnt 5843 3724 Pnt 5843 3770 Pnt 5856 280 Pnt 5856 326 Pnt 5856 372 Pnt 5856 418 Pnt 5856 464 Pnt 5856 510 Pnt 5856 556 Pnt 5856 601 Pnt 5856 647 Pnt 5856 693 Pnt 5856 739 Pnt 5856 785 Pnt 5856 831 Pnt 5856 877 Pnt 5856 923 Pnt 5856 969 Pnt 5856 1015 Pnt 5856 1061 Pnt 5856 1107 Pnt 5856 1152 Pnt 5856 1198 Pnt 5856 1244 Pnt 5856 1290 Pnt 5856 1336 Pnt 5856 1382 Pnt 5856 1428 Pnt 5856 1474 Pnt 5856 1520 Pnt 5856 1566 Pnt 5856 1612 Pnt 5856 1658 Pnt 5856 1704 Pnt 5856 1749 Pnt 5856 1795 Pnt 5856 1841 Pnt 5856 2806 Pnt 5856 2852 Pnt 5856 2897 Pnt 5856 2943 Pnt 5856 2989 Pnt 5856 3035 Pnt 5856 3081 Pnt 5856 3127 Pnt 5856 3173 Pnt 5856 3219 Pnt 5856 3448 Pnt 5856 3494 Pnt 5856 3540 Pnt 5856 3724 Pnt 5856 3770 Pnt 5869 280 Pnt 5869 326 Pnt 5869 372 Pnt 5869 418 Pnt 5869 464 Pnt 5869 510 Pnt 5869 556 Pnt 5869 601 Pnt 5869 647 Pnt 5869 693 Pnt 5869 739 Pnt 5869 785 Pnt 5869 831 Pnt 5869 877 Pnt 5869 923 Pnt 5869 969 Pnt 5869 1015 Pnt 5869 1061 Pnt 5869 1107 Pnt 5869 1152 Pnt 5869 1198 Pnt 5869 1244 Pnt 5869 1290 Pnt 5869 1336 Pnt 5869 1382 Pnt 5869 1428 Pnt 5869 1474 Pnt 5869 1520 Pnt 5869 1566 Pnt 5869 1612 Pnt 5869 1658 Pnt 5869 1704 Pnt 5869 1749 Pnt 5869 1795 Pnt 5869 1841 Pnt 5869 1887 Pnt 5869 2852 Pnt 5869 2897 Pnt 5869 2943 Pnt 5869 2989 Pnt 5869 3035 Pnt 5869 3081 Pnt 5869 3127 Pnt 5869 3173 Pnt 5869 3219 Pnt 5869 3265 Pnt 5869 3494 Pnt 5869 3540 Pnt 5869 3586 Pnt 5869 3724 Pnt 5869 3770 Pnt 5869 3816 Pnt 5882 280 Pnt 5882 326 Pnt 5882 372 Pnt 5882 418 Pnt 5882 464 Pnt 5882 510 Pnt 5882 556 Pnt 5882 601 Pnt 5882 647 Pnt 5882 693 Pnt 5882 739 Pnt 5882 785 Pnt 5882 831 Pnt 5882 877 Pnt 5882 923 Pnt 5882 969 Pnt 5882 1015 Pnt 5882 1061 Pnt 5882 1107 Pnt 5882 1152 Pnt 5882 1198 Pnt 5882 1244 Pnt 5882 1290 Pnt 5882 1336 Pnt 5882 1382 Pnt 5882 1428 Pnt 5882 1474 Pnt 5882 1520 Pnt 5882 1566 Pnt 5882 1612 Pnt 5882 1658 Pnt 5882 1704 Pnt 5882 1749 Pnt 5882 1795 Pnt 5882 1841 Pnt 5882 1887 Pnt 5882 2897 Pnt 5882 2943 Pnt 5882 2989 Pnt 5882 3035 Pnt 5882 3081 Pnt 5882 3173 Pnt 5882 3219 Pnt 5882 3265 Pnt 5882 3494 Pnt 5882 3540 Pnt 5882 3586 Pnt 5882 3770 Pnt 5882 3816 Pnt 5894 280 Pnt 5894 326 Pnt 5894 372 Pnt 5894 418 Pnt 5894 464 Pnt 5894 510 Pnt 5894 556 Pnt 5894 601 Pnt 5894 647 Pnt 5894 693 Pnt 5894 739 Pnt 5894 785 Pnt 5894 831 Pnt 5894 877 Pnt 5894 923 Pnt 5894 969 Pnt 5894 1015 Pnt 5894 1061 Pnt 5894 1107 Pnt 5894 1152 Pnt 5894 1198 Pnt 5894 1244 Pnt 5894 1290 Pnt 5894 1336 Pnt 5894 1382 Pnt 5894 1428 Pnt 5894 1474 Pnt 5894 1520 Pnt 5894 1566 Pnt 5894 1612 Pnt 5894 1658 Pnt 5894 1704 Pnt 5894 1749 Pnt 5894 1795 Pnt 5894 1841 Pnt 5894 1887 Pnt 5894 1933 Pnt 5894 2897 Pnt 5894 2943 Pnt 5894 2989 Pnt 5894 3035 Pnt 5894 3081 Pnt 5894 3127 Pnt 5894 3173 Pnt 5894 3219 Pnt 5894 3265 Pnt 5894 3311 Pnt 5894 3540 Pnt 5894 3586 Pnt 5894 3632 Pnt 5894 3770 Pnt 5894 3816 Pnt 5894 3862 Pnt 5907 280 Pnt 5907 326 Pnt 5907 372 Pnt 5907 418 Pnt 5907 464 Pnt 5907 510 Pnt 5907 556 Pnt 5907 601 Pnt 5907 647 Pnt 5907 693 Pnt 5907 739 Pnt 5907 785 Pnt 5907 831 Pnt 5907 877 Pnt 5907 923 Pnt 5907 969 Pnt 5907 1015 Pnt 5907 1061 Pnt 5907 1107 Pnt 5907 1152 Pnt 5907 1198 Pnt 5907 1244 Pnt 5907 1290 Pnt 5907 1336 Pnt 5907 1382 Pnt 5907 1428 Pnt 5907 1474 Pnt 5907 1520 Pnt 5907 1566 Pnt 5907 1612 Pnt 5907 1658 Pnt 5907 1704 Pnt 5907 1749 Pnt 5907 1795 Pnt 5907 1841 Pnt 5907 1887 Pnt 5907 1933 Pnt 5907 1979 Pnt 5907 2943 Pnt 5907 2989 Pnt 5907 3035 Pnt 5907 3081 Pnt 5907 3127 Pnt 5907 3219 Pnt 5907 3265 Pnt 5907 3311 Pnt 5907 3540 Pnt 5907 3586 Pnt 5907 3632 Pnt 5907 3816 Pnt 5907 3862 Pnt 5920 280 Pnt 5920 326 Pnt 5920 372 Pnt 5920 418 Pnt 5920 464 Pnt 5920 510 Pnt 5920 556 Pnt 5920 601 Pnt 5920 647 Pnt 5920 693 Pnt 5920 739 Pnt 5920 785 Pnt 5920 831 Pnt 5920 877 Pnt 5920 923 Pnt 5920 969 Pnt 5920 1015 Pnt 5920 1061 Pnt 5920 1107 Pnt 5920 1152 Pnt 5920 1198 Pnt 5920 1244 Pnt 5920 1290 Pnt 5920 1336 Pnt 5920 1382 Pnt 5920 1428 Pnt 5920 1474 Pnt 5920 1520 Pnt 5920 1566 Pnt 5920 1612 Pnt 5920 1658 Pnt 5920 1704 Pnt 5920 1749 Pnt 5920 1795 Pnt 5920 1841 Pnt 5920 1887 Pnt 5920 1933 Pnt 5920 1979 Pnt 5920 2025 Pnt 5920 2989 Pnt 5920 3035 Pnt 5920 3081 Pnt 5920 3127 Pnt 5920 3173 Pnt 5920 3219 Pnt 5920 3265 Pnt 5920 3311 Pnt 5920 3357 Pnt 5920 3586 Pnt 5920 3632 Pnt 5920 3678 Pnt 5920 3816 Pnt 5920 3862 Pnt 5920 3908 Pnt 5932 280 Pnt 5932 326 Pnt 5932 372 Pnt 5932 418 Pnt 5932 464 Pnt 5932 510 Pnt 5932 556 Pnt 5932 601 Pnt 5932 647 Pnt 5932 693 Pnt 5932 739 Pnt 5932 785 Pnt 5932 831 Pnt 5932 877 Pnt 5932 923 Pnt 5932 969 Pnt 5932 1015 Pnt 5932 1061 Pnt 5932 1107 Pnt 5932 1152 Pnt 5932 1198 Pnt 5932 1244 Pnt 5932 1290 Pnt 5932 1336 Pnt 5932 1382 Pnt 5932 1428 Pnt 5932 1474 Pnt 5932 1520 Pnt 5932 1566 Pnt 5932 1612 Pnt 5932 1658 Pnt 5932 1704 Pnt 5932 1749 Pnt 5932 1795 Pnt 5932 1841 Pnt 5932 1887 Pnt 5932 1933 Pnt 5932 1979 Pnt 5932 2025 Pnt 5932 2989 Pnt 5932 3035 Pnt 5932 3081 Pnt 5932 3127 Pnt 5932 3173 Pnt 5932 3219 Pnt 5932 3265 Pnt 5932 3311 Pnt 5932 3357 Pnt 5932 3632 Pnt 5932 3678 Pnt 5932 3862 Pnt 5932 3908 Pnt 5932 3954 Pnt 5945 280 Pnt 5945 326 Pnt 5945 372 Pnt 5945 418 Pnt 5945 464 Pnt 5945 510 Pnt 5945 556 Pnt 5945 601 Pnt 5945 647 Pnt 5945 693 Pnt 5945 739 Pnt 5945 785 Pnt 5945 831 Pnt 5945 877 Pnt 5945 923 Pnt 5945 969 Pnt 5945 1015 Pnt 5945 1061 Pnt 5945 1107 Pnt 5945 1152 Pnt 5945 1198 Pnt 5945 1244 Pnt 5945 1290 Pnt 5945 1336 Pnt 5945 1382 Pnt 5945 1428 Pnt 5945 1474 Pnt 5945 1520 Pnt 5945 1566 Pnt 5945 1612 Pnt 5945 1658 Pnt 5945 1704 Pnt 5945 1749 Pnt 5945 1795 Pnt 5945 1841 Pnt 5945 1887 Pnt 5945 1933 Pnt 5945 1979 Pnt 5945 2025 Pnt 5945 2071 Pnt 5945 3035 Pnt 5945 3081 Pnt 5945 3127 Pnt 5945 3173 Pnt 5945 3219 Pnt 5945 3311 Pnt 5945 3357 Pnt 5945 3403 Pnt 5945 3632 Pnt 5945 3678 Pnt 5945 3724 Pnt 5945 3862 Pnt 5945 3908 Pnt 5945 3954 Pnt 5958 280 Pnt 5958 326 Pnt 5958 372 Pnt 5958 418 Pnt 5958 464 Pnt 5958 510 Pnt 5958 556 Pnt 5958 601 Pnt 5958 647 Pnt 5958 693 Pnt 5958 739 Pnt 5958 785 Pnt 5958 831 Pnt 5958 877 Pnt 5958 923 Pnt 5958 969 Pnt 5958 1015 Pnt 5958 1061 Pnt 5958 1107 Pnt 5958 1152 Pnt 5958 1198 Pnt 5958 1244 Pnt 5958 1290 Pnt 5958 1336 Pnt 5958 1382 Pnt 5958 1428 Pnt 5958 1474 Pnt 5958 1520 Pnt 5958 1566 Pnt 5958 1612 Pnt 5958 1658 Pnt 5958 1704 Pnt 5958 1749 Pnt 5958 1795 Pnt 5958 1841 Pnt 5958 1887 Pnt 5958 1933 Pnt 5958 1979 Pnt 5958 2025 Pnt 5958 2071 Pnt 5958 2117 Pnt 5958 3035 Pnt 5958 3081 Pnt 5958 3127 Pnt 5958 3173 Pnt 5958 3219 Pnt 5958 3265 Pnt 5958 3311 Pnt 5958 3357 Pnt 5958 3403 Pnt 5958 3678 Pnt 5958 3724 Pnt 5958 3770 Pnt 5958 3908 Pnt 5958 3954 Pnt 5958 4000 Pnt 5971 280 Pnt 5971 326 Pnt 5971 372 Pnt 5971 418 Pnt 5971 464 Pnt 5971 510 Pnt 5971 556 Pnt 5971 601 Pnt 5971 647 Pnt 5971 693 Pnt 5971 739 Pnt 5971 785 Pnt 5971 831 Pnt 5971 877 Pnt 5971 923 Pnt 5971 969 Pnt 5971 1015 Pnt 5971 1061 Pnt 5971 1107 Pnt 5971 1152 Pnt 5971 1198 Pnt 5971 1244 Pnt 5971 1290 Pnt 5971 1336 Pnt 5971 1382 Pnt 5971 1428 Pnt 5971 1474 Pnt 5971 1520 Pnt 5971 1566 Pnt 5971 1612 Pnt 5971 1658 Pnt 5971 1704 Pnt 5971 1749 Pnt 5971 1795 Pnt 5971 1841 Pnt 5971 1887 Pnt 5971 1933 Pnt 5971 1979 Pnt 5971 2025 Pnt 5971 2071 Pnt 5971 2117 Pnt 5971 2163 Pnt 5971 3081 Pnt 5971 3127 Pnt 5971 3173 Pnt 5971 3219 Pnt 5971 3265 Pnt 5971 3357 Pnt 5971 3403 Pnt 5971 3448 Pnt 5971 3678 Pnt 5971 3724 Pnt 5971 3770 Pnt 5971 3908 Pnt 5971 3954 Pnt 5971 4000 Pnt 5983 280 Pnt 5983 326 Pnt 5983 372 Pnt 5983 418 Pnt 5983 464 Pnt 5983 510 Pnt 5983 556 Pnt 5983 601 Pnt 5983 647 Pnt 5983 693 Pnt 5983 739 Pnt 5983 785 Pnt 5983 831 Pnt 5983 877 Pnt 5983 923 Pnt 5983 969 Pnt 5983 1015 Pnt 5983 1061 Pnt 5983 1107 Pnt 5983 1152 Pnt 5983 1198 Pnt 5983 1244 Pnt 5983 1290 Pnt 5983 1336 Pnt 5983 1382 Pnt 5983 1428 Pnt 5983 1474 Pnt 5983 1520 Pnt 5983 1566 Pnt 5983 1612 Pnt 5983 1658 Pnt 5983 1704 Pnt 5983 1749 Pnt 5983 1795 Pnt 5983 1841 Pnt 5983 1887 Pnt 5983 1933 Pnt 5983 1979 Pnt 5983 2025 Pnt 5983 2071 Pnt 5983 2117 Pnt 5983 2163 Pnt 5983 3127 Pnt 5983 3173 Pnt 5983 3219 Pnt 5983 3265 Pnt 5983 3311 Pnt 5983 3357 Pnt 5983 3403 Pnt 5983 3448 Pnt 5983 3494 Pnt 5983 3724 Pnt 5983 3770 Pnt 5983 3816 Pnt 5983 3954 Pnt 5983 4000 Pnt 5983 4045 Pnt 5996 280 Pnt 5996 326 Pnt 5996 372 Pnt 5996 418 Pnt 5996 464 Pnt 5996 510 Pnt 5996 556 Pnt 5996 601 Pnt 5996 647 Pnt 5996 693 Pnt 5996 739 Pnt 5996 785 Pnt 5996 831 Pnt 5996 877 Pnt 5996 923 Pnt 5996 969 Pnt 5996 1015 Pnt 5996 1061 Pnt 5996 1107 Pnt 5996 1152 Pnt 5996 1198 Pnt 5996 1244 Pnt 5996 1290 Pnt 5996 1336 Pnt 5996 1382 Pnt 5996 1428 Pnt 5996 1474 Pnt 5996 1520 Pnt 5996 1566 Pnt 5996 1612 Pnt 5996 1658 Pnt 5996 1704 Pnt 5996 1749 Pnt 5996 1795 Pnt 5996 1841 Pnt 5996 1887 Pnt 5996 1933 Pnt 5996 1979 Pnt 5996 2025 Pnt 5996 2071 Pnt 5996 2117 Pnt 5996 2163 Pnt 5996 2209 Pnt 5996 3127 Pnt 5996 3173 Pnt 5996 3219 Pnt 5996 3265 Pnt 5996 3311 Pnt 5996 3403 Pnt 5996 3448 Pnt 5996 3494 Pnt 5996 3724 Pnt 5996 3770 Pnt 5996 3816 Pnt 5996 3954 Pnt 5996 4000 Pnt 5996 4045 Pnt 6009 280 Pnt 6009 326 Pnt 6009 372 Pnt 6009 418 Pnt 6009 464 Pnt 6009 510 Pnt 6009 556 Pnt 6009 601 Pnt 6009 647 Pnt 6009 693 Pnt 6009 739 Pnt 6009 785 Pnt 6009 831 Pnt 6009 877 Pnt 6009 923 Pnt 6009 969 Pnt 6009 1015 Pnt 6009 1061 Pnt 6009 1107 Pnt 6009 1152 Pnt 6009 1198 Pnt 6009 1244 Pnt 6009 1290 Pnt 6009 1336 Pnt 6009 1382 Pnt 6009 1428 Pnt 6009 1474 Pnt 6009 1520 Pnt 6009 1566 Pnt 6009 1612 Pnt 6009 1658 Pnt 6009 1704 Pnt 6009 1749 Pnt 6009 1795 Pnt 6009 1841 Pnt 6009 1887 Pnt 6009 1933 Pnt 6009 1979 Pnt 6009 2025 Pnt 6009 2071 Pnt 6009 2117 Pnt 6009 2163 Pnt 6009 2209 Pnt 6009 2255 Pnt 6009 3173 Pnt 6009 3219 Pnt 6009 3265 Pnt 6009 3311 Pnt 6009 3357 Pnt 6009 3403 Pnt 6009 3448 Pnt 6009 3494 Pnt 6009 3540 Pnt 6009 3770 Pnt 6009 3816 Pnt 6009 3862 Pnt 6009 4000 Pnt 6009 4045 Pnt 6009 4091 Pnt 6021 280 Pnt 6021 326 Pnt 6021 372 Pnt 6021 418 Pnt 6021 464 Pnt 6021 510 Pnt 6021 556 Pnt 6021 601 Pnt 6021 647 Pnt 6021 693 Pnt 6021 739 Pnt 6021 785 Pnt 6021 831 Pnt 6021 877 Pnt 6021 923 Pnt 6021 969 Pnt 6021 1015 Pnt 6021 1061 Pnt 6021 1107 Pnt 6021 1152 Pnt 6021 1198 Pnt 6021 1244 Pnt 6021 1290 Pnt 6021 1336 Pnt 6021 1382 Pnt 6021 1428 Pnt 6021 1474 Pnt 6021 1520 Pnt 6021 1566 Pnt 6021 1612 Pnt 6021 1658 Pnt 6021 1704 Pnt 6021 1749 Pnt 6021 1795 Pnt 6021 1841 Pnt 6021 1887 Pnt 6021 1933 Pnt 6021 1979 Pnt 6021 2025 Pnt 6021 2071 Pnt 6021 2117 Pnt 6021 2163 Pnt 6021 2209 Pnt 6021 2255 Pnt 6021 2300 Pnt 6021 3173 Pnt 6021 3219 Pnt 6021 3265 Pnt 6021 3311 Pnt 6021 3357 Pnt 6021 3403 Pnt 6021 3448 Pnt 6021 3494 Pnt 6021 3540 Pnt 6021 3770 Pnt 6021 3816 Pnt 6021 3862 Pnt 6021 4000 Pnt 6021 4045 Pnt 6021 4091 Pnt 6034 280 Pnt 6034 326 Pnt 6034 372 Pnt 6034 418 Pnt 6034 464 Pnt 6034 510 Pnt 6034 556 Pnt 6034 601 Pnt 6034 647 Pnt 6034 693 Pnt 6034 739 Pnt 6034 785 Pnt 6034 831 Pnt 6034 877 Pnt 6034 923 Pnt 6034 969 Pnt 6034 1015 Pnt 6034 1061 Pnt 6034 1107 Pnt 6034 1152 Pnt 6034 1198 Pnt 6034 1244 Pnt 6034 1290 Pnt 6034 1336 Pnt 6034 1382 Pnt 6034 1428 Pnt 6034 1474 Pnt 6034 1520 Pnt 6034 1566 Pnt 6034 1612 Pnt 6034 1658 Pnt 6034 1704 Pnt 6034 1749 Pnt 6034 1795 Pnt 6034 1841 Pnt 6034 1887 Pnt 6034 1933 Pnt 6034 1979 Pnt 6034 2025 Pnt 6034 2071 Pnt 6034 2117 Pnt 6034 2163 Pnt 6034 2209 Pnt 6034 2255 Pnt 6034 2300 Pnt 6034 3219 Pnt 6034 3265 Pnt 6034 3311 Pnt 6034 3357 Pnt 6034 3403 Pnt 6034 3448 Pnt 6034 3494 Pnt 6034 3540 Pnt 6034 3586 Pnt 6034 3816 Pnt 6034 3862 Pnt 6034 3908 Pnt 6034 4045 Pnt 6034 4091 Pnt 6034 4137 Pnt 6047 280 Pnt 6047 326 Pnt 6047 372 Pnt 6047 418 Pnt 6047 464 Pnt 6047 510 Pnt 6047 556 Pnt 6047 601 Pnt 6047 647 Pnt 6047 693 Pnt 6047 739 Pnt 6047 785 Pnt 6047 831 Pnt 6047 877 Pnt 6047 923 Pnt 6047 969 Pnt 6047 1015 Pnt 6047 1061 Pnt 6047 1107 Pnt 6047 1152 Pnt 6047 1198 Pnt 6047 1244 Pnt 6047 1290 Pnt 6047 1336 Pnt 6047 1382 Pnt 6047 1428 Pnt 6047 1474 Pnt 6047 1520 Pnt 6047 1566 Pnt 6047 1612 Pnt 6047 1658 Pnt 6047 1704 Pnt 6047 1749 Pnt 6047 1795 Pnt 6047 1841 Pnt 6047 1887 Pnt 6047 1933 Pnt 6047 1979 Pnt 6047 2025 Pnt 6047 2071 Pnt 6047 2117 Pnt 6047 2163 Pnt 6047 2209 Pnt 6047 2255 Pnt 6047 2300 Pnt 6047 2346 Pnt 6047 3265 Pnt 6047 3311 Pnt 6047 3357 Pnt 6047 3403 Pnt 6047 3448 Pnt 6047 3494 Pnt 6047 3540 Pnt 6047 3586 Pnt 6047 3816 Pnt 6047 3862 Pnt 6047 3908 Pnt 6047 4045 Pnt 6047 4091 Pnt 6060 280 Pnt 6060 326 Pnt 6060 372 Pnt 6060 418 Pnt 6060 464 Pnt 6060 510 Pnt 6060 556 Pnt 6060 601 Pnt 6060 647 Pnt 6060 693 Pnt 6060 739 Pnt 6060 785 Pnt 6060 831 Pnt 6060 877 Pnt 6060 923 Pnt 6060 969 Pnt 6060 1015 Pnt 6060 1061 Pnt 6060 1107 Pnt 6060 1152 Pnt 6060 1198 Pnt 6060 1244 Pnt 6060 1290 Pnt 6060 1336 Pnt 6060 1382 Pnt 6060 1428 Pnt 6060 1474 Pnt 6060 1520 Pnt 6060 1566 Pnt 6060 1612 Pnt 6060 1658 Pnt 6060 1704 Pnt 6060 1749 Pnt 6060 1795 Pnt 6060 1841 Pnt 6060 1887 Pnt 6060 1933 Pnt 6060 1979 Pnt 6060 2025 Pnt 6060 2071 Pnt 6060 2117 Pnt 6060 2163 Pnt 6060 2209 Pnt 6060 2255 Pnt 6060 2300 Pnt 6060 2346 Pnt 6060 2392 Pnt 6060 3265 Pnt 6060 3311 Pnt 6060 3357 Pnt 6060 3403 Pnt 6060 3448 Pnt 6060 3494 Pnt 6060 3540 Pnt 6060 3586 Pnt 6060 3632 Pnt 6060 3862 Pnt 6060 3908 Pnt 6060 3954 Pnt 6060 4091 Pnt 6060 4137 Pnt 6060 4183 Pnt 6072 280 Pnt 6072 326 Pnt 6072 372 Pnt 6072 418 Pnt 6072 464 Pnt 6072 510 Pnt 6072 556 Pnt 6072 601 Pnt 6072 647 Pnt 6072 693 Pnt 6072 739 Pnt 6072 785 Pnt 6072 831 Pnt 6072 877 Pnt 6072 923 Pnt 6072 969 Pnt 6072 1015 Pnt 6072 1061 Pnt 6072 1107 Pnt 6072 1152 Pnt 6072 1198 Pnt 6072 1244 Pnt 6072 1290 Pnt 6072 1336 Pnt 6072 1382 Pnt 6072 1428 Pnt 6072 1474 Pnt 6072 1520 Pnt 6072 1566 Pnt 6072 1612 Pnt 6072 1658 Pnt 6072 1704 Pnt 6072 1749 Pnt 6072 1795 Pnt 6072 1841 Pnt 6072 1887 Pnt 6072 1933 Pnt 6072 1979 Pnt 6072 2025 Pnt 6072 2071 Pnt 6072 2117 Pnt 6072 2163 Pnt 6072 2209 Pnt 6072 2255 Pnt 6072 2300 Pnt 6072 2346 Pnt 6072 2392 Pnt 6072 3311 Pnt 6072 3357 Pnt 6072 3403 Pnt 6072 3448 Pnt 6072 3494 Pnt 6072 3540 Pnt 6072 3586 Pnt 6072 3632 Pnt 6072 3862 Pnt 6072 3908 Pnt 6072 3954 Pnt 6072 4091 Pnt 6072 4137 Pnt 6085 280 Pnt 6085 326 Pnt 6085 372 Pnt 6085 418 Pnt 6085 464 Pnt 6085 510 Pnt 6085 556 Pnt 6085 601 Pnt 6085 647 Pnt 6085 693 Pnt 6085 739 Pnt 6085 785 Pnt 6085 831 Pnt 6085 877 Pnt 6085 923 Pnt 6085 969 Pnt 6085 1015 Pnt 6085 1061 Pnt 6085 1107 Pnt 6085 1152 Pnt 6085 1198 Pnt 6085 1244 Pnt 6085 1290 Pnt 6085 1336 Pnt 6085 1382 Pnt 6085 1428 Pnt 6085 1474 Pnt 6085 1520 Pnt 6085 1566 Pnt 6085 1612 Pnt 6085 1658 Pnt 6085 1704 Pnt 6085 1749 Pnt 6085 1795 Pnt 6085 1841 Pnt 6085 1887 Pnt 6085 1933 Pnt 6085 1979 Pnt 6085 2025 Pnt 6085 2071 Pnt 6085 2117 Pnt 6085 2163 Pnt 6085 2209 Pnt 6085 2255 Pnt 6085 2300 Pnt 6085 2346 Pnt 6085 2392 Pnt 6085 2438 Pnt 6085 3311 Pnt 6085 3357 Pnt 6085 3403 Pnt 6085 3448 Pnt 6085 3494 Pnt 6085 3586 Pnt 6085 3632 Pnt 6085 3678 Pnt 6085 3908 Pnt 6085 3954 Pnt 6085 4000 Pnt 6085 4137 Pnt 6085 4183 Pnt 6085 4229 Pnt 6098 280 Pnt 6098 326 Pnt 6098 372 Pnt 6098 418 Pnt 6098 464 Pnt 6098 510 Pnt 6098 556 Pnt 6098 601 Pnt 6098 647 Pnt 6098 693 Pnt 6098 739 Pnt 6098 785 Pnt 6098 831 Pnt 6098 877 Pnt 6098 923 Pnt 6098 969 Pnt 6098 1015 Pnt 6098 1061 Pnt 6098 1107 Pnt 6098 1152 Pnt 6098 1198 Pnt 6098 1244 Pnt 6098 1290 Pnt 6098 1336 Pnt 6098 1382 Pnt 6098 1428 Pnt 6098 1474 Pnt 6098 1520 Pnt 6098 1566 Pnt 6098 1612 Pnt 6098 1658 Pnt 6098 1704 Pnt 6098 1749 Pnt 6098 1795 Pnt 6098 1841 Pnt 6098 1887 Pnt 6098 1933 Pnt 6098 2025 Pnt 6098 2071 Pnt 6098 2117 Pnt 6098 2163 Pnt 6098 2209 Pnt 6098 2255 Pnt 6098 2300 Pnt 6098 2346 Pnt 6098 2392 Pnt 6098 2438 Pnt 6098 2484 Pnt 6098 3357 Pnt 6098 3403 Pnt 6098 3448 Pnt 6098 3494 Pnt 6098 3540 Pnt 6098 3586 Pnt 6098 3632 Pnt 6098 3678 Pnt 6098 3908 Pnt 6098 3954 Pnt 6098 4000 Pnt 6098 4137 Pnt 6098 4183 Pnt 6110 280 Pnt 6110 326 Pnt 6110 372 Pnt 6110 418 Pnt 6110 464 Pnt 6110 510 Pnt 6110 556 Pnt 6110 601 Pnt 6110 647 Pnt 6110 693 Pnt 6110 739 Pnt 6110 785 Pnt 6110 831 Pnt 6110 877 Pnt 6110 923 Pnt 6110 969 Pnt 6110 1015 Pnt 6110 1061 Pnt 6110 1107 Pnt 6110 1152 Pnt 6110 1198 Pnt 6110 1244 Pnt 6110 1290 Pnt 6110 1336 Pnt 6110 1382 Pnt 6110 1428 Pnt 6110 1474 Pnt 6110 1520 Pnt 6110 1566 Pnt 6110 1612 Pnt 6110 1658 Pnt 6110 1704 Pnt 6110 1749 Pnt 6110 1795 Pnt 6110 1841 Pnt 6110 1887 Pnt 6110 1933 Pnt 6110 1979 Pnt 6110 2071 Pnt 6110 2117 Pnt 6110 2163 Pnt 6110 2209 Pnt 6110 2255 Pnt 6110 2300 Pnt 6110 2346 Pnt 6110 2392 Pnt 6110 2438 Pnt 6110 2484 Pnt 6110 3357 Pnt 6110 3403 Pnt 6110 3448 Pnt 6110 3494 Pnt 6110 3540 Pnt 6110 3632 Pnt 6110 3678 Pnt 6110 3724 Pnt 6110 3954 Pnt 6110 4000 Pnt 6110 4045 Pnt 6110 4183 Pnt 6110 4229 Pnt 6110 4275 Pnt 6123 280 Pnt 6123 326 Pnt 6123 372 Pnt 6123 418 Pnt 6123 464 Pnt 6123 510 Pnt 6123 556 Pnt 6123 601 Pnt 6123 647 Pnt 6123 693 Pnt 6123 739 Pnt 6123 785 Pnt 6123 831 Pnt 6123 877 Pnt 6123 923 Pnt 6123 969 Pnt 6123 1015 Pnt 6123 1061 Pnt 6123 1107 Pnt 6123 1152 Pnt 6123 1198 Pnt 6123 1244 Pnt 6123 1290 Pnt 6123 1336 Pnt 6123 1382 Pnt 6123 1428 Pnt 6123 1474 Pnt 6123 1520 Pnt 6123 1566 Pnt 6123 1612 Pnt 6123 1658 Pnt 6123 1704 Pnt 6123 1749 Pnt 6123 1795 Pnt 6123 1841 Pnt 6123 1887 Pnt 6123 1933 Pnt 6123 1979 Pnt 6123 2025 Pnt 6123 2117 Pnt 6123 2163 Pnt 6123 2209 Pnt 6123 2255 Pnt 6123 2300 Pnt 6123 2346 Pnt 6123 2392 Pnt 6123 2438 Pnt 6123 2484 Pnt 6123 2530 Pnt 6123 3403 Pnt 6123 3448 Pnt 6123 3494 Pnt 6123 3540 Pnt 6123 3586 Pnt 6123 3632 Pnt 6123 3678 Pnt 6123 3724 Pnt 6123 3954 Pnt 6123 4000 Pnt 6123 4045 Pnt 6123 4183 Pnt 6123 4229 Pnt 6123 4275 Pnt 6136 280 Pnt 6136 326 Pnt 6136 372 Pnt 6136 418 Pnt 6136 464 Pnt 6136 510 Pnt 6136 556 Pnt 6136 601 Pnt 6136 647 Pnt 6136 693 Pnt 6136 739 Pnt 6136 785 Pnt 6136 831 Pnt 6136 877 Pnt 6136 923 Pnt 6136 969 Pnt 6136 1015 Pnt 6136 1061 Pnt 6136 1107 Pnt 6136 1152 Pnt 6136 1198 Pnt 6136 1244 Pnt 6136 1290 Pnt 6136 1336 Pnt 6136 1382 Pnt 6136 1428 Pnt 6136 1474 Pnt 6136 1520 Pnt 6136 1566 Pnt 6136 1612 Pnt 6136 1658 Pnt 6136 1704 Pnt 6136 1749 Pnt 6136 1795 Pnt 6136 1841 Pnt 6136 1887 Pnt 6136 1933 Pnt 6136 1979 Pnt 6136 2025 Pnt 6136 2071 Pnt 6136 2163 Pnt 6136 2209 Pnt 6136 2255 Pnt 6136 2300 Pnt 6136 2346 Pnt 6136 2392 Pnt 6136 2438 Pnt 6136 2484 Pnt 6136 2530 Pnt 6136 3448 Pnt 6136 3494 Pnt 6136 3540 Pnt 6136 3586 Pnt 6136 3678 Pnt 6136 3724 Pnt 6136 3770 Pnt 6136 4000 Pnt 6136 4045 Pnt 6136 4091 Pnt 6136 4229 Pnt 6136 4275 Pnt 6136 4321 Pnt 6149 280 Pnt 6149 326 Pnt 6149 372 Pnt 6149 418 Pnt 6149 464 Pnt 6149 510 Pnt 6149 556 Pnt 6149 601 Pnt 6149 647 Pnt 6149 693 Pnt 6149 739 Pnt 6149 785 Pnt 6149 831 Pnt 6149 877 Pnt 6149 923 Pnt 6149 969 Pnt 6149 1015 Pnt 6149 1061 Pnt 6149 1107 Pnt 6149 1152 Pnt 6149 1198 Pnt 6149 1244 Pnt 6149 1290 Pnt 6149 1336 Pnt 6149 1382 Pnt 6149 1428 Pnt 6149 1474 Pnt 6149 1520 Pnt 6149 1566 Pnt 6149 1612 Pnt 6149 1658 Pnt 6149 1704 Pnt 6149 1749 Pnt 6149 1795 Pnt 6149 1841 Pnt 6149 1887 Pnt 6149 1933 Pnt 6149 1979 Pnt 6149 2025 Pnt 6149 2071 Pnt 6149 2117 Pnt 6149 2163 Pnt 6149 2209 Pnt 6149 2255 Pnt 6149 2300 Pnt 6149 2346 Pnt 6149 2392 Pnt 6149 2438 Pnt 6149 2484 Pnt 6149 2530 Pnt 6149 2576 Pnt 6149 3448 Pnt 6149 3494 Pnt 6149 3540 Pnt 6149 3586 Pnt 6149 3632 Pnt 6149 3678 Pnt 6149 3724 Pnt 6149 3770 Pnt 6149 4000 Pnt 6149 4045 Pnt 6149 4091 Pnt 6149 4229 Pnt 6149 4275 Pnt 6149 4321 Pnt 6161 280 Pnt 6161 326 Pnt 6161 372 Pnt 6161 418 Pnt 6161 464 Pnt 6161 510 Pnt 6161 556 Pnt 6161 601 Pnt 6161 647 Pnt 6161 693 Pnt 6161 739 Pnt 6161 785 Pnt 6161 831 Pnt 6161 877 Pnt 6161 923 Pnt 6161 969 Pnt 6161 1015 Pnt 6161 1061 Pnt 6161 1107 Pnt 6161 1152 Pnt 6161 1198 Pnt 6161 1244 Pnt 6161 1290 Pnt 6161 1336 Pnt 6161 1382 Pnt 6161 1428 Pnt 6161 1474 Pnt 6161 1520 Pnt 6161 1566 Pnt 6161 1612 Pnt 6161 1658 Pnt 6161 1704 Pnt 6161 1749 Pnt 6161 1795 Pnt 6161 1841 Pnt 6161 1887 Pnt 6161 1933 Pnt 6161 1979 Pnt 6161 2025 Pnt 6161 2071 Pnt 6161 2117 Pnt 6161 2209 Pnt 6161 2255 Pnt 6161 2300 Pnt 6161 2346 Pnt 6161 2392 Pnt 6161 2438 Pnt 6161 2484 Pnt 6161 2530 Pnt 6161 2576 Pnt 6161 2622 Pnt 6161 3494 Pnt 6161 3540 Pnt 6161 3586 Pnt 6161 3632 Pnt 6161 3724 Pnt 6161 3770 Pnt 6161 3816 Pnt 6161 4045 Pnt 6161 4091 Pnt 6161 4137 Pnt 6161 4275 Pnt 6161 4321 Pnt 6161 4367 Pnt 6174 280 Pnt 6174 326 Pnt 6174 372 Pnt 6174 418 Pnt 6174 464 Pnt 6174 510 Pnt 6174 556 Pnt 6174 601 Pnt 6174 647 Pnt 6174 693 Pnt 6174 739 Pnt 6174 785 Pnt 6174 831 Pnt 6174 877 Pnt 6174 923 Pnt 6174 969 Pnt 6174 1015 Pnt 6174 1061 Pnt 6174 1107 Pnt 6174 1152 Pnt 6174 1198 Pnt 6174 1244 Pnt 6174 1290 Pnt 6174 1336 Pnt 6174 1382 Pnt 6174 1428 Pnt 6174 1474 Pnt 6174 1520 Pnt 6174 1566 Pnt 6174 1612 Pnt 6174 1658 Pnt 6174 1704 Pnt 6174 1749 Pnt 6174 1795 Pnt 6174 1841 Pnt 6174 1887 Pnt 6174 1933 Pnt 6174 1979 Pnt 6174 2025 Pnt 6174 2071 Pnt 6174 2117 Pnt 6174 2163 Pnt 6174 2255 Pnt 6174 2300 Pnt 6174 2346 Pnt 6174 2392 Pnt 6174 2438 Pnt 6174 2484 Pnt 6174 2530 Pnt 6174 2576 Pnt 6174 2622 Pnt 6174 3494 Pnt 6174 3540 Pnt 6174 3586 Pnt 6174 3632 Pnt 6174 3678 Pnt 6174 3724 Pnt 6174 3770 Pnt 6174 3816 Pnt 6174 4045 Pnt 6174 4091 Pnt 6174 4137 Pnt 6174 4275 Pnt 6174 4321 Pnt 6174 4367 Pnt 6187 280 Pnt 6187 326 Pnt 6187 372 Pnt 6187 418 Pnt 6187 464 Pnt 6187 510 Pnt 6187 556 Pnt 6187 601 Pnt 6187 647 Pnt 6187 693 Pnt 6187 739 Pnt 6187 785 Pnt 6187 831 Pnt 6187 877 Pnt 6187 923 Pnt 6187 969 Pnt 6187 1015 Pnt 6187 1061 Pnt 6187 1107 Pnt 6187 1152 Pnt 6187 1198 Pnt 6187 1244 Pnt 6187 1290 Pnt 6187 1336 Pnt 6187 1382 Pnt 6187 1428 Pnt 6187 1474 Pnt 6187 1520 Pnt 6187 1566 Pnt 6187 1612 Pnt 6187 1658 Pnt 6187 1704 Pnt 6187 1749 Pnt 6187 1795 Pnt 6187 1841 Pnt 6187 1887 Pnt 6187 1933 Pnt 6187 1979 Pnt 6187 2025 Pnt 6187 2071 Pnt 6187 2117 Pnt 6187 2163 Pnt 6187 2209 Pnt 6187 2300 Pnt 6187 2346 Pnt 6187 2392 Pnt 6187 2438 Pnt 6187 2484 Pnt 6187 2530 Pnt 6187 2576 Pnt 6187 2622 Pnt 6187 2668 Pnt 6187 3540 Pnt 6187 3586 Pnt 6187 3632 Pnt 6187 3678 Pnt 6187 3770 Pnt 6187 3816 Pnt 6187 3862 Pnt 6187 4091 Pnt 6187 4137 Pnt 6187 4183 Pnt 6187 4321 Pnt 6187 4367 Pnt 6187 4413 Pnt 6199 280 Pnt 6199 326 Pnt 6199 372 Pnt 6199 418 Pnt 6199 464 Pnt 6199 510 Pnt 6199 556 Pnt 6199 601 Pnt 6199 647 Pnt 6199 693 Pnt 6199 739 Pnt 6199 785 Pnt 6199 831 Pnt 6199 877 Pnt 6199 923 Pnt 6199 969 Pnt 6199 1015 Pnt 6199 1061 Pnt 6199 1107 Pnt 6199 1152 Pnt 6199 1198 Pnt 6199 1244 Pnt 6199 1290 Pnt 6199 1336 Pnt 6199 1382 Pnt 6199 1428 Pnt 6199 1474 Pnt 6199 1520 Pnt 6199 1566 Pnt 6199 1612 Pnt 6199 1658 Pnt 6199 1704 Pnt 6199 1749 Pnt 6199 1795 Pnt 6199 1841 Pnt 6199 1887 Pnt 6199 1933 Pnt 6199 1979 Pnt 6199 2025 Pnt 6199 2071 Pnt 6199 2117 Pnt 6199 2163 Pnt 6199 2209 Pnt 6199 2255 Pnt 6199 2346 Pnt 6199 2392 Pnt 6199 2438 Pnt 6199 2484 Pnt 6199 2530 Pnt 6199 2576 Pnt 6199 2622 Pnt 6199 2668 Pnt 6199 3540 Pnt 6199 3586 Pnt 6199 3632 Pnt 6199 3678 Pnt 6199 3724 Pnt 6199 3770 Pnt 6199 3816 Pnt 6199 3862 Pnt 6199 4091 Pnt 6199 4137 Pnt 6199 4183 Pnt 6199 4321 Pnt 6199 4367 Pnt 6199 4413 Pnt 6212 280 Pnt 6212 326 Pnt 6212 372 Pnt 6212 418 Pnt 6212 464 Pnt 6212 510 Pnt 6212 556 Pnt 6212 601 Pnt 6212 647 Pnt 6212 693 Pnt 6212 739 Pnt 6212 785 Pnt 6212 831 Pnt 6212 877 Pnt 6212 923 Pnt 6212 969 Pnt 6212 1015 Pnt 6212 1061 Pnt 6212 1107 Pnt 6212 1152 Pnt 6212 1198 Pnt 6212 1244 Pnt 6212 1290 Pnt 6212 1336 Pnt 6212 1382 Pnt 6212 1428 Pnt 6212 1474 Pnt 6212 1520 Pnt 6212 1566 Pnt 6212 1612 Pnt 6212 1658 Pnt 6212 1704 Pnt 6212 1749 Pnt 6212 1795 Pnt 6212 1841 Pnt 6212 1887 Pnt 6212 1933 Pnt 6212 1979 Pnt 6212 2025 Pnt 6212 2071 Pnt 6212 2117 Pnt 6212 2163 Pnt 6212 2209 Pnt 6212 2255 Pnt 6212 2346 Pnt 6212 2392 Pnt 6212 2438 Pnt 6212 2484 Pnt 6212 2530 Pnt 6212 2576 Pnt 6212 2622 Pnt 6212 2668 Pnt 6212 2714 Pnt 6212 3586 Pnt 6212 3632 Pnt 6212 3678 Pnt 6212 3724 Pnt 6212 3816 Pnt 6212 3862 Pnt 6212 3908 Pnt 6212 4137 Pnt 6212 4183 Pnt 6212 4229 Pnt 6212 4367 Pnt 6212 4413 Pnt 6212 4459 Pnt 6225 280 Pnt 6225 326 Pnt 6225 372 Pnt 6225 418 Pnt 6225 464 Pnt 6225 510 Pnt 6225 556 Pnt 6225 601 Pnt 6225 647 Pnt 6225 693 Pnt 6225 739 Pnt 6225 785 Pnt 6225 831 Pnt 6225 877 Pnt 6225 923 Pnt 6225 969 Pnt 6225 1015 Pnt 6225 1061 Pnt 6225 1107 Pnt 6225 1152 Pnt 6225 1198 Pnt 6225 1244 Pnt 6225 1290 Pnt 6225 1336 Pnt 6225 1382 Pnt 6225 1428 Pnt 6225 1474 Pnt 6225 1520 Pnt 6225 1566 Pnt 6225 1612 Pnt 6225 1658 Pnt 6225 1704 Pnt 6225 1749 Pnt 6225 1795 Pnt 6225 1841 Pnt 6225 1887 Pnt 6225 1933 Pnt 6225 1979 Pnt 6225 2025 Pnt 6225 2071 Pnt 6225 2117 Pnt 6225 2163 Pnt 6225 2209 Pnt 6225 2255 Pnt 6225 2300 Pnt 6225 2392 Pnt 6225 2438 Pnt 6225 2484 Pnt 6225 2530 Pnt 6225 2576 Pnt 6225 2622 Pnt 6225 2668 Pnt 6225 2714 Pnt 6225 2760 Pnt 6225 3586 Pnt 6225 3632 Pnt 6225 3678 Pnt 6225 3724 Pnt 6225 3770 Pnt 6225 3816 Pnt 6225 3862 Pnt 6225 3908 Pnt 6225 4137 Pnt 6225 4183 Pnt 6225 4229 Pnt 6225 4367 Pnt 6225 4413 Pnt 6225 4459 Pnt 6237 280 Pnt 6237 326 Pnt 6237 372 Pnt 6237 418 Pnt 6237 464 Pnt 6237 510 Pnt 6237 556 Pnt 6237 601 Pnt 6237 647 Pnt 6237 693 Pnt 6237 739 Pnt 6237 785 Pnt 6237 831 Pnt 6237 877 Pnt 6237 923 Pnt 6237 969 Pnt 6237 1015 Pnt 6237 1061 Pnt 6237 1107 Pnt 6237 1152 Pnt 6237 1198 Pnt 6237 1244 Pnt 6237 1290 Pnt 6237 1336 Pnt 6237 1382 Pnt 6237 1428 Pnt 6237 1474 Pnt 6237 1520 Pnt 6237 1566 Pnt 6237 1612 Pnt 6237 1658 Pnt 6237 1704 Pnt 6237 1749 Pnt 6237 1795 Pnt 6237 1841 Pnt 6237 1887 Pnt 6237 1933 Pnt 6237 1979 Pnt 6237 2025 Pnt 6237 2071 Pnt 6237 2117 Pnt 6237 2163 Pnt 6237 2209 Pnt 6237 2255 Pnt 6237 2300 Pnt 6237 2346 Pnt 6237 2438 Pnt 6237 2484 Pnt 6237 2530 Pnt 6237 2576 Pnt 6237 2622 Pnt 6237 2668 Pnt 6237 2714 Pnt 6237 2760 Pnt 6237 3632 Pnt 6237 3678 Pnt 6237 3724 Pnt 6237 3770 Pnt 6237 3862 Pnt 6237 3908 Pnt 6237 3954 Pnt 6237 4183 Pnt 6237 4229 Pnt 6237 4275 Pnt 6237 4413 Pnt 6237 4459 Pnt 6250 280 Pnt 6250 326 Pnt 6250 372 Pnt 6250 418 Pnt 6250 464 Pnt 6250 510 Pnt 6250 556 Pnt 6250 601 Pnt 6250 647 Pnt 6250 693 Pnt 6250 739 Pnt 6250 785 Pnt 6250 831 Pnt 6250 877 Pnt 6250 923 Pnt 6250 969 Pnt 6250 1015 Pnt 6250 1061 Pnt 6250 1107 Pnt 6250 1152 Pnt 6250 1198 Pnt 6250 1244 Pnt 6250 1290 Pnt 6250 1336 Pnt 6250 1382 Pnt 6250 1428 Pnt 6250 1474 Pnt 6250 1520 Pnt 6250 1566 Pnt 6250 1612 Pnt 6250 1658 Pnt 6250 1704 Pnt 6250 1749 Pnt 6250 1795 Pnt 6250 1841 Pnt 6250 1887 Pnt 6250 1933 Pnt 6250 1979 Pnt 6250 2025 Pnt 6250 2071 Pnt 6250 2117 Pnt 6250 2163 Pnt 6250 2209 Pnt 6250 2255 Pnt 6250 2300 Pnt 6250 2346 Pnt 6250 2392 Pnt 6250 2438 Pnt 6250 2484 Pnt 6250 2530 Pnt 6250 2576 Pnt 6250 2622 Pnt 6250 2668 Pnt 6250 2714 Pnt 6250 2760 Pnt 6250 2806 Pnt 6250 3632 Pnt 6250 3678 Pnt 6250 3724 Pnt 6250 3770 Pnt 6250 3816 Pnt 6250 3862 Pnt 6250 3908 Pnt 6250 3954 Pnt 6250 4183 Pnt 6250 4229 Pnt 6250 4275 Pnt 6250 4413 Pnt 6250 4459 Pnt 6250 4505 Pnt 6263 280 Pnt 6263 326 Pnt 6263 372 Pnt 6263 418 Pnt 6263 464 Pnt 6263 510 Pnt 6263 556 Pnt 6263 601 Pnt 6263 647 Pnt 6263 693 Pnt 6263 739 Pnt 6263 785 Pnt 6263 831 Pnt 6263 877 Pnt 6263 923 Pnt 6263 969 Pnt 6263 1015 Pnt 6263 1061 Pnt 6263 1107 Pnt 6263 1152 Pnt 6263 1198 Pnt 6263 1244 Pnt 6263 1290 Pnt 6263 1336 Pnt 6263 1382 Pnt 6263 1428 Pnt 6263 1474 Pnt 6263 1520 Pnt 6263 1566 Pnt 6263 1612 Pnt 6263 1658 Pnt 6263 1704 Pnt 6263 1749 Pnt 6263 1795 Pnt 6263 1841 Pnt 6263 1887 Pnt 6263 1933 Pnt 6263 1979 Pnt 6263 2025 Pnt 6263 2071 Pnt 6263 2117 Pnt 6263 2163 Pnt 6263 2209 Pnt 6263 2255 Pnt 6263 2300 Pnt 6263 2346 Pnt 6263 2392 Pnt 6263 2484 Pnt 6263 2530 Pnt 6263 2576 Pnt 6263 2622 Pnt 6263 2668 Pnt 6263 2714 Pnt 6263 2760 Pnt 6263 2806 Pnt 6263 3678 Pnt 6263 3724 Pnt 6263 3770 Pnt 6263 3816 Pnt 6263 3908 Pnt 6263 3954 Pnt 6263 4000 Pnt 6263 4229 Pnt 6263 4275 Pnt 6263 4321 Pnt 6263 4459 Pnt 6263 4505 Pnt 6276 280 Pnt 6276 326 Pnt 6276 372 Pnt 6276 418 Pnt 6276 464 Pnt 6276 510 Pnt 6276 556 Pnt 6276 601 Pnt 6276 647 Pnt 6276 693 Pnt 6276 739 Pnt 6276 785 Pnt 6276 831 Pnt 6276 877 Pnt 6276 923 Pnt 6276 969 Pnt 6276 1015 Pnt 6276 1061 Pnt 6276 1107 Pnt 6276 1152 Pnt 6276 1198 Pnt 6276 1244 Pnt 6276 1290 Pnt 6276 1336 Pnt 6276 1382 Pnt 6276 1428 Pnt 6276 1474 Pnt 6276 1520 Pnt 6276 1566 Pnt 6276 1612 Pnt 6276 1658 Pnt 6276 1704 Pnt 6276 1749 Pnt 6276 1795 Pnt 6276 1841 Pnt 6276 1887 Pnt 6276 1933 Pnt 6276 1979 Pnt 6276 2025 Pnt 6276 2071 Pnt 6276 2117 Pnt 6276 2163 Pnt 6276 2209 Pnt 6276 2255 Pnt 6276 2300 Pnt 6276 2346 Pnt 6276 2392 Pnt 6276 2438 Pnt 6276 2530 Pnt 6276 2576 Pnt 6276 2622 Pnt 6276 2668 Pnt 6276 2714 Pnt 6276 2760 Pnt 6276 2806 Pnt 6276 2852 Pnt 6276 3678 Pnt 6276 3724 Pnt 6276 3770 Pnt 6276 3816 Pnt 6276 3862 Pnt 6276 3908 Pnt 6276 3954 Pnt 6276 4000 Pnt 6276 4229 Pnt 6276 4275 Pnt 6276 4321 Pnt 6276 4459 Pnt 6276 4505 Pnt 6276 4551 Pnt 6288 280 Pnt 6288 326 Pnt 6288 372 Pnt 6288 418 Pnt 6288 464 Pnt 6288 510 Pnt 6288 556 Pnt 6288 601 Pnt 6288 647 Pnt 6288 693 Pnt 6288 739 Pnt 6288 785 Pnt 6288 831 Pnt 6288 877 Pnt 6288 923 Pnt 6288 969 Pnt 6288 1015 Pnt 6288 1061 Pnt 6288 1107 Pnt 6288 1152 Pnt 6288 1198 Pnt 6288 1244 Pnt 6288 1290 Pnt 6288 1336 Pnt 6288 1382 Pnt 6288 1428 Pnt 6288 1474 Pnt 6288 1520 Pnt 6288 1566 Pnt 6288 1612 Pnt 6288 1658 Pnt 6288 1704 Pnt 6288 1749 Pnt 6288 1795 Pnt 6288 1841 Pnt 6288 1887 Pnt 6288 1933 Pnt 6288 1979 Pnt 6288 2025 Pnt 6288 2071 Pnt 6288 2117 Pnt 6288 2163 Pnt 6288 2209 Pnt 6288 2255 Pnt 6288 2300 Pnt 6288 2346 Pnt 6288 2392 Pnt 6288 2438 Pnt 6288 2484 Pnt 6288 2576 Pnt 6288 2622 Pnt 6288 2668 Pnt 6288 2714 Pnt 6288 2760 Pnt 6288 2806 Pnt 6288 2852 Pnt 6288 2897 Pnt 6288 3724 Pnt 6288 3770 Pnt 6288 3816 Pnt 6288 3862 Pnt 6288 3954 Pnt 6288 4000 Pnt 6288 4045 Pnt 6288 4275 Pnt 6288 4321 Pnt 6288 4367 Pnt 6288 4505 Pnt 6288 4551 Pnt 6301 280 Pnt 6301 326 Pnt 6301 372 Pnt 6301 418 Pnt 6301 464 Pnt 6301 510 Pnt 6301 556 Pnt 6301 601 Pnt 6301 647 Pnt 6301 693 Pnt 6301 739 Pnt 6301 785 Pnt 6301 831 Pnt 6301 877 Pnt 6301 923 Pnt 6301 969 Pnt 6301 1015 Pnt 6301 1061 Pnt 6301 1107 Pnt 6301 1152 Pnt 6301 1198 Pnt 6301 1244 Pnt 6301 1290 Pnt 6301 1336 Pnt 6301 1382 Pnt 6301 1428 Pnt 6301 1474 Pnt 6301 1520 Pnt 6301 1566 Pnt 6301 1612 Pnt 6301 1658 Pnt 6301 1704 Pnt 6301 1749 Pnt 6301 1795 Pnt 6301 1841 Pnt 6301 1887 Pnt 6301 1933 Pnt 6301 1979 Pnt 6301 2025 Pnt 6301 2071 Pnt 6301 2117 Pnt 6301 2163 Pnt 6301 2209 Pnt 6301 2255 Pnt 6301 2300 Pnt 6301 2346 Pnt 6301 2392 Pnt 6301 2438 Pnt 6301 2484 Pnt 6301 2576 Pnt 6301 2622 Pnt 6301 2668 Pnt 6301 2714 Pnt 6301 2760 Pnt 6301 2806 Pnt 6301 2852 Pnt 6301 2897 Pnt 6301 3724 Pnt 6301 3770 Pnt 6301 3816 Pnt 6301 3862 Pnt 6301 3908 Pnt 6301 3954 Pnt 6301 4000 Pnt 6301 4045 Pnt 6301 4275 Pnt 6301 4321 Pnt 6301 4367 Pnt 6301 4505 Pnt 6301 4551 Pnt 6301 4596 Pnt 6314 280 Pnt 6314 326 Pnt 6314 372 Pnt 6314 418 Pnt 6314 464 Pnt 6314 510 Pnt 6314 556 Pnt 6314 601 Pnt 6314 647 Pnt 6314 693 Pnt 6314 739 Pnt 6314 785 Pnt 6314 831 Pnt 6314 877 Pnt 6314 923 Pnt 6314 969 Pnt 6314 1015 Pnt 6314 1061 Pnt 6314 1107 Pnt 6314 1152 Pnt 6314 1198 Pnt 6314 1244 Pnt 6314 1290 Pnt 6314 1336 Pnt 6314 1382 Pnt 6314 1428 Pnt 6314 1474 Pnt 6314 1520 Pnt 6314 1566 Pnt 6314 1612 Pnt 6314 1658 Pnt 6314 1704 Pnt 6314 1749 Pnt 6314 1795 Pnt 6314 1841 Pnt 6314 1887 Pnt 6314 1933 Pnt 6314 1979 Pnt 6314 2025 Pnt 6314 2071 Pnt 6314 2117 Pnt 6314 2163 Pnt 6314 2209 Pnt 6314 2255 Pnt 6314 2300 Pnt 6314 2346 Pnt 6314 2392 Pnt 6314 2438 Pnt 6314 2484 Pnt 6314 2530 Pnt 6314 2622 Pnt 6314 2668 Pnt 6314 2714 Pnt 6314 2760 Pnt 6314 2806 Pnt 6314 2852 Pnt 6314 2897 Pnt 6314 2943 Pnt 6314 3770 Pnt 6314 3816 Pnt 6314 3862 Pnt 6314 3908 Pnt 6314 4000 Pnt 6314 4045 Pnt 6314 4091 Pnt 6314 4321 Pnt 6314 4367 Pnt 6314 4413 Pnt 6314 4551 Pnt 6314 4596 Pnt 6326 280 Pnt 6326 326 Pnt 6326 372 Pnt 6326 418 Pnt 6326 464 Pnt 6326 510 Pnt 6326 556 Pnt 6326 601 Pnt 6326 647 Pnt 6326 693 Pnt 6326 739 Pnt 6326 785 Pnt 6326 831 Pnt 6326 877 Pnt 6326 923 Pnt 6326 969 Pnt 6326 1015 Pnt 6326 1061 Pnt 6326 1107 Pnt 6326 1152 Pnt 6326 1198 Pnt 6326 1244 Pnt 6326 1290 Pnt 6326 1336 Pnt 6326 1382 Pnt 6326 1428 Pnt 6326 1474 Pnt 6326 1520 Pnt 6326 1566 Pnt 6326 1612 Pnt 6326 1658 Pnt 6326 1704 Pnt 6326 1749 Pnt 6326 1795 Pnt 6326 1841 Pnt 6326 1887 Pnt 6326 1933 Pnt 6326 1979 Pnt 6326 2025 Pnt 6326 2071 Pnt 6326 2117 Pnt 6326 2163 Pnt 6326 2209 Pnt 6326 2255 Pnt 6326 2300 Pnt 6326 2346 Pnt 6326 2392 Pnt 6326 2438 Pnt 6326 2484 Pnt 6326 2530 Pnt 6326 2622 Pnt 6326 2668 Pnt 6326 2714 Pnt 6326 2760 Pnt 6326 2806 Pnt 6326 2852 Pnt 6326 2897 Pnt 6326 2943 Pnt 6326 3770 Pnt 6326 3816 Pnt 6326 3862 Pnt 6326 3908 Pnt 6326 3954 Pnt 6326 4000 Pnt 6326 4045 Pnt 6326 4091 Pnt 6326 4321 Pnt 6326 4367 Pnt 6326 4413 Pnt 6326 4551 Pnt 6326 4596 Pnt 6326 4642 Pnt 6339 280 Pnt 6339 326 Pnt 6339 372 Pnt 6339 418 Pnt 6339 464 Pnt 6339 510 Pnt 6339 556 Pnt 6339 601 Pnt 6339 647 Pnt 6339 693 Pnt 6339 739 Pnt 6339 785 Pnt 6339 831 Pnt 6339 877 Pnt 6339 923 Pnt 6339 969 Pnt 6339 1015 Pnt 6339 1061 Pnt 6339 1107 Pnt 6339 1152 Pnt 6339 1198 Pnt 6339 1244 Pnt 6339 1290 Pnt 6339 1336 Pnt 6339 1382 Pnt 6339 1428 Pnt 6339 1474 Pnt 6339 1520 Pnt 6339 1566 Pnt 6339 1612 Pnt 6339 1658 Pnt 6339 1704 Pnt 6339 1749 Pnt 6339 1795 Pnt 6339 1841 Pnt 6339 1887 Pnt 6339 1933 Pnt 6339 1979 Pnt 6339 2025 Pnt 6339 2071 Pnt 6339 2117 Pnt 6339 2163 Pnt 6339 2209 Pnt 6339 2255 Pnt 6339 2300 Pnt 6339 2346 Pnt 6339 2392 Pnt 6339 2438 Pnt 6339 2484 Pnt 6339 2530 Pnt 6339 2576 Pnt 6339 2668 Pnt 6339 2714 Pnt 6339 2760 Pnt 6339 2806 Pnt 6339 2852 Pnt 6339 2897 Pnt 6339 2943 Pnt 6339 2989 Pnt 6339 3816 Pnt 6339 3862 Pnt 6339 3908 Pnt 6339 3954 Pnt 6339 4045 Pnt 6339 4091 Pnt 6339 4137 Pnt 6339 4367 Pnt 6339 4413 Pnt 6339 4596 Pnt 6339 4642 Pnt 6352 280 Pnt 6352 326 Pnt 6352 372 Pnt 6352 418 Pnt 6352 464 Pnt 6352 510 Pnt 6352 556 Pnt 6352 601 Pnt 6352 647 Pnt 6352 693 Pnt 6352 739 Pnt 6352 785 Pnt 6352 831 Pnt 6352 877 Pnt 6352 923 Pnt 6352 969 Pnt 6352 1015 Pnt 6352 1061 Pnt 6352 1107 Pnt 6352 1152 Pnt 6352 1198 Pnt 6352 1244 Pnt 6352 1290 Pnt 6352 1336 Pnt 6352 1382 Pnt 6352 1428 Pnt 6352 1474 Pnt 6352 1520 Pnt 6352 1566 Pnt 6352 1612 Pnt 6352 1658 Pnt 6352 1704 Pnt 6352 1749 Pnt 6352 1795 Pnt 6352 1841 Pnt 6352 1887 Pnt 6352 1933 Pnt 6352 1979 Pnt 6352 2025 Pnt 6352 2117 Pnt 6352 2163 Pnt 6352 2209 Pnt 6352 2255 Pnt 6352 2300 Pnt 6352 2346 Pnt 6352 2392 Pnt 6352 2438 Pnt 6352 2484 Pnt 6352 2530 Pnt 6352 2576 Pnt 6352 2622 Pnt 6352 2714 Pnt 6352 2760 Pnt 6352 2806 Pnt 6352 2852 Pnt 6352 2897 Pnt 6352 2943 Pnt 6352 2989 Pnt 6352 3816 Pnt 6352 3862 Pnt 6352 3908 Pnt 6352 3954 Pnt 6352 4000 Pnt 6352 4045 Pnt 6352 4091 Pnt 6352 4137 Pnt 6352 4367 Pnt 6352 4413 Pnt 6352 4459 Pnt 6352 4596 Pnt 6352 4642 Pnt 6352 4688 Pnt 6365 280 Pnt 6365 326 Pnt 6365 372 Pnt 6365 418 Pnt 6365 464 Pnt 6365 510 Pnt 6365 556 Pnt 6365 601 Pnt 6365 647 Pnt 6365 693 Pnt 6365 739 Pnt 6365 785 Pnt 6365 831 Pnt 6365 877 Pnt 6365 923 Pnt 6365 969 Pnt 6365 1015 Pnt 6365 1061 Pnt 6365 1107 Pnt 6365 1152 Pnt 6365 1198 Pnt 6365 1244 Pnt 6365 1290 Pnt 6365 1336 Pnt 6365 1382 Pnt 6365 1428 Pnt 6365 1474 Pnt 6365 1520 Pnt 6365 1566 Pnt 6365 1612 Pnt 6365 1658 Pnt 6365 1704 Pnt 6365 1749 Pnt 6365 1795 Pnt 6365 1841 Pnt 6365 1887 Pnt 6365 1933 Pnt 6365 1979 Pnt 6365 2025 Pnt 6365 2071 Pnt 6365 2163 Pnt 6365 2209 Pnt 6365 2255 Pnt 6365 2300 Pnt 6365 2346 Pnt 6365 2392 Pnt 6365 2438 Pnt 6365 2484 Pnt 6365 2530 Pnt 6365 2576 Pnt 6365 2622 Pnt 6365 2714 Pnt 6365 2760 Pnt 6365 2806 Pnt 6365 2852 Pnt 6365 2897 Pnt 6365 2943 Pnt 6365 2989 Pnt 6365 3035 Pnt 6365 3862 Pnt 6365 3908 Pnt 6365 3954 Pnt 6365 4000 Pnt 6365 4091 Pnt 6365 4137 Pnt 6365 4183 Pnt 6365 4413 Pnt 6365 4459 Pnt 6365 4642 Pnt 6365 4688 Pnt 6377 280 Pnt 6377 326 Pnt 6377 372 Pnt 6377 418 Pnt 6377 464 Pnt 6377 510 Pnt 6377 556 Pnt 6377 601 Pnt 6377 647 Pnt 6377 693 Pnt 6377 739 Pnt 6377 785 Pnt 6377 831 Pnt 6377 877 Pnt 6377 923 Pnt 6377 969 Pnt 6377 1015 Pnt 6377 1061 Pnt 6377 1107 Pnt 6377 1152 Pnt 6377 1198 Pnt 6377 1244 Pnt 6377 1290 Pnt 6377 1336 Pnt 6377 1382 Pnt 6377 1428 Pnt 6377 1474 Pnt 6377 1520 Pnt 6377 1566 Pnt 6377 1612 Pnt 6377 1658 Pnt 6377 1704 Pnt 6377 1749 Pnt 6377 1795 Pnt 6377 1841 Pnt 6377 1887 Pnt 6377 1933 Pnt 6377 1979 Pnt 6377 2025 Pnt 6377 2071 Pnt 6377 2117 Pnt 6377 2209 Pnt 6377 2255 Pnt 6377 2300 Pnt 6377 2346 Pnt 6377 2392 Pnt 6377 2438 Pnt 6377 2484 Pnt 6377 2530 Pnt 6377 2576 Pnt 6377 2622 Pnt 6377 2668 Pnt 6377 2760 Pnt 6377 2806 Pnt 6377 2852 Pnt 6377 2897 Pnt 6377 2943 Pnt 6377 2989 Pnt 6377 3035 Pnt 6377 3862 Pnt 6377 3908 Pnt 6377 3954 Pnt 6377 4000 Pnt 6377 4045 Pnt 6377 4091 Pnt 6377 4137 Pnt 6377 4183 Pnt 6377 4413 Pnt 6377 4459 Pnt 6377 4505 Pnt 6377 4642 Pnt 6377 4688 Pnt 6377 4734 Pnt 6390 280 Pnt 6390 326 Pnt 6390 372 Pnt 6390 418 Pnt 6390 464 Pnt 6390 510 Pnt 6390 556 Pnt 6390 601 Pnt 6390 647 Pnt 6390 693 Pnt 6390 739 Pnt 6390 785 Pnt 6390 831 Pnt 6390 877 Pnt 6390 923 Pnt 6390 969 Pnt 6390 1015 Pnt 6390 1061 Pnt 6390 1107 Pnt 6390 1152 Pnt 6390 1198 Pnt 6390 1244 Pnt 6390 1290 Pnt 6390 1336 Pnt 6390 1382 Pnt 6390 1428 Pnt 6390 1474 Pnt 6390 1520 Pnt 6390 1566 Pnt 6390 1612 Pnt 6390 1658 Pnt 6390 1704 Pnt 6390 1749 Pnt 6390 1795 Pnt 6390 1841 Pnt 6390 1887 Pnt 6390 1933 Pnt 6390 1979 Pnt 6390 2025 Pnt 6390 2071 Pnt 6390 2117 Pnt 6390 2163 Pnt 6390 2255 Pnt 6390 2300 Pnt 6390 2346 Pnt 6390 2392 Pnt 6390 2438 Pnt 6390 2484 Pnt 6390 2530 Pnt 6390 2576 Pnt 6390 2622 Pnt 6390 2668 Pnt 6390 2714 Pnt 6390 2806 Pnt 6390 2852 Pnt 6390 2897 Pnt 6390 2943 Pnt 6390 2989 Pnt 6390 3035 Pnt 6390 3081 Pnt 6390 3908 Pnt 6390 3954 Pnt 6390 4000 Pnt 6390 4045 Pnt 6390 4137 Pnt 6390 4183 Pnt 6390 4229 Pnt 6390 4459 Pnt 6390 4505 Pnt 6390 4688 Pnt 6390 4734 Pnt 6403 280 Pnt 6403 326 Pnt 6403 372 Pnt 6403 418 Pnt 6403 464 Pnt 6403 510 Pnt 6403 556 Pnt 6403 601 Pnt 6403 647 Pnt 6403 693 Pnt 6403 739 Pnt 6403 785 Pnt 6403 831 Pnt 6403 877 Pnt 6403 923 Pnt 6403 969 Pnt 6403 1015 Pnt 6403 1061 Pnt 6403 1107 Pnt 6403 1152 Pnt 6403 1198 Pnt 6403 1244 Pnt 6403 1290 Pnt 6403 1336 Pnt 6403 1382 Pnt 6403 1428 Pnt 6403 1474 Pnt 6403 1520 Pnt 6403 1566 Pnt 6403 1612 Pnt 6403 1658 Pnt 6403 1704 Pnt 6403 1749 Pnt 6403 1795 Pnt 6403 1841 Pnt 6403 1887 Pnt 6403 1933 Pnt 6403 1979 Pnt 6403 2025 Pnt 6403 2071 Pnt 6403 2117 Pnt 6403 2163 Pnt 6403 2209 Pnt 6403 2255 Pnt 6403 2300 Pnt 6403 2346 Pnt 6403 2392 Pnt 6403 2438 Pnt 6403 2484 Pnt 6403 2530 Pnt 6403 2576 Pnt 6403 2622 Pnt 6403 2668 Pnt 6403 2714 Pnt 6403 2806 Pnt 6403 2852 Pnt 6403 2897 Pnt 6403 2943 Pnt 6403 2989 Pnt 6403 3035 Pnt 6403 3081 Pnt 6403 3127 Pnt 6403 3908 Pnt 6403 3954 Pnt 6403 4000 Pnt 6403 4045 Pnt 6403 4091 Pnt 6403 4137 Pnt 6403 4183 Pnt 6403 4229 Pnt 6403 4459 Pnt 6403 4505 Pnt 6403 4551 Pnt 6403 4688 Pnt 6403 4734 Pnt 6403 4780 Pnt 6415 280 Pnt 6415 326 Pnt 6415 372 Pnt 6415 418 Pnt 6415 464 Pnt 6415 510 Pnt 6415 556 Pnt 6415 601 Pnt 6415 647 Pnt 6415 693 Pnt 6415 739 Pnt 6415 785 Pnt 6415 831 Pnt 6415 877 Pnt 6415 923 Pnt 6415 969 Pnt 6415 1015 Pnt 6415 1061 Pnt 6415 1107 Pnt 6415 1152 Pnt 6415 1198 Pnt 6415 1244 Pnt 6415 1290 Pnt 6415 1336 Pnt 6415 1382 Pnt 6415 1428 Pnt 6415 1474 Pnt 6415 1520 Pnt 6415 1566 Pnt 6415 1612 Pnt 6415 1658 Pnt 6415 1704 Pnt 6415 1749 Pnt 6415 1795 Pnt 6415 1841 Pnt 6415 1887 Pnt 6415 1933 Pnt 6415 1979 Pnt 6415 2025 Pnt 6415 2071 Pnt 6415 2117 Pnt 6415 2163 Pnt 6415 2209 Pnt 6415 2300 Pnt 6415 2346 Pnt 6415 2392 Pnt 6415 2438 Pnt 6415 2484 Pnt 6415 2530 Pnt 6415 2576 Pnt 6415 2622 Pnt 6415 2668 Pnt 6415 2714 Pnt 6415 2760 Pnt 6415 2852 Pnt 6415 2897 Pnt 6415 2943 Pnt 6415 2989 Pnt 6415 3035 Pnt 6415 3081 Pnt 6415 3127 Pnt 6415 3954 Pnt 6415 4000 Pnt 6415 4045 Pnt 6415 4091 Pnt 6415 4183 Pnt 6415 4229 Pnt 6415 4275 Pnt 6415 4505 Pnt 6415 4551 Pnt 6415 4734 Pnt 6415 4780 Pnt 6428 280 Pnt 6428 326 Pnt 6428 372 Pnt 6428 418 Pnt 6428 464 Pnt 6428 510 Pnt 6428 556 Pnt 6428 601 Pnt 6428 647 Pnt 6428 693 Pnt 6428 739 Pnt 6428 785 Pnt 6428 831 Pnt 6428 877 Pnt 6428 923 Pnt 6428 969 Pnt 6428 1015 Pnt 6428 1061 Pnt 6428 1107 Pnt 6428 1152 Pnt 6428 1198 Pnt 6428 1244 Pnt 6428 1290 Pnt 6428 1336 Pnt 6428 1382 Pnt 6428 1428 Pnt 6428 1474 Pnt 6428 1520 Pnt 6428 1566 Pnt 6428 1612 Pnt 6428 1658 Pnt 6428 1704 Pnt 6428 1749 Pnt 6428 1795 Pnt 6428 1841 Pnt 6428 1887 Pnt 6428 1933 Pnt 6428 1979 Pnt 6428 2025 Pnt 6428 2071 Pnt 6428 2117 Pnt 6428 2163 Pnt 6428 2209 Pnt 6428 2255 Pnt 6428 2346 Pnt 6428 2392 Pnt 6428 2438 Pnt 6428 2484 Pnt 6428 2530 Pnt 6428 2576 Pnt 6428 2622 Pnt 6428 2668 Pnt 6428 2714 Pnt 6428 2760 Pnt 6428 2852 Pnt 6428 2897 Pnt 6428 2943 Pnt 6428 2989 Pnt 6428 3035 Pnt 6428 3081 Pnt 6428 3127 Pnt 6428 3173 Pnt 6428 3954 Pnt 6428 4000 Pnt 6428 4045 Pnt 6428 4091 Pnt 6428 4137 Pnt 6428 4183 Pnt 6428 4229 Pnt 6428 4275 Pnt 6428 4505 Pnt 6428 4551 Pnt 6428 4596 Pnt 6428 4734 Pnt 6428 4826 Pnt 6441 280 Pnt 6441 326 Pnt 6441 372 Pnt 6441 418 Pnt 6441 464 Pnt 6441 510 Pnt 6441 556 Pnt 6441 601 Pnt 6441 647 Pnt 6441 693 Pnt 6441 739 Pnt 6441 785 Pnt 6441 831 Pnt 6441 877 Pnt 6441 923 Pnt 6441 969 Pnt 6441 1015 Pnt 6441 1061 Pnt 6441 1107 Pnt 6441 1152 Pnt 6441 1198 Pnt 6441 1244 Pnt 6441 1290 Pnt 6441 1336 Pnt 6441 1382 Pnt 6441 1428 Pnt 6441 1474 Pnt 6441 1520 Pnt 6441 1566 Pnt 6441 1612 Pnt 6441 1658 Pnt 6441 1704 Pnt 6441 1749 Pnt 6441 1795 Pnt 6441 1841 Pnt 6441 1887 Pnt 6441 1933 Pnt 6441 1979 Pnt 6441 2025 Pnt 6441 2071 Pnt 6441 2117 Pnt 6441 2163 Pnt 6441 2209 Pnt 6441 2255 Pnt 6441 2300 Pnt 6441 2392 Pnt 6441 2438 Pnt 6441 2484 Pnt 6441 2530 Pnt 6441 2576 Pnt 6441 2622 Pnt 6441 2668 Pnt 6441 2714 Pnt 6441 2760 Pnt 6441 2806 Pnt 6441 2897 Pnt 6441 2943 Pnt 6441 2989 Pnt 6441 3035 Pnt 6441 3081 Pnt 6441 3127 Pnt 6441 3173 Pnt 6441 4000 Pnt 6441 4045 Pnt 6441 4091 Pnt 6441 4137 Pnt 6441 4229 Pnt 6441 4275 Pnt 6441 4321 Pnt 6441 4505 Pnt 6441 4551 Pnt 6441 4596 Pnt 6441 4780 Pnt 6441 4826 Pnt 6454 280 Pnt 6454 326 Pnt 6454 372 Pnt 6454 418 Pnt 6454 464 Pnt 6454 510 Pnt 6454 556 Pnt 6454 601 Pnt 6454 647 Pnt 6454 693 Pnt 6454 739 Pnt 6454 785 Pnt 6454 831 Pnt 6454 877 Pnt 6454 923 Pnt 6454 969 Pnt 6454 1015 Pnt 6454 1061 Pnt 6454 1107 Pnt 6454 1152 Pnt 6454 1198 Pnt 6454 1244 Pnt 6454 1290 Pnt 6454 1336 Pnt 6454 1382 Pnt 6454 1428 Pnt 6454 1474 Pnt 6454 1520 Pnt 6454 1566 Pnt 6454 1612 Pnt 6454 1658 Pnt 6454 1704 Pnt 6454 1749 Pnt 6454 1795 Pnt 6454 1841 Pnt 6454 1887 Pnt 6454 1933 Pnt 6454 1979 Pnt 6454 2025 Pnt 6454 2071 Pnt 6454 2117 Pnt 6454 2163 Pnt 6454 2209 Pnt 6454 2255 Pnt 6454 2300 Pnt 6454 2392 Pnt 6454 2438 Pnt 6454 2484 Pnt 6454 2530 Pnt 6454 2576 Pnt 6454 2622 Pnt 6454 2668 Pnt 6454 2714 Pnt 6454 2760 Pnt 6454 2806 Pnt 6454 2897 Pnt 6454 2943 Pnt 6454 2989 Pnt 6454 3035 Pnt 6454 3081 Pnt 6454 3127 Pnt 6454 3173 Pnt 6454 3219 Pnt 6454 4000 Pnt 6454 4045 Pnt 6454 4091 Pnt 6454 4137 Pnt 6454 4183 Pnt 6454 4229 Pnt 6454 4275 Pnt 6454 4321 Pnt 6454 4551 Pnt 6454 4596 Pnt 6454 4642 Pnt 6454 4780 Pnt 6454 4872 Pnt 6466 280 Pnt 6466 326 Pnt 6466 372 Pnt 6466 418 Pnt 6466 464 Pnt 6466 510 Pnt 6466 556 Pnt 6466 601 Pnt 6466 647 Pnt 6466 693 Pnt 6466 739 Pnt 6466 785 Pnt 6466 831 Pnt 6466 877 Pnt 6466 923 Pnt 6466 969 Pnt 6466 1015 Pnt 6466 1061 Pnt 6466 1107 Pnt 6466 1152 Pnt 6466 1198 Pnt 6466 1244 Pnt 6466 1290 Pnt 6466 1336 Pnt 6466 1382 Pnt 6466 1428 Pnt 6466 1474 Pnt 6466 1520 Pnt 6466 1566 Pnt 6466 1612 Pnt 6466 1658 Pnt 6466 1704 Pnt 6466 1749 Pnt 6466 1795 Pnt 6466 1841 Pnt 6466 1887 Pnt 6466 1933 Pnt 6466 1979 Pnt 6466 2025 Pnt 6466 2071 Pnt 6466 2117 Pnt 6466 2163 Pnt 6466 2209 Pnt 6466 2255 Pnt 6466 2300 Pnt 6466 2346 Pnt 6466 2438 Pnt 6466 2484 Pnt 6466 2530 Pnt 6466 2576 Pnt 6466 2622 Pnt 6466 2668 Pnt 6466 2714 Pnt 6466 2760 Pnt 6466 2806 Pnt 6466 2852 Pnt 6466 2943 Pnt 6466 2989 Pnt 6466 3035 Pnt 6466 3081 Pnt 6466 3127 Pnt 6466 3173 Pnt 6466 3219 Pnt 6466 4045 Pnt 6466 4091 Pnt 6466 4137 Pnt 6466 4183 Pnt 6466 4229 Pnt 6466 4275 Pnt 6466 4321 Pnt 6466 4367 Pnt 6466 4551 Pnt 6466 4596 Pnt 6466 4642 Pnt 6466 4826 Pnt 6466 4872 Pnt 6479 280 Pnt 6479 326 Pnt 6479 372 Pnt 6479 418 Pnt 6479 464 Pnt 6479 510 Pnt 6479 556 Pnt 6479 601 Pnt 6479 647 Pnt 6479 1290 Pnt 6479 1336 Pnt 6479 1382 Pnt 6479 1428 Pnt 6479 1474 Pnt 6479 1520 Pnt 6479 1566 Pnt 6479 1612 Pnt 6479 1658 Pnt 6479 1704 Pnt 6479 1749 Pnt 6479 1795 Pnt 6479 1841 Pnt 6479 1887 Pnt 6479 1933 Pnt 6479 1979 Pnt 6479 2025 Pnt 6479 2071 Pnt 6479 2117 Pnt 6479 2163 Pnt 6479 2209 Pnt 6479 2255 Pnt 6479 2300 Pnt 6479 2346 Pnt 6479 2392 Pnt 6479 2484 Pnt 6479 2530 Pnt 6479 2576 Pnt 6479 2622 Pnt 6479 2668 Pnt 6479 2714 Pnt 6479 2760 Pnt 6479 2806 Pnt 6479 2852 Pnt 6479 2897 Pnt 6479 2989 Pnt 6479 3035 Pnt 6479 3081 Pnt 6479 3127 Pnt 6479 3173 Pnt 6479 3219 Pnt 6479 3265 Pnt 6479 4045 Pnt 6479 4091 Pnt 6479 4137 Pnt 6479 4183 Pnt 6479 4229 Pnt 6479 4275 Pnt 6479 4321 Pnt 6479 4367 Pnt 6479 4596 Pnt 6479 4642 Pnt 6479 4688 Pnt 6479 4826 Pnt 6492 280 Pnt 6492 326 Pnt 6492 372 Pnt 6492 418 Pnt 6492 464 Pnt 6492 510 Pnt 6492 1428 Pnt 6492 1474 Pnt 6492 1520 Pnt 6492 1566 Pnt 6492 1612 Pnt 6492 1658 Pnt 6492 1704 Pnt 6492 1749 Pnt 6492 1795 Pnt 6492 1841 Pnt 6492 1887 Pnt 6492 1933 Pnt 6492 1979 Pnt 6492 2025 Pnt 6492 2071 Pnt 6492 2117 Pnt 6492 2163 Pnt 6492 2209 Pnt 6492 2255 Pnt 6492 2300 Pnt 6492 2346 Pnt 6492 2392 Pnt 6492 2484 Pnt 6492 2530 Pnt 6492 2576 Pnt 6492 2622 Pnt 6492 2668 Pnt 6492 2714 Pnt 6492 2760 Pnt 6492 2806 Pnt 6492 2852 Pnt 6492 2897 Pnt 6492 2989 Pnt 6492 3035 Pnt 6492 3081 Pnt 6492 3127 Pnt 6492 3173 Pnt 6492 3219 Pnt 6492 3265 Pnt 6492 4091 Pnt 6492 4137 Pnt 6492 4183 Pnt 6492 4229 Pnt 6492 4275 Pnt 6492 4321 Pnt 6492 4367 Pnt 6492 4596 Pnt 6492 4642 Pnt 6492 4688 Pnt 6492 4872 Pnt 6504 280 Pnt 6504 326 Pnt 6504 372 Pnt 6504 418 Pnt 6504 464 Pnt 6504 510 Pnt 6504 556 Pnt 6504 601 Pnt 6504 647 Pnt 6504 1520 Pnt 6504 1566 Pnt 6504 1612 Pnt 6504 1658 Pnt 6504 1704 Pnt 6504 1749 Pnt 6504 1795 Pnt 6504 1841 Pnt 6504 1887 Pnt 6504 1933 Pnt 6504 1979 Pnt 6504 2025 Pnt 6504 2071 Pnt 6504 2117 Pnt 6504 2163 Pnt 6504 2209 Pnt 6504 2255 Pnt 6504 2300 Pnt 6504 2346 Pnt 6504 2392 Pnt 6504 2438 Pnt 6504 2530 Pnt 6504 2576 Pnt 6504 2622 Pnt 6504 2668 Pnt 6504 2714 Pnt 6504 2760 Pnt 6504 2806 Pnt 6504 2852 Pnt 6504 2897 Pnt 6504 2943 Pnt 6504 3035 Pnt 6504 3081 Pnt 6504 3127 Pnt 6504 3173 Pnt 6504 3219 Pnt 6504 3265 Pnt 6504 3311 Pnt 6504 4091 Pnt 6504 4137 Pnt 6504 4183 Pnt 6504 4229 Pnt 6504 4275 Pnt 6504 4321 Pnt 6504 4367 Pnt 6504 4413 Pnt 6504 4642 Pnt 6504 4688 Pnt 6504 4734 Pnt 6504 4872 Pnt 6517 280 Pnt 6517 326 Pnt 6517 372 Pnt 6517 418 Pnt 6517 464 Pnt 6517 510 Pnt 6517 556 Pnt 6517 601 Pnt 6517 647 Pnt 6517 693 Pnt 6517 739 Pnt 6517 785 Pnt 6517 831 Pnt 6517 877 Pnt 6517 1612 Pnt 6517 1658 Pnt 6517 1704 Pnt 6517 1749 Pnt 6517 1795 Pnt 6517 1841 Pnt 6517 1887 Pnt 6517 1933 Pnt 6517 1979 Pnt 6517 2025 Pnt 6517 2071 Pnt 6517 2117 Pnt 6517 2163 Pnt 6517 2209 Pnt 6517 2255 Pnt 6517 2300 Pnt 6517 2346 Pnt 6517 2392 Pnt 6517 2438 Pnt 6517 2484 Pnt 6517 2576 Pnt 6517 2622 Pnt 6517 2668 Pnt 6517 2714 Pnt 6517 2760 Pnt 6517 2806 Pnt 6517 2852 Pnt 6517 2897 Pnt 6517 2943 Pnt 6517 3035 Pnt 6517 3081 Pnt 6517 3127 Pnt 6517 3173 Pnt 6517 3219 Pnt 6517 3265 Pnt 6517 3311 Pnt 6517 4137 Pnt 6517 4183 Pnt 6517 4229 Pnt 6517 4275 Pnt 6517 4321 Pnt 6517 4367 Pnt 6517 4413 Pnt 6517 4642 Pnt 6517 4688 Pnt 6517 4734 Pnt 6530 280 Pnt 6530 326 Pnt 6530 372 Pnt 6530 418 Pnt 6530 464 Pnt 6530 510 Pnt 6530 556 Pnt 6530 601 Pnt 6530 647 Pnt 6530 693 Pnt 6530 739 Pnt 6530 785 Pnt 6530 831 Pnt 6530 877 Pnt 6530 923 Pnt 6530 969 Pnt 6530 1658 Pnt 6530 1704 Pnt 6530 1749 Pnt 6530 1795 Pnt 6530 1841 Pnt 6530 1887 Pnt 6530 1933 Pnt 6530 1979 Pnt 6530 2025 Pnt 6530 2071 Pnt 6530 2117 Pnt 6530 2163 Pnt 6530 2209 Pnt 6530 2255 Pnt 6530 2300 Pnt 6530 2346 Pnt 6530 2392 Pnt 6530 2438 Pnt 6530 2484 Pnt 6530 2576 Pnt 6530 2622 Pnt 6530 2668 Pnt 6530 2714 Pnt 6530 2760 Pnt 6530 2806 Pnt 6530 2852 Pnt 6530 2897 Pnt 6530 2943 Pnt 6530 2989 Pnt 6530 3081 Pnt 6530 3127 Pnt 6530 3173 Pnt 6530 3219 Pnt 6530 3265 Pnt 6530 3311 Pnt 6530 3357 Pnt 6530 4137 Pnt 6530 4183 Pnt 6530 4229 Pnt 6530 4275 Pnt 6530 4321 Pnt 6530 4367 Pnt 6530 4413 Pnt 6530 4459 Pnt 6530 4688 Pnt 6530 4734 Pnt 6530 4780 Pnt 6543 280 Pnt 6543 326 Pnt 6543 372 Pnt 6543 418 Pnt 6543 464 Pnt 6543 510 Pnt 6543 556 Pnt 6543 601 Pnt 6543 647 Pnt 6543 693 Pnt 6543 739 Pnt 6543 785 Pnt 6543 831 Pnt 6543 877 Pnt 6543 923 Pnt 6543 969 Pnt 6543 1015 Pnt 6543 1061 Pnt 6543 1704 Pnt 6543 1749 Pnt 6543 1795 Pnt 6543 1841 Pnt 6543 1887 Pnt 6543 1933 Pnt 6543 1979 Pnt 6543 2025 Pnt 6543 2071 Pnt 6543 2117 Pnt 6543 2163 Pnt 6543 2209 Pnt 6543 2255 Pnt 6543 2300 Pnt 6543 2346 Pnt 6543 2392 Pnt 6543 2438 Pnt 6543 2484 Pnt 6543 2530 Pnt 6543 2622 Pnt 6543 2668 Pnt 6543 2714 Pnt 6543 2760 Pnt 6543 2806 Pnt 6543 2852 Pnt 6543 2897 Pnt 6543 2943 Pnt 6543 2989 Pnt 6543 3081 Pnt 6543 3127 Pnt 6543 3173 Pnt 6543 3219 Pnt 6543 3265 Pnt 6543 3311 Pnt 6543 3357 Pnt 6543 4183 Pnt 6543 4229 Pnt 6543 4275 Pnt 6543 4321 Pnt 6543 4367 Pnt 6543 4413 Pnt 6543 4459 Pnt 6543 4688 Pnt 6543 4734 Pnt 6543 4780 Pnt 6555 280 Pnt 6555 326 Pnt 6555 372 Pnt 6555 418 Pnt 6555 464 Pnt 6555 510 Pnt 6555 556 Pnt 6555 601 Pnt 6555 647 Pnt 6555 693 Pnt 6555 739 Pnt 6555 785 Pnt 6555 831 Pnt 6555 877 Pnt 6555 923 Pnt 6555 969 Pnt 6555 1015 Pnt 6555 1061 Pnt 6555 1107 Pnt 6555 1152 Pnt 6555 1795 Pnt 6555 1841 Pnt 6555 1887 Pnt 6555 1933 Pnt 6555 1979 Pnt 6555 2025 Pnt 6555 2071 Pnt 6555 2117 Pnt 6555 2163 Pnt 6555 2209 Pnt 6555 2255 Pnt 6555 2300 Pnt 6555 2346 Pnt 6555 2392 Pnt 6555 2438 Pnt 6555 2484 Pnt 6555 2530 Pnt 6555 2576 Pnt 6555 2668 Pnt 6555 2714 Pnt 6555 2760 Pnt 6555 2806 Pnt 6555 2852 Pnt 6555 2897 Pnt 6555 2943 Pnt 6555 2989 Pnt 6555 3035 Pnt 6555 3127 Pnt 6555 3173 Pnt 6555 3219 Pnt 6555 3265 Pnt 6555 3311 Pnt 6555 3357 Pnt 6555 3403 Pnt 6555 4183 Pnt 6555 4229 Pnt 6555 4275 Pnt 6555 4321 Pnt 6555 4413 Pnt 6555 4459 Pnt 6555 4505 Pnt 6555 4734 Pnt 6555 4780 Pnt 6555 4826 Pnt 6568 280 Pnt 6568 326 Pnt 6568 372 Pnt 6568 418 Pnt 6568 464 Pnt 6568 510 Pnt 6568 556 Pnt 6568 601 Pnt 6568 647 Pnt 6568 693 Pnt 6568 739 Pnt 6568 785 Pnt 6568 831 Pnt 6568 877 Pnt 6568 923 Pnt 6568 969 Pnt 6568 1015 Pnt 6568 1061 Pnt 6568 1107 Pnt 6568 1152 Pnt 6568 1198 Pnt 6568 1244 Pnt 6568 1841 Pnt 6568 1887 Pnt 6568 1933 Pnt 6568 1979 Pnt 6568 2025 Pnt 6568 2071 Pnt 6568 2117 Pnt 6568 2163 Pnt 6568 2209 Pnt 6568 2255 Pnt 6568 2300 Pnt 6568 2346 Pnt 6568 2392 Pnt 6568 2438 Pnt 6568 2484 Pnt 6568 2530 Pnt 6568 2576 Pnt 6568 2668 Pnt 6568 2714 Pnt 6568 2760 Pnt 6568 2806 Pnt 6568 2852 Pnt 6568 2897 Pnt 6568 2943 Pnt 6568 2989 Pnt 6568 3035 Pnt 6568 3127 Pnt 6568 3173 Pnt 6568 3219 Pnt 6568 3265 Pnt 6568 3311 Pnt 6568 3357 Pnt 6568 3403 Pnt 6568 4183 Pnt 6568 4229 Pnt 6568 4275 Pnt 6568 4321 Pnt 6568 4367 Pnt 6568 4413 Pnt 6568 4459 Pnt 6568 4505 Pnt 6568 4734 Pnt 6568 4780 Pnt 6568 4826 Pnt 6581 280 Pnt 6581 326 Pnt 6581 372 Pnt 6581 418 Pnt 6581 464 Pnt 6581 510 Pnt 6581 556 Pnt 6581 601 Pnt 6581 647 Pnt 6581 693 Pnt 6581 739 Pnt 6581 785 Pnt 6581 831 Pnt 6581 877 Pnt 6581 923 Pnt 6581 969 Pnt 6581 1015 Pnt 6581 1061 Pnt 6581 1107 Pnt 6581 1152 Pnt 6581 1198 Pnt 6581 1244 Pnt 6581 1290 Pnt 6581 1887 Pnt 6581 1933 Pnt 6581 1979 Pnt 6581 2025 Pnt 6581 2071 Pnt 6581 2117 Pnt 6581 2163 Pnt 6581 2209 Pnt 6581 2255 Pnt 6581 2300 Pnt 6581 2346 Pnt 6581 2392 Pnt 6581 2438 Pnt 6581 2484 Pnt 6581 2530 Pnt 6581 2576 Pnt 6581 2622 Pnt 6581 2714 Pnt 6581 2760 Pnt 6581 2806 Pnt 6581 2852 Pnt 6581 2897 Pnt 6581 2943 Pnt 6581 2989 Pnt 6581 3035 Pnt 6581 3081 Pnt 6581 3173 Pnt 6581 3219 Pnt 6581 3265 Pnt 6581 3311 Pnt 6581 3357 Pnt 6581 3403 Pnt 6581 3448 Pnt 6581 4229 Pnt 6581 4275 Pnt 6581 4321 Pnt 6581 4367 Pnt 6581 4459 Pnt 6581 4505 Pnt 6581 4551 Pnt 6581 4780 Pnt 6581 4826 Pnt 6581 4872 Pnt 6593 280 Pnt 6593 326 Pnt 6593 372 Pnt 6593 418 Pnt 6593 464 Pnt 6593 510 Pnt 6593 556 Pnt 6593 601 Pnt 6593 647 Pnt 6593 693 Pnt 6593 739 Pnt 6593 785 Pnt 6593 831 Pnt 6593 877 Pnt 6593 923 Pnt 6593 969 Pnt 6593 1015 Pnt 6593 1061 Pnt 6593 1107 Pnt 6593 1152 Pnt 6593 1198 Pnt 6593 1244 Pnt 6593 1290 Pnt 6593 1336 Pnt 6593 1933 Pnt 6593 1979 Pnt 6593 2025 Pnt 6593 2071 Pnt 6593 2117 Pnt 6593 2163 Pnt 6593 2209 Pnt 6593 2255 Pnt 6593 2300 Pnt 6593 2346 Pnt 6593 2392 Pnt 6593 2438 Pnt 6593 2484 Pnt 6593 2530 Pnt 6593 2576 Pnt 6593 2622 Pnt 6593 2714 Pnt 6593 2760 Pnt 6593 2806 Pnt 6593 2852 Pnt 6593 2897 Pnt 6593 2943 Pnt 6593 2989 Pnt 6593 3035 Pnt 6593 3081 Pnt 6593 3173 Pnt 6593 3219 Pnt 6593 3265 Pnt 6593 3311 Pnt 6593 3357 Pnt 6593 3403 Pnt 6593 3448 Pnt 6593 4229 Pnt 6593 4275 Pnt 6593 4321 Pnt 6593 4367 Pnt 6593 4413 Pnt 6593 4459 Pnt 6593 4505 Pnt 6593 4551 Pnt 6593 4780 Pnt 6593 4826 Pnt 6593 4872 Pnt 6606 280 Pnt 6606 326 Pnt 6606 372 Pnt 6606 418 Pnt 6606 464 Pnt 6606 510 Pnt 6606 556 Pnt 6606 601 Pnt 6606 647 Pnt 6606 693 Pnt 6606 739 Pnt 6606 785 Pnt 6606 831 Pnt 6606 877 Pnt 6606 923 Pnt 6606 969 Pnt 6606 1015 Pnt 6606 1061 Pnt 6606 1107 Pnt 6606 1152 Pnt 6606 1198 Pnt 6606 1244 Pnt 6606 1290 Pnt 6606 1336 Pnt 6606 1382 Pnt 6606 1979 Pnt 6606 2025 Pnt 6606 2071 Pnt 6606 2117 Pnt 6606 2163 Pnt 6606 2209 Pnt 6606 2255 Pnt 6606 2300 Pnt 6606 2346 Pnt 6606 2392 Pnt 6606 2438 Pnt 6606 2484 Pnt 6606 2530 Pnt 6606 2576 Pnt 6606 2622 Pnt 6606 2668 Pnt 6606 2760 Pnt 6606 2806 Pnt 6606 2852 Pnt 6606 2897 Pnt 6606 2943 Pnt 6606 2989 Pnt 6606 3035 Pnt 6606 3081 Pnt 6606 3127 Pnt 6606 3219 Pnt 6606 3265 Pnt 6606 3311 Pnt 6606 3357 Pnt 6606 3403 Pnt 6606 3448 Pnt 6606 3494 Pnt 6606 4275 Pnt 6606 4321 Pnt 6606 4367 Pnt 6606 4413 Pnt 6606 4505 Pnt 6606 4551 Pnt 6606 4596 Pnt 6606 4826 Pnt 6606 4872 Pnt 6619 280 Pnt 6619 326 Pnt 6619 372 Pnt 6619 418 Pnt 6619 464 Pnt 6619 510 Pnt 6619 556 Pnt 6619 601 Pnt 6619 647 Pnt 6619 693 Pnt 6619 739 Pnt 6619 785 Pnt 6619 831 Pnt 6619 877 Pnt 6619 923 Pnt 6619 969 Pnt 6619 1015 Pnt 6619 1061 Pnt 6619 1107 Pnt 6619 1152 Pnt 6619 1198 Pnt 6619 1244 Pnt 6619 1290 Pnt 6619 1336 Pnt 6619 1382 Pnt 6619 1428 Pnt 6619 1474 Pnt 6619 2025 Pnt 6619 2071 Pnt 6619 2117 Pnt 6619 2163 Pnt 6619 2209 Pnt 6619 2255 Pnt 6619 2300 Pnt 6619 2346 Pnt 6619 2392 Pnt 6619 2438 Pnt 6619 2484 Pnt 6619 2530 Pnt 6619 2576 Pnt 6619 2622 Pnt 6619 2668 Pnt 6619 2760 Pnt 6619 2806 Pnt 6619 2852 Pnt 6619 2897 Pnt 6619 2943 Pnt 6619 2989 Pnt 6619 3035 Pnt 6619 3081 Pnt 6619 3127 Pnt 6619 3219 Pnt 6619 3265 Pnt 6619 3311 Pnt 6619 3357 Pnt 6619 3403 Pnt 6619 3448 Pnt 6619 3494 Pnt 6619 4275 Pnt 6619 4321 Pnt 6619 4367 Pnt 6619 4413 Pnt 6619 4459 Pnt 6619 4505 Pnt 6619 4551 Pnt 6619 4596 Pnt 6619 4826 Pnt 6619 4872 Pnt 6632 280 Pnt 6632 326 Pnt 6632 372 Pnt 6632 418 Pnt 6632 464 Pnt 6632 510 Pnt 6632 556 Pnt 6632 601 Pnt 6632 647 Pnt 6632 693 Pnt 6632 739 Pnt 6632 785 Pnt 6632 831 Pnt 6632 877 Pnt 6632 923 Pnt 6632 969 Pnt 6632 1015 Pnt 6632 1061 Pnt 6632 1107 Pnt 6632 1152 Pnt 6632 1198 Pnt 6632 1244 Pnt 6632 1290 Pnt 6632 1336 Pnt 6632 1382 Pnt 6632 1428 Pnt 6632 1474 Pnt 6632 1520 Pnt 6632 2071 Pnt 6632 2117 Pnt 6632 2163 Pnt 6632 2209 Pnt 6632 2255 Pnt 6632 2300 Pnt 6632 2346 Pnt 6632 2392 Pnt 6632 2438 Pnt 6632 2484 Pnt 6632 2530 Pnt 6632 2576 Pnt 6632 2622 Pnt 6632 2668 Pnt 6632 2714 Pnt 6632 2806 Pnt 6632 2852 Pnt 6632 2897 Pnt 6632 2943 Pnt 6632 2989 Pnt 6632 3035 Pnt 6632 3081 Pnt 6632 3127 Pnt 6632 3173 Pnt 6632 3265 Pnt 6632 3311 Pnt 6632 3357 Pnt 6632 3403 Pnt 6632 3448 Pnt 6632 3494 Pnt 6632 3540 Pnt 6632 4321 Pnt 6632 4367 Pnt 6632 4413 Pnt 6632 4459 Pnt 6632 4505 Pnt 6632 4551 Pnt 6632 4596 Pnt 6632 4642 Pnt 6632 4826 Pnt 6632 4872 Pnt 6644 280 Pnt 6644 326 Pnt 6644 372 Pnt 6644 418 Pnt 6644 464 Pnt 6644 510 Pnt 6644 556 Pnt 6644 601 Pnt 6644 647 Pnt 6644 693 Pnt 6644 739 Pnt 6644 785 Pnt 6644 831 Pnt 6644 877 Pnt 6644 923 Pnt 6644 969 Pnt 6644 1015 Pnt 6644 1061 Pnt 6644 1107 Pnt 6644 1152 Pnt 6644 1198 Pnt 6644 1244 Pnt 6644 1290 Pnt 6644 1336 Pnt 6644 1382 Pnt 6644 1428 Pnt 6644 1474 Pnt 6644 1520 Pnt 6644 1566 Pnt 6644 2117 Pnt 6644 2163 Pnt 6644 2209 Pnt 6644 2255 Pnt 6644 2300 Pnt 6644 2346 Pnt 6644 2392 Pnt 6644 2438 Pnt 6644 2484 Pnt 6644 2530 Pnt 6644 2576 Pnt 6644 2622 Pnt 6644 2668 Pnt 6644 2714 Pnt 6644 2760 Pnt 6644 2852 Pnt 6644 2897 Pnt 6644 2943 Pnt 6644 2989 Pnt 6644 3035 Pnt 6644 3081 Pnt 6644 3127 Pnt 6644 3173 Pnt 6644 3265 Pnt 6644 3311 Pnt 6644 3357 Pnt 6644 3403 Pnt 6644 3448 Pnt 6644 3494 Pnt 6644 3540 Pnt 6644 4321 Pnt 6644 4367 Pnt 6644 4413 Pnt 6644 4459 Pnt 6644 4505 Pnt 6644 4551 Pnt 6644 4596 Pnt 6644 4642 Pnt 6644 4872 Pnt 6657 280 Pnt 6657 326 Pnt 6657 372 Pnt 6657 418 Pnt 6657 464 Pnt 6657 510 Pnt 6657 556 Pnt 6657 601 Pnt 6657 647 Pnt 6657 693 Pnt 6657 739 Pnt 6657 785 Pnt 6657 831 Pnt 6657 877 Pnt 6657 923 Pnt 6657 969 Pnt 6657 1015 Pnt 6657 1061 Pnt 6657 1107 Pnt 6657 1152 Pnt 6657 1198 Pnt 6657 1244 Pnt 6657 1290 Pnt 6657 1336 Pnt 6657 1382 Pnt 6657 1428 Pnt 6657 1474 Pnt 6657 1520 Pnt 6657 1566 Pnt 6657 1612 Pnt 6657 2117 Pnt 6657 2163 Pnt 6657 2209 Pnt 6657 2255 Pnt 6657 2300 Pnt 6657 2346 Pnt 6657 2392 Pnt 6657 2438 Pnt 6657 2484 Pnt 6657 2530 Pnt 6657 2576 Pnt 6657 2622 Pnt 6657 2668 Pnt 6657 2714 Pnt 6657 2760 Pnt 6657 2852 Pnt 6657 2897 Pnt 6657 2943 Pnt 6657 2989 Pnt 6657 3035 Pnt 6657 3081 Pnt 6657 3127 Pnt 6657 3173 Pnt 6657 3219 Pnt 6657 3311 Pnt 6657 3357 Pnt 6657 3403 Pnt 6657 3448 Pnt 6657 3494 Pnt 6657 3540 Pnt 6657 3586 Pnt 6657 4367 Pnt 6657 4413 Pnt 6657 4459 Pnt 6657 4505 Pnt 6657 4551 Pnt 6657 4596 Pnt 6657 4642 Pnt 6657 4872 Pnt 6670 280 Pnt 6670 326 Pnt 6670 372 Pnt 6670 418 Pnt 6670 464 Pnt 6670 510 Pnt 6670 556 Pnt 6670 601 Pnt 6670 647 Pnt 6670 693 Pnt 6670 739 Pnt 6670 785 Pnt 6670 831 Pnt 6670 877 Pnt 6670 923 Pnt 6670 969 Pnt 6670 1015 Pnt 6670 1061 Pnt 6670 1107 Pnt 6670 1152 Pnt 6670 1198 Pnt 6670 1244 Pnt 6670 1290 Pnt 6670 1336 Pnt 6670 1382 Pnt 6670 1428 Pnt 6670 1474 Pnt 6670 1520 Pnt 6670 1566 Pnt 6670 1612 Pnt 6670 1658 Pnt 6670 2163 Pnt 6670 2209 Pnt 6670 2255 Pnt 6670 2300 Pnt 6670 2346 Pnt 6670 2392 Pnt 6670 2438 Pnt 6670 2484 Pnt 6670 2530 Pnt 6670 2576 Pnt 6670 2622 Pnt 6670 2668 Pnt 6670 2714 Pnt 6670 2760 Pnt 6670 2806 Pnt 6670 2897 Pnt 6670 2943 Pnt 6670 2989 Pnt 6670 3035 Pnt 6670 3081 Pnt 6670 3127 Pnt 6670 3173 Pnt 6670 3219 Pnt 6670 3311 Pnt 6670 3357 Pnt 6670 3403 Pnt 6670 3448 Pnt 6670 3494 Pnt 6670 3540 Pnt 6670 3586 Pnt 6670 4367 Pnt 6670 4413 Pnt 6670 4459 Pnt 6670 4505 Pnt 6670 4596 Pnt 6670 4642 Pnt 6670 4688 Pnt 6682 280 Pnt 6682 326 Pnt 6682 372 Pnt 6682 418 Pnt 6682 464 Pnt 6682 510 Pnt 6682 556 Pnt 6682 601 Pnt 6682 647 Pnt 6682 693 Pnt 6682 739 Pnt 6682 785 Pnt 6682 831 Pnt 6682 877 Pnt 6682 923 Pnt 6682 969 Pnt 6682 1015 Pnt 6682 1061 Pnt 6682 1107 Pnt 6682 1152 Pnt 6682 1198 Pnt 6682 1244 Pnt 6682 1290 Pnt 6682 1336 Pnt 6682 1382 Pnt 6682 1428 Pnt 6682 1474 Pnt 6682 1520 Pnt 6682 1566 Pnt 6682 1612 Pnt 6682 1658 Pnt 6682 2209 Pnt 6682 2255 Pnt 6682 2300 Pnt 6682 2346 Pnt 6682 2392 Pnt 6682 2438 Pnt 6682 2484 Pnt 6682 2530 Pnt 6682 2576 Pnt 6682 2622 Pnt 6682 2668 Pnt 6682 2714 Pnt 6682 2760 Pnt 6682 2806 Pnt 6682 2897 Pnt 6682 2943 Pnt 6682 2989 Pnt 6682 3035 Pnt 6682 3081 Pnt 6682 3127 Pnt 6682 3173 Pnt 6682 3219 Pnt 6682 3265 Pnt 6682 3357 Pnt 6682 3403 Pnt 6682 3448 Pnt 6682 3494 Pnt 6682 3540 Pnt 6682 3586 Pnt 6682 4367 Pnt 6682 4413 Pnt 6682 4459 Pnt 6682 4505 Pnt 6682 4551 Pnt 6682 4596 Pnt 6682 4642 Pnt 6682 4688 Pnt 6695 280 Pnt 6695 326 Pnt 6695 372 Pnt 6695 418 Pnt 6695 464 Pnt 6695 510 Pnt 6695 556 Pnt 6695 601 Pnt 6695 647 Pnt 6695 693 Pnt 6695 739 Pnt 6695 785 Pnt 6695 831 Pnt 6695 877 Pnt 6695 923 Pnt 6695 969 Pnt 6695 1015 Pnt 6695 1061 Pnt 6695 1107 Pnt 6695 1152 Pnt 6695 1198 Pnt 6695 1244 Pnt 6695 1290 Pnt 6695 1336 Pnt 6695 1382 Pnt 6695 1428 Pnt 6695 1474 Pnt 6695 1520 Pnt 6695 1566 Pnt 6695 1612 Pnt 6695 1658 Pnt 6695 1704 Pnt 6695 2255 Pnt 6695 2300 Pnt 6695 2346 Pnt 6695 2392 Pnt 6695 2438 Pnt 6695 2484 Pnt 6695 2530 Pnt 6695 2576 Pnt 6695 2622 Pnt 6695 2668 Pnt 6695 2714 Pnt 6695 2760 Pnt 6695 2806 Pnt 6695 2852 Pnt 6695 2943 Pnt 6695 2989 Pnt 6695 3035 Pnt 6695 3081 Pnt 6695 3127 Pnt 6695 3173 Pnt 6695 3219 Pnt 6695 3265 Pnt 6695 3357 Pnt 6695 3403 Pnt 6695 3448 Pnt 6695 3494 Pnt 6695 3540 Pnt 6695 3586 Pnt 6695 3632 Pnt 6695 4413 Pnt 6695 4459 Pnt 6695 4505 Pnt 6695 4551 Pnt 6695 4642 Pnt 6695 4688 Pnt 6695 4734 Pnt 6708 280 Pnt 6708 326 Pnt 6708 372 Pnt 6708 418 Pnt 6708 464 Pnt 6708 510 Pnt 6708 556 Pnt 6708 601 Pnt 6708 647 Pnt 6708 693 Pnt 6708 739 Pnt 6708 785 Pnt 6708 831 Pnt 6708 877 Pnt 6708 923 Pnt 6708 969 Pnt 6708 1015 Pnt 6708 1061 Pnt 6708 1107 Pnt 6708 1152 Pnt 6708 1198 Pnt 6708 1244 Pnt 6708 1290 Pnt 6708 1336 Pnt 6708 1382 Pnt 6708 1428 Pnt 6708 1474 Pnt 6708 1520 Pnt 6708 1566 Pnt 6708 1612 Pnt 6708 1658 Pnt 6708 1704 Pnt 6708 1749 Pnt 6708 2300 Pnt 6708 2346 Pnt 6708 2392 Pnt 6708 2438 Pnt 6708 2484 Pnt 6708 2530 Pnt 6708 2576 Pnt 6708 2622 Pnt 6708 2668 Pnt 6708 2714 Pnt 6708 2760 Pnt 6708 2806 Pnt 6708 2852 Pnt 6708 2943 Pnt 6708 2989 Pnt 6708 3035 Pnt 6708 3081 Pnt 6708 3127 Pnt 6708 3173 Pnt 6708 3219 Pnt 6708 3265 Pnt 6708 3311 Pnt 6708 3403 Pnt 6708 3448 Pnt 6708 3494 Pnt 6708 3540 Pnt 6708 3586 Pnt 6708 3632 Pnt 6708 4413 Pnt 6708 4459 Pnt 6708 4505 Pnt 6708 4551 Pnt 6708 4596 Pnt 6708 4642 Pnt 6708 4688 Pnt 6708 4734 Pnt 6720 280 Pnt 6720 326 Pnt 6720 372 Pnt 6720 418 Pnt 6720 464 Pnt 6720 510 Pnt 6720 556 Pnt 6720 601 Pnt 6720 647 Pnt 6720 693 Pnt 6720 739 Pnt 6720 785 Pnt 6720 831 Pnt 6720 877 Pnt 6720 923 Pnt 6720 969 Pnt 6720 1015 Pnt 6720 1061 Pnt 6720 1107 Pnt 6720 1152 Pnt 6720 1198 Pnt 6720 1244 Pnt 6720 1290 Pnt 6720 1336 Pnt 6720 1382 Pnt 6720 1428 Pnt 6720 1474 Pnt 6720 1520 Pnt 6720 1566 Pnt 6720 1612 Pnt 6720 1658 Pnt 6720 1704 Pnt 6720 1749 Pnt 6720 1795 Pnt 6720 2300 Pnt 6720 2346 Pnt 6720 2392 Pnt 6720 2438 Pnt 6720 2484 Pnt 6720 2530 Pnt 6720 2576 Pnt 6720 2622 Pnt 6720 2668 Pnt 6720 2714 Pnt 6720 2760 Pnt 6720 2806 Pnt 6720 2852 Pnt 6720 2897 Pnt 6720 2989 Pnt 6720 3035 Pnt 6720 3081 Pnt 6720 3127 Pnt 6720 3173 Pnt 6720 3219 Pnt 6720 3265 Pnt 6720 3311 Pnt 6720 3403 Pnt 6720 3448 Pnt 6720 3494 Pnt 6720 3540 Pnt 6720 3586 Pnt 6720 3632 Pnt 6720 3678 Pnt 6720 4459 Pnt 6720 4505 Pnt 6720 4551 Pnt 6720 4596 Pnt 6720 4642 Pnt 6720 4688 Pnt 6720 4734 Pnt 6720 4780 Pnt 6733 280 Pnt 6733 326 Pnt 6733 372 Pnt 6733 418 Pnt 6733 464 Pnt 6733 510 Pnt 6733 556 Pnt 6733 601 Pnt 6733 647 Pnt 6733 693 Pnt 6733 739 Pnt 6733 785 Pnt 6733 831 Pnt 6733 877 Pnt 6733 923 Pnt 6733 969 Pnt 6733 1015 Pnt 6733 1061 Pnt 6733 1107 Pnt 6733 1152 Pnt 6733 1198 Pnt 6733 1244 Pnt 6733 1290 Pnt 6733 1336 Pnt 6733 1382 Pnt 6733 1428 Pnt 6733 1474 Pnt 6733 1520 Pnt 6733 1566 Pnt 6733 1612 Pnt 6733 1658 Pnt 6733 1704 Pnt 6733 1749 Pnt 6733 1795 Pnt 6733 1841 Pnt 6733 2346 Pnt 6733 2392 Pnt 6733 2438 Pnt 6733 2484 Pnt 6733 2530 Pnt 6733 2576 Pnt 6733 2622 Pnt 6733 2668 Pnt 6733 2714 Pnt 6733 2760 Pnt 6733 2806 Pnt 6733 2852 Pnt 6733 2897 Pnt 6733 2989 Pnt 6733 3035 Pnt 6733 3081 Pnt 6733 3127 Pnt 6733 3173 Pnt 6733 3219 Pnt 6733 3265 Pnt 6733 3311 Pnt 6733 3357 Pnt 6733 3448 Pnt 6733 3494 Pnt 6733 3540 Pnt 6733 3586 Pnt 6733 3632 Pnt 6733 3678 Pnt 6733 4459 Pnt 6733 4505 Pnt 6733 4551 Pnt 6733 4596 Pnt 6733 4642 Pnt 6733 4688 Pnt 6733 4734 Pnt 6733 4780 Pnt 6746 280 Pnt 6746 326 Pnt 6746 372 Pnt 6746 418 Pnt 6746 464 Pnt 6746 510 Pnt 6746 556 Pnt 6746 601 Pnt 6746 647 Pnt 6746 693 Pnt 6746 739 Pnt 6746 785 Pnt 6746 831 Pnt 6746 877 Pnt 6746 923 Pnt 6746 969 Pnt 6746 1015 Pnt 6746 1061 Pnt 6746 1107 Pnt 6746 1152 Pnt 6746 1198 Pnt 6746 1244 Pnt 6746 1290 Pnt 6746 1336 Pnt 6746 1382 Pnt 6746 1428 Pnt 6746 1474 Pnt 6746 1520 Pnt 6746 1566 Pnt 6746 1612 Pnt 6746 1658 Pnt 6746 1704 Pnt 6746 1749 Pnt 6746 1795 Pnt 6746 1841 Pnt 6746 1887 Pnt 6746 2392 Pnt 6746 2438 Pnt 6746 2484 Pnt 6746 2530 Pnt 6746 2576 Pnt 6746 2622 Pnt 6746 2668 Pnt 6746 2714 Pnt 6746 2760 Pnt 6746 2806 Pnt 6746 2852 Pnt 6746 2897 Pnt 6746 2943 Pnt 6746 3035 Pnt 6746 3081 Pnt 6746 3127 Pnt 6746 3173 Pnt 6746 3219 Pnt 6746 3265 Pnt 6746 3311 Pnt 6746 3357 Pnt 6746 3448 Pnt 6746 3494 Pnt 6746 3540 Pnt 6746 3586 Pnt 6746 3632 Pnt 6746 3678 Pnt 6746 3724 Pnt 6746 4459 Pnt 6746 4505 Pnt 6746 4551 Pnt 6746 4596 Pnt 6746 4642 Pnt 6746 4688 Pnt 6746 4734 Pnt 6746 4780 Pnt 6746 4826 Pnt 6759 280 Pnt 6759 326 Pnt 6759 372 Pnt 6759 418 Pnt 6759 464 Pnt 6759 510 Pnt 6759 556 Pnt 6759 601 Pnt 6759 647 Pnt 6759 693 Pnt 6759 739 Pnt 6759 785 Pnt 6759 831 Pnt 6759 877 Pnt 6759 923 Pnt 6759 969 Pnt 6759 1015 Pnt 6759 1061 Pnt 6759 1107 Pnt 6759 1152 Pnt 6759 1198 Pnt 6759 1244 Pnt 6759 1290 Pnt 6759 1336 Pnt 6759 1382 Pnt 6759 1428 Pnt 6759 1474 Pnt 6759 1520 Pnt 6759 1566 Pnt 6759 1612 Pnt 6759 1658 Pnt 6759 1704 Pnt 6759 1749 Pnt 6759 1795 Pnt 6759 1841 Pnt 6759 1887 Pnt 6759 2392 Pnt 6759 2438 Pnt 6759 2484 Pnt 6759 2530 Pnt 6759 2576 Pnt 6759 2622 Pnt 6759 2668 Pnt 6759 2714 Pnt 6759 2760 Pnt 6759 2806 Pnt 6759 2852 Pnt 6759 2897 Pnt 6759 2943 Pnt 6759 3035 Pnt 6759 3081 Pnt 6759 3127 Pnt 6759 3173 Pnt 6759 3219 Pnt 6759 3265 Pnt 6759 3311 Pnt 6759 3357 Pnt 6759 3403 Pnt 6759 3494 Pnt 6759 3540 Pnt 6759 3586 Pnt 6759 3632 Pnt 6759 3678 Pnt 6759 3724 Pnt 6759 4505 Pnt 6759 4551 Pnt 6759 4596 Pnt 6759 4642 Pnt 6759 4688 Pnt 6759 4734 Pnt 6759 4780 Pnt 6759 4826 Pnt 6771 280 Pnt 6771 326 Pnt 6771 372 Pnt 6771 418 Pnt 6771 464 Pnt 6771 510 Pnt 6771 556 Pnt 6771 601 Pnt 6771 647 Pnt 6771 693 Pnt 6771 739 Pnt 6771 785 Pnt 6771 831 Pnt 6771 877 Pnt 6771 923 Pnt 6771 969 Pnt 6771 1015 Pnt 6771 1061 Pnt 6771 1107 Pnt 6771 1152 Pnt 6771 1198 Pnt 6771 1244 Pnt 6771 1290 Pnt 6771 1336 Pnt 6771 1382 Pnt 6771 1428 Pnt 6771 1474 Pnt 6771 1520 Pnt 6771 1566 Pnt 6771 1612 Pnt 6771 1658 Pnt 6771 1704 Pnt 6771 1749 Pnt 6771 1795 Pnt 6771 1841 Pnt 6771 1887 Pnt 6771 1933 Pnt 6771 2438 Pnt 6771 2484 Pnt 6771 2530 Pnt 6771 2576 Pnt 6771 2622 Pnt 6771 2668 Pnt 6771 2714 Pnt 6771 2760 Pnt 6771 2806 Pnt 6771 2852 Pnt 6771 2897 Pnt 6771 2943 Pnt 6771 2989 Pnt 6771 3081 Pnt 6771 3127 Pnt 6771 3173 Pnt 6771 3219 Pnt 6771 3265 Pnt 6771 3311 Pnt 6771 3357 Pnt 6771 3403 Pnt 6771 3494 Pnt 6771 3540 Pnt 6771 3586 Pnt 6771 3632 Pnt 6771 3678 Pnt 6771 3724 Pnt 6771 3770 Pnt 6771 4505 Pnt 6771 4551 Pnt 6771 4596 Pnt 6771 4642 Pnt 6771 4688 Pnt 6771 4734 Pnt 6771 4780 Pnt 6771 4826 Pnt 6771 4872 Pnt 6784 280 Pnt 6784 326 Pnt 6784 372 Pnt 6784 418 Pnt 6784 464 Pnt 6784 510 Pnt 6784 556 Pnt 6784 601 Pnt 6784 647 Pnt 6784 693 Pnt 6784 739 Pnt 6784 785 Pnt 6784 831 Pnt 6784 877 Pnt 6784 923 Pnt 6784 969 Pnt 6784 1015 Pnt 6784 1061 Pnt 6784 1107 Pnt 6784 1152 Pnt 6784 1198 Pnt 6784 1244 Pnt 6784 1290 Pnt 6784 1336 Pnt 6784 1382 Pnt 6784 1428 Pnt 6784 1474 Pnt 6784 1520 Pnt 6784 1566 Pnt 6784 1612 Pnt 6784 1658 Pnt 6784 1704 Pnt 6784 1749 Pnt 6784 1795 Pnt 6784 1841 Pnt 6784 1887 Pnt 6784 1933 Pnt 6784 1979 Pnt 6784 2438 Pnt 6784 2484 Pnt 6784 2530 Pnt 6784 2576 Pnt 6784 2622 Pnt 6784 2668 Pnt 6784 2714 Pnt 6784 2760 Pnt 6784 2806 Pnt 6784 2852 Pnt 6784 2897 Pnt 6784 2943 Pnt 6784 2989 Pnt 6784 3081 Pnt 6784 3127 Pnt 6784 3173 Pnt 6784 3219 Pnt 6784 3265 Pnt 6784 3311 Pnt 6784 3357 Pnt 6784 3403 Pnt 6784 3448 Pnt 6784 3540 Pnt 6784 3586 Pnt 6784 3632 Pnt 6784 3678 Pnt 6784 3724 Pnt 6784 3770 Pnt 6784 4551 Pnt 6784 4596 Pnt 6784 4642 Pnt 6784 4688 Pnt 6784 4780 Pnt 6784 4826 Pnt 6784 4872 Pnt 6797 280 Pnt 6797 326 Pnt 6797 372 Pnt 6797 418 Pnt 6797 464 Pnt 6797 510 Pnt 6797 556 Pnt 6797 601 Pnt 6797 647 Pnt 6797 693 Pnt 6797 739 Pnt 6797 785 Pnt 6797 831 Pnt 6797 877 Pnt 6797 923 Pnt 6797 969 Pnt 6797 1015 Pnt 6797 1061 Pnt 6797 1107 Pnt 6797 1152 Pnt 6797 1198 Pnt 6797 1244 Pnt 6797 1290 Pnt 6797 1336 Pnt 6797 1382 Pnt 6797 1428 Pnt 6797 1474 Pnt 6797 1520 Pnt 6797 1566 Pnt 6797 1612 Pnt 6797 1658 Pnt 6797 1704 Pnt 6797 1749 Pnt 6797 1795 Pnt 6797 1841 Pnt 6797 1887 Pnt 6797 1933 Pnt 6797 1979 Pnt 6797 2484 Pnt 6797 2530 Pnt 6797 2576 Pnt 6797 2622 Pnt 6797 2668 Pnt 6797 2714 Pnt 6797 2760 Pnt 6797 2806 Pnt 6797 2852 Pnt 6797 2897 Pnt 6797 2943 Pnt 6797 2989 Pnt 6797 3035 Pnt 6797 3127 Pnt 6797 3173 Pnt 6797 3219 Pnt 6797 3265 Pnt 6797 3311 Pnt 6797 3357 Pnt 6797 3403 Pnt 6797 3448 Pnt 6797 3540 Pnt 6797 3586 Pnt 6797 3632 Pnt 6797 3678 Pnt 6797 3724 Pnt 6797 3770 Pnt 6797 3816 Pnt 6797 4551 Pnt 6797 4596 Pnt 6797 4642 Pnt 6797 4688 Pnt 6797 4734 Pnt 6797 4780 Pnt 6797 4826 Pnt 6797 4872 Pnt 6809 280 Pnt 6809 326 Pnt 6809 372 Pnt 6809 418 Pnt 6809 464 Pnt 6809 510 Pnt 6809 556 Pnt 6809 601 Pnt 6809 647 Pnt 6809 693 Pnt 6809 739 Pnt 6809 785 Pnt 6809 831 Pnt 6809 877 Pnt 6809 923 Pnt 6809 969 Pnt 6809 1015 Pnt 6809 1061 Pnt 6809 1107 Pnt 6809 1152 Pnt 6809 1198 Pnt 6809 1244 Pnt 6809 1290 Pnt 6809 1336 Pnt 6809 1382 Pnt 6809 1428 Pnt 6809 1474 Pnt 6809 1520 Pnt 6809 1566 Pnt 6809 1612 Pnt 6809 1658 Pnt 6809 1704 Pnt 6809 1749 Pnt 6809 1795 Pnt 6809 1841 Pnt 6809 1887 Pnt 6809 1933 Pnt 6809 1979 Pnt 6809 2025 Pnt 6809 2530 Pnt 6809 2576 Pnt 6809 2622 Pnt 6809 2668 Pnt 6809 2714 Pnt 6809 2760 Pnt 6809 2806 Pnt 6809 2852 Pnt 6809 2897 Pnt 6809 2943 Pnt 6809 2989 Pnt 6809 3035 Pnt 6809 3127 Pnt 6809 3173 Pnt 6809 3219 Pnt 6809 3265 Pnt 6809 3311 Pnt 6809 3357 Pnt 6809 3403 Pnt 6809 3448 Pnt 6809 3494 Pnt 6809 3540 Pnt 6809 3586 Pnt 6809 3632 Pnt 6809 3678 Pnt 6809 3724 Pnt 6809 3770 Pnt 6809 3816 Pnt 6809 4596 Pnt 6809 4642 Pnt 6809 4688 Pnt 6809 4734 Pnt 6809 4780 Pnt 6809 4826 Pnt 6809 4872 Pnt 6822 280 Pnt 6822 326 Pnt 6822 372 Pnt 6822 418 Pnt 6822 464 Pnt 6822 510 Pnt 6822 556 Pnt 6822 601 Pnt 6822 647 Pnt 6822 693 Pnt 6822 739 Pnt 6822 785 Pnt 6822 831 Pnt 6822 877 Pnt 6822 923 Pnt 6822 969 Pnt 6822 1015 Pnt 6822 1061 Pnt 6822 1107 Pnt 6822 1152 Pnt 6822 1198 Pnt 6822 1244 Pnt 6822 1290 Pnt 6822 1336 Pnt 6822 1382 Pnt 6822 1428 Pnt 6822 1474 Pnt 6822 1520 Pnt 6822 1566 Pnt 6822 1612 Pnt 6822 1658 Pnt 6822 1704 Pnt 6822 1749 Pnt 6822 1795 Pnt 6822 1841 Pnt 6822 1887 Pnt 6822 1933 Pnt 6822 1979 Pnt 6822 2025 Pnt 6822 2071 Pnt 6822 2530 Pnt 6822 2576 Pnt 6822 2622 Pnt 6822 2668 Pnt 6822 2714 Pnt 6822 2760 Pnt 6822 2806 Pnt 6822 2852 Pnt 6822 2897 Pnt 6822 2943 Pnt 6822 2989 Pnt 6822 3035 Pnt 6822 3081 Pnt 6822 3173 Pnt 6822 3219 Pnt 6822 3265 Pnt 6822 3311 Pnt 6822 3357 Pnt 6822 3403 Pnt 6822 3448 Pnt 6822 3494 Pnt 6822 3586 Pnt 6822 3632 Pnt 6822 3678 Pnt 6822 3724 Pnt 6822 3770 Pnt 6822 3816 Pnt 6822 4596 Pnt 6822 4642 Pnt 6822 4688 Pnt 6822 4734 Pnt 6822 4780 Pnt 6822 4826 Pnt 6822 4872 Pnt 6835 280 Pnt 6835 326 Pnt 6835 372 Pnt 6835 418 Pnt 6835 464 Pnt 6835 510 Pnt 6835 556 Pnt 6835 601 Pnt 6835 647 Pnt 6835 693 Pnt 6835 739 Pnt 6835 785 Pnt 6835 831 Pnt 6835 877 Pnt 6835 923 Pnt 6835 969 Pnt 6835 1015 Pnt 6835 1061 Pnt 6835 1107 Pnt 6835 1152 Pnt 6835 1198 Pnt 6835 1244 Pnt 6835 1290 Pnt 6835 1336 Pnt 6835 1382 Pnt 6835 1428 Pnt 6835 1474 Pnt 6835 1520 Pnt 6835 1566 Pnt 6835 1612 Pnt 6835 1658 Pnt 6835 1704 Pnt 6835 1749 Pnt 6835 1795 Pnt 6835 1841 Pnt 6835 1887 Pnt 6835 1933 Pnt 6835 1979 Pnt 6835 2025 Pnt 6835 2071 Pnt 6835 2576 Pnt 6835 2622 Pnt 6835 2668 Pnt 6835 2714 Pnt 6835 2760 Pnt 6835 2806 Pnt 6835 2852 Pnt 6835 2897 Pnt 6835 2943 Pnt 6835 2989 Pnt 6835 3035 Pnt 6835 3081 Pnt 6835 3173 Pnt 6835 3219 Pnt 6835 3265 Pnt 6835 3311 Pnt 6835 3357 Pnt 6835 3403 Pnt 6835 3448 Pnt 6835 3494 Pnt 6835 3586 Pnt 6835 3632 Pnt 6835 3678 Pnt 6835 3724 Pnt 6835 3770 Pnt 6835 3816 Pnt 6835 3862 Pnt 6835 4596 Pnt 6835 4642 Pnt 6835 4688 Pnt 6835 4734 Pnt 6835 4780 Pnt 6835 4826 Pnt 6835 4872 Pnt 6848 280 Pnt 6848 326 Pnt 6848 372 Pnt 6848 418 Pnt 6848 464 Pnt 6848 510 Pnt 6848 556 Pnt 6848 601 Pnt 6848 647 Pnt 6848 693 Pnt 6848 739 Pnt 6848 785 Pnt 6848 831 Pnt 6848 877 Pnt 6848 923 Pnt 6848 969 Pnt 6848 1015 Pnt 6848 1061 Pnt 6848 1107 Pnt 6848 1152 Pnt 6848 1198 Pnt 6848 1244 Pnt 6848 1290 Pnt 6848 1336 Pnt 6848 1382 Pnt 6848 1428 Pnt 6848 1474 Pnt 6848 1520 Pnt 6848 1566 Pnt 6848 1612 Pnt 6848 1658 Pnt 6848 1704 Pnt 6848 1749 Pnt 6848 1795 Pnt 6848 1841 Pnt 6848 1887 Pnt 6848 1933 Pnt 6848 1979 Pnt 6848 2025 Pnt 6848 2071 Pnt 6848 2117 Pnt 6848 2576 Pnt 6848 2622 Pnt 6848 2668 Pnt 6848 2714 Pnt 6848 2760 Pnt 6848 2806 Pnt 6848 2852 Pnt 6848 2897 Pnt 6848 2943 Pnt 6848 2989 Pnt 6848 3035 Pnt 6848 3081 Pnt 6848 3127 Pnt 6848 3219 Pnt 6848 3265 Pnt 6848 3311 Pnt 6848 3357 Pnt 6848 3403 Pnt 6848 3448 Pnt 6848 3494 Pnt 6848 3540 Pnt 6848 3632 Pnt 6848 3678 Pnt 6848 3724 Pnt 6848 3770 Pnt 6848 3816 Pnt 6848 3862 Pnt 6848 4642 Pnt 6848 4688 Pnt 6848 4734 Pnt 6848 4780 Pnt 6848 4826 Pnt 6848 4872 Pnt 6860 280 Pnt 6860 326 Pnt 6860 372 Pnt 6860 418 Pnt 6860 464 Pnt 6860 510 Pnt 6860 556 Pnt 6860 601 Pnt 6860 647 Pnt 6860 693 Pnt 6860 739 Pnt 6860 785 Pnt 6860 831 Pnt 6860 877 Pnt 6860 923 Pnt 6860 969 Pnt 6860 1015 Pnt 6860 1061 Pnt 6860 1107 Pnt 6860 1152 Pnt 6860 1198 Pnt 6860 1244 Pnt 6860 1290 Pnt 6860 1336 Pnt 6860 1382 Pnt 6860 1428 Pnt 6860 1474 Pnt 6860 1520 Pnt 6860 1566 Pnt 6860 1612 Pnt 6860 1658 Pnt 6860 1704 Pnt 6860 1749 Pnt 6860 1795 Pnt 6860 1841 Pnt 6860 1887 Pnt 6860 1933 Pnt 6860 1979 Pnt 6860 2025 Pnt 6860 2071 Pnt 6860 2117 Pnt 6860 2163 Pnt 6860 2622 Pnt 6860 2668 Pnt 6860 2714 Pnt 6860 2760 Pnt 6860 2806 Pnt 6860 2852 Pnt 6860 2897 Pnt 6860 2943 Pnt 6860 2989 Pnt 6860 3035 Pnt 6860 3081 Pnt 6860 3127 Pnt 6860 3219 Pnt 6860 3265 Pnt 6860 3311 Pnt 6860 3357 Pnt 6860 3403 Pnt 6860 3448 Pnt 6860 3494 Pnt 6860 3540 Pnt 6860 3632 Pnt 6860 3678 Pnt 6860 3724 Pnt 6860 3770 Pnt 6860 3816 Pnt 6860 3862 Pnt 6860 3908 Pnt 6860 4642 Pnt 6860 4688 Pnt 6860 4734 Pnt 6860 4780 Pnt 6860 4826 Pnt 6860 4872 Pnt 6873 280 Pnt 6873 326 Pnt 6873 372 Pnt 6873 418 Pnt 6873 464 Pnt 6873 510 Pnt 6873 556 Pnt 6873 601 Pnt 6873 647 Pnt 6873 693 Pnt 6873 739 Pnt 6873 785 Pnt 6873 831 Pnt 6873 877 Pnt 6873 923 Pnt 6873 969 Pnt 6873 1015 Pnt 6873 1061 Pnt 6873 1107 Pnt 6873 1152 Pnt 6873 1198 Pnt 6873 1244 Pnt 6873 1290 Pnt 6873 1336 Pnt 6873 1382 Pnt 6873 1428 Pnt 6873 1474 Pnt 6873 1520 Pnt 6873 1566 Pnt 6873 1612 Pnt 6873 1658 Pnt 6873 1704 Pnt 6873 1749 Pnt 6873 1795 Pnt 6873 1841 Pnt 6873 1887 Pnt 6873 1933 Pnt 6873 1979 Pnt 6873 2025 Pnt 6873 2071 Pnt 6873 2117 Pnt 6873 2163 Pnt 6873 2622 Pnt 6873 2668 Pnt 6873 2714 Pnt 6873 2760 Pnt 6873 2806 Pnt 6873 2852 Pnt 6873 2897 Pnt 6873 2943 Pnt 6873 2989 Pnt 6873 3035 Pnt 6873 3081 Pnt 6873 3127 Pnt 6873 3173 Pnt 6873 3219 Pnt 6873 3265 Pnt 6873 3311 Pnt 6873 3357 Pnt 6873 3403 Pnt 6873 3448 Pnt 6873 3494 Pnt 6873 3540 Pnt 6873 3586 Pnt 6873 3678 Pnt 6873 3724 Pnt 6873 3770 Pnt 6873 3816 Pnt 6873 3862 Pnt 6873 3908 Pnt 6873 4688 Pnt 6873 4734 Pnt 6873 4780 Pnt 6873 4826 Pnt 6886 280 Pnt 6886 326 Pnt 6886 372 Pnt 6886 418 Pnt 6886 464 Pnt 6886 510 Pnt 6886 556 Pnt 6886 601 Pnt 6886 647 Pnt 6886 693 Pnt 6886 739 Pnt 6886 785 Pnt 6886 831 Pnt 6886 877 Pnt 6886 923 Pnt 6886 969 Pnt 6886 1015 Pnt 6886 1061 Pnt 6886 1107 Pnt 6886 1152 Pnt 6886 1198 Pnt 6886 1244 Pnt 6886 1290 Pnt 6886 1336 Pnt 6886 1382 Pnt 6886 1428 Pnt 6886 1474 Pnt 6886 1520 Pnt 6886 1566 Pnt 6886 1612 Pnt 6886 1658 Pnt 6886 1704 Pnt 6886 1749 Pnt 6886 1795 Pnt 6886 1841 Pnt 6886 1887 Pnt 6886 1933 Pnt 6886 1979 Pnt 6886 2025 Pnt 6886 2071 Pnt 6886 2117 Pnt 6886 2163 Pnt 6886 2209 Pnt 6886 2668 Pnt 6886 2714 Pnt 6886 2760 Pnt 6886 2806 Pnt 6886 2852 Pnt 6886 2897 Pnt 6886 2943 Pnt 6886 2989 Pnt 6886 3035 Pnt 6886 3081 Pnt 6886 3127 Pnt 6886 3173 Pnt 6886 3265 Pnt 6886 3311 Pnt 6886 3357 Pnt 6886 3403 Pnt 6886 3448 Pnt 6886 3494 Pnt 6886 3540 Pnt 6886 3586 Pnt 6886 3678 Pnt 6886 3724 Pnt 6886 3770 Pnt 6886 3816 Pnt 6886 3862 Pnt 6886 3908 Pnt 6886 3954 Pnt 6886 4688 Pnt 6886 4734 Pnt 6886 4780 Pnt 6886 4826 Pnt 6886 4872 Pnt 6898 280 Pnt 6898 326 Pnt 6898 372 Pnt 6898 418 Pnt 6898 464 Pnt 6898 510 Pnt 6898 556 Pnt 6898 601 Pnt 6898 647 Pnt 6898 693 Pnt 6898 739 Pnt 6898 785 Pnt 6898 831 Pnt 6898 877 Pnt 6898 923 Pnt 6898 969 Pnt 6898 1015 Pnt 6898 1061 Pnt 6898 1107 Pnt 6898 1152 Pnt 6898 1198 Pnt 6898 1244 Pnt 6898 1290 Pnt 6898 1336 Pnt 6898 1382 Pnt 6898 1428 Pnt 6898 1474 Pnt 6898 1520 Pnt 6898 1566 Pnt 6898 1612 Pnt 6898 1658 Pnt 6898 1704 Pnt 6898 1749 Pnt 6898 1795 Pnt 6898 1841 Pnt 6898 1887 Pnt 6898 1933 Pnt 6898 1979 Pnt 6898 2025 Pnt 6898 2071 Pnt 6898 2117 Pnt 6898 2163 Pnt 6898 2209 Pnt 6898 2668 Pnt 6898 2714 Pnt 6898 2760 Pnt 6898 2806 Pnt 6898 2852 Pnt 6898 2897 Pnt 6898 2943 Pnt 6898 2989 Pnt 6898 3035 Pnt 6898 3081 Pnt 6898 3127 Pnt 6898 3173 Pnt 6898 3219 Pnt 6898 3265 Pnt 6898 3311 Pnt 6898 3357 Pnt 6898 3403 Pnt 6898 3448 Pnt 6898 3494 Pnt 6898 3540 Pnt 6898 3586 Pnt 6898 3632 Pnt 6898 3678 Pnt 6898 3724 Pnt 6898 3770 Pnt 6898 3816 Pnt 6898 3862 Pnt 6898 3908 Pnt 6898 3954 Pnt 6898 4688 Pnt 6898 4734 Pnt 6898 4780 Pnt 6898 4826 Pnt 6898 4872 Pnt 6911 280 Pnt 6911 326 Pnt 6911 372 Pnt 6911 418 Pnt 6911 464 Pnt 6911 510 Pnt 6911 556 Pnt 6911 601 Pnt 6911 647 Pnt 6911 693 Pnt 6911 739 Pnt 6911 785 Pnt 6911 831 Pnt 6911 877 Pnt 6911 923 Pnt 6911 969 Pnt 6911 1015 Pnt 6911 1061 Pnt 6911 1107 Pnt 6911 1152 Pnt 6911 1198 Pnt 6911 1244 Pnt 6911 1290 Pnt 6911 1336 Pnt 6911 1382 Pnt 6911 1428 Pnt 6911 1474 Pnt 6911 1520 Pnt 6911 1566 Pnt 6911 1612 Pnt 6911 1658 Pnt 6911 1704 Pnt 6911 1749 Pnt 6911 1795 Pnt 6911 1841 Pnt 6911 1887 Pnt 6911 1933 Pnt 6911 1979 Pnt 6911 2025 Pnt 6911 2071 Pnt 6911 2117 Pnt 6911 2163 Pnt 6911 2209 Pnt 6911 2255 Pnt 6911 2714 Pnt 6911 2760 Pnt 6911 2806 Pnt 6911 2852 Pnt 6911 2897 Pnt 6911 2943 Pnt 6911 2989 Pnt 6911 3035 Pnt 6911 3081 Pnt 6911 3127 Pnt 6911 3173 Pnt 6911 3219 Pnt 6911 3311 Pnt 6911 3357 Pnt 6911 3403 Pnt 6911 3448 Pnt 6911 3494 Pnt 6911 3540 Pnt 6911 3586 Pnt 6911 3632 Pnt 6911 3724 Pnt 6911 3770 Pnt 6911 3816 Pnt 6911 3862 Pnt 6911 3908 Pnt 6911 3954 Pnt 6911 4000 Pnt 6911 4734 Pnt 6911 4780 Pnt 6911 4826 Pnt 6911 4872 Pnt 6924 280 Pnt 6924 326 Pnt 6924 372 Pnt 6924 418 Pnt 6924 464 Pnt 6924 510 Pnt 6924 556 Pnt 6924 601 Pnt 6924 647 Pnt 6924 693 Pnt 6924 739 Pnt 6924 785 Pnt 6924 831 Pnt 6924 877 Pnt 6924 923 Pnt 6924 969 Pnt 6924 1015 Pnt 6924 1061 Pnt 6924 1107 Pnt 6924 1152 Pnt 6924 1198 Pnt 6924 1244 Pnt 6924 1290 Pnt 6924 1336 Pnt 6924 1382 Pnt 6924 1428 Pnt 6924 1474 Pnt 6924 1520 Pnt 6924 1566 Pnt 6924 1612 Pnt 6924 1658 Pnt 6924 1704 Pnt 6924 1749 Pnt 6924 1795 Pnt 6924 1841 Pnt 6924 1887 Pnt 6924 1933 Pnt 6924 1979 Pnt 6924 2025 Pnt 6924 2071 Pnt 6924 2117 Pnt 6924 2163 Pnt 6924 2209 Pnt 6924 2255 Pnt 6924 2300 Pnt 6924 2714 Pnt 6924 2760 Pnt 6924 2806 Pnt 6924 2852 Pnt 6924 2897 Pnt 6924 2943 Pnt 6924 2989 Pnt 6924 3035 Pnt 6924 3081 Pnt 6924 3127 Pnt 6924 3173 Pnt 6924 3219 Pnt 6924 3311 Pnt 6924 3357 Pnt 6924 3403 Pnt 6924 3448 Pnt 6924 3494 Pnt 6924 3540 Pnt 6924 3586 Pnt 6924 3632 Pnt 6924 3724 Pnt 6924 3770 Pnt 6924 3816 Pnt 6924 3862 Pnt 6924 3908 Pnt 6924 3954 Pnt 6924 4000 Pnt 6924 4734 Pnt 6924 4780 Pnt 6924 4826 Pnt 6924 4872 Pnt 6937 280 Pnt 6937 326 Pnt 6937 372 Pnt 6937 418 Pnt 6937 464 Pnt 6937 510 Pnt 6937 556 Pnt 6937 601 Pnt 6937 647 Pnt 6937 693 Pnt 6937 739 Pnt 6937 785 Pnt 6937 831 Pnt 6937 877 Pnt 6937 923 Pnt 6937 969 Pnt 6937 1015 Pnt 6937 1061 Pnt 6937 1107 Pnt 6937 1152 Pnt 6937 1198 Pnt 6937 1244 Pnt 6937 1290 Pnt 6937 1336 Pnt 6937 1382 Pnt 6937 1428 Pnt 6937 1474 Pnt 6937 1520 Pnt 6937 1566 Pnt 6937 1612 Pnt 6937 1658 Pnt 6937 1704 Pnt 6937 1749 Pnt 6937 1795 Pnt 6937 1841 Pnt 6937 1887 Pnt 6937 1933 Pnt 6937 1979 Pnt 6937 2025 Pnt 6937 2071 Pnt 6937 2117 Pnt 6937 2163 Pnt 6937 2209 Pnt 6937 2255 Pnt 6937 2300 Pnt 6937 2760 Pnt 6937 2806 Pnt 6937 2852 Pnt 6937 2897 Pnt 6937 2943 Pnt 6937 2989 Pnt 6937 3035 Pnt 6937 3081 Pnt 6937 3127 Pnt 6937 3173 Pnt 6937 3219 Pnt 6937 3265 Pnt 6937 3357 Pnt 6937 3403 Pnt 6937 3448 Pnt 6937 3494 Pnt 6937 3540 Pnt 6937 3586 Pnt 6937 3632 Pnt 6937 3678 Pnt 6937 3770 Pnt 6937 3816 Pnt 6937 3862 Pnt 6937 3908 Pnt 6937 3954 Pnt 6937 4000 Pnt 6937 4734 Pnt 6937 4780 Pnt 6937 4826 Pnt 6937 4872 Pnt 6949 280 Pnt 6949 326 Pnt 6949 372 Pnt 6949 418 Pnt 6949 464 Pnt 6949 510 Pnt 6949 556 Pnt 6949 601 Pnt 6949 647 Pnt 6949 693 Pnt 6949 739 Pnt 6949 785 Pnt 6949 831 Pnt 6949 877 Pnt 6949 923 Pnt 6949 969 Pnt 6949 1015 Pnt 6949 1061 Pnt 6949 1107 Pnt 6949 1152 Pnt 6949 1198 Pnt 6949 1244 Pnt 6949 1290 Pnt 6949 1336 Pnt 6949 1382 Pnt 6949 1428 Pnt 6949 1474 Pnt 6949 1520 Pnt 6949 1566 Pnt 6949 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b(tongues)f(in)f(a)i(Hill's)d(equation)i (with)f(quasi-p)s(erio)s(dic)e(forcing,)354 2127 y(\(3.30\),)50 b(with)42 b(t)m(w)m(o)j(frequencies.)80 b(V)-8 b(alue)44 b Fn(a)g FB(is)e(on)i(the)g(horizon)m(tal)f(direction)f(and)i Fn(b)f FB(is)354 2240 y(on)f(v)m(ertical.)74 b(Mark)m(ed)42 b(p)s(oin)m(ts)e(corresp)s(ond)h(to)h(the)f(complemen)m(tary)h(of)g (tongues)g(\(the)354 2353 y(sp)s(ectrum\).)c(P)m(airs)25 b(of)g(the)g(form)g(\()p Fn(k)1588 2367 y FC(1)1628 2353 y Fn(;)15 b(k)1715 2367 y FC(2)1755 2353 y FB(\))26 b(in)d(the)j Fn(b)f FB(=)g(0)g(axis)g(indicate)f(that)h(the)h(resonance)354 2466 y(tongue)31 b(whic)m(h)e(emanates)j(from)e(this)f(p)s(oin)m(t)g (has)i(rotation)f(n)m(um)m(b)s(er)f(\()p Fn(k)2865 2480 y FC(1)2905 2466 y Fn(!)2962 2480 y FC(1)3022 2466 y FB(+)20 b Fn(k)3160 2480 y FC(2)3199 2466 y Fn(!)3256 2480 y FC(2)3295 2466 y FB(\))p Fn(=)p FB(2)28 2768 y Fy(The)41 b(rotation)d(n)m(um)m(b)s(er)j(has)f(v)m(ery)h(go)s(o)s(d)e 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Fs(\024)g Fx(\013)q Fy(\()p Fx(\025)21 b Fy(+)i Fx(\016)t Fy(;)17 b Fx(q)2642 4954 y FC(2)2681 4939 y Fy(\))p Fx(:)-118 5159 y Fy(No)m(w)33 b(the)g(result)g(follo)m(ws)e(from)g(the)i(already)f(pro)m(v)m(ed)i (con)m(tin)m(uit)m(y)f(in)f Fx(\025)p Fy(.)p Fh(\003)28 5280 y Fy(As)42 b(w)m(e)h(are)f(dealing)e(with)i(quasi-p)s(erio)s(dic)d (p)s(oten)m(tials,)k(this)e(result)h(ma)m(y)g(seem)g(a)f(bit)g (restrictiv)m(e,)j(as,)-118 5400 y(for)c(instance,)j(it)d(cannot)h(b)s (e)g(applied)f(if)f Fx(q)1521 5415 y FC(1)1561 5400 y Fy(\()p Fx(t)p Fy(\))j(=)f Fx(Q)p Fy(\()p Fx(!)2007 5415 y FC(1)2047 5400 y Fx(t)p Fy(\))g(and)g Fx(q)2402 5415 y FC(2)2441 5400 y Fy(\()p Fx(t)p Fy(\))h(=)g Fx(Q)p Fy(\()p Fx(!)2888 5415 y FC(2)2927 5400 y Fx(t)p Fy(\),)h(where)f Fx(Q)g Fy(:)g Fw(T)3611 5364 y Fr(d)3696 5400 y Fs(!)f Fw(R)52 b Fy(is)-118 5521 y(con)m(tin)m(uous)35 b(and)f Fx(!)624 5536 y FC(1)663 5521 y Fx(;)17 b(!)768 5536 y FC(2)841 5521 y Fy(are)34 b(t)m(w)m(o)g(irrational)d(frequency)36 b(v)m(ectors)f(as)g(close)f(as)g(w)m(e)h(w)m(an)m(t.)48 b(T)-8 b(o)34 b(w)m(ork)h(this)f(out)-118 5641 y(w)m(e)i(state)g(the)g (follo)m(wing)d(theorem,)j(whic)m(h)g(giv)m(es)f(m)m(uc)m(h)h(b)s (etter)g(con)m(tin)m(uit)m(y)g(prop)s(erties)f(for)g(quasi-p)s(erio)s (dic)-118 5761 y(p)s(oten)m(tials)1900 6155 y(62)p eop %%Page: 63 67 63 66 bop -118 219 a Fv(Theorem)37 b(3.3.9)h(\([4)o(],)g([47]\))49 b FA(L)-5 b(et)47 b Fx(Q)1379 234 y Fr(n)1426 219 y FA(,)j(for)c Fx(n)k Fs(\025)h Fy(1)p FA(,)e(b)-5 b(e)46 b(a)h(se)-5 b(quenc)g(e)46 b(of)g(c)-5 b(ontinuous)47 b(functions)f(on)h(the)-118 339 y(torus)39 b Fw(T)199 303 y Fr(d)281 339 y FA(which)e(c)-5 b(onver)g(ge)37 b(uniformly)h(to)g(a)g(function)g Fx(Q)h FA(\(henc)-5 b(e)37 b(c)-5 b(ontinuous\).)54 b(L)-5 b(et)39 b Fx(!)3240 354 y Fr(n)3320 339 y Fs(2)c Fw(R)3487 303 y Fr(d)3571 339 y FA(c)-5 b(onver)g(ging)-118 459 y(to)40 b(a)f(r)-5 b(ational)5 b(ly)40 b(indep)-5 b(endent)38 b(ve)-5 b(ctor)39 b Fx(!)h Fs(2)d Fw(R)1645 423 y Fr(d)1691 459 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b(onver)g(genc)g(e)33 b(is)i(uniform)f(in)h Fy(\()p Fx(\025;)17 b(\036)p Fy(\))27 b Fs(2)h Fx(J)j Fs(\002)23 b Fw(T)1968 1441 y Fr(d)2046 1477 y FA(for)35 b(every)g(c)-5 b(omp)g(act)34 b(subset)h Fx(J)i Fs(\032)28 b Fw(R)5 b FA(.)-118 1699 y Fv(Remark)37 b(3.3.10)49 b FA(Note)30 b(that)f(in)g(the)g(right)g(hand)f(side)h(of)g(e)-5 b(quation)28 b(\(3.31\))g(we)h(haven)-10 b('t)29 b(written)g(any)g(dep) -5 b(en-)-118 1820 y(denc)g(e)34 b(on)h(the)h(initial)e(phase)h Fx(\036)p FA(.)46 b(This)34 b(is)h(b)-5 b(e)g(c)g(ause)35 b(the)g(r)-5 b(otation)36 b(numb)-5 b(er)35 b(with)g(initial)g(phase)f Fx(\036)h FA(is)g(c)-5 b(onstant)-118 1940 y(over)44 b(the)g(closur)-5 b(e)43 b(in)h Fw(T)806 1904 y Fr(d)894 1940 y FA(of)g(the)g(orbit)g Fx(!)t(t)28 b Fy(+)h Fx(\036)44 b FA(for)g Fx(t)h Fs(2)g Fw(R)5 b FA(,)52 b(which)43 b(turns)h(to)h(b)-5 b(e)43 b(the)i(whole)e(sp)-5 b(ac)g(e)43 b(if)h(the)-118 2060 y(c)-5 b(omp)g(onents)33 b(of)i Fx(!)j FA(ar)-5 b(e)35 b(r)-5 b(ational)5 b(ly)34 b(indep)-5 b(endent.)28 2282 y Fy(W)d(e)42 b(don't)f(giv)m(e)g(a)g(pro)s(of)f(of)h (this)g(theorem,)i(b)s(ecause)f(it)f(is)f(similar)e(to)j(the)h(one)f (of)g(the)g(con)m(tin)m(uit)m(y)g(in)-118 2403 y Fx(\025)p Fy(.)52 b(Nev)m(ertheless,)39 b(it)c(is)g(imp)s(ortan)m(t)f(to)h (stress)i(that)f(all)d(these)k(results)f(can)g(b)s(e)g(obtained)f(b)s (ecause)i(the)f(\015o)m(w)-118 2523 y(for)i(the)h(angles)f(is)g(on)h(a) f(compact)g(space,)j(top)s(ologically)35 b Fw(T)2178 2487 y Fr(d)p FC(+1)2312 2523 y Fy(.)61 b(Indeed,)42 b(the)d(compactness)h(of)e(the)h(space)-118 2644 y Fx(B)d Fy(=)31 b Fw(P)158 2607 y FC(1)200 2644 y Fy(\()p Fw(R)5 b Fy(\))29 b Fs(\002)24 b Fw(T)535 2607 y Fr(d)614 2644 y Fy(implies)32 b(that)i(the)h(set)h(of)e(all)f(normalized)f(Borel)i (measures)i(is)e(compact)g(with)g(the)h(w)m(eak)-118 2764 y(top)s(ology)-8 b(.)42 b(The)33 b(other)g(argumen)m(ts)g(are)f (similar)e(to)i(the)h(already)f(men)m(tioned)g(pro)s(of.)28 2884 y(Finally)e(w)m(e)k(w)m(an)m(t)g(to)f(relate)f(the)h(rotation)e(n) m(um)m(b)s(er)i(to)g(a)g(limit,)c(called)j(usually)g(the)h FA(inte)-5 b(gr)g(ate)g(d)35 b(density)-118 3005 y(of)f(states)41 b Fy(and)33 b(whic)m(h)g(is)f(often)h(used)h(in)d(this)i(con)m(text.)44 b(Recall)31 b(\014rst)i(that)g(w)m(e)g(sa)m(w)h(that)1116 3269 y Fx(\013)q Fy(\()p Fx(\025)p Fy(\))27 b(=)1452 3201 y Fx(\031)t(N)10 b Fy(\()p Fx(a;)17 b(b)p Fy(;)g Fx(\025)p Fy(\))p 1452 3246 461 4 v 1575 3337 a Fx(b)23 b Fs(\000)g Fx(a)1955 3269 y Fy(when)34 b Fx(b)22 b Fs(\000)h Fx(a)28 b Fs(!)f Fy(+)p Fs(1)p Fx(;)-118 3522 y Fy(where)k Fx(N)10 b Fy(\()p Fx(a;)17 b(b)p Fy(;)g Fx(\025)p Fy(\))30 b(is)g(the)g(n)m(um)m(b)s(er)h(of)e(zero)s(es)i(of)f(a)g(solution)e(of) i(equation)f(\(3.21\))h(in)f(the)h(in)m(terv)-5 b(al)29 b([)p Fx(a;)17 b(b)p Fy(].)43 b(No)m(w)-118 3643 y(consider)33 b(the)g(regular)e(eigen)m(v)-5 b(alue)32 b(problem)g(on)g([)p Fx(a;)17 b(b)p Fy(])1172 3775 y Fq(\032)1288 3854 y Fs(\000)p Fx(x)1420 3818 y Ft(0)q(0)1486 3854 y Fy(+)22 b Fx(q)t Fy(\()p Fx(t)p Fy(\))p Fx(x)28 b Fy(=)f Fx(\025x)61 b Fy(on)f Fx(a)28 b Fs(\024)g Fx(t)g Fs(\024)g Fx(b)1288 3975 y(x)p Fy(\()p Fx(a)p Fy(;)17 b Fx(\025)p Fy(\))28 b(=)g Fx(x)p Fy(\()p Fx(b)p Fy(;)17 b Fx(\025)p Fy(\))28 b(=)f(0)p Fx(:)3766 3915 y Fy(\(3.32\))28 4188 y(It)j(follo)m(ws)f (from)g(Sturm's)i(comparison)e(theorem)h(\(see,)i(for)d(instance,)i ([17]\))f(that)g(the)h(n)m(um)m(b)s(er)f Fx(\027)6 b Fy(\()p Fx(a;)17 b(b)p Fy(;)g Fx(\025)p Fy(\))-118 4308 y(of)32 b(eigen)m(v)-5 b(alues)33 b Fx(\025)555 4323 y Fr(j)619 4308 y Fs(\024)28 b Fx(\025)k Fy(of)g(the)h(ab)s(o)m(v)m(e)h (problem)d(di\013ers)i(from)e Fx(N)10 b Fy(\()p Fx(a;)17 b(b)p Fy(;)g Fx(\025)p Fy(\))33 b(b)m(y)g Fs(\006)p Fy(1,)g(so)1492 4580 y(lim)1428 4643 y Fr(b)p Ft(\000)p Fr(a)p Ft(!1)1718 4513 y Fx(\027)6 b Fy(\()p Fx(a;)17 b(b)p Fy(;)g Fx(\025)p Fy(\))p 1718 4558 368 4 v 1795 4649 a Fx(b)23 b Fs(\000)f Fx(a)2123 4580 y Fy(=)2237 4513 y Fx(\013)q Fy(\()p Fx(\025)p Fy(\))p 2237 4558 196 4 v 2305 4649 a Fx(\031)2442 4580 y(;)-118 4832 y Fy(and)33 b(the)g(left)f(hand)i(side)e(is)h(often)g (called)f(the)h FA(inte)-5 b(gr)g(ate)g(d)35 b(density)g(of)g(states)p Fy(,)e(used)h(in)e(quan)m(tum)h(mec)m(hanics)-118 4952 y(\(see)i([4])g(and)f(references)j(therein\))d(and)h(written)f(as)h Fx(k)s Fy(\()p Fx(\025)p Fy(\))f(or)h Fx(N)10 b Fy(\()p Fx(\025)p Fy(\).)49 b(This)35 b(is)f(the)h(distribution)d(function)i (of)-118 5072 y(a)e(measure)h Fx(dk)s Fy(,)f(whic)m(h)i(is)e(the)h FA(density)h(of)h(states)p Fy(.)-118 5360 y Fg(3.3.2)136 b(Extension)46 b(to)f(complex)g Ff(\025)h Fg(and)e(relation)j(with)e(W) -11 b(eyl's)46 b Ff(m)p Fg(-functions)-118 5545 y Fy(As)29 b(w)m(e)g(will)d(see)j(the)g(rotation)d(n)m(um)m(b)s(er)j(can)f(b)s(e)g (view)m(ed)i(as)e(the)h(imaginary)c(part)j(of)g(a)g(Herglotz)f (function,)i(an)-118 5666 y(analytic)34 b(function)g(in)g(the)i(upp)s (er)f(half-plane.)49 b(T)-8 b(o)35 b(de\014ne)i(suc)m(h)f(a)f(function) f(it)g(will)f(b)s(e)i(v)m(ery)i(imp)s(ortan)m(t)c(to)-118 5786 y(use)d(the)g(prop)s(erties)f(of)g(quasi-p)s(erio)s(dicit)m(y)f (that)h(w)m(e)h(pro)m(v)m(ed)h(for)e(W)-8 b(eyl's)30 b Fx(m)g Fy(functions)f(and)h(their)e(extensions)-118 5906 y(to)k(the)h(torus)g Fw(T)482 5870 y Fr(d)526 5906 y Fy(.)1900 6155 y(63)p eop %%Page: 64 68 64 67 bop 28 219 a Fy(Recall)28 b(that)h(in)g(theorem)g(3.2.26)f(w)m(e) j(constructed)g(extensions)f Fx(M)10 b Fy(\()p Fs(\001)p Fy(;)17 b Fx(\025)p Fy(\))28 b(:)g Fw(T)2877 182 y Fr(d)2948 219 y Fs(!)g Fw(R)5 b Fy(,)36 b(as)29 b(regular)f(as)i Fx(Q)p Fy(,)g(for)-118 339 y Fx(\025)f Fy(in)f(the)h(resolv)m(en)m(t)h (set)g(\(w)m(ell,)f(for)f(real)g Fx(\025)h Fy(in)f(this)h(set,)h(w)m(e) g(had)f(to)g(sligh)m(tly)e(mo)s(dify)g Fx(M)40 b Fy(if)28 b(w)m(e)i(didn't)e(w)m(an)m(t)i(to)-118 459 y(pro)5 b(jectivize)32 b(an)m(ything\).)43 b(Let)32 b(us)g(consider)h(for)e(the)h(momen)m(t)f (Im)h Fx(\025)c(>)f Fy(0.)43 b(T)-8 b(o)32 b(an)m(y)g(quasi-p)s(erio)s (dic)e(function)-118 580 y(\(or)k(rather)h(to)f(its)g(con)m(tin)m(uous) h(extension)g(to)f Fw(T)1702 544 y Fr(d)1746 580 y Fy(\))g(it)g(is)g (naturally)f(asso)s(ciated)h(the)h(a)m(v)m(erage,)h(whic)m(h)f(alw)m(a) m(ys)-118 700 y(exists.)44 b(W)-8 b(e)33 b(therefore)g(de\014ne)1083 944 y Fx(w)s Fy(\()p Fx(\025)p Fy(\))27 b(=)g([)p Fx(M)1540 959 y FC(+)1600 944 y Fy(\()p Fs(\001)p Fx(;)17 b(\025)p Fy(\)])1831 973 y Fk(T)1881 954 y Fj(d)1943 944 y Fy(=)2047 808 y Fq(Z)2102 1034 y Fk(T)2152 1015 y Fj(d)2203 944 y Fx(M)2297 959 y FC(+)2357 944 y Fy(\()p Fx(\022)s Fy(;)g Fx(\025)p Fy(\))p Fx(d\026)p Fy(\()p Fx(\022)s Fy(\))-118 1211 y(where)47 b Fx(\026)e Fy(stands)i(for)e(the)h(usual)f(Haar)g (measure)h(on)g(the)g(torus.)83 b(W)-8 b(e)46 b(recall)e(that)i(in)f (section)h(3.2.4)f(w)m(e)-118 1331 y(established)31 b(sev)m(eral)h (iden)m(tities)f(b)s(et)m(w)m(een)i Fx(M)1588 1346 y Ft(\000)1647 1331 y Fy(\()p Fs(\001)p Fy(;)17 b Fx(\025)p Fy(\),)31 b Fx(M)2004 1346 y FC(+)2063 1331 y Fy(\()p 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b(\025)p Fy(\)])2084 2287 y Fk(T)2134 2268 y Fj(d)2191 2258 y Fs(\000)23 b Fy([)p Fx(M)2412 2273 y FC(+)2471 2258 y Fy(\()p Fs(\001)p Fx(;)17 b(\025)p Fy(\)])2703 2287 y Fk(T)2753 2268 y Fj(d)2804 2258 y Fx(:)-118 2534 y Fy(W)-8 b(e)33 b(will)d(no)m(w)j(sho)m(w)h(that)e(the)h(follo)m(wing) d(iden)m(tit)m(y)j(holds)1318 2754 y([)p Fx(M)1439 2769 y Ft(\000)1498 2754 y Fy(\()p Fs(\001)p Fx(;)17 b(\025)p Fy(\)])1730 2783 y Fk(T)1780 2764 y Fj(d)1842 2754 y Fy(=)28 b Fs(\000)17 b Fy([)p Fx(M)2161 2769 y FC(+)2220 2754 y Fy(\()p Fs(\001)p Fx(;)g(\025)p Fy(\)])2451 2783 y Fk(T)2501 2764 y Fj(d)2553 2754 y Fx(:)-118 2974 y Fy(T)-8 b(o)33 b(this)f(end)h(note)g(that)747 3224 y Fx(m)832 3239 y Ft(\000)914 3224 y Fy(+)22 b Fx(m)1097 3239 y FC(+)1184 3224 y Fy(=)1297 3157 y(\()p Fx(\037)1396 3172 y Ft(\000)1456 3157 y Fx(\037)1517 3172 y FC(+)1576 3157 y Fy(\))1614 3120 y Ft(0)p 1297 3201 340 4 v 1347 3292 a Fx(\037)1408 3307 y Ft(\000)1467 3292 y Fx(\037)1528 3307 y FC(+)1675 3224 y Fy(=)1806 3157 y Fx(d)p 1788 3201 86 4 v 1788 3292 a(dt)1901 3224 y Fy(log)16 b(\()p Fx(\037)2142 3239 y Ft(\000)2201 3224 y Fx(\037)2262 3239 y FC(+)2321 3224 y Fy(\))28 b(=)2518 3157 y Fx(d)p 2500 3201 V 2500 3292 a(dt)2613 3224 y Fy(log)17 b Fx(G)p Fy(\()p Fx(t;)g(t)p Fy(;)g Fx(\025)p Fy(\))p Fx(:)-118 3497 y Fy(No)m(w,)33 b Fx(G)p Fy(\()p Fx(t;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))32 b(is)g(quasi-p)s(erio)s(dic)f(and)i(its)f (imaginary)d(part)k(is)f(b)s(ounded)h(a)m(w)m(a)m(y)h(from)d(zero)i (and,)g(since)-118 3617 y(log)16 b Fx(G)p Fy(\()p Fx(t;)h(t)p Fy(;)g Fx(\025)p Fy(\))37 b(is)f(b)s(ounded,)j(w)m(e)f(conclude)g(that) e Fx(m)1816 3632 y Ft(\000)1901 3617 y Fy(+)25 b Fx(m)2087 3632 y Ft(\000)2183 3617 y Fy(has)38 b(zero)f(a)m(v)m(erage)h(and)f (the)g(desired)h(iden)m(tities)-118 3738 y(follo)m(w.)k(Summing)30 b(up,)j(w)m(e)h(ha)m(v)m(e)g(pro)m(v)m(ed)g(the)f(follo)m(wing)d(prop)s (osition)-118 3966 y Fv(Prop)s(osition)35 b(3.3.11)j(\([47]\))49 b FA(F)-7 b(or)33 b(Im)i Fx(\025)27 b Fs(6)p Fy(=)h(0)34 b FA(we)h(have)f(that)784 4243 y Fs(\000)878 4103 y Fq(\024)1073 4176 y Fy(1)p 940 4220 314 4 v 940 4311 a(2\000\()p Fs(\001)p Fy(;)17 b Fx(\025)p Fy(\))1264 4103 y Fq(\025)1317 4342 y Fk(T)1367 4323 y Fj(d)1429 4243 y Fy(=)27 b Fs(\000)17 b Fy([)q Fx(M)1748 4258 y Ft(\000)1807 4243 y Fy(\()p Fs(\001)p Fx(;)g(\025)p Fy(\)])2038 4272 y Fk(T)2088 4253 y Fj(d)2151 4243 y Fy(=)27 b([)p Fx(M)2375 4258 y FC(+)2435 4243 y Fy(\()p Fs(\001)p Fx(;)17 b(\025)p Fy(\)])2666 4272 y Fk(T)2716 4253 y Fj(d)2778 4243 y Fy(=)28 b Fx(w)s Fy(\()p Fx(\025)p Fy(\))p Fx(:)28 4540 y Fy(W)-8 b(e)26 b(ha)m(v)m(e)g(seen)h(that)e(for)f(an)m(y)i Fx(\022)31 b Fs(2)d Fw(T)1366 4504 y Fr(d)1410 4540 y Fy(,)e(the)g(functions)f Fx(M)2131 4555 y Ft(\006)2190 4540 y Fy(\()p Fs(\001)p Fy(;)17 b Fx(\025)p Fy(\))25 b(w)m(ere)h(Herglotz)f(functions.)41 b(W)-8 b(e)25 b(therefore)-118 4661 y(conclude)36 b(that)f(the)g(function)g Fx(w)f Fy(=)e Fx(w)s Fy(\()p Fx(\025)p Fy(\))j(is)f(a)h(Herglotz)g(function)g(and,)h (in)e(particular,)g(it)g(is)h(holomorphic)-118 4781 y(in)i(the)i(upp)s (er)g(half-plane.)59 b(Moreo)m(v)m(er,)41 b(from)c(prop)s(osition)f (3.2.29)i(w)m(e)h(get)g(the)f(follo)m(wing)e(inequalities)g(for)-118 4902 y(Im)c Fx(\025)27 b Fs(6)p Fy(=)h(0)1282 5004 y(Im)k Fx(w)s Fy(\()p Fx(\025)p Fy(\))p 1282 5048 355 4 v 1356 5140 a(Im)g Fx(\025)1674 5071 y(>)c Fy(0)p Fx(;)211 b Fy(Re)33 b Fx(w)s Fy(\()p Fx(\025)p Fy(\))27 b Fx(<)g Fy(0)p Fx(:)-118 5277 y Fy(As)33 b(a)f(consequence,)j(the)e(function)f Fx(h)p Fy(\()p Fx(\025)p Fy(\))c(=)f(Im)32 b Fx(w)s Fy(\()p Fx(\025)p Fy(\))g(is)g(harmonic)e(in)i(the)h(upp)s(er)g(half-plane,)d (hence)k(it)d(has)-118 5397 y(a)h(represen)m(tation)i(of)e(the)h(form) 1385 5674 y Fx(h)p Fy(\()p Fx(\025)p Fy(\))28 b(=)f(Im)32 b Fx(\025)1928 5538 y Fq(Z)2027 5565 y FC(+)p Ft(1)1983 5764 y(\0001)2227 5606 y Fx(d\033)t Fy(\()p Fx(s)p Fy(\))p 2183 5651 320 4 v 2183 5742 a Fs(j)p Fx(s)22 b Fs(\000)h Fx(\025)p Fs(j)2464 5713 y FC(2)3766 5674 y Fy(\(3.33\))1900 6155 y(64)p eop %%Page: 65 69 65 68 bop -118 219 a Fy(where)43 b Fx(\033)t Fy(\()p Fx(s)p Fy(\))e(is)g(a)g(monotone)g(increasing)g(function)g(on)g Fw(R)53 b Fy(and)41 b Fx(d\033)k Fy(is)d(the)g(asso)s(ciated)f (exterior)g(measure.)-118 339 y(Moreo)m(v)m(er,)34 b(at)e(the)h(p)s (oin)m(ts)g(of)f(con)m(tin)m(uit)m(y)g(of)g Fx(\033)37 b Fy(one)c(has)1144 588 y Fx(\033)t Fy(\()p Fx(s)1287 603 y FC(2)1326 588 y Fy(\))22 b Fs(\000)h Fx(\033)t Fy(\()p Fx(s)1629 603 y FC(1)1668 588 y Fy(\))28 b(=)1852 521 y(1)p 1847 565 59 4 v 1847 656 a Fx(\031)1934 588 y Fy(lim)1933 648 y Fr(")p Ft(!)p FC(0)2088 588 y Fy(Im)k Fx(w)s Fy(\()p Fx(s)21 b Fy(+)h Fx(i")p Fy(\))p Fx(ds:)-118 831 y Fy(Also,)32 b(for)g(an)m(y)h(con)m(tin)m(uous)h(function)e Fx(f)11 b Fy(\()p Fx(s)p Fy(\))32 b(with)g(compact)g(supp)s(ort,)h(w)m (e)h(ha)m(v)m(e)g(that)940 968 y Fq(Z)1040 994 y FC(+)p Ft(1)995 1193 y(\0001)1186 1103 y Fx(f)11 b Fy(\()p Fx(s)p Fy(\))p Fx(d\033)t Fy(\()p Fx(s)p Fy(\))27 b(=)54 b(lim)1729 1171 y Fr(")p Ft(!)p 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Fx(t)c Fs(!)f(1)36 b FA(exists)h(and)f(is)g(indep)-5 b(endent)36 b(of)g(the)h(solution)g (chosen)e(\(as)i(long)f(as)g(\(3.34\))g(holds\).)50 b(F)-7 b(or)35 b(r)-5 b(e)g(al)37 b Fx(\025)-118 2906 y FA(we)e(have)h Fx(h)p Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))29 b Fs(!)h Fx(\013)q Fy(\()p Fx(\025)p Fy(\))35 b FA(as)h Fx(t)30 b Fs(!)f(1)p FA(.)48 b(Mor)-5 b(e)g(over,)36 b(for)g(Im)e Fx(\025)c(>)f Fy(0)p FA(,)36 b Fx(h)h FA(is)e(the)h (imaginary)g(p)-5 b(art)36 b(of)f Fx(w)s FA(,)h(in)g(the)-118 3026 y(ab)-5 b(ove)34 b(notation.)28 3243 y Fy(F)-8 b(rom)31 b(this)h(theorem)h(w)m(e)g(can)g(also)f(deriv)m(e)h(the)g(follo)m (wing.)-118 3460 y Fv(Theorem)k(3.3.13)h(\(cf.[47]\))48 b FA(The)34 b(function)g Fx(h)p Fy(\()p Fx(\025)p Fy(\))28 b(=)f FA(Im)35 b Fx(w)s Fy(\()p Fx(\025)p Fy(\))e FA(is)i(c)-5 b(ontinuous)34 b(in)g(the)h(close)-5 b(d)33 b(upp)-5 b(er)34 b(half-)-118 3581 y(plane)g(Im)g Fx(\025)28 b Fs(\025)g Fy(0)p FA(,)34 b(and)1515 3701 y Fy(lim)1489 3769 y Fr(")p Ft(!)p FC(0)1628 3750 y Fp(+)1694 3701 y Fx(h)p Fy(\()p Fx(\025)22 b Fy(+)h Fx(i")p Fy(\))k(=)h Fx(\013)q Fy(\()p Fx(\025)p Fy(\))-118 3906 y FA(for)35 b(r)-5 b(e)g(al)34 b Fx(\025)p FA(.)28 4123 y Fy(The)g(pro)s(ofs)e(can) h(b)s(e)g(found)f(in)g(the)h(same)f(reference.)46 b(T)-8 b(o)32 b(pro)m(v)m(e)i(theorem)e(3.3.13,)g(w)m(e)i(de\014ne)1349 4333 y Fx(\036)p Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))27 b(=)g Fx(\036)1807 4348 y FC(1)1846 4333 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))22 b Fs(\000)h Fx(i\036)2271 4348 y FC(2)2310 4333 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))p Fx(;)-118 4544 y Fy(b)s(eing)29 b Fx(\036)200 4559 y FC(1)269 4544 y Fy(and)i Fx(\036)515 4559 y FC(2)584 4544 y Fy(t)m(w)m(o)f(linearly)e(indep)s(enden)m(t)k(solutions)d(with)g (W)-8 b(ronskian)30 b(one,)h(whic)m(h)g(are)f(real)f(for)h(real)f Fx(\025)p Fy(.)-118 4664 y(Then)h(\(3.34\))f(holds)g(and)h Fx(\036)f Fy(is)g(en)m(tire)g(with)g(no)g(zero)s(es)i(if)d Fx(t)g Fs(\025)g Fy(0)h(and)h(Im)i Fx(\025)27 b Fs(\025)h Fy(0.)42 b(Hence)31 b(the)f(function)f Fx(h)p Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))-118 4784 y(is)35 b(harmonic)f(in)g (the)i(set)g(Im)c Fx(\025)g Fs(\025)h Fy(0)i(for)g(ev)m(ery)i Fx(t)32 b Fs(\025)h Fy(0)i(and)h(w)m(e)g(can)f(therefore)h(represen)m (t)i(it)c(in)g(the)i(form)e(of)-118 4905 y(\(3.33\))e(for)g(a)g (suitable)g(monotone)f(increasing)h(function)g Fx(\033)t Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))33 b(whic)m(h)g(satis\014es)g (that)1179 5183 y Fx(\033)t Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)1412 5198 y FC(2)1451 5183 y Fy(\))22 b Fs(\000)h Fx(\033)t Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)1844 5198 y FC(1)1883 5183 y Fy(\))27 b(=)2067 5116 y(1)p 2062 5160 59 4 v 2062 5252 a Fx(\031)2147 5048 y Fq(Z)2247 5074 y Fr(\025)2288 5083 y Fp(2)2203 5273 y Fr(\025)2244 5282 y Fp(1)2344 5183 y Fx(h)p Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))p Fx(d\025)-118 5455 y Fy(for)31 b FA(al)5 b(l)41 b Fy(real)30 b Fx(\025)421 5470 y FC(1)460 5455 y Fy(,)i Fx(\025)576 5470 y FC(2)615 5455 y Fy(,)g(b)s(ecause)h(w) m(e)f(ha)m(v)m(e)g(already)f(seen)i(that)e Fx(h)p Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))31 b(is)g(con)m(tin)m(uous)h(in)e(Im)i Fx(\025)c Fs(\025)g Fy(0.)43 b(No)m(w,)32 b(due)-118 5576 y(to)g(theorem)g(3.3.12,)g(w)m(e)i(ha)m(v)m(e)g(that)e Fx(h)p Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))28 b Fs(!)f Fx(h)p Fy(\()p Fx(\025)p Fy(\))33 b(when)g Fx(t)28 b Fs(!)f(1)33 b Fy(in)e(the)i(op)s(en)g(upp)s(er)g(plane)f(Im)g Fx(\025)c(>)g Fy(0.)28 5696 y(Therefore,)34 b(for)e(the)h Fx(\033)k Fy(of)32 b(the)h(limit)c(function)j Fx(h)p Fy(\()p Fx(\025)p Fy(\),)g(w)m(e)i(ha)m(v)m(e)g(that,)e(at)h(the)g(p)s (oin)m(ts)f(of)g(con)m(tin)m(uit)m(y)g(of)h Fx(\033)t Fy(,)1208 5906 y Fx(\033)t Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)1441 5921 y FC(2)1480 5906 y Fy(\))22 b Fs(\000)h Fx(\033)t Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)1873 5921 y FC(1)1912 5906 y Fy(\))28 b Fs(!)f Fx(\033)t Fy(\()p Fx(\025)2259 5921 y FC(2)2299 5906 y Fy(\))22 b Fs(\000)g Fx(\033)t Fy(\()p Fx(\025)2612 5921 y FC(1)2652 5906 y Fy(\))1900 6155 y(65)p eop %%Page: 66 70 66 69 bop -118 219 a Fy(as)33 b Fx(t)28 b Fs(!)f Fy(+)p Fs(1)p Fy(;)32 b(i.e.,)1139 402 y Fx(\033)t Fy(\()p Fx(\025)1293 417 y FC(2)1332 402 y Fy(\))22 b Fs(\000)h Fx(\033)t Fy(\()p Fx(\025)1646 417 y FC(1)1685 402 y Fy(\))k(=)71 b(lim)1854 461 y Fr(t)p Ft(!)p FC(+)p Ft(1)2107 334 y Fy(1)p 2102 379 59 4 v 2102 470 a Fx(\031)2188 266 y Fq(Z)2287 292 y Fr(\025)2328 301 y Fp(2)2243 492 y Fr(\025)2284 501 y Fp(1)2384 402 y Fx(h)p Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))p Fx(d\025)-118 631 y Fy(at)36 b(suc)m(h)i(p)s(oin)m (ts)e(of)g(con)m(tin)m(uit)m(y)-8 b(.)55 b(On)37 b(the)g(other)f(hand,) i(using)e(theorem)h(3.3.12)e(it)h(can)h(b)s(e)f(sho)m(wn)i(that)e(the) -118 752 y(con)m(v)m(ergence)g Fx(h)p Fy(\()p Fx(t)p Fy(;)17 b Fx(\025)p Fy(\))29 b Fs(!)g Fx(\013)q Fy(\()p Fx(\025)p Fy(\))j(for)h(real)g Fx(\025)g Fy(is)g(uniform)f(on)h (compact)g(in)m(terv)-5 b(als)33 b([)p Fx(\025)2976 767 y FC(1)3015 752 y Fx(;)17 b(\025)3116 767 y FC(2)3155 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1979 y(F)-8 b(rom)29 b(this)i(represen)m(tation)g(and) g(from)e(the)j(con)m(tin)m(uit)m(y)e(of)h Fx(\013)g Fy(it)f(follo)m(ws) f(that)i Fx(h)p Fy(\()p Fx(\025)18 b Fy(+)g Fx(i")p Fy(\))28 b Fs(!)f Fx(\013)q Fy(\()p Fx(\025)p Fy(\))j(as)h Fx(")d Fs(!)f Fy(0)3957 1943 y FC(+)-118 2100 y Fy(and)33 b(for)f(real)f Fx(\025)p Fy(.)44 b(So)32 b Fx(h)h Fy(is)f(con)m(tin)m(uous)h(in)f(the) h(upp)s(er)g(half-plane,)e(as)h(w)m(e)i(w)m(an)m(ted)g(to)e(pro)m(v)m (e.)p Fh(\003)28 2220 y Fy(Inciden)m(tally)-8 b(,)32 b(from)f(\(3.35\))h(w)m(e)i(\014nd)f(the)g(represen)m(tation)571 2433 y Fx(w)s Fy(\()p Fx(\025)p Fy(\))21 b Fs(\000)h Fx(w)s Fy(\()p Fx(\025)1065 2448 y FC(0)1104 2433 y Fy(\))p 571 2477 572 4 v 719 2568 a Fx(\025)g Fs(\000)g Fx(\025)954 2583 y FC(0)1180 2500 y Fy(=)1298 2433 y(1)p 1293 2477 59 4 v 1293 2568 a Fx(\031)1379 2365 y Fq(Z)1478 2391 y FC(+)p Ft(1)1434 2590 y(\0001)1862 2433 y Fx(\013)q Fy(\()p Fx(s)p Fy(\))p 1635 2477 641 4 v 1635 2568 a(\()p Fx(s)g Fs(\000)g Fx(\025)p 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b(It)33 b(is)f(easily)g(seen)i(that)e(the)h(follo)m(wing)d (is)i(true)1026 3829 y Fx(w)s Fy(\()1141 3803 y(\026)1137 3829 y Fx(\025)p Fy(\))27 b(=)p 1363 3743 206 4 v 28 w Fx(w)s Fy(\()p Fx(\025)p Fy(\))o Fx(;)1677 3762 y Fy(Im)32 b Fx(w)s Fy(\()p Fx(\025)p Fy(\))p 1677 3807 355 4 v 1751 3898 a(Im)g Fx(\025)2069 3829 y(>)27 b Fy(0)88 b(for)32 b(Im)g Fx(\025)c Fs(6)p Fy(=)f(0)p Fx(:)895 b Fy(\(3.36\))28 4073 y(Moreo)m(v)m(er,)41 b(as)d(w)m(e)h(ha)m(v)m(e)g(sho)m(wn)g(that)f (for)f Fx(\025)f Fs(\024)h Fx(\025)1930 4036 y Ft(\003)2007 4073 y Fy(the)i(rotation)d(n)m(um)m(b)s(er)i(is)f(zero,)j(w)m(e)f (conclude)f(from)-118 4193 y(theorem)22 b(3.3.13)f(and)i(\(3.36\))e (that)h Fx(w)s Fy(\()p Fx(z)t Fy(\))g(is)g(real)f(for)h(real)f Fx(\025)28 b Fs(\024)g Fx(\025)2226 4157 y Ft(\003)2265 4193 y Fy(.)40 b(In)23 b(other)f(w)m(ords)h Fx(w)s Fy(\()p Fx(z)t Fy(\))f(extends)j(analytically)-118 4313 y(through)44 b(\()p Fs(\0001)p Fx(;)17 b(\025)578 4277 y Ft(\003)617 4313 y Fy(\).)77 b(Ho)m(w)m(ev)m(er,)49 b Fx(w)e Fy(cannot)d(b)s(e)g (seen)h(as)f(a)g(one-v)-5 b(alued)43 b(function)h(on)f(the)i(resolv)m (en)m(t)g(set.)-118 4434 y(Indeed,)34 b(if)d Fx(I)41 b Fy(is)32 b(an)g(in)m(terv)-5 b(al)32 b(in)f(a)i(sp)s(ectral)f(gap,)g (then)856 4651 y(Im)g Fx(w)s Fy(\()p Fx(\025)22 b Fy(+)g Fx(i")p Fy(\))27 b Fs(!)h Fx(\013)q Fy(\()p Fx(\025)p Fy(\))f(=)g Fx(\013)1953 4666 y Fr(I)2021 4651 y Fs(2)h(M)p Fy(\()p Fx(!)t Fy(\))p Fx(;)48 b Fy(when)34 b Fx(")27 b Fs(!)g Fy(0)2955 4610 y FC(+)3014 4651 y Fx(:)-118 4869 y Fy(Moreo)m(v)m(er,)34 b(b)m(y)g(\(3.36\))d(w)m(e)j(ha)m(v)m(e)g (that)456 5087 y Fx(w)s Fy(\()p Fx(\025)22 b Fy(+)g Fx(i")p Fy(\))g Fs(\000)g Fx(w)s Fy(\()p Fx(\025)g Fs(\000)g Fx(i")p Fy(\))28 b(=)g(2)p Fx(i)p Fy(Im)k Fx(w)s Fy(\()p Fx(\025)21 b Fy(+)h Fx(i")p Fy(\))28 b Fs(!)f Fy(2)p Fx(\013)2421 5102 y Fr(I)2461 5087 y Fx(;)77 b Fy(if)31 b Fx(\025)d Fs(2)g Fx(I)40 b Fy(and)33 b Fx(")27 b Fs(!)g Fy(0)3355 5046 y FC(+)3414 5087 y Fy(;)-118 5304 y(i.e.,)k Fx(w)s 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b(it)e(is)h(not)g(lik)m(ely)e (that)i(w)m(e)h(ha)m(v)m(e)1900 6155 y(66)p eop %%Page: 67 71 67 70 bop -118 219 a Fy(alw)m(a)m(ys)26 b(di\013eren)m(tiabilit)m(y)c (for)j Fx(\013)q Fy(\()p Fx(\025)p Fy(\))f(outside)h(the)h(sp)s(ectral) f(gaps,)i(where)f(w)m(e)g(kno)m(w)h(that)e(the)g(deriv)-5 b(ativ)m(e)25 b(exists)-118 339 y(and)j(is)g(equal)g(to)g(zero)h(\(see) g([69])f(for)g(some)g(results)h(and)f(references)j(ab)s(out)c(this)h (problem\).)41 b(Ho)m(w)m(ev)m(er)31 b(w)m(e)e(will)-118 459 y(consider)k(di\013eren)m(tiation)d(in)i(the)h(follo)m(wing)d (sense.)28 580 y(Consider)k(again)d(Sc)m(hr\177)-49 b(odinger)34 b(equation)e(\(3.21\))h(and)g(let)f Fx(p)h Fy(b)s(e)g(another)h (quasi-p)s(erio)s(dic)d(function)h(with)-118 700 y(frequency)d(mo)s (dule)d(con)m(tained)i(in)e(the)i(frequency)i(mo)s(dule)25 b(of)i Fx(q)t Fy(.)42 b(Assume)28 b(moreo)m(v)m(er)g(that)f(Im)32 b Fx(\025)c Fs(6)p Fy(=)f(0.)42 b(Then)-118 820 y(w)m(e)34 b(ma)m(y)e(talk)g(ab)s(out)g(the)h(deriv)-5 b(ativ)m(e)32 b Fx(\016)t(w)s(=\016)t(q)j Fy(if)d(it)f(satis\014ed)i(that)1328 1030 y Fx(d)p 1306 1075 97 4 v 1306 1166 a(d")1412 1098 y(w)s Fy(\()p Fx(\025)p Fy(;)17 b Fx(q)25 b Fy(+)d Fx("p)p Fy(\))1922 1170 y Fr(")p FC(=0)2077 1098 y Fy(=)2180 957 y Fq(\024)2243 1030 y Fx(\016)t(w)p 2243 1075 120 4 v 2256 1166 a(\016)t(q)2372 1098 y(p)2421 957 y Fq(\025)2474 1197 y Fk(T)2524 1178 y Fj(d)2575 1098 y Fx(:)-118 1379 y Fy(This)33 b(will)d(b)s(e)j(the)g(con)m(ten)m(t)h(of)e(the)h(follo)m (wing)c(theorem,)k(whose)h(pro)s(of)d(can)i(b)s(e)g(found)g(in)f([47)o (],)-118 1607 y Fv(Theorem)37 b(3.3.14)h(\([47]\))48 b FA(F)-7 b(or)34 b(Im)g Fx(\025)28 b Fs(6)p Fy(=)f(0)p FA(,)35 b(the)g(functional)f(derivative)2704 1568 y Fr(\016)r(w)p 2704 1585 87 4 v 2713 1642 a(\016)r(q)2835 1607 y FA(exists)g(and)h(it) g(is)f(given)h(by)1545 1832 y Fx(\016)t(w)p 1545 1876 120 4 v 1558 1968 a(\016)t(q)1675 1899 y Fy(\()p Fx(t)p Fy(\))28 b(=)f Fs(\000)p Fx(G)p Fy(\()p Fx(t;)17 b(t)p Fy(;)g Fx(\025)p Fy(\))-118 2164 y 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b(Cantor)h(sp)-5 b(e)g(ctrum)34 b(of)h(some)f(generic)g (Schr\177)-50 b(odinger)34 b(e)-5 b(quations.)-118 3332 y Fg(3.3.3)136 b(The)46 b(sp)t(ectral)h(functions)g(for)f(the)h(half)g (and)f(whole)i(line.)66 b(Con)l(v)l(erse)47 b(of)293 3481 y(theorem)f(3.3.4)-118 3666 y Fy(W)-8 b(e)32 b(no)m(w)h(w)m(an)m (t)g(to)f(use)h(the)g(represen)m(tation)g(for)f(harmonic)f(functions)h (to)g(deriv)m(e)g(an)h(imp)s(ortan)m(t)d(relation)g(of)-118 3786 y(the)39 b(rotation)f(n)m(um)m(b)s(er)h(with)g(the)h(sp)s(ectrum)f (of)g(the)h(Sc)m(hr\177)-49 b(odinger)39 b(op)s(erator)f(on)h(the)h (whole)f(line.)62 b(T)-8 b(o)39 b(this)-118 3907 y(end)32 b(recall)e(that)i(ev)m(ery)h(p)s(ositiv)m(e)f(harmonic)e(function)h Fx(h)p Fy(\()p Fx(\025)p Fy(\))g(in)g(Im)h Fx(\025)c(>)g Fy(0)j(can)h(b)s(e)g(represen)m(ted)i(in)d(the)h(form)1200 4189 y Fx(h)p Fy(\()p Fx(\025)p Fy(\))c(=)g(Im)k Fx(\025)1744 4048 y Fq(\022)1816 4053 y(Z)1916 4080 y FC(+)p Ft(1)1872 4279 y(\0001)2120 4122 y Fx(d\045)p Fy(\()p Fx(s)p Fy(\))p 2072 4166 320 4 v 2072 4257 a Fs(j)p Fx(\025)22 b Fs(\000)g Fx(s)p Fs(j)2352 4228 y FC(2)2424 4189 y Fy(+)g Fx(c)2564 4048 y Fq(\023)2654 4189 y Fx(;)1085 b Fy(\(3.37\))-118 4463 y(where)40 b Fx(\045)f Fy(is)f(a)g(monotone)g(increasing)g (function)g(and)h Fx(c)f Fs(\025)g Fy(0)h(is)f(a)g(constan)m(t.)63 b(Moreo)m(v)m(er,)42 b(at)c(the)h(p)s(oin)m(ts)f(of)-118 4583 y(con)m(tin)m(uit)m(y)33 b(of)f Fx(\045)p Fy(,)h(w)m(e)g(ha)m(v)m (e)h(that)1114 4866 y Fx(\045)p Fy(\()p Fx(\025)1259 4881 y FC(2)1299 4866 y Fy(\))22 b Fs(\000)h Fx(\045)p Fy(\()p Fx(\025)1604 4881 y FC(1)1643 4866 y Fy(\))28 b(=)54 b(lim)1812 4933 y Fr(")p Ft(!)p FC(0)1951 4914 y Fp(+)2033 4799 y Fy(1)p 2028 4843 59 4 v 2028 4934 a Fx(\031)2114 4730 y Fq(Z)2213 4757 y Fr(\025)2254 4766 y Fp(2)2169 4956 y Fr(\025)2210 4965 y Fp(1)2310 4866 y Fx(h)p Fy(\()p Fx(s)22 b Fy(+)g Fx(i")p Fy(\))p Fx(ds)-118 5142 y Fy(and)1647 5293 y Fx(c)27 b Fy(=)44 b(lim)1820 5352 y Fr(t)p Ft(!1)2013 5225 y Fx(h)p Fy(\()p Fx(it)p Fy(\))p 2013 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219 y Fx(;)17 b(\025)3583 234 y FC(2)3670 219 y FA(two)48 b(r)-5 b(e)g(al)-118 339 y(numb)g(ers.)55 b(Then,)38 b(for)h(almost)f(al)5 b(l)38 b Fx(\022)f Fs(2)e Fw(T)1472 303 y Fr(d)1516 339 y FA(,)k(the)g(function)f Fx(\045)2188 354 y FC(+)2247 339 y Fy(\()p Fx(\022)s(;)17 b(\025)p Fy(\))38 b FA(is)g(c)-5 b(ontinuous)39 b(at)f Fx(\025)3290 354 y FC(1)3368 339 y FA(and)g Fx(\025)3618 354 y FC(2)3696 339 y FA(and)g(the)-118 459 y(r)-5 b(elation)782 623 y Fx(\045)832 638 y FC(+)891 623 y Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)1078 638 y FC(2)1117 623 y Fy(\))22 b Fs(\000)h Fx(\045)1327 638 y FC(+)1386 623 y Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)1573 638 y FC(1)1612 623 y Fy(\))28 b(=)54 b(lim)1782 690 y Fr(")p Ft(!)p FC(0)1921 671 y Fp(+)2002 556 y Fy(1)p 1997 600 59 4 v 1997 691 a Fx(\031)2083 487 y Fq(Z)2182 514 y Fr(\025)2223 523 y Fp(2)2138 713 y Fr(\025)2179 722 y Fp(1)2279 623 y FA(Im)34 b Fx(M)2525 638 y FC(+)2585 623 y Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)k Fy(+)h Fx(i")p Fy(\))p Fx(d\025)-118 848 y FA(holds)34 b(with)h(b)-5 b(ounde)g(d)34 b(c)-5 b(onver)g(genc)g(e)33 b(in)i Fx(\022)s FA(.)28 1061 y Fy(With)44 b(this)h(prop)s(osition)e(in)h(mind)g(w)m(e)i(can)f(relate)f (the)h(functions)g Fx(\045)p Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\),)48 b(whic)m(h)e(w)m(e)g(shall)d(call)g(the)-118 1182 y FA(half-line)33 b(sp)-5 b(e)g(ctr)g(al)35 b(functions)p Fy(,)d(to)g(the)h Fx(w)s Fy(-function.)42 b(Recall)31 b(that,)i(for)f(Im)g Fx(\025)27 b(>)h Fy(0,)1511 1389 y Fx(w)s Fy(\()p Fx(\025)p Fy(\))e(=)i([)p Fx(M)1968 1404 y FC(+)2028 1389 y Fy(\()p Fs(\001)p Fy(;)17 b Fx(\025)p Fy(\)])2259 1418 y Fk(T)2309 1399 y Fj(d)2360 1389 y Fx(:)-118 1596 y Fy(No)m(w,)40 b(if)c Fx(f)48 b Fy(:)36 b Fw(R)48 b Fs(!)36 b Fw(R)49 b Fy(is)37 b(a)h(con)m(tin)m(uous)g (function)f(with)h(compact)f(supp)s(ort,)j(w)m(e)f(ha)m(v)m(e)g(that,)g (b)m(y)g(the)f(ab)s(o)m(v)m(e)-118 1716 y(prop)s(osition,)905 1893 y(lim)878 1961 y Fr(")p Ft(!)p FC(0)1017 1942 y Fp(+)1099 1826 y Fy(1)p 1093 1870 V 1093 1962 a Fx(\031)1179 1758 y Fq(Z)1279 1784 y FC(+)p Ft(1)1234 1983 y(\0001)1425 1893 y Fx(f)11 b Fy(\()p Fx(\025)p Fy(\))1634 1753 y Fq(\022)1706 1758 y(Z)1762 1983 y Fk(T)1812 1964 y Fj(d)1863 1893 y Fy(Im)32 b Fx(M)2106 1908 y FC(+)2165 1893 y Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)22 b Fy(+)g Fx(i")p Fy(\))p Fx(d\026)p Fy(\()p Fx(\022)s Fy(\))2823 1753 y Fq(\023)2912 1893 y Fx(d\025)542 2196 y Fy(=)54 b(lim)645 2264 y Fr(")p Ft(!)p FC(0)784 2245 y Fp(+)866 2129 y Fy(1)p 861 2173 V 861 2265 a Fx(\031)946 2061 y Fq(Z)1046 2087 y FC(+)p Ft(1)1002 2286 y(\0001)1192 2196 y Fx(f)11 b Fy(\()p Fx(\025)p Fy(\)Im)32 b Fx(w)s Fy(\()p Fx(\025)21 b Fy(+)h Fx(i")p Fy(\))p Fx(d\025)27 b Fy(=)2176 2061 y Fq(Z)2231 2286 y Fk(T)2281 2267 y Fj(d)2332 2061 y Fq(Z)2432 2087 y FC(+)p Ft(1)2388 2286 y(\0001)2578 2196 y Fx(f)11 b Fy(\()p Fx(\025)p Fy(\))p Fx(d\045)p Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))p Fx(d\026)p Fy(\()p Fx(\022)s Fy(\))p Fx(:)28 2424 y Fy(Since)33 b(Im)f Fx(w)s Fy(\()p Fx(\025)21 b Fy(+)h Fx(i")p Fy(\))28 b Fs(!)f Fx(\013)q Fy(\()p Fx(\025)p Fy(\))32 b(uniformly)f(on)h(compact)g Fx(\025)p Fy(-in)m(terv)-5 b(als,)31 b(w)m(e)j(get)923 2558 y Fq(Z)1023 2584 y FC(+)p Ft(1)978 2784 y(\0001)1169 2694 y Fx(f)11 b Fy(\()p Fx(\025)p Fy(\))p Fx(\013)q Fy(\()p Fx(\025)p Fy(\))p Fx(d\025)26 b Fy(=)1795 2558 y Fq(Z)1850 2784 y Fk(T)1900 2765 y Fj(d)1951 2558 y Fq(Z)2051 2584 y FC(+)p Ft(1)2006 2784 y(\0001)2197 2694 y Fx(f)11 b Fy(\()p Fx(\025)p Fy(\))p Fx(d\045)p Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))p Fx(d\026)p Fy(\()p Fx(\022)s Fy(\))p Fx(:)790 b Fy(\(3.38\))-118 2955 y(W)-8 b(e)33 b(will)d(use)k(then)f(the)g(notation)1314 3049 y Fq(Z)1369 3275 y Fk(T)1419 3256 y Fj(d)1471 3185 y Fy(\()p Fx(d\045)p Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\)\))f Fx(d\026)p Fy(\()p Fx(\022)s Fy(\))27 b(=)g Fx(\013)q Fy(\()p Fx(\025)p Fy(\))p Fx(d\025;)-118 3445 y Fy(where)37 b Fx(d\025)e Fy(is)g(the)h(Leb)s(esgue)h(measure)f(on)f Fw(R)5 b Fy(,)43 b(to)35 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b(b)m(y)g Fx(\013)q Fy(.)-118 5036 y Fv(Lemma)j(3.3.17)h(\([47]\))48 b FA(If)34 b(Im)h Fx(\025)27 b(>)h Fy(0)34 b FA(then)1317 5156 y Fq(Z)1372 5382 y Fk(T)1422 5363 y Fj(d)1473 5292 y Fy(\()p Fx(d)9 b Fy(^)-58 b Fx(\045)p Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\)\))f Fx(d\026)p Fy(\()p Fx(\022)s Fy(\))27 b(=)2271 5224 y(1)p 2266 5269 59 4 v 2266 5360 a Fx(\031)2335 5292 y(d\013)q Fy(\()p Fx(\025)p Fy(\))-118 5546 y FA(in)i(the)h(sense) e(that,)j(for)f(al)5 b(l)29 b(c)-5 b(ontinuous)29 b(functions)g Fx(f)39 b Fy(:)28 b Fw(R)38 b Fs(!)27 b Fw(R)41 b FA(with)29 b(c)-5 b(omp)g(act)29 b(supp)-5 b(ort,)31 b(the)e(fol)5 b(lowing)29 b(holds)904 5674 y Fq(Z)959 5899 y Fk(T)1009 5880 y Fj(d)1060 5674 y Fq(Z)1160 5700 y FC(+)p Ft(1)1116 5899 y(\0001)1306 5809 y Fx(f)11 b Fy(\()p Fx(\025)p Fy(\))p Fx(d)e Fy(^)-58 b Fx(\045)p Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))p Fx(d\026)p Fy(\()p Fx(\022)s Fy(\))26 b(=)2203 5742 y(1)p 2198 5787 V 2198 5878 a Fx(\031)2283 5674 y Fq(Z)2383 5700 y FC(+)p Ft(1)2339 5899 y(\0001)2529 5809 y Fx(f)11 b Fy(\()p Fx(\025)p Fy(\))p Fx(d\013)q Fy(\()p Fx(\025)p Fy(\))p Fx(:)1900 6155 y Fy(68)p eop %%Page: 69 73 69 72 bop 28 219 a Fy(No)m(w)33 b(w)m(e)h(can)f(pro)m(v)m(e)h(the)f (con)m(v)m(erse)i(of)d(theorem)g(3.3.4)-118 421 y Fv(Theorem)37 b(3.3.18)h(\([47]\))48 b FA(The)e(supp)-5 b(ort)48 b(of)e(the)i(me)-5 b(asur)g(e)46 b Fx(d\013)q Fy(\()p Fx(\025)p Fy(\))g FA(agr)-5 b(e)g(es)47 b(with)g(the)g(sp)-5 b(e)g(ctrum)47 b Fx(\033)t Fy(\()p Fx(H)8 b Fy(\))46 b FA(of)-118 542 y(e)-5 b(quation)34 b(\(3.21\))g(on)h Fx(L)770 506 y FC(2)810 542 y Fy(\()p Fs(\0001)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))p FA(.)28 745 y Fv(Pro)s(of:)54 b Fy(By)38 b(theorem)g(3.3.4)f (it)g(is)h(clear)f(that)h(w)m(e)h(only)e(ha)m(v)m(e)i(to)f(pro)m(v)m(e) h(that)f(if)e Fx(I)46 b Fy(is)37 b(a)h(b)s(ounded)h(op)s(en)-118 865 y(in)m(terv)-5 b(al)31 b(on)i(whic)m(h)g(the)g(rotation)e(n)m(um)m (b)s(er)i Fx(\013)q Fy(\()p Fx(\025)p Fy(\))f(is)g(constan)m(t,)h(then) 1666 1063 y Fx(\033)t Fy(\()p Fx(H)8 b Fy(\))21 b Fs(\\)i Fx(I)35 b Fy(=)28 b Fs(;)1534 b Fy(\(3.40\))-118 1260 y(F)-8 b(rom)31 b(the)i(in)m(tegration)e(of)h(measures)h(of)f(the)h (previous)g(lemma)e(w)m(e)i(obtain)f(that)1250 1365 y Fq(Z)1306 1591 y Fk(T)1356 1572 y Fj(d)1407 1365 y Fq(Z)1462 1591 y Fr(I)1527 1501 y Fy(^)-57 b Fx(\045)p Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))p Fx(d\026)p Fy(\()p Fx(\025)p Fy(\))26 b(=)2182 1434 y(1)p 2177 1478 59 4 v 2177 1569 a Fx(\031)2262 1365 y Fq(Z)2318 1591 y Fr(I)2374 1501 y Fx(d\013)q Fy(\()p Fx(\025)p Fy(\))p Fx(;)-118 1746 y Fy(considering)33 b(the)h(in)m(tegration)e(with)i(resp)s(ect)h (to)e(a)g(b)s(ounded)i(in)m(terv)-5 b(al)32 b(as)i(the)g(m)m (ultiplication)29 b(b)m(y)35 b(a)f(c)m(harac-)-118 1866 y(teristic)29 b(function)h(and)h(appro)m(ximating)d(b)m(y)j(con)m(tin)m (uous)h(functions)e(with)g(compact)g(supp)s(ort.)43 b(Therefore)32 b(w)m(e)-118 1987 y(ha)m(v)m(e)i(that,)e(for)g(almost)f(all)g Fx(\022)f Fs(2)f Fw(T)1178 1950 y Fr(d)1221 1987 y Fy(,)1627 2014 y Fq(Z)1682 2240 y Fr(I)1738 2150 y Fx(d)9 b Fy(^)-58 b Fx(\045)p Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))28 b(=)f(0)p Fx(:)-118 2367 y Fy(No)m(w)33 b(let)f(us)h(\014x)g(a)g(v)-5 b(alue)32 b(of)g Fx(\022)j Fy(in)d(this)g(set)i(of)e(measure)h(one.)44 b(Then)33 b(it)f(follo)m(ws)f(from)g(\(3.39\))h(that)1066 2565 y(Im)g(\000\()p Fx(\022)s Fy(;)17 b Fx(\025)22 b Fy(+)g Fx(i")p Fy(\))27 b Fs(!)g Fy(0)88 b(for)f Fx(\025)28 b Fs(2)g Fx(I)8 b(;)44 b(")28 b Fs(!)f Fy(0)2746 2524 y FC(+)2805 2565 y Fx(:)-118 2763 y Fy(W)-8 b(e)33 b(can)g(apply)f(the) h(re\015ection)g(principle)e(to)h(get)h(the)g(analyticit)m(y)e(of)h (\000\()p Fx(\022)s Fy(;)17 b Fs(\001)p Fy(\))32 b(through)g Fx(I)8 b Fy(,)33 b(and)f(th)m(us,)1381 2937 y Fs(\000)p Fy(1)p 1301 2982 286 4 v 1301 3073 a(\000\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))1625 3005 y(=)27 b Fx(M)1822 3020 y FC(+)1882 3005 y Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))k Fs(\000)i Fx(M)2322 3020 y Ft(\000)2381 3005 y Fy(\()p Fx(\022)s Fy(;)17 b 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b(the)-5 b(or)g(em)37 b(in)h(c)-5 b(ombination)37 b(with)h(\(3.3.4\))e(shows)i(that)g(a)g(p)-5 b(oint)38 b Fx(\025)33 b Fs(2)h Fw(R)49 b FA(is)38 b(in)-118 5786 y(the)g(sp)-5 b(e)g(ctrum)37 b Fx(\033)t Fy(\()p Fx(H)8 b Fy(\))37 b FA(of)g(e)-5 b(quation)37 b(\(3.21\))g(on)g Fx(L)1733 5750 y FC(2)1773 5786 y Fy(\()p Fs(\0001)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))37 b FA(if)g(and)g(only)h(if)f(the)h(r)-5 b(otation)37 b(numb)-5 b(er)37 b(is)h(not)-118 5906 y(c)-5 b(onstant)34 b(at)h Fx(\025)p FA(.)1900 6155 y Fy(69)p eop %%Page: 70 74 70 73 bop -118 219 a Fg(3.3.4)136 b(The)55 b(real)i(part)g(of)f Ff(w)s Fg(.)93 b(The)56 b(Ly)l(apuno)l(v)g(exp)t(onen)l(ts)h(for)f(Sc)l (hr\177)-67 b(odinger)293 368 y(equation)46 b(with)f(quasi-p)t(erio)t (dic)g(p)t(oten)l(tial)-118 553 y Fy(In)34 b(the)g(previous)g (sections,)h(the)f(imaginary)d(part)j(of)f(function)g Fx(w)k Fy(has)d(receiv)m(ed)h(a)e(great)h(deal)f(of)g(atten)m(tion,) -118 673 y(mainly)40 b(b)s(ecause)k(of)e(its)g(go)s(o)s(d)f(prop)s (erties.)73 b(This)43 b(section)f(is)g(dev)m(oted)i(to)e(a)h(short)f (surv)m(ey)j(of)d(results)h(on)-118 793 y(the)g FA(r)-5 b(e)g(al)53 b Fy(part)43 b(of)g(this)f(function,)k(in)c(principle)f (de\014ned)k(on)e(the)g(resolv)m(en)m(t)h(set,)j(but)c(that,)i(with)e (certain)-118 914 y(restrictions,)48 b(can)d(b)s(e)g(extended)i(to)e (the)g(whole)g(real)f(line.)79 b(It)45 b(turns)h(out)f(that)f(this)h (ob)5 b(ject)46 b(is)e(strongly)-118 1034 y(related)36 b(to)f(the)i(Ly)m(apuno)m(v)h(exp)s(onen)m(t.)55 b(The)37 b(approac)m(h)g(to)f(the)g(problem)f(in)g(this)h(section)g(and)h(the)f (results)-118 1155 y(are)c(mostly)g(tak)m(en)i(from)d([44].)28 1275 y(F)-8 b(or)34 b(the)g(discussion)h(on)f(reducibilit)m(y)e(it)h (will)f(b)s(e)i(v)m(ery)i(imp)s(ortan)m(t)c(to)i(distinguish)f(b)s(et)m (w)m(een)j(the)e(Sc)m(hr\177)-49 b(o-)-118 1395 y(dinger)33 b(equation)g(with)g(a)h FA(\014xe)-5 b(d)43 b Fy(quasi-p)s(erio)s(dic) 31 b(p)s(oten)m(tial,)h(and)i(the)g(Sc)m(hr\177)-49 b(odinger)34 b(equation)f(with)g 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Fx(!)t(;)1027 b Fy(\(3.42\))-118 2516 y(where)32 b Fx(Q)f Fy(is)f(the)h(extension)g(of)g(the)g(previous) g(quasi-p)s(erio)s(dic)e(function)h(\(i.e.,)g(there)i(exists)f(an)g (initial)c(phase)-118 2637 y Fx(\036)g Fs(2)h Fw(T)124 2600 y Fr(d)201 2637 y Fy(suc)m(h)34 b(that)e Fx(q)t Fy(\()p Fx(t)p Fy(\))c(=)f Fx(Q)p Fy(\()p Fx(!)t(t)22 b Fy(+)g Fx(\036)p Fy(\)\).)28 2757 y(W)-8 b(e)33 b(b)s(egin)f(with)g (the)h(de\014nition)f(of)g(the)h(main)e(ob)5 b(ject)33 b(of)f(in)m(terest)h(in)f(this)g(section)-118 2985 y Fv(De\014nition)k(3.3.20)49 b FA(We)33 b(de\014ne)e(the)39 b Fy(upp)s(er)31 b(Ly)m(apuno)m(v)g(exp)s(onen)m(t)j FA(for)e(r)-5 b(e)g(al)31 b Fx(\025)p FA(,)i Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))p FA(,)32 b(for)g(e)-5 b(quation)32 b(\(3.41\))-118 3106 y(as)1017 3258 y Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))27 b(=)h(sup)1505 3118 y Fq(\032)1580 3258 y Fy(lim)17 b(sup)1619 3338 y Fr(t)p Ft(!)p FC(+)p Ft(1)1923 3191 y Fy(1)p 1906 3235 85 4 v 1906 3327 a(2)p Fx(t)2016 3258 y Fy(log)2159 3178 y Fq(\000)2205 3258 y Fx(x)p Fy(\()p Fx(t)p Fy(\))2371 3217 y FC(2)2433 3258 y Fy(+)22 b Fx(x)2586 3217 y Ft(0)2609 3258 y Fy(\()p Fx(t)p Fy(\))2720 3217 y FC(2)2760 3178 y Fq(\001)2806 3118 y(\033)-118 3491 y FA(wher)-5 b(e)34 b(the)g(supr)-5 b(emum)34 b(is)g(taken)g(over)g(al)5 b(l)34 b(solutions)f(of)h(the)h (system)f(\(3.42\).)44 b(This)33 b(implies)g(c)-5 b(onsidering)33 b(al)5 b(l)-118 3612 y(initial)34 b(c)-5 b(onditions)34 b(for)h Fx(x)p Fy(\(0\))p Fx(;)17 b(x)1069 3576 y Ft(0)1092 3612 y Fy(\(0\))28 b Fs(2)g Fw(R)45 b FA(and)35 b Fx(\036)27 b Fs(2)h Fw(T)1877 3576 y Fr(d)1921 3612 y FA(.)-118 3840 y Fv(Remark)37 b(3.3.21)49 b FA(Due)41 b(to)g(the)g(line)-5 b(ar)40 b(char)-5 b(acter)40 b(of)h(the)g(\015ow)g(one)f(c)-5 b(an)40 b(always)h(take)g(the)g(supr)-5 b(emum)40 b(on)-118 3960 y(the)35 b(set)g(of)f(initial)h(c)-5 b(onditions)33 b(given)i(by)g Fx(x)p Fy(\(0\))1624 3924 y FC(2)1685 3960 y Fy(+)22 b Fx(x)1838 3924 y Ft(0)1862 3960 y Fy(\(0\))1987 3924 y FC(2)2054 3960 y Fy(=)27 b(1)p FA(.)28 4189 y Fy(W)-8 b(e)32 b(will)c(see)k(later)e(on)h(that)g(the)g(function)f Fx(\014)37 b Fy(can)31 b(also)f(b)s(e)h(considered)h(for)e(complex)g Fx(\025)p Fy(.)43 b(Note)31 b(that)g(if)f(w)m(e)-118 4309 y(write)i(equation)h(\(3.42\))e(as)i(a)f(\014rst-order)h(system) 1069 4587 y Fx(u)1125 4546 y Ft(0)1148 4587 y Fy(\()p Fx(t)p Fy(\))28 b(=)1390 4447 y Fq(\022)1709 4526 y Fy(0)286 b(1)1505 4647 y Fs(\000)p Fx(\025)23 b Fy(+)f Fx(Q)p Fy(\()p Fx(\022)s Fy(\))83 b(0)2134 4447 y Fq(\023)2224 4587 y Fx(u)p Fy(\()p Fx(t)p Fy(\))p Fx(;)100 b(\022)2566 4546 y Ft(0)2617 4587 y Fy(=)27 b Fx(!)t(;)954 b Fy(\(3.43\))-118 4870 y(where)34 b Fx(u)27 b Fs(2)h Fw(R)407 4833 y FC(2)452 4870 y Fy(,)33 b(then)g(the)g(upp)s(er)g(Ly)m(apuno)m(v)h(exp)s(onen)m (t)g(can)f(equiv)-5 b(alen)m(tly)32 b(b)s(e)h(written)f(as)669 5141 y Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))27 b(=)h(sup)1157 5001 y Fq(\032)1232 5141 y Fy(lim)17 b(sup)1271 5221 y Fr(t)p Ft(!)p FC(+)p Ft(1)1558 5074 y Fy(1)p 1558 5118 49 4 v 1565 5210 a Fx(t)1633 5141 y Fy(log)f Fs(k)p Fx(u)p Fy(\()p Fx(t)p Fy(\))p Fs(k)p Fy(;)44 b Fx(u)32 b Fy(is)g(a)h(solution) e(of)h(\(3.43\))3110 5001 y Fq(\033)3201 5141 y Fx(:)538 b Fy(\(3.44\))28 5428 y(Consider)44 b(no)m(w,)j(for)42 b(complex)h Fx(\025)p Fy(,)j(the)d(\015o)m(w)h(on)f(the)h(complex)f (bundle)g(\006)j(=)g Fw(P)3109 5391 y FC(1)3150 5428 y Fy(\()p Fw(C)20 b Fy(\))36 b Fs(\002)30 b Fw(T)3498 5391 y Fr(d)3541 5428 y Fy(,)46 b(using)d(the)-118 5548 y(standard)24 b(pro)5 b(jectivization)22 b(exp)s(osed)j(in)e(the)i(b)s (eginning)d(of)h(the)h(c)m(hapter.)42 b(Consider)24 b(the)g(follo)m (wing)d(function,)-118 5668 y(for)32 b Fx(\025)c Fs(2)g Fw(C)20 b Fy(,)1103 5838 y(\()p Fx(l)r(;)d(\022)s Fy(\))28 b Fs(2)g Fy(\006)g Fs(7!)g Fx(f)1698 5853 y Fr(\025)1743 5838 y Fy(\()p Fx(l)r(;)17 b(\022)s Fy(\))28 b(=)f(Re)2231 5771 y Fs(h)p Fx(A)2343 5786 y Fr(\025)2388 5771 y Fy(\()p Fx(\022)s Fy(\))p Fx(u)2568 5786 y FC(0)2607 5771 y Fx(;)17 b(u)2707 5786 y FC(0)2746 5771 y Fs(i)p 2231 5815 554 4 v 2352 5907 a(h)p Fx(u)2447 5922 y FC(0)2486 5907 y Fx(;)g(u)2586 5922 y FC(0)2624 5907 y Fs(i)1900 6155 y Fy(70)p eop %%Page: 71 75 71 74 bop -118 219 a Fy(where)36 b Fx(u)222 234 y FC(0)291 219 y Fs(2)c Fw(C)455 182 y FC(2)535 219 y Fy(is)i(an)m(y)h(nonzero)g (v)m(ector)g(whic)m(h)g(represen)m(ts)i(the)e(pro)5 b(jectiv)m(e)35 b(complex)f(line)g Fx(l)f Fs(2)e Fw(P)3635 182 y FC(1)3676 219 y Fy(\()p Fw(C)20 b Fy(\))41 b(and)-118 339 y Fx(A)-45 354 y Fr(\025)33 339 y Fy(is)32 b(the)h(matrix)1331 507 y Fx(A)1404 522 y Fr(\025)1449 507 y Fy(\()p Fx(\022)s Fy(\))28 b(=)1704 367 y Fq(\022)2024 446 y Fy(0)287 b(1)1819 567 y Fs(\000)p Fx(\025)23 b Fs(\000)f Fx(Q)p Fy(\()p Fx(\022)s Fy(\))84 b(0)2450 367 y Fq(\023)2540 507 y Fx(:)-118 730 y Fy(Then,)34 b(if)d Fx(u)p Fy(\()p Fx(t)p Fy(\))h(satis\014es)i(\(3.43\))d(with)i Fx(u)p Fy(\(0\))26 b(=)i Fx(u)1683 745 y FC(0)1754 730 y Fy(and)33 b(initial)c(phase)k Fx(\036)p Fy(,)g(one)g(has)884 921 y(1)p 884 966 49 4 v 891 1057 a Fx(t)959 989 y Fy(\(log)17 b Fs(k)p Fx(u)p Fy(\()p Fx(t)p Fy(\))p Fs(k)k(\000)i Fy(log)17 b Fs(k)p Fx(u)p Fy(\(0\))p Fs(k)p Fy(\))26 b(=)2130 921 y(1)p 2130 966 V 2137 1057 a Fx(t)2205 853 y Fq(Z)2305 880 y Fr(t)2260 1079 y FC(0)2351 989 y Fx(f)2399 1004 y Fr(\025)2444 989 y Fy(\()p Fx(l)r(;)17 b(!)t(s)k Fy(+)h Fx(\036)p Fy(\))p Fx(ds;)759 b Fy(\(3.45\))-118 1238 y(and)24 b(therefore)h(the)f (exp)s(onen)m(tial)f(gro)m(wth)i(of)e Fx(u)p Fy(\()p Fx(t)p Fy(\))h(is)f(determined)h(b)m(y)h(a)f(time)e(a)m(v)m(erage)j(of) f Fx(f)3245 1253 y Fr(\025)3290 1238 y Fy(.)41 b(W)-8 b(e)24 b(will)e(also)h(use)-118 1358 y(the)29 b(pro)5 b(jectiv)m(e)30 b(\015o)m(w)g(on)e(\(\006)934 1373 y Fk(R)987 1358 y Fx(;)17 b Fw(R)5 b Fy(\))34 b(where)c(\006)1517 1373 y Fk(R)1598 1358 y Fy(=)d Fw(P)1760 1322 y FC(1)1802 1358 y Fy(\()p Fw(R)t Fy(\))15 b Fs(\002)g Fw(T)2117 1322 y Fr(d)2163 1358 y Fy(,)30 b(with)e(the)h(notation)f(from)g(the)h (in)m(tro)s(duction)e(in)-118 1478 y(this)j(c)m(hapter.)44 b(Recall)29 b(that)h(the)h(function)f Fx(M)1598 1493 y FC(+)1658 1478 y Fy(\()p Fx(\036)p Fy(;)17 b Fx(\025)p Fy(\),)30 b(for)g(Im)i Fx(\025)27 b Fs(6)p Fy(=)h(0)i(giv)m(es)h(the)g (direction)f(of)g(the)h(pro)5 b(jectiv)m(e)-118 1599 y(lines)44 b(of)g(solutions)g(of)h(\(3.42\))f(with)g(initial)d(phase)46 b Fx(\036)f Fy(that)f(b)s(elong)g(to)h Fx(L)2738 1563 y FC(2)2777 1599 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))44 b(in)g(the)i(sense)g(that)f(if)-118 1719 y Fx(\037)-57 1734 y FC(+)2 1719 y Fy(\()p Fs(\001)p Fy(;)17 b Fx(\036)p Fy(\))32 b(is)g(a)g(solution)f(b)s(elonging)g(to)h Fx(L)1415 1683 y FC(2)1455 1719 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\),)31 b(then)927 1978 y Fx(\037)988 1993 y FC(+)1047 1978 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))28 b(=)f Fx(\037)1452 1993 y FC(+)1511 1978 y Fy(\(0;)17 b Fx(\036)p Fy(\))g(exp)1920 1838 y Fq(\022)1993 1843 y(Z)2093 1869 y Fr(t)2048 2068 y FC(0)2139 1978 y Fx(M)2233 1993 y FC(+)2292 1978 y Fy(\()p Fx(!)t(s)22 b Fy(+)g Fx(\036)p Fy(;)17 b Fx(\025)p Fy(\))p Fx(ds)2855 1838 y Fq(\023)2943 1978 y Fx(;)-118 2226 y Fy(so)33 b(that)f(the)h(real)f(part)g(of)g Fx(w)j Fy(can)e(b)s(e)g(expressed)i(b)m(y)f(means)e(of)g(these)i (solutions)901 2461 y(Re)f Fx(w)s Fy(\()p Fx(\025)p Fy(\))27 b(=)70 b(lim)1385 2521 y Fr(t)p Ft(!)p FC(+)p Ft(1)1651 2394 y Fy(1)p 1633 2439 85 4 v 1633 2530 a(2)p Fx(t)1744 2461 y Fy(log)1886 2381 y Fq(\000)1932 2461 y Fs(j)p Fx(\037)2021 2476 y FC(+)2080 2461 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))p Fs(j)2321 2420 y FC(2)2382 2461 y Fy(+)22 b Fs(j)p Fx(\037)2569 2420 y Ft(0)2569 2486 y FC(+)2628 2461 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))p Fs(j)2869 2420 y FC(2)2907 2381 y Fq(\001)2969 2461 y Fx(:)-118 2704 y Fy(Th)m(us)40 b(Re)33 b Fx(w)s Fy(\()p Fx(\025)p Fy(\))k(is)i(the)g(exp)s(onen)m(tial)f(rate)g(of)g(deca)m(y)i (of)e(the)h(solutions)f Fx(u)p Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))37 b(=)h(\()p Fx(\037)3158 2719 y FC(+)3217 2704 y Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))p Fx(;)g(\037)3535 2668 y Ft(0)3535 2728 y FC(+)3593 2704 y Fy(\()p Fx(t)p Fy(;)g Fx(\036)p Fy(\)\))3844 2668 y Fr(T)3937 2704 y Fy(of)-118 2824 y(\(3.42\))46 b(in)f Fx(L)371 2788 y FC(2)411 2824 y Fy(\(0)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))45 b(and)i(this)f(rate)g(is)g(indep)s(enden)m(t)i(of)e(the)h(initial)42 b(phase.)86 b(Using)46 b(the)h(exp)s(onen)m(tial)-118 2945 y(dic)m(hotom)m(y)32 b(of)g(the)h(system)h(for)e(Im)g Fx(\025)27 b Fs(6)p Fy(=)h(0)k(it)g(is)g(easily)g(sho)m(wn)i(that)1558 3141 y Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))27 b(=)g Fs(\000)p Fy(Re)34 b Fx(w)s Fy(\()p Fx(\025)p Fy(\))p Fx(;)-118 3337 y Fy(where)g Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))32 b(is)g(de\014ned)i(as)e(in)g(\(3.44\).)-118 3538 y Fv(Remark)37 b(3.3.22)49 b FA(We)33 b(c)-5 b(an)33 b(use)g(similar)g(te)-5 b(chniques)32 b(to)i(those)e(in)h(the)h(pr)-5 b(o)g(of)32 b(of)h(lemma)f(3.3.1)h(to)g(pr)-5 b(ove)33 b(the)-118 3658 y(fol)5 b(lowing)33 b(in)i(the)g(r)-5 b(e)g(al)34 b(c)-5 b(ase:)10 3841 y(If)48 b Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))27 b(=)g(0)35 b FA(and)f Fx(\025)28 b Fs(2)g Fw(R)45 b FA(then)1225 3761 y Fq(R)1272 3876 y FC(\006)1344 3841 y Fx(f)1392 3856 y Fr(\025)1437 3841 y Fx(d\026)27 b Fy(=)h(0)34 b FA(for)h(every)f(invariant)g(me)-5 b(asur)g(e)34 b Fx(\026)h FA(on)f Fy(\006)p FA(,)h(b)-5 b(e)g(c)g(ause)34 b(we)g(have)126 3962 y(that)j(the)g(time-aver)-5 b(age)35 b(\(3.45\))h(is)g(zer)-5 b(o)36 b(for)h(al)5 b(l)36 b(solutions)g(of)h (\(3.43\).)49 b(Thus)36 b(using)h(lemma)e(3.3.1)h(we)126 4082 y(c)-5 b(onclude)34 b(that)1203 4246 y Fy(lim)1092 4312 y Ft(j)p Fr(b)p Ft(\000)p Fr(a)p Ft(j!)p FC(+)p Ft(1)1559 4179 y Fy(1)p 1476 4223 215 4 v 1476 4314 a Fx(b)23 b Fs(\000)f Fx(a)1717 4110 y Fq(Z)1817 4137 y Fr(b)1772 4336 y(a)1868 4246 y Fx(f)1916 4261 y Fr(\025)1961 4246 y Fy(\()p Fx(l)r Fy(\()p Fx(s)p Fy(;)17 b Fx(l)2187 4261 y FC(0)2226 4246 y Fx(;)g(\036)p Fy(\);)g Fx(!)t(s)k Fy(+)h Fx(\036)p Fy(\)\))p Fx(ds)27 b Fy(=)g(0)126 4473 y FA(for)34 b(al)5 b(l)35 b Fx(\033)d Fy(=)27 b(\()p Fx(l)678 4488 y FC(0)718 4473 y Fx(;)17 b(\036)p Fy(\))27 b Fs(2)h Fy(\006)g(=)f Fw(P)1239 4437 y FC(1)1281 4473 y Fy(\()p Fw(C)20 b Fy(\))28 b Fs(\002)22 b Fw(T)1613 4437 y Fr(d)1657 4473 y FA(,)35 b(and)f(the)h(c)-5 b(onver)g(genc)g(e) 33 b(is)i(uniform)f(in)h Fx(a)p FA(,)g Fx(b)g FA(and)f Fx(\033)t FA(.)10 4670 y(If)48 b Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))27 b Fx(>)g Fy(0)k FA(and)f Fx(\025)d Fs(2)h Fw(R)42 b FA(then)30 b(ther)-5 b(e)31 b(ar)-5 b(e)30 b(exactly)g(two)h(er)-5 b(go)g(dic)29 b(me)-5 b(asur)g(es)30 b(on)g Fy(\006)p FA(,)i Fx(\026)3169 4685 y FC(+)3258 4670 y FA(and)e Fx(\026)3502 4685 y Ft(\000)3592 4670 y FA(\(se)-5 b(e)30 b([42)o(]\);)126 4790 y(one)k(has)g(that)i Fx(\026)743 4805 y Ft(\006)801 4790 y Fy(\(\006)909 4805 y Fk(R)962 4790 y Fy(\))28 b(=)f(1)p FA(,)35 b(and)1303 4899 y Fq(Z)1359 5125 y FC(\006)1430 5035 y Fx(f)1478 5050 y Fr(\025)1524 5035 y Fx(d\026)1634 5050 y FC(+)1720 5035 y Fy(=)27 b Fs(\000)1917 4899 y Fq(Z)1973 5125 y FC(\006)2045 5035 y Fx(f)2093 5050 y Fr(\025)2138 5035 y Fx(d\026)2248 5050 y Ft(\000)2334 5035 y Fy(=)2452 5009 y(^)2438 5035 y Fx(\014)5 b Fy(\()p Fx(\025)p Fy(\))28 b Fx(>)f Fy(0)p Fx(:)126 5304 y FA(Mor)-5 b(e)g(over,)596 5278 y Fy(^)581 5304 y Fx(\014)37 b FA(is)32 b(the)f(right)g(end-p)-5 b(oint)31 b(of)g(the)g(Sacker-Sel)5 b(l)30 b(sp)-5 b(e)g(ctrum)32 b(of)f(e)-5 b(quations)31 b(\(3.43\).)42 b(It)32 b(is)f(very)126 5425 y(imp)-5 b(ortant)28 b(to)g(note)g(that)g(this)g(do)-5 b(es)28 b(not)g(imply)f(an)h(exp)-5 b(onential)27 b(dichotomy,)h(b)-5 b(e)g(c)g(ause)28 b(it)g(c)-5 b(ould)28 b(happ)-5 b(en,)126 5545 y(and)27 b(it)i(c)-5 b(ertainly)28 b(do)-5 b(es,)28 b(that)h(the)f(p)-5 b(oints)28 b Fs(\006)1749 5519 y Fy(^)1735 5545 y Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))28 b FA(ar)-5 b(e)28 b(the)g(endp)-5 b(oints)27 b(of)h(the)g(same)g (Sacker-Sel)5 b(l)26 b(sp)-5 b(e)g(ctr)g(al)126 5666 y(interval,)43 b(ther)-5 b(efor)g(e)41 b(c)-5 b(orr)g(esp)g(onding)40 b(to)i(the)g(same)f(invariant)g(subbund)5 b(le)41 b(\(the)h(whole)f(sp) -5 b(ac)g(e\).)64 b(This)126 5786 y(should)41 b(not)h(c)-5 b(ome)41 b(as)h(a)g(surprise)f(b)-5 b(e)g(c)g(ause)41 b(p)-5 b(ositive)42 b(Lyapunov)f(exp)-5 b(onents)41 b(do)h(not)g(imply) f(uniform)126 5906 y(hyp)-5 b(erb)g(olicity.)1900 6155 y Fy(71)p eop %%Page: 72 76 72 75 bop -118 219 a Fy(The)33 b(follo)m(wing)d(result)j(giv)m(es)g (more)f(details)f(in)h(the)h(case)g(of)g(p)s(ositiv)m(e)e(upp)s(er)j (Ly)m(apuno)m(v)g(exp)s(onen)m(t.)-118 447 y Fv(Prop)s(osition)h (3.3.23)j(\([44]\))49 b FA(If)42 b Fx(\025)h Fs(2)g Fw(R)55 b FA(and)42 b Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))43 b Fx(>)f Fy(0)p FA(,)j(then)e Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))43 b(=)2828 421 y(^)2814 447 y Fx(\014)5 b Fy(\()p Fx(\025)p Fy(\))p FA(.)70 b(Mor)-5 b(e)g(over,)44 b(for)f(almost)-118 567 y(every)33 b Fx(\036)27 b Fs(2)h Fw(T)380 531 y Fr(d)457 567 y FA(\(wher)-5 b(e)32 b(the)g(me)-5 b(asur)g(e)32 b(her)-5 b(e)33 b(is)f(the)h(usual)g(Haar)g(me)-5 b(asur)g(e\),)32 b(ther)-5 b(e)33 b(exist)f(unique)h(solutions)f Fx(u)3957 582 y Ft(\006)-118 688 y FA(of)i(\(3.43\))g(\(up)h(to)g(a)g(c)-5 b(onstant)34 b(multiple\))h(with)1395 952 y Fy(lim)1380 1012 y Fr(t)p Ft(!1)1573 885 y Fy(1)p 1573 929 49 4 v 1580 1020 a Fx(t)1648 952 y Fy(log)17 b Fs(k)p Fx(u)1897 967 y Ft(\006)1955 952 y Fy(\()p Fx(t)p Fy(\))p Fs(k)28 b Fy(=)f Fs(\006)p Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))28 1213 y(The)33 b(pro)s(of)d(is)h(similar)d(to)j(the)h(construction)g(of) f(the)g(rotation)f(n)m(um)m(b)s(er)i(and)f(can)h(b)s(e)g(found)f(in)g ([44].)43 b(The)-118 1333 y FA(almost)34 b(everywher)-5 b(e)39 b Fy(part)33 b(comes)g(from)e(an)h(application)e(of)j (Birkho\013)e(ergo)s(dic)h(theorem.)-118 1561 y Fv(Remark)37 b(3.3.24)49 b FA(It)27 b(is)f(an)h(e)-5 b(asy)27 b(c)-5 b(onse)g(quenc)g(e)25 b(of)i(the)g(discussion)e(pr)-5 b(evious)27 b(to)g(the)g(lemma)e(that)j(if)e Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))27 b(=)h(0)-118 1682 y FA(then)35 b(ne)-5 b(c)g(essarily)600 1656 y Fy(^)586 1682 y Fx(\014)33 b Fy(=)27 b(0)35 b FA(and)f(the)h(Sacker-Sel)5 b(l)34 b(sp)-5 b(e)g(ctrum)34 b(r)-5 b(e)g(duc)g(es)35 b(to)g(a)f(single)g(p) -5 b(oint.)28 1910 y Fy(The)44 b(rotation)e(n)m(um)m(b)s(er)i Fx(\013)g Fy(and)f(the)h(Ly)m(apuno)m(v)h(exp)s(onen)m(t)f Fx(\014)49 b Fy(are)43 b(closely)g(related)g(b)m(y)h(means)f(of)g(the) -118 2030 y(Thouless)33 b(form)m(ula,)e(whic)m(h)i(w)m(e)h(no)m(w)f (state)-118 2259 y Fv(Theorem)k(3.3.25)h(\(Thouless)f(form)m(ula,)g (\([4)o(],)h(also)f([85])h(and)g([3]\)\))49 b FA(L)-5 b(et)35 b Fx(\014)2993 2274 y FC(0)3032 2259 y Fy(\()p Fx(\025)p Fy(\))28 b(=)3296 2174 y Fq(p)p 3396 2174 485 4 v 85 x Fy(max)o(\(0)p Fx(;)17 b Fs(\000)p Fx(\025)p Fy(\))-118 2391 y FA(and)44 b Fx(\013)143 2406 y FC(0)183 2391 y Fy(\()p Fx(\025)p Fy(\))h(=)483 2305 y Fq(p)p 583 2305 407 4 v 86 x Fy(max)o(\(0)p Fx(;)17 b(\025)p Fy(\))p FA(.)74 b(Then)44 b(for)h(any)g(quasi-p)-5 b(erio)g(dic)43 b(function)h Fx(q)49 b FA(on)44 b Fw(R)56 b FA(we)44 b(have)h(that)g(for)f(a.e.)-118 2511 y Fx(\025)27 b Fs(2)i Fw(R)1193 2678 y Fy(lim)1137 2740 y Fr(T)10 b Ft(!)p FC(+)p Ft(1)1416 2611 y Fy(1)p 1411 2655 59 4 v 1411 2746 a Fx(\031)1496 2542 y Fq(Z)1596 2569 y Fr(T)1552 2768 y Ft(\0001)1698 2678 y Fy(log)16 b(\()p Fx(\025)22 b Fs(\000)h Fx(\025)2114 2637 y Ft(0)2137 2678 y Fy(\))17 b Fx(d)p Fy(\()p Fx(\013)22 b Fs(\000)h Fx(\013)2527 2693 y FC(0)2566 2678 y Fy(\)\()p Fx(\025)2699 2637 y Ft(0)2722 2678 y Fy(\))-118 2919 y FA(exists)34 b(and)h(for)f(a.e.)45 b(p)-5 b(air)34 b Fy(\()p Fx(\036;)17 b(\025)p Fy(\))27 b Fs(2)h Fw(T)1312 2883 y Fr(d)1378 2919 y Fs(\002)23 b Fw(R)5 b FA(,)40 b(the)35 b(limit)1169 3184 y Fx(\014)6 b Fy(\()p Fx(\025;)17 b(\036)p Fy(\))27 b(=)75 b(lim)1595 3250 y Ft(j)p Fr(T)10 b Ft(j!1)1892 3116 y Fy(1)p 1853 3161 127 4 v 1853 3252 a Fs(j)p Fx(T)k Fs(j)2006 3184 y Fy(log)i Fs(k)p Fx(M)10 b Fy(\()p Fx(T)k Fy(;)j Fx(\025;)g(\036)p Fy(\))p Fs(k)o Fx(;)-118 3460 y FA(b)-5 b(eing)34 b Fx(M)10 b Fy(\()p Fx(T)k Fy(;)j Fx(\025;)g(\036)p Fy(\))34 b FA(the)h(fundamental)f(matrix)h(of)f(\(3.43\),)g(exists)g(and)h(it)g (is)g(e)-5 b(qual)34 b(to)897 3742 y Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))27 b(=)h Fx(\014)1277 3757 y FC(0)1316 3742 y Fy(\()p Fx(\025)p Fy(\))22 b(+)1584 3675 y(1)p 1579 3719 59 4 v 1579 3810 a Fx(\031)1665 3606 y Fq(Z)1764 3633 y FC(+)p Ft(1)1720 3832 y(\0001)1911 3742 y Fy(log)16 b(\()p Fx(\025)22 b Fs(\000)h Fx(\025)2327 3701 y Ft(0)2350 3742 y Fy(\))17 b Fx(d)p Fy(\()p Fx(\013)22 b Fs(\000)h Fx(\013)2740 3757 y FC(0)2779 3742 y Fy(\)\()p Fx(\025)2912 3701 y Ft(0)2935 3742 y Fy(\))p Fx(:)28 4024 y Fy(In)33 b(previous)g(sections)g(w)m(e)h(sho)m(w)m(ed)g(that,)f(for)f(real)g Fx(\025)p Fy(,)1419 4244 y(lim)1392 4312 y Fr(")p Ft(!)p FC(0)1531 4293 y Fp(+)1598 4244 y Fy(Im)g Fx(w)s Fy(\()p Fx(\025)22 b Fy(+)g Fx(i")p Fy(\))27 b(=)h Fx(\013)q Fy(\()p Fx(\025)p Fy(\))p Fx(;)-118 4499 y Fy(the)48 b(rotation)e(n)m(um)m(b)s(er.)89 b(No)m(w)48 b(w)m(e)h(are)e(going)g (to)g(state)h(a)f(similar)e(result,)51 b(but)d(with)f(some)h (non-trivial)-118 4619 y(restrictions,)-118 4847 y Fv(Theorem)37 b(3.3.26)h(\([44]\))134 b FA(\(i\))48 b(If)27 b Fx(\025)g Fs(2)h Fw(R)39 b FA(and)27 b Fx(z)32 b Fs(!)27 b Fx(\025)h FA(non-tangential)5 b(ly)26 b(for)h(Im)35 b Fx(z)d(>)c Fy(0)p FA(,)g(then)f Fs(\000)p FA(R)-5 b(e)36 b Fx(w)s Fy(\()p Fx(z)t Fy(\))126 4968 y FA(tends)42 b(to)h Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))p FA(.)67 b(If)42 b Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))42 b(=)g(0)p FA(,)i(then)f Fx(\014)48 b FA(is)42 b(c)-5 b(ontinuous)43 b(at)f Fx(\025)p FA(,)j(and)d Fs(\000)p FA(R)-5 b(e)35 b Fx(w)s Fy(\()p Fx(z)t Fy(\))42 b Fs(!)g Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))42 b FA(whenever)126 5088 y Fx(z)32 b Fs(!)27 b Fx(\025)p FA(,)35 b(non-tangential)5 b(ly)34 b(or)h(not.)-62 5291 y(\(ii\))48 b Fx(\025)27 b Fs(2)h Fw(R)39 b Fs(!)27 b Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))35 b FA(is)f(upp)-5 b(er)35 b(semi-c)-5 b(ontinuous)34 b(on)g Fw(R)5 b FA(.)-92 5495 y(\(iii\))48 b(On)34 b Fw(R)5 b FA(,)41 b Fx(\014)f FA(is)35 b(non-ne)-5 b(gative,)33 b(of)i(\014rst)g(Bair)-5 b(e)34 b(class,)g(and)g(has)h(the)f(me)-5 b(an)34 b(value)h(pr)-5 b(op)g(erty.)28 5723 y Fy(W)d(e)37 b(can)f(no)m(w)g(pro)m(v)m(e)i(a)d (nice)h(regularit)m(y)f(result)h(for)f(the)i(upp)s(er)f(Ly)m(apuno)m(v) i(exp)s(onen)m(t)f(in)e(the)i(resolv)m(en)m(t)-118 5844 y(set)c Fx(\032)p Fy(\()p Fx(H)8 b Fy(\))32 b(of)g(the)i(Sc)m(hr\177) -49 b(odinger)32 b(op)s(erator)g(with)g(quasi-p)s(erio)s(dic)f(p)s (oten)m(tial)g(on)h Fx(L)2949 5807 y FC(2)2989 5844 y Fy(\()p Fs(\0001)p Fx(;)17 b Fy(+)p Fs(1)p Fy(\).)1900 6155 y(72)p eop %%Page: 73 77 73 76 bop -118 219 a Fv(Prop)s(osition)35 b(3.3.27)j(\([44]\))49 b FA(The)42 b(upp)-5 b(er)42 b(Lyapunov)h(exp)-5 b(onent,)44 b Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))p FA(,)44 b(is)f(harmonic)e(on)i (the)f(r)-5 b(esolvent)-118 339 y(set)35 b(of)f(the)h(whole-line)f(op) -5 b(er)g(ator)34 b Fx(H)8 b FA(.)28 554 y Fv(Pro)s(of:)42 b Fy(Consider)30 b(the)g(function)f Fx(w)s Fy(,)h(for)f(whic)m(h)h(w)m (e)h(already)e(sa)m(w)i(that)e Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))27 b(=)h Fs(\000)p Fy(Re)33 b Fx(w)s Fy(\()p Fx(\025)p Fy(\),)c(for)g(Im)j Fx(\025)c Fs(6)p Fy(=)-118 674 y(0.)43 b(Recall)31 b(that,)i(from)e(\(3.36\),)p 1073 587 206 4 v 32 w Fx(w)s Fy(\()p Fx(\025)p Fy(\))c(=)g Fx(w)s Fy(\()1524 648 y(\026)1520 674 y Fx(\025)o Fy(\).)44 b(No)m(w,)33 b(if)f Fx(I)40 b Fy(is)32 b(an)g(in)m(terv)-5 b(al)32 b(in)g(the)h(resolv)m(en)m(t)g(set,)h(then)1268 882 y Fs(\000)p Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))22 b(+)g Fx(i\013)q Fy(\()p Fx(\025)p Fy(\))28 b(=)54 b(lim)2019 950 y Fr(")p Ft(!)p FC(0)2158 931 y Fp(+)2225 882 y Fx(w)s Fy(\()p Fx(\025)21 b Fy(+)h Fx(i")p Fy(\))-118 1130 y(and)35 b Fx(\013)q Fy(\()p Fx(\025)p Fy(\))f(is)g(constan)m(t)h(for)f Fx(\025)d Fs(2)h Fx(I)8 b Fy(,)35 b(sa)m(y)g Fx(\013)1481 1145 y Fr(I)1521 1130 y Fy(.)50 b(W)-8 b(e)34 b(also)g(ha)m(v)m(e)i (that,)f(when)h(w)m(e)f(tend)h(to)e(the)h(real)e(axis)i(on)f(the)-118 1251 y(other)f(side)f(then)1396 1371 y(lim)1369 1439 y Fr(")p Ft(!)p FC(0)1508 1420 y Fi(\000)1576 1371 y Fx(w)s Fy(\()p Fx(\025)21 b Fy(+)i Fx(i")p Fy(\))k(=)h Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))21 b Fs(\000)i Fx(\013)2489 1386 y Fr(I)-118 1575 y Fy(for)32 b Fx(\025)c Fs(2)g Fx(I)8 b Fy(.)43 b(So,)32 b(if)g(w)m(e)i(de\014ne)1171 1896 y Fx(w)1244 1855 y Ft(\003)1282 1896 y Fy(\()p Fx(\025)p Fy(\))28 b(=)1546 1692 y Fq(8)1546 1781 y(<)1546 1961 y(:)1812 1775 y Fx(w)s Fy(\()p Fx(\025)p Fy(\))p Fx(;)218 b Fy(Im)32 b Fx(\025)27 b(>)h Fy(0)p Fx(;)1690 1895 y(\014)6 b Fy(\()p Fx(\025)p Fy(\))22 b(+)g Fx(i\013)2099 1910 y Fr(I)2139 1895 y Fx(;)175 b(\025)27 b Fs(2)i Fx(I)8 b(;)1676 2015 y(w)s Fy(\()p Fx(\025)p Fy(\))22 b(+)g(2)p Fx(\013)2113 2030 y Fr(I)2153 2015 y Fx(;)83 b Fy(Im)32 b Fx(\025)27 b(<)h Fy(0)p Fx(;)-118 2222 y Fy(then)g Fx(w)172 2186 y Ft(\003)238 2222 y Fy(is)f(holomorphic)e(on)j Fs(f)p Fy(Im)k Fx(\025)27 b Fs(6)p Fy(=)h(0)p Fs(g)12 b([)g Fx(I)35 b Fy(b)m(y)28 b(the)g(re\015ection)g(principle.)40 b(It)28 b(follo)m(ws)e(that)h Fx(\014)33 b Fy(is)27 b(harmonic)-118 2343 y(on)32 b(the)h(resolv)m(en)m(t)h(set,)f(b)s(ecause)h(it)e(is)g (the)h(real)f(part)g(of)g(an)h(analytic)e(function.)43 b Fh(\003)p Fy(.)28 2463 y(As)33 b(a)g(corollary)d(w)m(e)k(get)f(that)f (the)h(sp)s(ectrum)g(cannot)g(b)s(e)f(to)s(o)g(small)-118 2657 y Fv(Corollary)k(3.3.28)i(\([44)o(]\))49 b FA(L)-5 b(et)25 b Fx(I)36 b Fs(\032)28 b Fw(R)36 b FA(an)25 b(op)-5 b(en)25 b(interval)g(such)g(that)g Fx(I)9 b Fs(\\)q Fx(\033)t Fy(\()p Fx(H)f Fy(\))27 b Fs(6)p Fy(=)h Fs(;)p FA(.)41 b(Then)24 b(the)i(lo)-5 b(garithmic)-118 2777 y(c)g(ap)g(acity)34 b(of)h Fx(\033)t Fy(\()p Fx(H)8 b Fy(\))22 b Fs(\\)g Fx(I)43 b FA(is)35 b(p)-5 b(ositive.)-118 2992 y Fv(Remark)37 b(3.3.29)49 b FA(This)30 b(do)-5 b(esn)-10 b('t)30 b(imply)g(that)i (the)e(sp)-5 b(e)g(ctrum)31 b(c)-5 b(annot)30 b(have)g(zer)-5 b(o)30 b(me)-5 b(asur)g(e,)31 b(as,)g(for)g(instanc)-5 b(e,)-118 3112 y(the)35 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Fy(,)i(as)h(usual.)-118 4144 y(The)i(orien)m(tation)e(of)h Fw(P)748 4108 y FC(1)790 4144 y Fy(\()p Fw(R)5 b Fy(\))38 b(is)32 b(tak)m(en)i(to)e(agree)h (with)f(an)g(increase)h(in)f Fx(')p Fy(.)28 4264 y(No)m(w)41 b(\014x)h Fx(\022)i Fs(2)e Fw(T)665 4228 y Fr(d)709 4264 y Fy(.)67 b(It)41 b(can)g(b)s(e)f(seen)i(that)f(if)e Fx(\025)i Fy(increases)g(through)g Fx(I)8 b Fy(,)42 b(then)f Fx(M)3115 4279 y FC(+)3175 4264 y Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))40 b(and)h Fx(M)3732 4279 y Ft(\000)3791 4264 y Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))-118 4384 y(mo)m(v)m(e)36 b(in)f(opp)s(osite)g(directions)f(on)i Fw(P)1290 4348 y FC(1)1331 4384 y Fy(\()p Fw(R)5 b Fy(\))42 b(\(see,)37 b(for)e(instance,)i([17)o(]\).)53 b(Moreo)m(v)m(er,)38 b Fx(M)3126 4399 y FC(+)3185 4384 y Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))35 b(and)h Fx(M)3732 4399 y Ft(\000)3791 4384 y Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)p Fy(\))-118 4505 y(can)33 b(nev)m(er)h(coincide)e(if)f Fx(\025)d Fs(2)g Fx(I)8 b Fy(.)28 4625 y(Therefore,)34 b(as)f Fx(\025)674 4640 y Fr(n)748 4625 y Fs(!)28 b Fx(\025)933 4640 y FC(0)1004 4625 y Fy(with)33 b(the)g(terms)f(of)g(the)h(sequence) j(in)31 b Fx(I)8 b Fy(,)33 b(the)g(limits)1693 4833 y(lim)1669 4893 y Fr(n)p Ft(!1)1869 4833 y Fx(l)1898 4848 y Ft(\006)1957 4833 y Fy(\()p Fx(\022)s Fy(;)17 b Fx(\025)2144 4848 y Fr(n)2191 4833 y Fy(\))-118 5078 y(m)m(ust)33 b(exist)f(in)g Fw(P)527 5042 y FC(1)569 5078 y Fy(\()p Fw(R)t Fy(\))39 b(b)s(ecause)34 b Fw(P)1169 5042 y FC(1)1210 5078 y Fy(\()p Fw(R)5 b Fy(\))39 b(is)32 b(compact.)43 b(Call)31 b(these)j(limits)29 b Fx(l)2673 5093 y Ft(\006)2732 5078 y Fy(\()p Fx(\022)s Fy(\).)44 b(The)33 b(sets)1274 5286 y Fx(S)1334 5301 y Ft(\006)1421 5286 y Fy(=)1524 5206 y Fq(\010)1582 5286 y Fy(\()p Fx(l)1649 5301 y Ft(\006)1708 5286 y Fy(\()p Fx(\022)s Fy(\))p Fx(;)17 b(\022)s Fy(\);)44 b Fx(\022)31 b Fs(2)d Fw(T)2266 5245 y Fr(d)2310 5206 y Fq(\011)2396 5286 y Fs(\032)g Fy(\006)2571 5301 y Fk(R)-118 5495 y Fy(are)c(measurable)g(subsections)i(of)e(\006)1214 5510 y Fk(R)1294 5495 y Fy(=)j Fw(P)1456 5459 y FC(1)1498 5495 y Fy(\()p Fw(R)5 b Fy(\))g Fs(\002)g Fw(T)1794 5459 y Fr(d)1840 5495 y Fy(,)26 b(b)s(ecause)g(they)f(are)g(the)f(limit)d (of)j(measurable)g(functions.)-118 5615 y(Hence)34 b(they)f(de\014ne)h (ergo)s(dic)e(measures)h Fx(\026)1489 5579 y Ft(\006)1580 5615 y Fy(on)g(\006)g(via)f(the)h(form)m(ulae)1368 5731 y Fq(Z)1423 5956 y FC(\006)1495 5866 y Fx(g)t(d\026)1656 5881 y Ft(\006)1741 5866 y Fy(=)1845 5731 y Fq(Z)1900 5956 y Fk(T)1950 5937 y Fj(d)2001 5866 y Fx(g)t Fy(\()p Fx(\022)s Fy(;)17 b Fx(l)2211 5881 y Ft(\006)2269 5866 y Fy(\()p Fx(\022)s Fy(\)\))p Fx(d\022)1900 6155 y Fy(73)p eop %%Page: 74 78 74 77 bop -118 219 a Fy(for)32 b(con)m(tin)m(uous)h(functions)g Fx(g)e Fy(:)c(\006)i Fs(!)e Fw(R)5 b Fy(.)49 b(W)-8 b(e)33 b(ha)m(v)m(e)h(that,)f(with)f(the)h(previous)g(de\014nitions)482 501 y Fs(\000)p Fx(\014)6 b Fy(\()p Fx(\025)715 516 y Fr(n)762 501 y Fy(\))28 b(=)931 366 y Fq(Z)1031 392 y Fr(d)987 591 y Fk(T)1088 501 y Fx(f)1136 516 y Fr(\025)1177 524 y Fj(n)1224 501 y Fy(\()p Fx(\022)s(;)17 b(M)1448 516 y FC(+)1507 501 y Fy(\()p Fx(\022)s Fy(;)g Fx(\025)1694 516 y Fr(n)1741 501 y Fy(\)\))p Fx(d\022)30 b Fs(!)2070 366 y Fq(Z)2126 591 y Fk(T)2176 572 y Fj(d)2227 501 y Fx(f)2275 516 y Fr(\025)2316 525 y Fp(0)2355 501 y Fy(\()p Fx(\022)s Fy(;)17 b Fx(l)2514 516 y FC(+)2573 501 y Fy(\()p Fx(\022)s Fy(\)\))p Fx(d\022)30 b Fy(=)2965 366 y Fq(Z)3020 591 y FC(\006)3092 501 y Fx(f)3140 516 y Fr(\025)3181 525 y Fp(0)3220 501 y Fx(d\026)3330 516 y FC(+)3388 501 y Fx(:)-118 768 y Fy(Using)i(remark)g(3.3.22,)1516 796 y Fq(Z)1572 1022 y FC(\006)1643 932 y Fx(f)1691 947 y Fr(\025)1732 956 y Fp(0)1771 932 y Fx(d\026)1881 947 y FC(+)1967 932 y Fy(=)c Fs(\000)p Fx(\014)6 b Fy(\()p Fx(\025)2304 947 y FC(0)2343 932 y Fy(\))-118 1153 y(whic)m(h)33 b(completes)f(the)h(pro)s(of)f(of)g(the)h(theorem.)43 b Fh(\003)-118 1381 y Fv(Remark)37 b(3.3.31)49 b FA(R.)41 b(Johnson)e(pr)-5 b(ovide)g(d)40 b(in)g([44])g(an)h(example)e(of)h(a)h (Schr\177)-50 b(odinger)39 b(e)-5 b(quation)41 b(with)f(limit-)-118 1502 y(p)-5 b(erio)g(dic)35 b(p)-5 b(otential)35 b(\(a)g(function)h (which)e(is)i(uniform)f(limit)g(of)h(p)-5 b(erio)g(dic)35 b(functions)g(with)g(incr)-5 b(e)g(asing)35 b(p)-5 b(erio)g(d\))-118 1622 y(such)41 b(that)h(the)g(upp)-5 b(er)41 b(Lyapunov)g(exp)-5 b(onent,)42 b Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))p FA(,)43 b(is)e(not)h(c)-5 b(ontinuous)41 b(for)g(a)g(c)-5 b(ertain)42 b(value)f(of)g Fx(\025)p FA(.)64 b(This)-118 1742 y(function)40 b(is)g(not)h(quasi-p)-5 b(erio)g(dic,)40 b(and)g(in)g(some)g(sense,)h (this)g(example)e(is)h(the)h(opp)-5 b(osite)40 b(of)g(those)g(systems) -118 1863 y(that)35 b(we)g(ar)-5 b(e)34 b(inter)-5 b(este)g(d)35 b(in.)-118 2091 y Fv(Remark)i(3.3.32)49 b FA(V)-7 b(ery)35 b(r)-5 b(e)g(c)g(ently,)35 b(J.)g(Bour)-5 b(gain)35 b(and)f(S.)h (Jitomirskaya)f(\([9]\))h(have)f(shown)g(the)h(c)-5 b(ontinuity)-118 2212 y(of)31 b(the)g(Lyapunov)g(exp)-5 b(onent)30 b(with)h(r)-5 b(esp)g(e)g(ct)31 b(to)h(the)f(ener)-5 b(gy)31 b Fx(\025)g FA(for)g(the)g(discr)-5 b(ete)31 b(analo)-5 b(gue)30 b(of)h(the)g(Schr\177)-50 b(odinger)-118 2332 y(e)-5 b(quation)49 b(when)f(the)h(p)-5 b(otential)48 b(is)h(analytic)g (\(with)f(no)h(assumptions)f(on)h(the)g(fr)-5 b(e)g(quency\))48 b(and)h(only)f(one)-118 2452 y(fr)-5 b(e)g(quency)31 b(is)f(c)-5 b(onsider)g(e)g(d.)42 b(Mor)-5 b(e)g(over,)32 b(they)f(also)f(show)g(c)-5 b(ontinuity)32 b(with)e(r)-5 b(esp)g(e)g(ct)31 b(to)g(the)g(fr)-5 b(e)g(quency)31 b(at)g(every)-118 2573 y(irr)-5 b(ational)39 b(p)-5 b(oint.)57 b(When)39 b(the)g(Lyapunov)g(exp)-5 b(onent)38 b(is)h(p)-5 b(ositive)39 b(in)g(a)g(c)-5 b(omp)g(act)38 b(interval)h(for)g(the)g (ener)-5 b(gies)-118 2693 y(then)30 b(the)g(Lyapunov)g(exp)-5 b(onent)29 b(and)h(the)g(r)-5 b(otation)30 b(numb)-5 b(er)30 b(c)-5 b(an)30 b(b)-5 b(e)29 b(shown)h(to)g(b)-5 b(e)30 b(jointly)g(H\177)-50 b(older)30 b(c)-5 b(ontinuous)-118 2813 y(in)34 b(this)h(interval)g(\([34)o(]\).)-118 3102 y Fg(3.3.5)136 b(Application)45 b(to)h(Can)l(tor)f(sp)t(ectrum)-118 3287 y Fy(T)-8 b(o)40 b(conclude)h(this)f(section,)j(w)m(e)f(shall)d (sp)s(eak)i(a)f(little)e(bit)i(ab)s(out)g(the)h(existence)h(of)e(Can)m (tor)h(sp)s(ectrum)f(in)-118 3407 y(Sc)m(hr\177)-49 b(odinger)39 b(equation)f(with)g(quasi-p)s(erio)s(dic)f(p)s(oten)m(tial.)60 b(Recall)37 b(that)h(w)m(e)i(sa)m(w)f(that)g(this)f(sp)s(ectrum)h(w)m (as)-118 3528 y(a)d(closed)g(subset)i(of)d(the)i(real)e(line.)52 b(It)36 b(will)e(turn)i(out)g(that)g(quite)g(t)m(ypically)f(this)h(set) g(is)g(no)m(where)h(dense,)i(a)-118 3648 y(situation)31 b(that)h(will)f(b)s(e)h(referred)i(as)e FA(Cantor)j(sp)-5 b(e)g(ctrum)p Fy(.)28 3769 y(It)45 b(is)g(not)g(immediate)d(from)h(the) j(discussion)f(and)g(prop)s(erties)g(of)g(the)g(rotation)e(n)m(um)m(b)s (er)j(\(increasing)-118 3889 y(strictly)36 b(on)h(the)h(sp)s(ectrum)f (and)h(constan)m(t)g(o)m(v)m(er)g(the)f(in)m(terv)-5 b(als)37 b(of)f(the)i(resolv)m(en)m(t)g(set\))g(that)f(the)g(sp)s (ectrum)-118 4009 y(is)f(a)g(Can)m(tor)h(set,)h(b)s(ecause)g(this)e(do) s(es)h(not)g(imply)d(that)j(the)g(resolv)m(en)m(t)g(set)h(is)e(non)g(v) m(oid.)56 b(Indeed,)39 b(it)c(could)-118 4130 y(happ)s(en)30 b(\(and)g(it)f(certainly)h(do)s(es\))g(that)g(the)g(sp)s(ectral)g(gap)f (corresp)s(onding)h(to)g(a)g(certain)f(resonance)i(\(that)f(is)-118 4250 y(the)35 b(in)m(terior)d(of)i(the)h(set)g(of)f(p)s(oin)m(ts)g (with)f(rotation)g(n)m(um)m(b)s(er)i(in)e(the)i(resonance)h(mo)s (dule\))c(is)i(v)m(oid,)h(in)e(whic)m(h)-118 4370 y(case)d(w)m(e)h (will)c(sp)s(eak)k(of)e(a)g FA(c)-5 b(ol)5 b(lapse)-5 b(d)31 b(gap)p Fy(.)42 b(This)30 b(phenomenon)g(also)e(tak)m(es)j (place)f(in)e(the)i(p)s(erio)s(dic)e(case,)j(and,)-118 4491 y(though)39 b(is)f(not)h(p)s(ersisten)m(t)g(under)h(generic)f (\(ev)m(en)h(quasi-p)s(erio)s(dic\))d(p)s(erturbations)h(\(see)i([65])f (for)f(this)g FA(gap)-118 4611 y(op)-5 b(ening)9 b Fy(\),)34 b(it)g(can)h(b)s(e)g(sho)m(wn)h(\(using)f(reducibilit)m(y\))e(that)h (in)h(some)f(cases)j(the)e(analysis)f(for)g(its)h(existence)h(is)-118 4732 y(analogous)31 b(to)h(the)h(p)s(erio)s(dic)e(case)j(\([12)o(]\).) 28 4852 y(One)c(w)m(a)m(y)h(to)e(pro)m(v)m(e)i(the)e(existence)i(of)e (Can)m(tor)h(sp)s(ectrum)g(is)f(based)h(on)g(the)g(appro)m(ximation)d (b)m(y)j(p)s(erio)s(dic)-118 4972 y(p)s(oten)m(tials.)66 b(W)-8 b(e)41 b(shall)e(fo)s(cus)h(on)h(t)m(w)m(o)g(pap)s(ers.)67 b(The)42 b(\014rst,)h(due)e(to)f(J.)h(Moser)g(\([64]\),)h(constructs)g (a)e(limit)-118 5093 y(p)s(erio)s(dic)31 b(p)s(oten)m(tial)g(of)h(the)h (t)m(yp)s(e)937 5383 y Fx(q)t Fy(\()p Fx(t)p Fy(\))27 b(=)h Fx(a)1277 5398 y FC(0)1338 5383 y Fy(+)1487 5259 y Ft(1)1450 5288 y Fq(X)1437 5498 y Fr(j;s)p FC(=1)1624 5302 y Fq(\000)1670 5383 y Fx(a)1721 5398 y Fr(j)t(s)1807 5383 y Fy(cos\()p Fx(s)p Fy(2)2070 5342 y Ft(\000)p Fr(j)2161 5383 y Fx(t)p Fy(\))23 b(+)f Fx(b)2396 5398 y Fr(j)t(s)2482 5383 y Fy(sin\()p Fx(s)p Fy(2)2735 5342 y Ft(\000)p Fr(j)2826 5383 y Fx(t)p Fy(\))2899 5302 y Fq(\001)3766 5383 y Fy(\(3.46\))-118 5704 y(suc)m(h)32 b(that)e(it)f(op)s(ens)i(all)d(gaps)i(\(corresp)s (onding)g(to)g(iterations)f(of)h(the)g(P)m(oincar)m(\023)-46 b(e)30 b(map\))g(of)g(a)g(certain)f(p)s(erio)s(dic)-118 5824 y(p)s(oten)m(tial.)42 b(More)33 b(precisely)-8 b(,)33 b(the)g(main)d(statemen)m(t)j(in)f([64])h(is)1900 6155 y(74)p eop %%Page: 75 79 75 78 bop -118 219 a Fv(Theorem)37 b(3.3.33)h(\([64]\))48 b FA(Given)42 b Fx(\021)k(>)41 b Fy(0)h FA(and)g Fx(q)1825 234 y FC(0)1907 219 y FA(a)g(c)-5 b(ontinuous)43 b(function)f(of)g(p)-5 b(erio)g(d)41 b Fx(\031)t FA(,)k(ther)-5 b(e)42 b(exists)g(a)-118 339 y(limit-p)-5 b(erio)g(dic)33 b(analytic)i(function)f Fx(q)39 b FA(with)34 b(b)-5 b(asic)34 b(fr)-5 b(e)g(quencies)34 b Fy(2)2308 303 y Ft(\000)p Fr(j)2433 339 y FA(\()p Fx(j)g Fy(=)28 b(0)p Fx(;)17 b Fy(1)p Fx(;)g(:)g(:)g(:)e FA(\))34 b(with)h Fs(k)p Fx(q)25 b Fs(\000)d Fx(q)3513 354 y FC(0)3553 339 y Fs(k)3603 354 y Ft(1)3705 339 y Fx(<)28 b(\021)38 b FA(for)-118 459 y(which)28 b(the)h(Schr\177)-50 b(odinger)27 b(e)-5 b(quation)29 b(with)g(p)-5 b(otential)28 b Fx(q)33 b FA(has)28 b(al)5 b(l)29 b(sp)-5 b(e)g(ctr)g(al)28 b(gaps)g(op)-5 b(en)28 b(and,)i(henc)-5 b(e,)29 b(the)g(sp)-5 b(e)g(ctrum)-118 580 y(is)34 b(a)h(Cantor)g(set.)28 796 y Fy(There)42 b(is)d(a)h(v)m(ery)i(nice)e(geometrical)e(discussion)j(b)s(ehind)f (this)g(result,)i(whic)m(h)e(comes)h(from)e(a)h(detailed)-118 916 y(analysis)f(of)g(the)h(p)s(erio)s(dic)d(case.)65 b(In)40 b(this)f(case)i(w)m(e)f(kno)m(w)h(that)e(the)h(sp)s(ectral)f (gaps)h(are)f(lab)s(eled)f(with)h(the)-118 1037 y(p)s(ositiv)m(e)h(in)m (tegers,)j(b)s(ecause)f(they)f(corresp)s(ond)g(to)g(those)g(v)-5 b(alues)40 b(of)g Fx(\025)h Fy(for)f(whic)m(h)h Fs(j)p Fy(\001\()p Fx(\025)p Fy(\))p Fs(j)f Fx(>)h Fy(2,)i(where)e(\001)-118 1157 y(is)f(the)g(trace)g(of)g(the)h(P)m(oincar)m(\023)-46 b(e)40 b(map.)65 b(On)40 b(the)h(other)f(hand,)i(one)f(can)f(also)f (consider)i Fx(q)i Fy(as)e(a)f(function)f(of)-118 1277 y(p)s(erio)s(d)h(2)p Fx(\031)t(;)17 b Fy(3)p Fx(\031)t(;)g(:)g(:)g(:)e (;)i(m\031)46 b Fy(and)41 b(the)h(corresp)s(onding)g(sp)s(ectral)f (gaps.)70 b(Ho)m(w)m(ev)m(er,)46 b(it)41 b(turns)h(out)g(that)f(all)e (the)-118 1398 y(sp)s(ectral)h(gaps)h(corresp)s(onding)g(to)f(higher)h (p)s(erio)s(ds)f(of)g(the)h(P)m(oincar)m(\023)-46 b(e)41 b(map)f(are)h(collapsed,)h(b)s(ecause)g(they)-118 1518 y(are)h(all)e(double)i(ro)s(ots)g(of)g(the)h Fx(m\031)t Fy(-p)s(erio)s(d)e(time)f(map.)75 b(The)44 b(idea)f(of)g([64)o(])h(is)f (to)f(recurren)m(tly)j(de\014ne)f(the)-118 1639 y(p)s(erturbation)38 b(\(3.46\))h(so)g(that)g(all)f(gaps)h(corresp)s(onding)g(to)g(higher)g (p)s(erio)s(ds)g(are)g(op)s(ened.)64 b(T)-8 b(o)39 b(sho)m(w)i(that) -118 1759 y(previously)29 b(op)s(ened)i(gaps)e(are)h(not)f(closed)h (again)e(it)g(is)h(used)i(the)e(analog)f(of)h(the)h(functional)e(deriv) -5 b(ativ)m(e)29 b(from)-118 1879 y(theorem)g(3.3.14)f(for)h(p)s(erio)s (dic)f(p)s(oten)m(tials)g(whic)m(h)h(is)g(also)g(v)-5 b(alid)27 b(for)i(the)g(sp)s(ectral)g(in)m(terv)-5 b(als)29 b(\(i.e.)42 b(those)30 b(op)s(en)-118 2000 y(in)m(terv)-5 b(als)32 b(con)m(tained)g(in)g(the)h(sp)s(ectrum)g(of)f(the)h FA(p)-5 b(erio)g(dic)37 b Fy(op)s(erator\).)28 2120 y(In)31 b([46],)f(Moser)i(tec)m(hniques)g(are)e(hea)m(vily)g(used)i(b)m(y)f(R.) f(Johnson,)i(together)e(with)g(the)h(functional)e(deriv)-5 b(a-)-118 2240 y(tiv)m(e)46 b(\(and)g(sp)s(ecially)f(its)g(p)s(ositiv)m (eness\))i(to)e(pro)m(vide)i(a)e(more)g(general)h(result)g(using)f (again)g(the)h(p)s(erio)s(dic)-118 2361 y(appro)m(ximation.)41 b(The)34 b(main)d(statemen)m(t)i(of)f(this)g(pap)s(er)h(is)f(the)h (follo)m(wing)-118 2577 y Fv(Theorem)k(3.3.34)h(\([46]\))48 b FA(Ther)-5 b(e)47 b(is)g(a)g(r)-5 b(esidual)47 b(subset)h(\(i.e.,)i (c)-5 b(ountable)47 b(interse)-5 b(ction)47 b(of)g(op)-5 b(en)47 b(dense)-118 2697 y(sets\))37 b Fs(F)j(\032)32 b Fw(R)403 2661 y Fr(d)487 2697 y FA(such)k(that,)i(if)f Fx(!)d Fs(2)e(F)10 b FA(,)37 b(then)g(the)g(fol)5 b(lowing)35 b(statement)i(holds.)50 b(Ther)-5 b(e)36 b(is)h(a)f(r)-5 b(esidual)37 b(subset)-118 2818 y Fx(V)49 b Fy(=)28 b Fx(V)21 b Fy(\()p Fx(!)t Fy(\))27 b Fs(\032)h Fx(C)520 2782 y Fr(\016)558 2818 y Fy(\()p Fw(T)659 2782 y Fr(d)703 2818 y Fy(\))p FA(,)35 b(with)f Fy(0)28 b Fs(\024)g Fx(\016)j(<)d Fy(1)p FA(,)35 b(such)f(that,)h(if)g Fx(Q)28 b Fs(2)g Fx(V)56 b FA(and)35 b Fx(\036)27 b Fs(2)h Fw(T)2780 2782 y Fr(d)2824 2818 y FA(,)35 b(then)g(the)f(op)-5 b(er)g(ator)1323 3086 y Fx(H)35 b Fy(=)28 b Fx(H)1624 3101 y Fr(\036)1697 3086 y Fy(=)g Fs(\000)1906 3019 y Fx(d)1957 2983 y FC(2)p 1888 3063 126 4 v 1888 3155 a Fx(dt)1974 3126 y FC(2)2046 3086 y Fy(+)22 b Fx(Q)p Fy(\()p Fx(!)t(t)g Fy(+)g Fx(\036)p Fy(\))1191 b(\(3.47\))-118 3321 y FA(has)34 b(Cantor)h(sp)-5 b(e)g(ctrum.)28 3537 y Fy(The)34 b(core)f(of)f(the)h(pro)s(of)e(is)i (the)g(follo)m(wing)c(theorem,)-118 3753 y Fv(Theorem)37 b(3.3.35)h(\([46]\))48 b FA(L)-5 b(et)36 b Fx(A)30 b Fy(=)f Fw(R)1434 3717 y Fr(d)1503 3753 y Fs(\002)23 b Fx(C)1680 3717 y Fr(\016)1718 3753 y Fy(\()p Fw(T)1819 3717 y Fr(d)1863 3753 y Fy(\))p FA(,)36 b(wher)-5 b(e)35 b Fy(0)29 b Fs(\024)g Fx(\016)34 b(<)29 b Fy(1)p FA(.)47 b(Ther)-5 b(e)35 b(is)g(an)h(op)-5 b(en)35 b(dense)g(subset)-118 3874 y Fx(W)52 b Fs(\032)39 b Fx(A)i FA(such)f(that,)i(if)f Fy(\()p Fx(!)t(;)17 b(Q)p Fy(\))37 b Fs(2)i Fx(W)14 b FA(,)42 b(then)e(the)h(two-dimensional)d(system)j(asso)-5 b(ciate)g(d)40 b(to)g(\(3.47\))g(has)g(an)-118 3994 y(exp)-5 b(onential)33 b(dichotomy)i(for)f(al)5 b(l)35 b Fx(\036)27 b Fs(2)h Fw(T)1398 3958 y Fr(d)1442 3994 y FA(.)28 4210 y Fy(Finally)-8 b(,)30 b(note)i(that)g(the)g(existence)i(of)e(Can)m (tor)g(sp)s(ectrum)g(can)h(also)e(b)s(e)h(obtained)g(in)f(com)m (bination)f(with)-118 4331 y(reducibilit)m(y)35 b(results,)j(as)e(it)g (is)g(the)g(case)i(in)e([29)o(])h(and)f(the)h(references)i(therein.)55 b(This)36 b(will)f(b)s(e)h(discussed)i(in)-118 4451 y(section)33 b(3.4.3.)-118 4782 y Fu(3.4)161 b(Reducibilit)l(y)77 b(in)g(Sc)l(hr\177)-81 b(odinger)74 b(equation)i(with)g(quasi-p)t(e-) 250 4965 y(riodic)54 b(p)t(otential)-118 5184 y Fy(No)m(w)36 b(w)m(e)g(come)f(to)g(the)h(main)e(section)h(in)g(this)g(c)m(hapter.)53 b(Here)36 b(w)m(e)g(shall)e(mak)m(e)i(free)f(use)i(of)e(the)g (notations,)-118 5304 y(de\014nitions)42 b(and)g(results)h(from)f (previous)h(sections.)73 b(Up)43 b(to)f(no)m(w)h(w)m(e)h(ha)m(v)m(e)g (seen)g(that)e(the)h(b)s(eha)m(viour)f(of)-118 5425 y(Sc)m(hr\177)-49 b(odinger)30 b(equation)h(with)e(quasi-p)s(erio)s(dic)g(p)s(oten)m (tial)f(can)j(b)s(e)f(quite)h(accurately)f(describ)s(ed)h(in)f(the)h (resol-)-118 5545 y(v)m(en)m(t)40 b(set,)i(and)d(the)g(underlying)f (reason)i(is)e(that,)i(there,)i(the)d(system)h(is)e(uniformly)f(h)m(yp) s(erb)s(olic.)62 b(W)-8 b(e)39 b(will)-118 5666 y(therefore)h (distinguish)f(b)s(et)m(w)m(een)j(reducibilit)m(y)c(in)h(the)i(resolv)m (en)m(t)g(set)f(and)g(reducibilit)m(y)f(in)g(the)h(sp)s(ectrum.)-118 5786 y(Moreo)m(v)m(er,)h(in)c(the)i(latter)e(case,)j(the)e(rotation)f (n)m(um)m(b)s(er)h(\(whic)m(h)h(will)d(corresp)s(ond)i(to)g(the)h (imaginary)c(part)-118 5906 y(eigen)m(v)-5 b(alues)32 b(of)f(the)i(reduced)g(matrix\),)e(together)h(with)g(its)f (arithmetical)e(prop)s(erties,)j(will)e(pla)m(y)h(a)h(k)m(ey)h(role.) 1900 6155 y(75)p eop %%Page: 76 80 76 79 bop -118 219 a Fg(3.4.1)136 b(Reducibilit)l(y)49 b(in)f(the)g(resolv)l(en)l(t)h(set.)70 b(Uniformly)48 b(h)l(yp)t(erb)t(olic)g(reduci-)293 368 y(bilit)l(y)-118 553 y Fy(The)40 b(reducibilit)m(y)d(in)h(this)h(set)h(under)g(the)f(t)m (ypical)g(Diophan)m(tine)e(assumptions)i(on)g(the)g(frequency)j(v)m (ector)-118 673 y Fx(!)t Fy(,)33 b(has,)g(in)g(fact,)g(already)f(b)s (een)i(established,)g(as)f(w)m(e)h(ha)m(v)m(e)h(sho)m(wn)f(that)f(if)f Fx(\025)h Fy(is)g(in)f(the)i(resolv)m(en)m(t)g(set)g(of)e(the)-118 793 y(op)s(erator)1452 967 y Fx(H)j Fy(=)28 b Fs(\000)1777 900 y Fx(d)1828 864 y FC(2)p 1759 945 126 4 v 1759 1036 a Fx(dt)1845 1007 y FC(2)1917 967 y Fy(+)22 b Fx(Q)p Fy(\()p Fx(!)t(t)g Fy(+)g Fx(\036)p Fy(\))-118 1167 y(then)28 b(the)h(corresp)s(onding)e(t)m(w)m(o)i(dimensional)c(system)k(with)e (this)h(v)-5 b(alue)27 b(of)g Fx(\025)h Fy(has)g(an)g(exp)s(onen)m (tial)f(dic)m(hotom)m(y)-118 1287 y(with)35 b(t)m(w)m(o)h (one-dimensional)d(subbundles.)54 b(Therefore)37 b(w)m(e)f(are)g(under) g(the)g(full)e(sp)s(ectrum)i(h)m(yp)s(othesis)h(and)-118 1408 y(the)29 b(results)f(on)h(reducibilit)m(y)d(b)m(y)j(R.)g(Johnson)g (and)f(G.)g(Sell)f(\(see)i(theorem)f(2.5.1\))g(can)h(b)s(e)f(applied)f (if)h(suitable)-118 1528 y(arithmetic)i(prop)s(erties)j(on)f Fx(!)k Fy(and)d(regularit)m(y)e(prop)s(erties)i(on)f Fx(Q)h Fy(are)g(satis\014ed.)28 1649 y(F)-8 b(or)22 b(the)h(sak)m(e)i (of)d(self-con)m(tainedness)i(w)m(e)g(no)m(w)f(giv)m(e)g(a)f(pro)s(of)g (of)h(this)f(fact)h(using)f(what)h(w)m(e)h(ha)m(v)m(e)h(in)m(tro)s (duced)-118 1769 y(during)40 b(this)f(c)m(hapter.)68 b(W)-8 b(e)41 b(follo)m(w)d(the)j(approac)m(h)g(of)f([47)o(],)j (although)c(a)h(similar)d(result)j(can)h(b)s(e)f(found)g(in)-118 1889 y(man)m(y)32 b(other)h(places.)44 b(The)33 b(result)g(that)f(w)m (e)i(w)m(an)m(t)f(to)g(pro)m(v)m(e)g(is)f(the)h(follo)m(wing:)-118 2117 y Fv(Theorem)k(3.4.1)h(\([65]\))48 b FA(L)-5 b(et)41 b Fs(F)50 b FA(the)40 b(sp)-5 b(ac)g(e)40 b(of)g(quasi-p)-5 b(erio)g(dic)39 b(functions)i(whose)e(extension)h(to)g Fw(T)3741 2081 y Fr(d)3826 2117 y FA(is)g(of)-118 2237 y(class)34 b Fx(\013)h FA(\(with)f Fx(\013)29 b Fy(=)e Fx(r)m(;)17 b Fs(1)p Fx(;)g(a)p FA(,)34 b Fx(r)d Fs(\025)d Fy(0)p FA(\))34 b(and)g(with)h(fr)-5 b(e)g(quency)34 b Fx(!)d Fs(2)d Fw(R)2387 2201 y Fr(d)2434 2237 y FA(.)44 b(Then,)34 b(ther)-5 b(e)35 b(exists)f(a)h(tr)-5 b(ansformation)-118 2358 y Fx(T)41 b Fy(=)28 b Fx(T)14 b Fy(\()p Fx(t)p Fy(;)j Fx(\025)p Fy(\))34 b FA(such)h(that)g(the)g(elements)f(of)h Fx(T)8 b(;)17 b(T)1680 2322 y Ft(\000)p FC(1)1808 2358 y FA(b)-5 b(elong)34 b(to)h Fs(F)45 b FA(and)34 b(the)h(tr)-5 b(ansformation)1563 2495 y Fq(\022)1689 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(frequencies)i(of)e(the)h(forcing)f(\(the)h(frequency)h(v)m(ector)g Fx(!)t Fy(\))d(and)i(the)g(in)m(ternal)e(frequencies)j(of)-118 3026 y(the)j(system)h(\(the)f(rotation)e(n)m(um)m(b)s(er\).)56 b(W)-8 b(e)38 b(shall)d(use)j(the)f(follo)m(wing)d(notation.)55 b(Assume)37 b(that)g Fx(!)h Fs(2)d Fw(R)3867 2990 y Fr(d)3950 3026 y Fy(is)-118 3147 y(the)k(frequency)h(v)m(ector)g(of)e(the)h (forcing,)g(that)f(will)e(b)s(e)j(tak)m(en)g(\014xed)h(for)e(the)h (rest)g(of)f(the)h(section.)61 b(W)-8 b(e)39 b(sa)m(y)-118 3267 y(that)32 b(it)g(is)g FA(Diophantine)38 b Fy(whenev)m(er)d(there)f (exist)f(p)s(ositiv)m(e)e(constan)m(ts)j Fx(N)5 b(;)17 b(\034)39 b(>)28 b Fy(0)k(suc)m(h)i(that)1190 3455 y Fs(jh)p Fv(k)p Fs(ij)27 b(\025)h Fx(N)1603 3414 y Ft(\000)p FC(1)1698 3455 y Fs(j)p Fv(k)p Fs(j)1813 3414 y Ft(\000)p Fr(\034)1911 3455 y Fx(;)211 b Fv(k)28 b Fs(2)g Fw(Z)2399 3414 y Fr(d)2459 3455 y Fs(\000)23 b(f)p Fy(0)p Fs(g)-118 3643 y Fy(b)s(eing)h Fs(h)p Fv(k)p Fs(i)j Fy(=)g Fs(h)p Fv(k)p Fx(;)17 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b(that)f(it)f(is)i FA(r)-5 b(ational)34 b(with)h(r)-5 b(esp)g(e)g(ct)34 b(to)h Fx(!)h Fy(if)c Fx(\032)c Fs(2)g(M)p Fy(.)28 4483 y(The)33 b(h)m(yp)s(othesis)g(on)f(the)g(real)g (analyticit)m(y)e(and)i(the)g(smallness)g(of)f(the)h(p)s(oten)m(tial)f (will)e(b)s(e)j(stated)h(in)e(the)-118 4603 y(follo)m(wing)j(form.)55 b(Assume)38 b(that)f Fx(Q)e Fy(:)g Fw(T)1398 4567 y Fr(d)1477 4603 y Fs(!)g Fw(R)47 b Fy(is)37 b(the)g(lift)e(to)i Fw(T)2341 4567 y Fr(d)2421 4603 y Fy(of)g(the)g(p)s(oten)m(tial)e Fx(q)t Fy(,)j(whic)m(h)f(w)m(e)h(consider)-118 4724 y(analytic)30 b(in)g(a)h(complex)g(neigh)m(b)s(ourho)s(o)s(d)f Fs(j)p Fy(Im)i Fx(\022)s Fs(j)c Fx(<)f(r)34 b Fy(of)d Fw(T)2125 4687 y Fr(d)2169 4724 y Fy(.)43 b(T)-8 b(o)31 b(con)m(trol)g(the)g (size)h(of)f(this)g(function)f(w)m(e)j(use)-118 4844 y(the)g(norm)1526 4964 y Fs(j)p Fx(Q)p Fs(j)1659 4993 y Fr(r)1724 4964 y Fy(=)90 b(sup)1828 5050 y Ft(j)p FC(Im)22 b Fr(\022)r Ft(j)p Fr()31 b Fy(0.)48 b(These)36 b(conditions)e(are)g(not)g(ful\014lled)f 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Fy(=)31 b Fx(A)490 5801 y FC(0)553 5786 y Fy(+)675 5761 y(^)652 5786 y Fx(B)40 b Fy(and)35 b Fx(F)1021 5801 y FC(1)1095 5786 y Fy(is)f(of)g(the)h(size)g Fx(")1721 5722 y Fp(3)p 1721 5734 31 3 v 1721 5775 a(2)1765 5786 y Fy(,)h(that)e(is,)h(m)m(uc)m(h)g(smaller)e(than)h Fx(F)14 b Fy(.)50 b(Then)35 b(w)m(e)h(rep)s(eat)f(this)-118 5906 y(pro)s(cedure)42 b(with)f Fx(A)647 5921 y FC(1)728 5906 y Fy(\(whose)h(eigen)m(v)-5 b(alues)41 b Fs(\006)p Fx(i\013)1747 5921 y FC(1)1828 5906 y Fy(also)g(dep)s(end)h(on)f Fx(\025)p Fy(\))g(instead)g(of)g Fx(A)3198 5921 y FC(0)3237 5906 y Fy(.)70 b(In)41 b(this)g(w)m(a)m(y)h(one)1900 6155 y(79)p eop %%Page: 80 84 80 83 bop -118 219 a Fy(constructs)31 b(a)f(sequence)i(of)e (transformations)e Fx(Y)1692 234 y Fr(j)1745 219 y Fs(\001)17 b(\001)g(\001)g(\001)d(\001)j Fx(Y)2024 234 y FC(1)2062 219 y Fy(,)31 b(for)e Fx(j)34 b Fy(=)27 b(1)p Fx(;)17 b(:)g(:)g(:)f(;)h Fs(1)p Fy(,)29 b(that)h(transforms)f(the)i(system) -118 339 y(\(3.48\))j(more)g(and)h(more)f(closely)h(to)f(a)h(constan)m (t)g(co)s(e\016cien)m(t)h(system.)51 b(Since)35 b(the)g Fx(Y)3075 354 y Fr(j)3146 339 y Fy(are)g(closer)g(and)g(closer)-118 459 y(to)c(the)h(iden)m(tit)m(y)-8 b(,)31 b(it)g(is)g(seen)i(\(after)e (some)g(w)m(ork\))i(that)e(the)h(sequence)i(con)m(v)m(erges)g(to)d(a)g (transformation)e Fx(Y)22 b Fy(\()p Fx(\022)s Fy(\))-118 580 y(that)32 b(solv)m(es)i(the)f(non-linear)d(equation)j(\(3.51\).)28 700 y(This)44 b(approac)m(h,)i(p)s(erhaps)e(the)g(most)e(natural)g (one,)47 b(w)m(as)d(used)g(b)m(y)g(Dinauburg)e(and)h(Sinai)f(\([22],)k (see)-118 820 y(also)40 b([76])h(and)g([15)o(]\))g(to)g(pro)m(v)m(e)h (the)g(reducibilit)m(y)d(in)h(a)h(big)f(set)i(\(in)e(a)h(measure)g (sense\))i(close)e(to)g(constan)m(t)-118 941 y(co)s(e\016cien)m(ts.)i (Ho)m(w)m(ev)m(er,)31 b(this)c(pro)s(cedure)h(requires)g(that)f(one)h (restricts)g(the)f(parameters)h Fx(\025)p Fy(,)g(and)f(an)h(essen)m (tial)-118 1061 y(p)s(oin)m(t)36 b(is)h(to)g(con)m(trol)g(the)g(dep)s (endence)j(on)d Fx(\025)h Fy(in)e(order)i(to)e(ensure)j(that)e(w)m(e)i (do)e(not)g(exclude)h(to)s(o)f(man)m(y)g(or)-118 1182 y(ev)m(en)d(all)d Fx(\025)p Fy(.)28 1302 y(The)f(restriction)d(of)i (parameters)f(turns)h(out)g(to)f(b)s(e)h(imp)s(osed)e(only)i(b)m(y)g (the)g(use)h(of)e(transformations)f(close)-118 1422 y(to)44 b(the)h(iden)m(tit)m(y)-8 b(,)47 b(and)e(the)g(w)m(a)m(y)g(to)g(o)m(v)m (ercome)g(this)f(restriction)f(\(as)i(it)f(is)g(done)g(in)g([65])g(and) h([29]\))f(is)g(to)-118 1543 y(enlarge)37 b(the)h(class)g(of)f (transformations.)57 b(An)38 b FA(exp)-5 b(onential)38 b(over)h(the)g(torus)47 b Fy(is)37 b(a)g(matrix-v)-5 b(alued)35 b(mapping)-118 1663 y Fx(Z)f Fy(:)28 b Fw(T)101 1627 y Fr(d)173 1663 y Fs(!)f Fx(S)6 b(L)p Fy(\(2)p Fx(;)17 b Fw(R)5 b Fy(\))38 b(of)32 b(the)h(form)883 1893 y Fx(Z)7 b Fy(\()p Fx(\022)s Fy(\))27 b(=)h Fx(C)1289 1852 y Ft(\000)p FC(1)1383 1893 y Fx(e)1428 1852 y Fr(z)s FC(\()p Fr(\022)r FC(\))1558 1893 y Fx(C)r(;)211 b(z)t Fy(\()p Fx(\022)s Fy(\))29 b(=)2183 1826 y Fs(h)p Fv(k)p Fx(;)17 b(\022)s Fs(i)p 2183 1870 229 4 v 2273 1962 a Fy(2)2438 1753 y Fq(\022)2553 1832 y Fy(+)p Fx(i)114 b Fy(0)2583 1953 y(0)f Fs(\000)p Fx(i)2898 1753 y Fq(\023)2988 1893 y Fx(;)-118 2131 y Fy(where)40 b Fx(C)46 b Fy(is)38 b(a)h(non-singular)e (matrix)h(and)h Fv(k)f Fs(2)h Fw(Z)1834 2095 y Fr(d)1911 2131 y Fy(is)f(\014xed.)64 b(The)40 b(e\013ect)g(of)e(transforming)f (the)j(constan)m(t)-118 2252 y(matrix)31 b(system)1507 2372 y Fx(u)1563 2331 y Ft(0)1613 2372 y Fy(=)d Fx(A)1790 2387 y FC(0)1829 2372 y Fx(u;)212 b(\022)2172 2331 y Ft(0)2223 2372 y Fy(=)27 b Fx(!)-118 2526 y Fy(b)m(y)33 b(an)g(exp)s(onen)m(tial)f Fx(u)27 b Fs(7!)g Fx(Z)7 b Fy(\()p Fx(\022)s Fy(\))p Fx(u)32 b Fy(that)h(comm)m(utes)f(with)g Fx(A)2137 2541 y FC(0)2209 2526 y Fy(is)h(simply)e(to)h(replace)h Fx(A)3143 2541 y FC(0)3215 2526 y Fy(b)m(y)1397 2730 y(~)1372 2755 y Fx(A)1445 2770 y FC(0)1512 2755 y Fy(=)1615 2615 y Fq(\022)1689 2755 y Fx(\013)1751 2770 y FC(0)1812 2755 y Fy(+)1920 2688 y Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)p 1920 2732 245 4 v 2018 2824 a Fy(2)2175 2615 y Fq(\023)2301 2688 y Fy(1)p 2275 2732 102 4 v 2275 2824 a Fx(\013)2337 2839 y FC(0)2386 2755 y Fx(A)2459 2770 y FC(0)2499 2755 y Fx(:)-118 2985 y Fy(In)37 b(particular,)f(if)f(2)p Fx(\013)694 3000 y FC(0)758 2985 y Fy(+)25 b Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)33 b(\031)i Fy(0)h(so)h(that)f(condition)f(\(3.54\)) h(is)g(violated)g(for)g Fx(A)3096 3000 y FC(0)3135 2985 y Fy(,)i(then)f(this)f(condition)-118 3105 y(will)30 b(hold)i(for)458 3080 y(~)432 3105 y Fx(A)505 3120 y FC(0)545 3105 y Fy(.)28 3225 y(Hence,)45 b(using)40 b(transformations)f Fx(Y)1387 3240 y Fr(j)1423 3225 y Fy(\()p Fx(\022)s Fy(\))j(=)g Fx(Z)1774 3240 y Fr(j)1810 3225 y Fy(\()p Fx(\022)s Fy(\))28 b(+)2080 3200 y(^)2066 3225 y Fx(Y)2123 3240 y Fr(j)2159 3225 y Fy(\()p Fx(\022)s Fy(\))41 b(that)f(are)h(close)g(to)f(exp)s (onen)m(tials,)j(one)e(can)-118 3346 y(o)m(v)m(ercome)j(the)g (restriction)e(imp)s(osed)g(b)m(y)i(\(3.54\).)75 b(The)44 b(p)s(erturbation)e(theory)i(of)f(course)h(b)s(ecomes)f(more)-118 3466 y(complicated)27 b(since)h(the)h(estimates)f(are)g(less)h(go)s(o)s (d,)f(but)h(this)f(can)g(b)s(e)h(handled.)42 b(The)29 b(approac)m(h)g(w)m(orks)g(up)g(to)-118 3587 y(an)m(y)j(order)f(for)g (all)e Fx(\025)j Fy(without)e(exception,)j(but)e(when)i(w)m(e)f(w)m(an) m(t)g(to)f(go)g(to)g(the)h(limit)c(w)m(e)k(run)g(in)m(to)e(problems.) -118 3707 y(The)i(reason)f(is)g(that)g(an)g(in\014nite)f(pro)s(duct)h (of)g(exp)s(onen)m(tials)g(ma)m(y)f(not)h(con)m(v)m(erge,)j(and)d(it)f (is)g(only)h(for)f(almost)-118 3827 y(ev)m(ery)f(rotation)d(n)m(um)m(b) s(er)i Fx(\032)p Fy(\()p Fx(\025)p Fy(\))f(that)g(one)h(can)g(pro)m(v)m (e)h(the)e(con)m(v)m(ergence)j(of)d(the)h(sequence)i(of)d (transformations)1602 4000 y Fx(Y)1659 4015 y Fr(j)1717 4000 y Fs(\001)22 b(\001)17 b(\001)g(\001)k(\001)g Fx(Y)2012 4015 y FC(1)2079 4000 y Fs(!)27 b Fx(Y)5 b(:)-118 4172 y Fy(In)29 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Fw(R)t Fy(\))46 b(since)39 b(one)h(can)g(use)g(the)g(rotation)e(n)m(um)m(b)s(er)h(together)h(with) f(some)-118 4894 y(information)29 b(on)k(the)g(function)f Fx(\025)27 b Fs(7!)h Fx(\032)p Fy(\()p Fx(\025)p Fy(\).)28 5014 y(The)33 b(exp)s(onen)m(tial)f(transformations)e(w)m(ere)k (originally)28 b(in)m(tro)s(duced)33 b(b)m(y)g(Moser)g(and)f(P\177)-49 b(osc)m(hel)33 b(in)f([65)o(])h(and)-118 5135 y(they)g(w)m(ere)h(used)g (b)m(y)g(Eliasson)d(\([29]\))h(in)g(a)h(systematic)f(w)m(a)m(y)i(to)e (pro)m(v)m(e)i(the)f(follo)m(wing)d(theorem)-118 5308 y Fv(Theorem)37 b(3.4.4)h(\([29]\))48 b FA(Assume)41 b(that)h Fx(!)h Fs(2)d Fx(D)s(C)7 b Fy(\()p Fx(N)e(;)17 b(\034)11 b Fy(\))41 b FA(and)f(that)i Fx(Q)e Fy(:)f Fw(T)2826 5272 y Fr(d)2910 5308 y Fs(!)g Fw(R)52 b FA(is)41 b(r)-5 b(e)g(al)41 b(analytic)g(in)g(a)-118 5429 y(strip)35 b(of)f(width)h Fx(r)s FA(.)44 b(Then)34 b(ther)-5 b(e)35 b(exists)g(a)f(c)-5 b(onstant)35 b Fx(C)g Fy(=)27 b Fx(C)7 b Fy(\()p Fx(\034)f(;)17 b(r)m(;)g(N)10 b Fy(\))35 b FA(such)g(that)g(if)g(we)f(de\014ne)1418 5668 y Fx(\025)1475 5683 y FC(0)1514 5668 y Fy(\()p Fx(s)p Fy(\))28 b(=)1767 5527 y Fq(\032)1884 5534 y(\000)1950 5576 y Fr(s)p 1939 5592 55 4 v 1939 5649 a(C)2004 5534 y Fq(\001)2050 5557 y FC(2)2172 5615 y Fx(s)g Fs(\025)g Fx(C)1884 5735 y Fs(\0001)113 b Fx(s)28 b(<)f(C)-118 5906 y FA(then)35 b(the)f(fol)5 b(lowing)34 b(holds)g(for)h Fx(\025)27 b(>)h(\025)1324 5921 y FC(0)1363 5906 y Fy(\()p Fs(j)p Fx(Q)p Fs(j)1534 5921 y Fr(r)1572 5906 y Fy(\))p FA(.)1900 6155 y Fy(80)p eop %%Page: 81 85 81 84 bop -33 219 a FA(\(i\))49 b(If)32 b(the)g(r)-5 b(otation)33 b(numb)-5 b(er)32 b Fx(\032)p Fy(\()p Fx(\025)p Fy(\))h FA(is)f(Diophantine)f(or)h(r)-5 b(ational,)33 b(then)f(ther)-5 b(e)33 b(exists)f(a)g(matrix)g Fx(B)h Fy(=)28 b Fx(B)5 b Fy(\()p Fx(\025)p Fy(\))126 339 y FA(in)32 b Fx(sl)r Fy(\(2)p Fx(;)17 b Fw(R)5 b Fy(\))39 b FA(and)32 b(an)g(analytic)h(matrix-value)-5 b(d)32 b(function)h Fx(Y)49 b Fy(:)27 b Fw(T)2497 303 y Fr(d)2569 339 y Fs(!)g Fx(GL)p Fy(\(2)p Fx(;)17 b Fw(R)5 b Fy(\))p FA(,)39 b(also)32 b(dep)-5 b(ending)31 b(on)i Fx(\025)p FA(,)126 459 y(such)h(that)1344 612 y Fx(X)8 b Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)p Fy(\))27 b(=)h Fx(Y)1872 472 y Fq(\022)1955 545 y Fx(!)t(t)p 1955 589 100 4 v 1981 680 a Fy(2)2087 612 y(+)22 b Fx(\036)2243 472 y Fq(\023)2332 612 y Fx(e)2377 571 y Fr(B)s(t)2464 612 y Fx(Y)f Fy(\()p Fx(\036)p Fy(\))2676 571 y Ft(\000)p FC(1)2770 612 y Fx(:)-62 885 y FA(\(ii\))48 b(If)34 b Fx(\032)p Fy(\()p Fx(\025)p Fy(\))h FA(is)g(neither)f(Diophantine)f(nor)i(r)-5 b(ational,)34 b(then)479 1160 y Fy(lim)17 b(inf)511 1227 y Ft(j)p Fr(t)p Ft(j!1)766 1160 y Fs(j)p Fx(X)8 b Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)1058 1175 y FC(0)1096 1160 y Fy(\))22 b Fs(\000)h Fx(X)8 b Fy(\(0;)17 b Fx(\036)1534 1175 y FC(0)1572 1160 y Fy(\))p Fs(j)28 b Fx(<)1779 1093 y Fy(1)p 1779 1137 49 4 v 1779 1228 a(2)1838 1160 y Fs(j)p Fx(X)8 b Fy(\(0;)17 b Fx(\036)2144 1175 y FC(0)2182 1160 y Fy(\))p Fs(j)199 b FA(and)250 b Fy(lim)2817 1227 y Ft(j)p Fr(t)p Ft(j!1)3049 1093 y Fs(j)p Fx(X)8 b Fy(\()p Fx(t)p Fy(;)17 b Fx(\036)3341 1108 y FC(0)3380 1093 y Fy(\))p Fs(j)p 3049 1137 397 4 v 3230 1228 a Fx(t)3483 1160 y Fy(=)28 b(0)p Fx(;)126 1443 y FA(for)34 b(al)5 b(l)35 b Fx(\036)479 1458 y FC(0)546 1443 y Fs(2)28 b Fw(T)703 1407 y Fr(d)747 1443 y FA(.)-118 1672 y Fv(Remark)37 b(3.4.5)49 b FA(The)27 b(r)-5 b(esult)29 b(gives)e(no)g(information)g (on)h(the)g(dep)-5 b(endenc)g(e)26 b(of)i(the)g(matrix)f Fx(B)5 b Fy(\()p Fx(\025)p Fy(\))28 b FA(with)g(r)-5 b(esp)g(e)g(ct)-118 1792 y(to)39 b Fx(\025)f FA(when)g Fx(\025)g FA(is)h(in)f(a)g(sp)-5 b(e)g(ctr)g(al)38 b(gap.)56 b(However,)39 b(an)f(iter)-5 b(ative)38 b(scheme)g(c)-5 b(an)38 b(b)-5 b(e)38 b(done)g(\(se)-5 b(e)38 b([65)o(]\))g(to)h (obtain)-118 1912 y(a)c(family)f(of)h(matric)-5 b(es)35 b(which)f(ar)-5 b(e)35 b(r)-5 b(e)g(al)34 b(analytic)h(in)g(the)g (interior)g(of)f(the)i(gap)e(and)h(have)f Fx(C)3353 1876 y Ft(1)3463 1912 y FA(extensions)g(to)-118 2033 y(the)j(b)-5 b(oundaries)36 b(of)h(the)g(gap.)51 b(On)37 b(the)g(other)g(hand,)g (such)f(r)-5 b(esults)38 b(c)-5 b(an)36 b(b)-5 b(e)37 b(obtaine)-5 b(d)36 b(fr)-5 b(om)37 b(the)g(ab)-5 b(ove)36 b(r)-5 b(esult)-118 2153 y Fy(a)32 b(p)s(osteriori)h FA(\([12]\).)-118 2381 y Fv(Remark)k(3.4.6)49 b FA(In)34 b(the)h(se)-5 b(c)g(ond)33 b(item)i(of)f(the)-5 b(or)g(em)34 b(\(3.4.4\))g(it)g(is)h(essential)f(to)g(ask)g(only)h(for)f(p)-5 b(oint)35 b(c)-5 b(onver-)-118 2502 y(genc)g(e)24 b(in)h Fx(\036)304 2517 y FC(0)344 2502 y FA(,)i(b)-5 b(e)g(c)g(ause)24 b(the)i(r)-5 b(esult)25 b(c)-5 b(omes)25 b(fr)-5 b(om)25 b(a)g(p)-5 b(erturb)g(ation)25 b(metho)-5 b(d)25 b(which)f(is)h(not)h (absolutely)f(c)-5 b(onver)g(gent.)28 2730 y Fy(The)29 b(pro)s(of)e(of)h(this)g(theorem)g(as)g(already)f(sk)m(etc)m(hed)k (consists)e(of)f(three)g(main)f(steps.)43 b(This)28 b(will)e(b)s(e)i(a) g(little)-118 2851 y(bit)k(more)g(dev)m(elop)s(ed)h(in)f(the)h(follo)m (wing)d(subsections.)-118 3110 y Fv(The)38 b(small)d(divisor)h(lemma) -118 3295 y Fy(Let)f Fs(B)124 3310 y Fr(r)198 3295 y Fy(b)s(e)g(the)g(space)i(of)d(all)f(analytic)h(functions)h Fx(F)45 b Fy(:)33 b Fw(T)2046 3259 y Fr(d)2121 3295 y Fs(!)f Fx(g)t(l)r Fy(\(2)p Fx(;)17 b Fw(C)i Fy(\))41 b(in)34 b(a)h(complex)f(strip)h(of)g(width)g Fx(r)i Fy(for)-118 3415 y(whic)m(h)1398 3536 y Fs(j)p Fx(F)14 b Fs(j)1531 3551 y Fr(r)1595 3536 y Fy(=)90 b(sup)1699 3622 y Ft(j)p FC(Im)22 b Fr(\022)r Ft(j)p Fr()p FC(0)2054 4023 y Fs(B)2119 4038 y Fr(r)2157 4023 y Fx(:)28 4338 y Fy(If)i Fx(F)48 b Fy(dep)s(ends)36 b(on)f(a)f(parameter)f Fx(\025)e Fs(2)g Fy(\001)g Fs(\032)g Fw(R)5 b Fy(,)41 b(w)m(e)35 b(sa)m(y)g(that)f(it)g(is)f Fx(C)2648 4302 y FC(2)2722 4338 y Fy(in)h Fx(\025)g Fy(whenev)m(er)j Fx(\025)30 b Fs(7!)g Fx(F)3639 4353 y Fr(\025)3715 4338 y Fs(2)i(B)3878 4353 y Fr(r)3950 4338 y Fy(is)-118 4459 y Fx(C)-41 4422 y FC(2)-2 4459 y Fy(.)43 b(As)29 b(a)g(notation,)f(and) h(assuming)f(that)h(the)g(parameter)g(is)f(kno)m(wn,)j(sa)m(y)f Fx(\025)p Fy(,)g(w)m(e)f(will)e(write)i Fx(@)3469 4422 y Fr(\027)3513 4459 y Fx(F)42 b Fy(to)29 b(denote)-118 4579 y(the)j(deriv)-5 b(ativ)m(e)30 b(with)h(resp)s(ect)i(to)e(this)f (v)-5 b(ariable.)42 b(W)-8 b(e)31 b(sa)m(y)i(that)e(it)f(is)g FA(pie)-5 b(c)g(ewise)38 b Fx(C)2994 4543 y FC(2)3064 4579 y Fy(\(p)m(w.-)p Fx(C)3360 4543 y FC(2)3400 4579 y Fy(\))31 b(in)g Fx(\025)g Fy(on)g(some)-118 4699 y(set)k(\001)c Fs(\032)h Fw(R)45 b Fy(if)33 b(there)j(exists)f(a)f(\014nite)g(set)h Fs(f)p Fx(\025)1572 4714 y Fr(i)1600 4699 y Fs(g)g Fy(in)e(\001)i(suc)m (h)h(that)e Fx(F)48 b Fy(is)34 b Fx(C)2639 4663 y FC(2)2713 4699 y Fy(in)g(\001)23 b Fs(\000)h(f)p Fx(\025)3141 4714 y Fr(i)3169 4699 y Fs(g)34 b Fy(and)h(suc)m(h)h(that)e(the)-118 4820 y(righ)m(t)39 b(and)h(left)f(limits)e(of)i Fx(\016)944 4784 y Fr(\027)987 4820 y Fx(F)14 b Fy(,)42 b(for)d Fx(\027)46 b Fy(=)40 b(0)p Fx(;)17 b Fy(1)p Fx(;)g Fy(2,)41 b(exist)f(at)f(all)f (p)s(oin)m(ts)i Fx(\025)2667 4835 y Fr(i)2695 4820 y Fy(,)h(whenev)m(er)i(suc)m(h)e(a)f(limit)c(mak)m(es)-118 4940 y(sense.)28 5061 y(F)-8 b(or)32 b(eac)m(h)h Fx(F)42 b Fs(2)28 b(B)686 5076 y Fr(r)757 5061 y Fy(w)m(e)33 b(de\014ne)1333 5298 y(^)-54 b Fx(\033)t Fy(\()p Fx(F)14 b Fy(\))27 b(=)1670 5187 y Fq(n)1737 5298 y Fv(k)h Fs(2)g Fw(Z)1987 5257 y Fr(d)2025 5298 y Fy(;)2090 5273 y(^)2069 5298 y Fx(F)13 b Fy(\()p Fv(k)p Fy(\))28 b Fs(6)p Fy(=)f(0)2460 5187 y Fq(o)2543 5298 y Fx(;)-118 5565 y Fy(where)186 5540 y(^)164 5565 y Fx(F)13 b Fy(\()p Fv(k)p Fy(\))33 b(is)f(the)h Fv(k)p Fy(-th)f(F)-8 b(ourier)32 b(co)s(e\016cien)m(t)h (of)f Fx(F)14 b Fy(.)1900 6155 y(81)p eop %%Page: 82 86 82 85 bop -118 219 a Fv(Lemma)37 b(3.4.7)h(\(Small)d(divisor)h(lemma,)g ([29]\))49 b FA(L)-5 b(et)37 b Fx(A)30 b Fy(=)h Fx(A)p Fy(\()p Fx(\025)p Fy(\))f Fs(2)h Fx(sl)r Fy(\(2)p Fx(;)17 b Fw(C)i Fy(\))43 b FA(have)35 b(eigenvalues)g Fs(\006)p Fx(e)p Fy(\()p Fx(\025)p Fy(\))p FA(,)-118 339 y(and)f(assume)g(that)i Fs(j)p Fx(A)p Fy(\()p Fx(\025)p Fy(\))21 b Fs(\000)971 313 y Fy(~)967 339 y Fx(\025J)9 b Fs(j)27 b Fx(<)h Fy(3)35 b FA(for)f(some)1738 313 y Fy(~)1734 339 y Fx(\025)p Fy(\()p Fx(\025)p Fy(\))p FA(.)45 b(L)-5 b(et)35 b Fx(F)41 b Fs(2)28 b(B)2430 354 y Fr(r)2504 339 y FA(and)34 b(assume)1068 546 y Fs(j)p Fx(i)p Fs(h)p Fv(k)p Fs(i)21 b(\006)i Fy(2)p Fx(e)p Fy(\()p Fx(\025)p Fy(\))p Fs(j)k(\025)h Fx(K)1864 505 y Ft(\000)p FC(1)1959 546 y Fs(j)p Fv(k)p Fs(j)2074 505 y Ft(\000)p FC(1)2168 546 y Fx(;)215 b Fv(k)28 b Fs(2)33 b Fy(^)-54 b Fx(\033)t Fy(\()p Fx(F)14 b Fy(\))p Fx(:)-118 753 y FA(Then)34 b(ther)-5 b(e)35 b(exists)f(a)h(unique)g Fx(Y)49 b Fs(2)28 b(B)38 b FA(such)d(that)1001 960 y Fx(@)1052 975 y Fr(!)1103 960 y Fx(Y)49 b Fy(=)28 b([)p Fx(A;)17 b(Y)k Fy(])h(+)g Fx(F)213 b FA(and)204 b Fy(^)-54 b Fx(\033)t Fy(\()p Fx(Y)21 b Fy(\))28 b Fs(\032)33 b Fy(^)-54 b Fx(\033)t Fy(\()p Fx(F)14 b Fy(\))p Fx(:)-118 1167 y FA(The)34 b(matrix)h Fx(Y)56 b FA(satis\014es)34 b(the)h(estimate)1093 1428 y Fs(j)p Fx(Y)21 b Fs(j)1227 1443 y Fr(s)1291 1428 y Fs(\024)28 b Fx(c)1567 1361 y(K)1657 1325 y FC(2)p 1448 1405 369 4 v 1448 1497 a Fy(\()p Fx(r)d Fs(\000)d Fx(s)p Fy(\))1738 1468 y FC(3)p Fr(\034)1826 1428 y Fs(j)p Fx(F)14 b Fs(j)1959 1443 y Fr(r)1996 1428 y Fx(;)216 b FA(for)199 b Fx(s)28 b(<)g(r)m(;)-118 1695 y FA(wher)-5 b(e)32 b(the)h(c)-5 b(onstant)33 b Fx(c)g FA(dep)-5 b(ends)32 b(only)h(on)f Fx(\034)11 b FA(.)45 b(Mor)-5 b(e)g(over,)33 b(if)g Fx(F)s(;)17 b(A)27 b Fs(2)h(B)2533 1710 y Fr(r)2605 1695 y FA(ar)-5 b(e)32 b Fx(C)2845 1659 y FC(2)2918 1695 y FA(or)h(pw.-)p Fx(C)3299 1659 y FC(2)3371 1695 y FA(in)f Fx(\025)p FA(,)h(then)g(so)g(is)-118 1816 y Fx(Y)49 b Fs(2)28 b(B)147 1831 y Fr(s)184 1816 y FA(,)35 b(and)f(the)h(fol)5 b(lowing)34 b(estimates)g(hold)g(for)h Fx(s)27 b(<)h(r)818 2105 y Fs(j)p Fx(@)5 b(Y)22 b Fs(j)1009 2134 y Fr(s)1073 2105 y Fs(\024)28 b Fx(c)1237 1934 y Fq(")1425 2037 y Fx(K)1515 2001 y FC(2)p 1305 2082 V 1305 2173 a Fy(\()p Fx(r)d Fs(\000)d Fx(s)p Fy(\))1595 2144 y FC(3)p Fr(\034)1684 2105 y Fs(j)p Fx(@)5 b(F)14 b Fs(j)1873 2120 y Fr(r)1933 2105 y Fy(+)2031 1964 y Fq(\022)2234 2037 y Fx(K)2324 2001 y FC(2)p 2114 2082 V 2114 2173 a Fy(\()p Fx(r)25 b Fs(\000)d Fx(s)p Fy(\))2404 2144 y FC(3)p Fr(\034)2493 1964 y Fq(\023)2566 1985 y FC(2)2622 2105 y Fs(j)p Fx(@)5 b(A)p Fs(jj)p Fx(F)14 b Fs(j)2940 2120 y Fr(r)2978 1934 y Fq(#)3052 2105 y Fx(;)-104 2414 y Fq(\014)-104 2474 y(\014)-71 2499 y Fx(@)-15 2457 y FC(2)26 2499 y Fx(Y)104 2414 y Fq(\014)104 2474 y(\014)137 2538 y Fr(s)202 2499 y Fs(\024)28 b Fx(c)366 2328 y Fq(")553 2431 y Fx(K)643 2395 y FC(2)p 434 2476 V 434 2567 a Fy(\()p Fx(r)c Fs(\000)f Fx(s)p Fy(\))724 2538 y FC(3)p Fr(\034)812 2499 y Fs(j)p Fx(@)896 2457 y FC(2)936 2499 y Fx(F)14 b Fs(j)1041 2514 y Fr(r)1101 2499 y Fy(+)1199 2358 y Fq(\022)1402 2431 y Fx(K)1492 2395 y FC(2)p 1282 2476 V 1282 2567 a Fy(\()p Fx(r)25 b Fs(\000)d Fx(s)p Fy(\))1572 2538 y FC(3)p Fr(\034)1661 2358 y Fq(\023)1734 2378 y FC(2)1790 2499 y Fy(\()p Fs(j)p Fx(@)1912 2457 y FC(2)1952 2499 y Fx(A)p Fs(jj)p Fx(F)14 b Fs(j)2186 2514 y Fr(r)2245 2499 y Fy(+)22 b Fs(j)p Fx(@)5 b(A)p Fs(jj)p Fx(@)g(F)14 b Fs(j)2717 2514 y Fr(r)2755 2499 y Fy(\))22 b(+)2913 2358 y Fq(\022)3116 2431 y Fx(K)3206 2395 y FC(2)p 2997 2476 V 2997 2567 a Fy(\()p Fx(r)i Fs(\000)f Fx(s)p Fy(\))3287 2538 y FC(3)p Fr(\034)3375 2358 y Fq(\023)3449 2378 y FC(3)3505 2499 y Fs(j)p Fx(@)5 b(A)p Fs(j)3690 2457 y FC(2)3730 2499 y Fs(j)p Fx(F)14 b Fs(j)3863 2514 y Fr(r)3900 2328 y Fq(#)3974 2499 y Fx(:)28 2794 y Fy(The)41 b(pro)s(of)e(uses)i(standard)g(estimates)e (for)h(the)g(con)m(v)m(ergence)i(on)e(a)g(complex)f(strip)h(of)f(the)h (torus)h(of)e(a)-118 2914 y(formal)30 b(solution)h(of)h(a)h(system)g (of)f(linear)f(homological)e(equations)j(of)h(the)g(t)m(yp)s(e)1351 3121 y(\()p Fx(@)1440 3136 y Fr(!)1490 3121 y Fx(u)1546 3136 y FC(2)p Fr(l)1607 3121 y Fy(\)\()p Fx(\022)s Fy(\))22 b(+)g Fx(u)1945 3136 y FC(1)p Fr(l)2006 3121 y Fy(\()p Fx(\022)s Fy(\))g(+)g Fx(u)2306 3136 y FC(0)p Fr(l)2395 3121 y Fy(=)27 b(0)-118 3328 y(for)32 b(1)27 b Fs(\024)h Fx(l)i Fs(\024)e Fx(l)405 3343 y FC(0)477 3328 y Fy(and)k(for)g(all)e Fx(\022)h Fs(2)d Fw(T)1183 3292 y Fr(d)1259 3328 y Fy(using)k(the)h (formal)d(solution)g(in)i(terms)g(of)g(the)h(F)-8 b(ourier)31 b(series)i(as)f(w)m(e)h(did)-118 3449 y(in)f(the)h(\014rst)g(c)m (hapter.)-118 3707 y Fv(The)38 b(inductiv)m(e)e(lemma)-118 3891 y Fy(Let)i Fx(A)f Fs(2)h Fx(sl)r Fy(\(2)p Fx(;)17 b Fw(R)t Fy(\).)66 b(In)39 b(particular,)f(this)f(means)i(that)e(the)i (t)m(w)m(o)g(eigen)m(v)-5 b(alues)38 b(of)g Fx(A)g Fy(coincide)f(up)i (to)e(a)h(sign)-118 4012 y(and,)j(since)f Fx(A)g Fy(is)f(real,)h(they)h (are)e(b)s(oth)h(real)e(or)h(purely)h(imaginary)-8 b(.)62 b(Hence)41 b(the)f(imaginary)c(parts)k(of)f(the)-118 4132 y(eigen)m(v)-5 b(alues)31 b(are)h Fs(\006)p Fx(i\013)g Fy(and)g(w)m(e)g(can)g(\014x)g(the)g(sign)e(of)h Fx(\013)h Fy(inducing)f(an)g(orien)m(tation.)41 b(More)32 b(precisely)g(w)m(e)g (shall)-118 4253 y(require)h(that)1618 4373 y Fx(A)28 b Fy(=)g 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2195 y(will)g(do\).)28 2315 y(Let)40 b(no)m(w)g Fx(A)493 2330 y FC(1)532 2315 y Fy(\()p Fx(\025)p Fy(\))f Fs(2)g Fx(sl)r Fy(\(2)p Fx(;)17 b Fw(R)5 b Fy(\))45 b(and)39 b Fx(F)1425 2330 y FC(1)1465 2315 y Fy(\()p Fs(\001)p Fx(;)17 b(\025)p Fy(\))38 b Fs(2)h(B)1878 2330 y Fr(r)1956 2315 y Fy(b)s(e)g(real)g(and)g(p)m(w.-)p Fx(C)2746 2279 y FC(2)2825 2315 y Fy(in)g Fx(\025)f Fs(2)i Fy(\001.)64 b(W)-8 b(e)39 b(assume)h(that)-86 2436 y(tr)49 b([)p Fx(F)129 2451 y FC(1)169 2436 y Fy(\()p Fs(\001)p Fx(;)17 b(\025)p Fy(\)])400 2465 y Fk(T)450 2446 y Fj(d)513 2436 y Fy(=)27 b(0,)33 b(and)903 2749 y Fq(8)903 2839 y(>)903 2869 y(<)903 3048 y(>)903 3078 y(:)1033 2747 y(\014)1033 2807 y(\014)1033 2867 y(\014)1066 2862 y Fx(A)1139 2877 y FC(1)1179 2862 y Fy(\()p Fx(\025)p Fy(\))22 b Fs(\000)1438 2835 y Fy(~)1434 2862 y Fx(\025)1491 2877 y FC(1)1530 2862 y Fy(\()p Fx(\025)p Fy(\))p Fx(J)1726 2747 y Fq(\014)1726 2807 y(\014)1726 2867 y(\014)1787 2862 y Fx(<)27 b Fy(2)p Fx(;)83 b Fy(for)32 b(some)2447 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5480 y Fq([)1931 5696 y FC(0)p Fr(<)p Ft(j)p Fl(k)p Ft(j\024)p Fr(N)2215 5706 y Fj(j)2263 5575 y Fy(\003)2331 5590 y Fr(j)2368 5575 y Fy(\()p Fv(k)p Fy(\))p Fx(;)1900 6155 y Fy(83)p eop %%Page: 84 88 84 87 bop -118 219 a FA(and)42 b Fx(r)123 234 y Fr(j)188 219 y Fs(\000)29 b Fx(r)338 234 y Fr(j)t FC(+1)507 219 y Fy(=)43 b Fx(r)670 234 y Fr(j)706 219 y Fx(=)p Fy(2)p FA(,)i(if)d Fx(\025)h Fs(62)g Fy(\003)1258 234 y Fr(j)1294 219 y Fy(\(0\))p FA(,)i(and)d Fx(r)1735 234 y Fr(j)1800 219 y Fs(\000)29 b Fx(r)1950 234 y Fr(j)t FC(+1)2119 219 y Fy(=)43 b(2)2287 182 y Ft(\000)p Fr(j)2378 219 y Fx(r)2422 234 y FC(1)2461 219 y Fx(=)p Fy(2)g FA(if)g Fx(\025)f Fs(2)i Fy(\003)2982 234 y Fr(j)3018 219 y Fy(\(0\))p FA(.)69 b(Then)42 b(we)g(have)h(the)-118 339 y(fol)5 b(lowing)33 b(estimates:)142 466 y Fq(\014)142 526 y(\014)142 586 y(\014)142 646 y(\014)175 611 y Fx(@)231 570 y Fr(\027)292 470 y Fq(\022)365 611 y Fx(Y)422 626 y Fr(j)t FC(+1)549 611 y Fy(\()p Fs(\001)p Fx(;)17 b(\025)p Fy(\))k Fs(\000)i 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b(the)g(other)g(hand)h(the)f(set)h(\003)2017 2968 y Fr(j)2053 2953 y Fy(\(0\))e(will)f(con)m(v)m(erge)k(to)d(a)h (subset)h(of)f(the)g(sp)s(ectrum)-118 3073 y(and)f(of)f(\001,)i(whic)m (h,)h(as)d(w)m(e)i(will)c(see,)41 b(will)35 b(b)s(e)j(the)g(set)g(of)f (those)i Fx(\025)p Fy('s)f(with)f(Diophan)m(tine)f(rotation)g(n)m(um)m (b)s(er.)-118 3193 y(All)31 b(this)h(mak)m(es)h(it)f(necessary)j(to)d (distinguish)f(b)s(et)m(w)m(een)k(the)e(cases)h Fx(\025)27 b Fs(2)h Fy(\003)2704 3208 y Fr(j)2741 3193 y Fy(\(0\))k(and)g Fx(\025)c Fs(62)g Fy(\003)3334 3208 y Fr(j)3370 3193 y Fy(\(0\).)28 3314 y Fv(Case)43 b(1.)55 b Fy(Supp)s(ose)37 b(that)f Fx(\025)e Fs(2)h Fy(\003)1304 3329 y Fr(j)1340 3314 y Fy(\(0\).)54 b(Then,)39 b(from)c(the)i(de\014nition)e(of)h(this) g(set,)i(w)m(e)f(ha)m(v)m(e)h(that)e(for)g(all)-118 3434 y(0)27 b Fx(<)h Fs(j)p Fv(k)p Fs(j)f Fx(<)h(N)386 3449 y Fr(j)422 3434 y Fy(,)1526 3555 y Fs(j)o Fy(2)p Fx(\013)1664 3570 y Fr(j)1701 3555 y Fy(\()p Fx(\025)p Fy(\))22 b Fs(\000)g(h)p 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Fy(\(3.62\))1900 6155 y(84)p eop %%Page: 85 89 85 88 bop -118 219 a Fy(b)m(y)33 b(the)g(assumption)f(\(3.61\))g(for)g Fx(k)s Fy(,)h(b)s(ecause)h Fx(\025)27 b Fs(2)h Fy(\003)1853 234 y Fr(k)1896 219 y Fy(\()p Fv(k)p Fy(\))k(implies)e(that,)j(if)e Fv(k)d Fs(6)p Fy(=)g(0,)k(then)1346 483 y Fs(j)p Fy(2)p Fx(\013)1485 498 y Fr(k)1527 483 y Fs(j)27 b(\025)h Fx(N)1775 442 y Ft(\000)p Fr(\034)1765 511 y(k)1896 483 y Fs(\000)23 b Fy(2)p Fx(")2091 442 y Fr(\033)2091 508 y(k)2165 483 y Fs(\025)2280 416 y Fy(1)p 2280 460 49 4 v 2280 551 a(2)2338 483 y Fx(N)2426 442 y Ft(\000)p Fr(\034)2416 511 y(k)2525 483 y Fx(:)28 728 y Fy(Let)33 b(no)m(w)g Fx(G)g Fy(b)s(e)f(the)h(truncated)h(F)-8 b(ourier)31 b(series)1369 948 y Fx(G)p Fy(\()p Fx(\022)s Fy(\))d(=)1788 854 y Fq(X)1701 1070 y FC(0)p Fr(<)p Ft(j)p Fl(n)p Ft(j\024)p Fr(N)1987 1080 y Fj(j)2058 923 y Fy(^)2036 948 y Fx(F)2099 963 y Fr(j)2135 948 y Fy(\()p Fv(n)p Fy(\))p Fx(e)2318 907 y Fr(i)p Ft(h)p Fl(n)p Fr(;\022)r Ft(i)2502 948 y Fx(;)-118 1277 y 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Fk(T)3276 3104 y Fj(d)3338 3083 y Fy(=)28 b(0)p Fx(;)-118 3291 y Fy(so)33 b(w)m(e)g(only)f(need)i(to)e(consider)h(the)g(estimates)f(in)m(v)m(olv) m(ed)h(in)f(the)h(lemma.)28 3411 y(In)g(order)g(to)f(do)h(so,)f(w)m(e)i (note)f(that,)f(reducing)h(the)g(analyticit)m(y)e(band)1012 3664 y Fs(j)p Fx(@)1096 3623 y Fr(\027)1140 3664 y Fx(G)p Fs(j)1245 3679 y Fr(s)1309 3664 y Fx(<)d(c")1501 3679 y Fr(j)1537 3664 y Fx(N)1625 3623 y Fr(\034)8 b FC(+1)1615 3690 y Fr(j)1759 3664 y Fx(;)211 b(s)28 b Fy(=)g Fx(r)2219 3679 y Fr(j)t FC(+1)2367 3664 y Fy(+)2475 3597 y Fx(r)2519 3612 y Fr(j)2578 3597 y Fs(\000)23 b Fx(r)2722 3612 y Fr(j)t FC(+1)p 2475 3641 373 4 v 2637 3732 a Fy(2)2858 3664 y Fx(:)-118 3915 y Fy(Then)34 b(b)m(y)f(lemma)d(3.4.7)i(and)h (\(3.61\))f(for)g Fx(j)38 b Fy(w)m(e)c(get)f(the)g(estimate)843 4149 y Fs(j)p Fx(@)927 4107 y Fr(\027)971 4149 y Fx(Y)21 b Fs(j)1077 4164 y Fr(r)1109 4174 y Fj(j)1141 4164 y FC(+1)1263 4149 y Fx(<)27 b(c")1454 4098 y FC(1)p Ft(\000)p FC(\(2+3)p Fr(\027)t 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FC(1)1982 4918 y Fq(\014)1982 4977 y(\014)2015 5041 y Fr(r)2047 5051 y Fj(j)s Fp(+1)2189 5002 y Fx(<)27 b Fy(2)p Fx(;)-118 5192 y Fy(whic)m(h,)33 b(di\013eren)m(tiating)e Fx(F)880 5207 y Fr(j)t FC(+1)1039 5192 y Fy(and)i(after)f(some)h(manipulation)c(giv)m (es)k(the)g(desired)g(b)s(ound.)28 5313 y(The)25 b(estimates)e (\(3.61\))f(and)i(the)f(\014rst)h(one)g(of)f(\(3.61\))f(for)h Fx(j)9 b Fy(+)s(1)24 b(are)f(an)g(easy)i(consequence)h(of)d(the)h (de\014nition)-118 5433 y(of)e Fx(A)56 5448 y Fr(j)t FC(+1)205 5433 y Fy(and)g(\(3.60\),)h(already)f(sho)m(wn.)42 b(The)23 b(second)g(estimate)f(of)f(\(3.61\))h(m)m(ust)g(only)g(b)s(e)g (considered)h(whenev)m(er)-118 5554 y Fs(j)p Fx(A)-17 5569 y Fr(j)t FC(+1)109 5554 y Fy(\()p Fx(\025)p Fy(\))p Fs(j)28 b(\025)g Fy(8,)i(similarly)c(to)j(the)h(\014rst)h(step)f(of)f (the)i(iteration.)40 b(In)30 b(this)g(case)g(w)m(e)h(ha)m(v)m(e)g(that) f Fx(\025)d Fs(2)h(\\)3620 5506 y Fr(j)3620 5581 y(k)r FC(=1)3754 5554 y Fy(\003)3822 5569 y Fr(k)3864 5554 y Fy(\(0\),)-118 5674 y(i.e.)1248 5814 y Fs(j)p Fx(A)1349 5829 y Fr(j)t FC(+1)1476 5814 y Fy(\()p Fx(\025)p Fy(\))22 b Fs(\000)1735 5788 y Fy(~)1730 5814 y Fx(\025)1787 5829 y FC(1)1827 5814 y Fx(J)9 b Fs(j)27 b Fx(<)h Fy(2)22 b(+)g(2)p Fx(")2313 5763 y FC(2)p Fr(=)p FC(3)2313 5838 y(1)2450 5814 y Fx(<)2563 5747 y Fy(5)p 2563 5791 49 4 v 2563 5882 a(2)2622 5814 y Fx(:)1900 6155 y Fy(85)p eop %%Page: 86 90 86 89 bop -118 228 a Fy(Hence)34 b Fs(j)204 202 y Fy(~)200 228 y Fx(\025)257 243 y FC(1)296 228 y Fs(j)27 b Fx(>)h Fy(5.)43 b(But)33 b(then)g Fs(j)p Fx(\013)1080 243 y Fr(j)t FC(+1)1206 228 y Fy(\()p Fx(\025)p Fy(\))p Fs(j)27 b(\025)i(j)1532 202 y Fy(~)1528 228 y Fx(\025)1585 243 y FC(1)1624 228 y Fs(j)21 b(\000)1783 189 y FC(5)p 1783 206 36 4 v 1783 263 a(2)1856 228 y Fx(>)1969 181 y Ft(j)1992 163 y FC(~)1989 181 y Fr(\025)2030 190 y Fp(1)2065 181 y Ft(j)p 1969 206 115 4 v 2009 263 a FC(2)2094 228 y Fy(.)44 b(Hence)1398 371 y Fq(\014)1398 431 y(\014)1398 490 y(\014)1398 550 y(\014)1441 448 y Fx(A)1514 463 y Fr(j)t FC(+1)1641 448 y Fy(\()p Fx(\025)p Fy(\))p 1441 492 333 4 v 1447 584 a Fx(\013)1509 599 y Fr(j)t FC(+1)1636 584 y Fy(\()p Fx(\025)p Fy(\))1784 371 y Fq(\014)1784 431 y(\014)1784 490 y(\014)1784 550 y(\014)1845 515 y Fs(\024)28 b Fy(2)22 b(+)2180 448 y(5)p 2129 492 152 4 v 2129 597 a Fs(j)2161 570 y Fy(~)2157 597 y Fx(\025)2214 612 y FC(1)2253 597 y Fs(j)2318 515 y(\024)28 b Fy(3)p Fx(;)-118 804 y Fy(whic)m(h)33 b(pro)m(v)m(es)h(the)f(second)h (estimate)e(of)g(\(3.61\))g(for)g Fx(j)c Fy(+)22 b(1)32 b(and)h(case)g(1.)28 925 y Fv(Case)38 b(2:)44 b Fy(Supp)s(ose)33 b(no)m(w)h(that)e Fx(\025)c Fs(2)g Fy(\003)1471 940 y Fr(j)1507 925 y Fy(\()p Fv(k)p Fy(\),)33 b(with)f Fv(k)c Fs(6)p Fy(=)f(0,)33 b(and)f(let)1334 1205 y Fx(Z)7 b Fy(\()p Fx(\022)s Fy(\))27 b(=)h(exp)1828 1064 y Fq(\032)1939 1137 y Fs(h)p Fv(k)p Fx(;)17 b(\022)s Fs(i)p 1913 1182 281 4 v 1913 1273 a Fy(2)p Fx(\013)2024 1288 y Fr(j)2060 1273 y Fy(\()p Fx(\025)p Fy(\))2203 1205 y Fx(A)2276 1220 y Fr(j)2313 1205 y Fy(\()p Fx(\025)p Fy(\))2446 1064 y Fq(\033)2537 1205 y Fx(;)-118 1478 y Fy(that)27 b(is,)h Fx(Z)34 b Fy(is)26 b(an)i(exp)s(onen)m(tial)e(in)h(the)g(sense) i(de\014ned)g(b)s(efore.)42 b(Then)28 b(the)g(homological)23 b(equation,)28 b(as)f(already)-118 1599 y(stated,)33 b(b)s(ecomes)1234 1719 y Fx(@)1285 1734 y Fr(!)1336 1719 y Fx(Z)i Fy(=)27 b(\()p Fx(A)1652 1734 y Fr(j)1711 1719 y Fy(+)22 b Fx(F)1872 1734 y Fr(j)1909 1719 y Fy(\))p Fx(Z)29 b Fs(\000)22 b Fx(Z)7 b Fy(\()p Fx(B)2328 1734 y Fr(j)2387 1719 y Fy(+)22 b Fx(G)2562 1734 y Fr(j)2598 1719 y Fy(\))p Fx(;)-118 1893 y Fy(where)1099 2046 y Fx(B)1173 2061 y Fr(j)1237 2046 y Fy(=)1341 1906 y Fq(\022)1414 2046 y Fy(1)g Fs(\000)1600 1979 y(h)p Fv(k)p Fs(i)p 1595 2023 148 4 v 1595 2114 a Fy(2)p Fx(\013)1706 2129 y Fr(j)1752 1906 y Fq(\023)1842 2046 y Fx(A)1915 2061 y Fr(j)1952 2046 y Fx(;)211 b(G)2267 2061 y Fr(j)2332 2046 y Fy(=)27 b Fx(Z)2509 2005 y Ft(\000)p FC(1)2603 2046 y Fx(F)2666 2061 y Fr(j)2703 2046 y Fx(Z)r(:)-118 2287 y Fy(Note)37 b(that)f Fx(G)414 2302 y Fr(j)487 2287 y Fy(can)h(b)s(e)f(de\014ned)i (on)f Fw(T)1349 2251 y Fr(d)1429 2287 y Fy(instead)g(of)f(\(2)p Fw(T)p Fy(\))2073 2251 y Fr(d)2116 2287 y Fy(,)i(b)s(ecause)g(it)d(can) i(b)s(e)g(expressed)i(in)d(terms)g(of)g Fx(F)3979 2302 y Fr(j)-118 2407 y Fy(\(whic)m(h)g(is)f(de\014ned)i(on)e Fw(T)843 2371 y Fr(d)887 2407 y Fy(\))g(and)h(of)f(pro)s(ducts)h(of)f (t)m(w)m(o)h(elemen)m(ts)g(of)e Fx(Z)2557 2422 y Fr(j)2629 2407 y Fy(\(whic)m(h)i(are)f(de\014ned)i(on)f(\(2)p Fw(T)p Fy(\))3780 2371 y Fr(d)3859 2407 y Fy(and)-118 2541 y(th)m(us)i(pro)s (ducts)g(of)f(t)m(w)m(o)h(elemen)m(ts)f(are)h(de\014ned)g(on)f Fw(T)1929 2504 y Fr(d)1973 2541 y Fy(\).)58 b(The)38 b(rotation)e(n)m(um)m(b)s(er)h(of)g Fx(B)3229 2556 y Fr(j)3303 2541 y Fy(is)g Fx(\014)3461 2556 y Fr(j)3533 2541 y Fy(=)e Fx(\013)3706 2556 y Fr(j)3768 2541 y Fs(\000)3881 2493 y Ft(h)p Fl(k)p Ft(i)p 3881 2518 98 4 v 3912 2575 a FC(2)3989 2541 y Fy(,)-118 2661 y(and)e(satis\014es)787 2781 y Fs(j)p Fy(2)p Fx(\014)919 2796 y Fr(j)977 2781 y Fs(\000)22 b(h)p Fv(n)p Fs(ij)28 b(\025)g Fy(\(5)p Fx(N)1542 2796 y Fr(j)1578 2781 y Fy(\))1616 2740 y Ft(\000)p Fr(\034)1736 2781 y Fs(\000)23 b Fy(2)p Fx(")1931 2740 y Fr(\033)1931 2806 y(j)2005 2781 y Fs(\025)28 b Fy(2)p Fx(")2205 2740 y Fr(\033)2205 2806 y(j)2251 2781 y Fx(;)212 b Fy(0)27 b Fx(<)h Fs(j)p Fv(n)p Fs(j)f(\024)h Fy(5)p Fx(N)3047 2796 y Fr(j)3084 2781 y Fx(;)-118 2956 y Fy(b)s(ecause)38 b(of)e(the)h(de\014nition)f(of)g Fx(\014)1142 2971 y Fr(j)1215 2956 y Fy(and)g(the)h(condition)f Fx(\025)e Fs(2)h Fy(\003)2273 2971 y Fr(j)2309 2956 y Fy(\()p Fv(k)p Fy(\).)56 b(Moreo)m(v)m(er,)39 b(b)m(y)e(\(3.61\))f(for)g Fx(j)6 b Fy(,)38 b(w)m(e)f(ha)m(v)m(e)-118 3076 y(that)1501 3229 y Fs(j)p Fx(B)1603 3244 y Fr(j)1639 3229 y Fs(j)28 b Fx(<)f Fy(2)p Fx(")1893 3188 y Fr(\033)1893 3253 y(j)1956 3084 y Fq(\014)1956 3144 y(\014)1956 3204 y(\014)1956 3264 y(\014)2018 3161 y Fx(A)2091 3176 y Fr(j)p 1999 3206 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Fx(G)3402 4956 y Fr(j)3439 4941 y Fy(\(0\)\))3602 4830 y Fq(i)3665 4941 y Fx(:)28 5183 y Fy(As)38 b(in)f(the)h(\014rst)f (case,)j(all)35 b(the)j(requiremen)m(ts)g(are)g(ful\014lled.)55 b(The)39 b(estimates,)f(ho)m(w)m(ev)m(er,)j(b)s(ecome)c(a)h(bit)-118 5303 y(more)32 b(tedious,)g(b)s(ecause)i(of)e(the)h(transformation)e (to)h(an)h(exp)s(onen)m(tial.)1900 6155 y(86)p eop %%Page: 87 91 87 90 bop -118 219 a Fv(Sk)m(etc)m(h)37 b(of)h(pro)s(of)-118 403 y Fy(W)-8 b(e)41 b(no)m(w)g(sk)m(etc)m(h)i(the)e(pro)s(of)f(of)g (the)h(theorem)g(3.4.4.)67 b(Eac)m(h)41 b Fx(\025)g Fy(b)s(elongs)f(to) g(a)h(unique)g(set)g Fs(\\)p Fy(\003)3532 418 y Fr(j)3569 403 y Fy(\()p Fv(k)3666 418 y Fr(j)3702 403 y Fy(\),)i(for)d(a)-118 524 y(sequence)h(0)e Fs(\024)f(j)p Fv(k)582 539 y Fr(j)619 524 y Fs(j)g(\024)g Fx(N)878 539 y Fr(j)954 524 y Fy(\(b)s(ecause)i(of) e(the)i(estimates)e(in)g(the)h(inductiv)m(e)g(lemma\))e(whic)m(h)i(ma)m (y)g(b)s(e)g(v)m(oid.)-118 644 y(Moreo)m(v)m(er,)34 b(starting)e(the)h 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Fy(is)f(not)g(of)g(this)g(t)m(yp)s(e,)j (the)e(con)m(v)m(ergence)i(is)d(unclear.)55 b(Ho)m(w)m(ev)m(er)39 b(w)m(e)-118 1911 y(ha)m(v)m(e)34 b(that)1583 2031 y Fs(j)o Fx(Y)1667 2046 y Fr(j)1703 2031 y Fy(\(0\))22 b Fs(\000)h Fx(I)8 b Fs(j)27 b Fx(<)g(")2205 1981 y FC(1)p Fr(=)p FC(2)2205 2057 y Fr(j)-118 2206 y Fy(for)32 b(all)e Fx(j)6 b Fy(,)33 b(so)g(the)g(pro)s(duct)g(ab)s(o)m(v)m(e)g(is)f(con)m (v)m(ergen)m(t)j(at)d(zero.)44 b(But)33 b(more)f(is)g(true.)43 b(F)-8 b(or)32 b(eac)m(h)i Fx(\025)p Fy(,)e(the)h(pro)s(duct)1703 2353 y Fq(Y)1847 2448 y Fx(Y)1904 2463 y Fr(j)1956 2337 y Fq(\020)2026 2380 y Fx(!)p 2026 2425 65 4 v 2034 2516 a Fy(2)2100 2448 y Fx(t)2135 2337 y Fq(\021)-118 2693 y Fy(con)m(v)m(erges)i(uniformly)30 b(on)j(compact)f(in)m(terv)-5 b(als)32 b(of)g Fw(R)5 b Fy(.)49 b(In)33 b(order)g(to)f(see)i(this)e(w) m(e)h(only)f(need)i(to)e(note)h(that)1139 2820 y Fq(\014)1139 2880 y(\014)1139 2940 y(\014)1172 2935 y Fx(Y)1229 2950 y Fr(j)t FC(+1)1372 2824 y Fq(\020)1441 2868 y Fx(!)p 1441 2912 V 1449 3003 a Fy(2)1516 2935 y Fx(t)1551 2824 y Fq(\021)1633 2935 y Fs(\000)22 b Fx(I)1783 2820 y Fq(\014)1783 2880 y(\014)1783 2940 y(\014)1844 2935 y Fs(\024)28 b Fx(")1995 2884 y FC(1)p Fr(=)p FC(2)1995 2961 y Fr(j)2127 2935 y Fy(+)22 b Fs(j)p Fx(Z)2320 2950 y Fr(j)2356 2935 y Fy(\()p Fx(!)t(t)p Fy(\))f Fs(\000)i Fx(I)8 b Fs(j)p Fx(;)-118 3180 y Fy(where)656 3333 y Fx(Z)723 3348 y Fr(j)759 3333 y Fy(\()p Fx(!)t(t)p Fy(\))27 b(=)h(exp)1231 3193 y Fq(\022)1315 3266 y Fs(h)p Fv(k)1413 3281 y Fr(j)1449 3266 y Fs(i)p 1315 3310 174 4 v 1328 3402 a Fy(2)p Fx(\013)1439 3417 y Fr(j)1498 3333 y Fx(A)1571 3348 y Fr(j)1608 3333 y Fx(t)1643 3193 y Fq(\023)1744 3333 y Fy(=)f(cos)1995 3193 y Fq(\022)2078 3266 y Fs(h)p Fv(k)2176 3281 y Fr(j)2212 3266 y Fs(i)p 2078 3310 V 2140 3402 a Fy(2)2261 3333 y Fx(t)2296 3193 y Fq(\023)2386 3333 y Fx(I)j Fy(+)22 b(sin)2693 3193 y Fq(\022)2777 3266 y Fs(h)p Fv(k)2875 3281 y Fr(j)2911 3266 y Fs(i)p 2777 3310 V 2839 3402 a Fy(2)2960 3333 y Fx(t)2995 3193 y Fq(\023)3095 3266 y Fx(A)3168 3281 y Fr(j)p 3095 3310 110 4 v 3101 3402 a Fx(\013)3163 3417 y Fr(j)3215 3333 y Fx(;)-118 3566 y Fy(b)m(y)33 b(the)g(\014rst)g(estimate)f(of)g(\(3.58\).)28 3687 y(No)m(w,)47 b(if)42 b Fv(k)450 3702 y Fr(j)533 3687 y Fy(=)k(0)e(then)g(the)g(b)s(ound)f(will)f(b)s(e)h(for)g(all)f Fx(t)p Fy(,)k(and)e(if)e Fv(k)2585 3702 y Fr(j)2668 3687 y Fs(6)p Fy(=)k(0,)g(then,)h(also)c(b)m(y)h(the)g(lemma,)-118 3807 y Fs(j)p Fy(2)p Fx(\013)21 3822 y Fr(j)72 3807 y Fs(\000)15 b(h)p Fv(k)262 3822 y Fr(j)299 3807 y Fs(ij)27 b Fx(<)h Fy(2)p Fx(")592 3771 y Fr(\033)592 3832 y(j)638 3807 y Fy(,)i(whic)m(h)g(implies)c(that)j Fs(j)p Fx(A)1607 3822 y Fr(j)1644 3807 y Fs(j)e Fx(<)h Fy(32)p Fs(j)p Fx(\013)1991 3822 y Fr(j)2026 3807 y Fs(j)p Fx(N)2142 3771 y Fr(\034)2132 3832 y(j)2215 3807 y Fy(b)m(y)i(\(3.61\).)41 b(Let)30 b(no)m(w)g Fx(j)3077 3822 y FC(1)3144 3807 y Fx(<)e(j)3288 3822 y FC(2)3355 3807 y Fx(<)g(:)17 b(:)g(:)45 b Fy(b)s(e)29 b(all)e(the)-118 3928 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b(up)h(to)g(no)m(w,)h(the)f(conditions)f(on)h(the)g(rotation)e(n)m(um)m (b)s(er)i Fx(\032)g Fy(ha)m(v)m(e)h(only)e(b)s(een)i(written)e(in)g (terms)h(of)f(the)-118 4849 y(sequence)j Fx(\013)350 4864 y Fr(j)421 4849 y Fy(and)e(more)f(sp)s(eci\014cally)g(in)g(terms)g (of)h(the)g(sequence)i Fv(k)2493 4864 y Fr(j)2564 4849 y Fy(for)d(whic)m(h)i Fx(\025)31 b Fs(2)g Fy(\003)3250 4864 y Fr(j)3286 4849 y Fy(\()p Fv(k)3383 4864 y Fr(k)3426 4849 y Fy(\).)50 b(Recall)33 b(that)-118 4970 y(w)m(e)i(w)m(ere)h(only) e(able)g(to)g(sho)m(w)h(uniform)e(con)m(v)m(ergence)k(on)d(a)g(strip)g (of)g(the)h(torus)g(whenev)m(er)i Fv(k)3425 4985 y Fr(j)3492 4970 y Fy(=)31 b(0)j(for)f(all)g Fx(j)-118 5090 y Fy(large)f(enough.)44 b(W)-8 b(e)33 b(m)m(ust)g(no)m(w)g(relate)f(this)h(condition)e(with)i (the)g(arithmetical)c(prop)s(erties)k(of)f(the)h(rotation)-118 5210 y(n)m(um)m(b)s(er)26 b Fx(\032)p Fy(.)41 b(The)27 b(second)g(item)d(whic)m(h)j(remains)d(to)i(do)f(is)g(to)h(p)s(erform)e (the)j(initial)21 b(transformations)j(to)i(render)-118 5331 y(the)37 b(original)d(system,)39 b(either)d(with)h(large)e(energy) j(or)f(close)g(to)f(constan)m(t)i(co)s(e\016cien)m(ts,)h(to)d(a)h (suitable)f(form)-118 5451 y(b)s(efore)d(applying)e(the)i(iterativ)m(e) f(pro)s(cess.)28 5571 y(Let)h(no)m(w)g Fx(X)487 5586 y FC(1)559 5571 y Fy(b)s(e)g(a)f(solution)f(of)h(the)h(system)1337 5791 y Fx(X)1426 5750 y Ft(0)1418 5816 y FC(1)1457 5791 y Fy(\()p Fx(t)p Fy(\))28 b(=)f(\()p Fx(A)1810 5806 y FC(1)1872 5791 y Fy(+)22 b Fx(F)2033 5806 y FC(1)2073 5791 y Fy(\()p Fx(!)t(t)p Fy(\)\))15 b Fx(X)2383 5806 y FC(1)2423 5791 y Fy(\()p Fx(t)p Fy(\))p Fx(:)1900 6155 y Fy(87)p eop %%Page: 88 92 88 91 bop -118 219 a Fy(Then,)34 b(b)m(y)f(lemma)e(3.4.8)h(w)m(e)h(ha)m (v)m(e)h(a)e(represen)m(tation)1515 443 y Fx(X)1596 458 y FC(1)1635 443 y Fy(\()p Fx(t)p Fy(\))c(=)g Fx(Y)1973 333 y Fq(\020)2042 376 y Fx(!)p 2042 420 65 4 v 2050 511 a Fy(2)2116 443 y Fx(t)2151 333 y Fq(\021)2228 443 y Fx(e)2273 402 y Fr(At)2355 443 y 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FC(+1)1423 3234 y Fy(\()p Fx(\025)p Fy(\))22 b Fs(\000)1678 3094 y Fq(\022)1751 3234 y Fx(\013)1813 3249 y Fr(j)1850 3234 y Fy(\()p Fx(\025)p Fy(\))g Fs(\000)2114 3167 y(h)p Fv(k)p Fs(i)p 2114 3211 137 4 v 2158 3303 a Fy(2)2261 3094 y Fq(\023)2334 3090 y(\014)2334 3150 y(\014)2334 3209 y(\014)2334 3269 y(\014)2395 3234 y Fx(<)28 b(c")2587 3183 y FC(1)p Fr(=)p FC(4)2587 3260 y Fr(j)-118 3494 y Fy(for)k Fx(\025)c Fs(2)g Fy(\003)278 3509 y Fr(j)314 3494 y Fy(\()p Fv(k)p Fy(\),)k(whic)m(h)i(follo)m(ws)d (from)g(the)i(condition)e(\(3.61\))h(for)g Fx(j)c Fy(+)22 b(1.)28 3615 y(F)-8 b(or)32 b(\(ii\))f(w)m(e)i(pro)s(ceed)h(b)m(y)f (con)m(tradiction.)42 b(Supp)s(ose)34 b(that)1116 3818 y Fs(j)p Fy(2)9 b(~)-58 b Fx(\032)p Fy(\()p Fx(\025)p Fy(\))22 b Fs(\000)g(h)p Fv(k)p Fs(ij)27 b(\025)i Fx(K)1885 3776 y Ft(\000)p FC(1)1979 3818 y Fs(j)p Fv(k)p Fs(j)2094 3776 y Ft(\000)p Fr(s)2185 3818 y Fx(;)212 b Fs(j)p Fv(k)p Fs(j)27 b(\025)h Fx(N)5 b(;)-118 4020 y Fy(for)35 b(some)g Fx(N)10 b Fy(,)37 b Fx(K)7 b Fy(,)36 b(as)g Fx(s)c(>)g Fy(0.)52 b(Supp)s(ose)36 b(also)f(that)g(there)h(exists)g(an)g (increasing)e(sequence)k(\()p Fx(j)3427 4035 y Fr(k)3470 4020 y Fy(\))3508 4035 y Fr(k)3551 4020 y Fy(,)e(suc)m(h)h(that)-118 4141 y Fv(k)-59 4156 y Fr(j)-30 4168 y Fj(k)40 4141 y Fs(6)p Fy(=)27 b(0.)43 b(Hence,)34 b(b)m(y)g(the)f(de\014nition)e(of)h (the)h(sets)h(\003)1854 4156 y Fr(j)1890 4141 y Fy(\()p Fv(k)p Fy(\),)1438 4343 y Fs(j)p Fy(2)p Fx(\013)1577 4358 y Fr(j)1606 4370 y Fj(k)1647 4343 y Fy(\()p Fx(\025)p Fy(\))22 b Fs(\000)g(h)p Fv(k)1999 4358 y Fr(j)2028 4370 y Fj(k)2070 4343 y Fs(ij)27 b Fx(<)h Fy(2)p Fx(")2363 4302 y Fr(\033)2363 4368 y(j)2392 4380 y Fj(k)2433 4343 y Fx(:)-118 4546 y Fy(Then)616 4563 y Fq(\014)616 4622 y(\014)616 4682 y(\014)616 4742 y(\014)616 4802 y(\014)650 4737 y Fy(2)9 b(~)-58 b Fx(\032)p Fy(\()p Fx(\025)p Fy(\))22 b Fs(\000)1042 4608 y Fr(j)1071 4620 y Fj(k)1003 4642 y Fq(X)1057 4852 y FC(1)1147 4737 y Fs(h)p Fv(k)1245 4752 y Fr(l)1271 4737 y Fs(i)1310 4563 y Fq(\014)1310 4622 y(\014)1310 4682 y(\014)1310 4742 y(\014)1310 4802 y(\014)1371 4737 y Fs(\024)28 b Fy(2)p Fs(j)9 b Fy(~)-58 b Fx(\032)p Fy(\()p Fx(\025)p Fy(\))21 b Fs(\000)i Fx(\032)1907 4752 y Fr(j)1936 4764 y Fj(k)1978 4737 y Fy(\()p Fx(\025)p Fy(\))p Fs(j)f Fy(+)g Fs(j)p Fy(2)p Fx(\013)2398 4752 y Fr(j)2427 4764 y Fj(k)2468 4737 y Fy(\()p Fx(\025)p Fy(\))g Fs(\000)h(h)p Fv(k)2821 4752 y Fr(j)2850 4764 y Fj(k)2891 4737 y Fs(ij)k Fx(<)h Fy(4)p Fx(")3184 4696 y Fr(\033)3184 4762 y(j)3213 4774 y Fj(k)3254 4737 y Fx(:)-118 4986 y Fy(On)k(the)h(other)g(hand,)g(due)g(to)g(the)g (Diophan)m(tine)e(assumptions,)741 5104 y Fq(\014)741 5164 y(\014)741 5223 y(\014)741 5283 y(\014)741 5343 y(\014)783 5278 y Fy(~)-58 b Fx(\032)q Fy(\()p Fx(\025)p Fy(\))22 b Fs(\000)1118 5150 y Fr(j)1147 5162 y Fj(k)1079 5183 y Fq(X)1134 5393 y FC(1)1223 5278 y Fs(h)p Fv(k)1321 5293 y Fr(l)1347 5278 y Fs(i)1386 5104 y Fq(\014)1386 5164 y(\014)1386 5223 y(\014)1386 5283 y(\014)1386 5343 y(\014)1447 5278 y Fs(\025)28 b Fx(K)1642 5237 y Ft(\000)p FC(1)1736 5278 y Fy(\()p Fx(N)1852 5293 y Fr(j)1881 5302 y Fp(1)1942 5278 y Fy(+)22 b Fs(\001)17 b(\001)g(\001)j Fy(+)i Fx(N)2354 5293 y Fr(j)2383 5305 y Fj(k)2425 5278 y Fy(\))2463 5237 y Ft(\000)p Fr(s)2582 5278 y Fs(\025)28 b Fx(cK)2819 5237 y Ft(\000)p FC(1)2914 5278 y Fx(N)3002 5237 y Ft(\000)p FC(2)p Fr(s)2992 5304 y(j)3021 5316 y Fj(k)3129 5278 y Fx(;)-118 5572 y Fy(b)s(ecause)34 b(it)d(can)i(b)s(e)g(sho)m(wn)h(that)e Fx(N)1247 5536 y FC(3)1237 5597 y Fr(j)1266 5609 y Fj(l)1322 5572 y Fx(<)27 b(cN)1545 5587 y Fr(j)1574 5599 y Fj(l)p Fp(+1)1679 5572 y Fy(.)44 b(No)m(w)33 b(this)f(implies)e(that)1149 5861 y Fx(")1195 5820 y Ft(\000)p Fr(\033)1195 5887 y(j)1224 5899 y Fj(k)1324 5861 y Fs(\024)e Fy(const.)1696 5721 y Fq(\022)1769 5861 y Fy(log)1912 5721 y Fq(\022)2013 5794 y Fy(1)p 1995 5839 86 4 v 1995 5930 a Fx(")2041 5945 y FC(1)2090 5721 y Fq(\023)2180 5861 y Fy(\(2)22 b(+)g(2)p Fx(\033)t Fy(\))2533 5820 y Fr(j)2562 5832 y Fj(k)2603 5721 y Fq(\023)2677 5743 y FC(2)p Fr(s)1900 6155 y Fy(88)p eop %%Page: 89 93 89 92 bop -118 219 a Fy(for)31 b(in\014nitely)e(man)m(y)j Fx(k)s Fy(.)43 b(This)31 b(is)g(incompatible)e(with)39 b(~)-58 b Fx(\032)q Fy(\()p Fx(\025)p Fy(\))31 b(b)s(eing)f(either)h (rational)e(of)i(Diophan)m(tine,)f(due)i(to)-118 339 y(condition)f(\(3.55\))h(in)g(the)h(setup.)44 b Fh(\003)28 459 y Fy(T)-8 b(o)35 b(end)g(the)f(pro)s(of)g(of)g(theorem)g(3.4.4)g(w) m(e)h(m)m(ust)g(distinguish)e(b)s(et)m(w)m(een)j(the)f(case)g(of)f (high)g(energies)h(and)-118 580 y(the)f(case)g(of)f(a)g(small)e(p)s (oten)m(tial)g(in)i(order)g(to)g(p)s(erform)f(the)i(preliminary)d (transformations.)44 b(W)-8 b(e)34 b(sk)m(etc)m(h)h(the)-118 700 y(case)e(of)f(a)h(small)d(p)s(oten)m(tial.)28 820 y(Let)j Fx(\025)260 835 y FC(0)299 820 y Fy(\()p Fx(s)p Fy(\))g(as)f(in)g(theorem)h(3.4.4,)f(and)g(de\014ne)1382 1030 y Fv(C)p Fy(\()p Fx(\034)6 b(;)17 b(r)s Fy(\))27 b(=)h Fx(C)7 b Fy(\()p Fx(\034)f(;)17 b(\033)t Fy(\))p Fx(r)2160 989 y FC(\(\(4)p Fr(\034)8 b(=\033)r FC(\)+1\))-118 1240 y Fy(for)35 b Fx(\033)j Fy(=)33 b(1)p Fx(=)p Fy(33)i(sa)m(y)-8 b(,)38 b(where)f Fx(C)43 b Fy(is)36 b(the)g(constan)m(t)h(of)f(lemma)d (3.4.8.)54 b(Supp)s(ose)37 b(that)f Fs(j)p Fx(Q)p Fs(j)3175 1255 y Fr(r)3246 1240 y Fx(<)d Fv(C)p Fy(\()p Fx(r)m(;)17 b(\034)11 b Fy(\),)37 b(and)f(let)-118 1361 y Fx(\025)27 b Fs(2)i Fy(\()p Fs(\000)p Fy(1)p Fx(;)17 b Fy(1\).)43 b(No)m(w)33 b(equation)f(\(3.48\))g(can)h(b)s(e)g(expressed)i(as)1220 1570 y Fx(X)1309 1529 y Ft(0)1301 1595 y FC(1)1340 1570 y Fy(\()p Fx(t)p Fy(\))28 b(=)g(\()p Fx(A)1694 1585 y FC(1)1733 1570 y Fy(\()p Fx(\025)p Fy(\))22 b(+)g Fx(F)2049 1585 y FC(1)2089 1570 y Fy(\()p Fx(!)t(t;)17 b(\025)p Fy(\)\))e Fx(X)2500 1585 y FC(1)2540 1570 y Fy(\()p Fx(t)p Fy(\))p Fx(;)-118 1780 y Fy(with)32 b Fx(A)177 1795 y FC(1)245 1780 y Fy(=)c Fx(B)37 b Fy(and)c Fx(F)713 1795 y FC(1)785 1780 y Fy(as)g(de\014ned)h(in)e(the)i(preliminaries)29 b(and)k Fx(X)2375 1795 y FC(1)2447 1780 y Fy(ha)m(ving)g(rotation)e(n)m (um)m(b)s(er)i Fx(\032)p Fy(\()p Fx(\025)p Fy(\))28 b(=)36 b(~)-57 b Fx(\032)p Fy(\()p Fx(\025)p Fy(\).)-118 1901 y(Moreo)m(v)m(er)691 1915 y Fq(8)691 2005 y(<)691 2184 y(:)822 1993 y Fs(j)o Fx(A)922 2008 y FC(1)962 1993 y Fy(\()p Fx(\025)p Fy(\))p Fs(j)27 b Fx(<)g Fy(2)822 2032 y Fq(\014)822 2092 y(\014)855 2036 y(\000)931 2078 y Fr(@)p 910 2094 83 4 v 910 2151 a(@)t(\025)1003 2036 y Fq(\001)1048 2058 y Fr(\027)1108 2117 y Fx(A)1181 2132 y FC(1)1221 2117 y Fy(\()p Fx(\025)p Fy(\))1354 2032 y Fq(\014)1354 2092 y(\014)1414 2117 y Fx(<)h(")1564 2076 y Ft(\000)p Fr(\027)t(\033)1564 2141 y FC(1)1704 2117 y Fx(;)212 b(\027)34 b Fy(=)28 b(1)p Fx(;)17 b Fy(2)822 2156 y Fq(\014)822 2216 y(\014)855 2160 y(\000)931 2201 y Fr(@)p 910 2218 V 910 2275 a(@)t(\025)1003 2160 y Fq(\001)1048 2182 y Fr(\027)1108 2241 y Fx(F)1171 2256 y FC(1)1211 2241 y Fy(\()p Fs(\001)p Fx(;)g(\025)p Fy(\))1416 2156 y Fq(\014)1416 2216 y(\014)1448 2280 y Fr(r)1514 2241 y Fx(<)27 b(")1663 2256 y FC(1)1730 2241 y Fy(=)g Fx(C)7 b Fy(\()p Fx(\034)f(;)17 b(\033)t Fy(\))p Fx(r)2184 2205 y FC(\(\(4)p Fr(\034)8 b(=\033)r FC(\)+1\))2735 2241 y Fx(\027)34 b Fy(=)28 b(0)p Fx(;)17 b Fy(1)p Fx(;)g Fy(2)-118 2407 y(and)32 b(no)m(w)g(the)g(result)g(follo)m(ws)e(from)g (an)i(application)d(of)i(lemmas)f(3.4.8)h(and)h(3.4.9.)42 b(If)32 b Fx(\025)27 b Fs(\024)i(\000)p Fy(1,)j(then)g(w)m(e)h(are)-118 2527 y(in)i(the)h(resolv)m(en)m(t)h(set)f(if)e Fs(j)p Fx(Q)p Fs(j)963 2542 y Fr(r)1034 2527 y Fx(<)f Fy(1)i(and)h(the)g (result)f(is)g(also)g(true.)53 b(If)36 b Fx(\025)c Fs(\025)i Fy(1,)i(then)g(this)g(case)g(is)f(co)m(v)m(ered)j(b)m(y)-118 2648 y(the)33 b(large)e(p)s(oten)m(tial,)g(whic)m(h)i(is)g(treated)g(p) s(erforming)d(the)j(transformations)e(de\014ned)j(in)e(the)h (preliminaries.)28 2768 y(It)44 b(only)g(remains)f(to)g(pro)m(v)m(e)j (the)e FA(non-absolutely)g(c)-5 b(onver)g(gent)53 b Fy(part)44 b(of)f(the)h(theorem,)j(whic)m(h)e(giv)m(es)f(a)-118 2888 y(b)s(ound)36 b(for)g(the)h(rate)f(of)f(escap)s(e)j(of)e(p)s(oin)m (ts)f(in)h(the)g(sp)s(ectrum)h(with)f(rotation)e(n)m(um)m(b)s(er)46 b(~)-58 b Fx(\032)p Fy(\()p Fx(\025)p Fy(\))36 b(b)s(eing)g(neither) -118 3009 y(Diophan)m(tine)k(nor)h(rational.)66 b(In)42 b(this)f(case,)j(w)m(e)e(let)f Fx(\025)h Fs(2)h Fy(\003)2205 3024 y Fr(j)2241 3009 y Fy(\()p Fv(k)2338 3024 y Fr(j)2374 3009 y Fy(\).)69 b(If)42 b Fv(k)2674 3024 y Fr(j)2753 3009 y Fy(=)g(0)f(for)f Fx(j)47 b Fy(large)40 b(enough,)k(then)-118 3129 y Fx(X)8 b Fy(,)31 b(the)h(fundamen)m(tal)f(matrix,)f(has)i(a)f (Flo)s(quet)f(represen)m(tation,)j(so)e(the)h(result)g(trivially)c (follo)m(ws.)42 b(Assume,)-118 3249 y(therefore,)31 b(that)e(there)h (exists)h(an)e(increasing)g(sequence)j(of)d(in)m(tegers)h(1)e Fs(\024)g Fx(j)2711 3264 y FC(1)2778 3249 y Fx(<)g(j)2922 3264 y FC(2)2989 3249 y Fx(<)g(:)17 b(:)g(:)45 b Fy(suc)m(h)31 b(that)f Fv(k)3738 3264 y Fr(j)3767 3276 y Fj(k)3836 3249 y Fs(6)p Fy(=)e(0.)-118 3370 y(Then)38 b(w)m(e)h(can)e(apply)h(an) 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Fx(l)r Fy(.)44 b(Cho)s(ose)1731 3998 y Fx(t)1766 4013 y Fr(k)1836 3998 y Fy(=)2000 3931 y(4)p Fx(\031)p 1950 3975 208 4 v 1950 4066 a Fs(h)p Fv(k)2048 4081 y Fr(j)2077 4093 y Fj(k)2118 4066 y Fs(i)-118 4261 y Fy(and)33 b Fx(t)107 4276 y Fr(k)r Ft(\000)p FC(1)272 4261 y Fy(so)g(that)1159 4423 y Fx(t)1194 4438 y Fr(k)r Ft(\000)p FC(1)1350 4423 y Fy(+)22 b Fx(t)1483 4438 y Fr(k)1553 4423 y Fy(=)28 b Fx(n)1814 4355 y Fy(4)p Fx(\031)p 1725 4400 286 4 v 1725 4491 a Fs(h)p Fv(k)1823 4506 y Fr(j)1852 4518 y Fj(k)q Fi(\000)p Fp(1)1972 4491 y Fs(i)2021 4423 y Fx(;)211 b Fy(for)33 b(some)f Fx(n;)-118 4657 y Fy(and)h(so)f(on.)44 b(Then)33 b(w)m(e)h(get)f(that,)f(b)m(y)i (construction,)e Fx(Z)1952 4672 y Fr(j)1981 4681 y Fp(1)2020 4657 y Fy(\()p Fx(!)t(t)p Fy(\))27 b(=)g Fx(I)41 b Fy(and,)32 b(for)g(all)f Fx(j)2951 4672 y Fr(l)2977 4657 y Fy(,)i(with)f Fx(l)e Fs(\025)e Fy(1,)981 4867 y Fs(j)p Fx(Z)1076 4882 y Fr(j)1105 4894 y Fj(l)p Fp(+1)1209 4867 y Fy(\()p Fx(!)t(t)p Fy(\))21 b Fs(\000)i Fx(I)8 b Fs(j)27 b Fy(=)h Fs(j)p Fx(Z)1811 4882 y Fr(j)1840 4894 y Fj(l)p Fp(+1)1944 4867 y Fy(\()p Fx(!)t(T)2104 4882 y Fr(l)2129 4867 y Fy(\))23 b Fs(\000)f Fx(I)8 b Fs(j)27 b Fx(<)h Fy(70)p Fx(T)2654 4882 y Fr(l)2680 4867 y Fx(N)2768 4826 y Fr(\034)2758 4892 y(j)2787 4904 y Fj(l)2815 4867 y Fx(")2861 4826 y Fr(\033)2861 4892 y(j)2890 4904 y Fj(l)-118 5077 y Fy(as)33 b(in)e(\(3.63\),)h(where)1267 5256 y Fx(T)1324 5271 y Fr(l)1378 5256 y Fy(=)1482 5115 y Fq(\032)1598 5195 y Fx(t)1633 5210 y FC(1)1695 5195 y Fy(+)22 b Fs(\001)17 b(\001)g(\001)j Fy(+)i Fx(t)2064 5210 y Fr(l)2190 5195 y Fx(l)30 b Fs(\024)e Fx(k)d Fs(\000)e Fy(1)1598 5315 y Fx(t)1633 5330 y FC(1)1695 5315 y Fy(+)f Fs(\001)17 b(\001)g(\001)j Fy(+)i Fx(t)2064 5330 y Fr(k)2190 5315 y Fx(l)30 b Fs(\025)e Fx(k)s(:)-118 5477 y Fy(Since)1238 5597 y Fs(j)p Fx(T)1323 5612 y Fr(l)1349 5597 y Fs(j)f(\024)i Fy(4)p Fx(\031)t Fy(\()p Fx(N)1744 5556 y Fr(\034)1734 5622 y(j)1763 5631 y Fp(1)1823 5597 y Fy(+)22 b Fx(:)17 b(:)g(:)f(N)2140 5556 y Fr(\034)2130 5622 y(j)2159 5634 y Fj(l)2187 5597 y Fy(\))27 b Fs(\024)i Fy(4)p Fx(\031)t(N)2554 5556 y FC(2)p Fr(\034)2544 5622 y(j)2573 5634 y Fj(l)2632 5597 y Fx(;)-118 5767 y Fy(and)1467 5773 y Fq(\014)1467 5833 y(\014)1467 5893 y(\014)1500 5888 y Fx(Y)1557 5903 y Fr(j)t FC(+1)1700 5777 y Fq(\020)1769 5820 y Fx(!)p 1769 5865 65 4 v 1777 5956 a Fy(2)1844 5888 y Fx(t)1879 5777 y Fq(\021)1960 5888 y Fs(\000)23 b Fx(I)2111 5773 y Fq(\014)2111 5833 y(\014)2111 5893 y(\014)2172 5888 y Fx(<)k(")2321 5837 y FC(1)p Fr(=)p FC(2)2321 5913 y Fr(j)1900 6155 y Fy(89)p eop %%Page: 90 94 90 93 bop -118 219 a Fy(for)32 b(all)e Fx(j)k Fs(6)p Fy(=)28 b Fx(j)384 234 y Fr(l)442 219 y Fy(it)k(follo)m(ws)f(that)1402 246 y Fq(\014)1402 306 y(\014)1402 366 y(\014)1435 266 y(Y)1579 361 y Fx(Y)1636 376 y Fr(j)1688 250 y Fq(\020)1758 293 y Fx(!)p 1758 338 65 4 v 1766 429 a Fy(2)1832 361 y Fx(t)1867 250 y Fq(\021)1949 361 y Fs(\000)23 b Fx(I)2100 246 y Fq(\014)2100 306 y(\014)2100 366 y(\014)2160 361 y Fx(<)28 b(c")2352 310 y Fr(\033)r(=)p FC(4)2352 385 y(1)2469 361 y Fx(;)-118 549 y Fy(and)33 b(this)f(giv)m(es)h(the)g (\014rst)g(part)f(of)g(the)h(theorem)g(3.4.4\(ii\).)41 b(W)-8 b(e)33 b(no)m(w)g(go)f(to)g(the)h(second)h(part.)28 669 y(Let)f Fx(t)g Fy(b)s(e)f(suc)m(h)j(that)1358 742 y(1)p 1251 786 263 4 v 1251 877 a Fs(jh)p Fv(k)1377 892 y Fr(j)1406 904 y Fj(k)1447 877 y Fs(ij)1551 809 y(\024)29 b Fx(t)e(<)1965 742 y Fy(1)p 1833 786 313 4 v 1833 877 a Fs(jh)p Fv(k)1959 892 y Fr(j)1988 904 y Fj(k)q Fp(+1)2106 877 y Fs(i)2155 809 y Fx(;)212 b(k)31 b Fs(\025)d Fy(2)p Fx(:)-118 1030 y Fy(Then,)34 b(b)m(y)f(\(3.60\))f(and)h(\(3.63\))e(w)m (e)j(ha)m(v)m(e)g(the)f(follo)m(wing)d(p)s(ossibilities)593 1287 y Fq(\014)593 1347 y(\014)593 1406 y(\014)626 1401 y Fx(Y)683 1416 y Fr(j)t FC(+1)826 1291 y Fq(\020)895 1334 y Fx(!)p 895 1378 65 4 v 903 1470 a Fy(2)970 1401 y Fx(t)1005 1291 y Fq(\021)1087 1401 y Fs(\000)22 b Fx(I)1237 1287 y Fq(\014)1237 1347 y(\014)1237 1406 y(\014)1298 1401 y Fx(<)27 b(c)1460 1137 y Fq(8)1460 1227 y(>)1460 1257 y(>)1460 1287 y(<)1460 1466 y(>)1460 1496 y(>)1460 1526 y(:)1590 1223 y Fx(")1636 1173 y FC(1)p Fr(=)p FC(2)1636 1249 y Fr(j)2406 1223 y Fx(j)33 b Fs(6)p Fy(=)28 b Fx(j)2623 1238 y Fr(l)1590 1346 y Fx(N)1678 1310 y FC(2)p Fr(\034)1668 1371 y(j)1697 1383 y Fj(l)p Fi(\000)p Fp(1)1803 1346 y Fx(")1849 1310 y Fr(\033)1849 1371 y(j)1878 1383 y Fj(l)p Fi(\000)p Fp(1)2406 1346 y Fx(j)33 b Fy(=)28 b Fx(j)2623 1361 y Fr(l)2649 1346 y Fx(;)190 b(l)30 b Fs(\025)e Fx(k)e Fy(+)c(2)1590 1474 y(\(1)g(+)g Fs(j)p Fx(A)1898 1489 y Fr(j)1934 1474 y Fs(jjh)p Fv(k)2088 1489 y Fr(j)2124 1474 y Fs(ij)2191 1438 y Ft(\000)p FC(1)2285 1474 y Fy(\))83 b Fx(j)33 b Fy(=)28 b Fx(j)2623 1489 y Fr(l)2649 1474 y Fx(;)190 b(l)30 b Fs(\024)e Fx(k)1590 1594 y Fy(\(1)22 b(+)g Fs(j)p Fx(tA)1933 1609 y Fr(j)1970 1594 y Fs(j)p Fy(\))370 b Fx(j)33 b Fy(=)28 b Fx(j)2623 1609 y Fr(k)r FC(+1)2756 1594 y Fx(;)-118 1767 y Fy(Therefore,)476 1860 y Fq(\014)476 1919 y(\014)476 1979 y(\014)476 2039 y(\014)476 2099 y(\014)509 1909 y Fr(k)r FC(+1)510 1939 y Fq(Y)556 2149 y FC(2)655 2034 y Fx(Y)712 2049 y Fr(j)741 2061 y Fj(l)p Fp(+1)862 1923 y Fq(\020)931 1967 y Fx(!)p 931 2011 V 939 2102 a Fy(2)1005 2034 y Fx(t)1040 1923 y Fq(\021)1100 1860 y(\014)1100 1919 y(\014)1100 1979 y(\014)1100 2039 y(\014)1100 2099 y(\014)1161 2034 y Fs(\024)g Fx(c)1308 1993 y Fr(k)1412 1909 y(k)1367 1939 y Fq(Y)1413 2149 y FC(2)1511 1894 y Fq(\022)1585 1890 y(\014)1585 1949 y(\014)1585 2009 y(\014)1585 2069 y(\014)1667 1967 y Fs(h)p Fv(k)1765 1982 y Fr(j)1794 1994 y Fj(l)1822 1967 y Fs(i)p 1628 2011 272 4 v 1628 2102 a(h)p Fv(k)1726 2117 y Fr(j)1755 2129 y Fj(l)p Fi(\000)p Fp(1)1861 2102 y Fs(i)1910 1890 y Fq(\014)1910 1949 y(\014)1910 2009 y(\014)1910 2069 y(\014)1965 2034 y Fy(+)2063 1890 y Fq(\014)2063 1949 y(\014)2063 2009 y(\014)2063 2069 y(\014)2177 1967 y Fx(A)2250 1982 y Fr(j)2279 1994 y Fj(l)p 2106 2011 V 2106 2102 a Fs(h)p Fv(k)2204 2117 y Fr(j)2233 2129 y Fj(l)p Fi(\000)p Fp(1)2339 2102 y Fs(i)2388 1890 y Fq(\014)2388 1949 y(\014)2388 2009 y(\014)2388 2069 y(\014)2421 1894 y(\023)2511 1890 y(\014)2511 1949 y(\014)2511 2009 y(\014)2511 2069 y(\014)2561 1967 y Fs(h)p Fv(k)2659 1982 y Fr(j)2688 1994 y Fj(l)2716 1967 y Fs(i)p 2554 2011 208 4 v 2554 2102 a(h)p Fv(k)2652 2117 y Fr(j)2681 2129 y Fj(k)2723 2102 y Fs(i)2772 1890 y Fq(\014)2772 1949 y(\014)2772 2009 y(\014)2772 2069 y(\014)2822 2034 y Fy(\(1)21 b(+)h Fs(j)p Fx(tA)3164 2049 y Fr(j)3193 2061 y Fj(k)q Fp(+1)3312 2034 y Fs(j)p Fy(\))p Fx(;)361 b Fy(\(3.64\))-118 2295 y(and)33 b(since)1128 2310 y Fq(\014)1128 2370 y(\014)1128 2430 y(\014)1128 2489 y(\014)1211 2387 y Fs(h)p Fv(k)1309 2402 y Fr(j)1338 2414 y Fj(l)1365 2387 y Fs(i)p 1171 2432 272 4 v 1171 2523 a(h)p Fv(k)1269 2538 y Fr(j)1298 2550 y Fj(l)p Fi(\000)p Fp(1)1405 2523 y Fs(i)1453 2310 y Fq(\014)1453 2370 y(\014)1453 2430 y(\014)1453 2489 y(\014)1514 2454 y Fs(\024)1629 2378 y Fy(2)p Fx(")1724 2342 y Fr(\033)1724 2403 y(j)1753 2415 y Fj(l)1803 2378 y Fy(+)22 b(2)p Fs(j)p Fx(A)2051 2393 y Fr(j)2080 2405 y Fj(l)2107 2378 y Fs(j)p 1629 2432 506 4 v 1718 2523 a(jh)p Fv(k)1844 2538 y Fr(j)1873 2550 y Fj(l)p Fi(\000)p Fp(1)1979 2523 y Fs(ij)2172 2454 y(\024)29 b Fy(70)p Fx(N)2464 2413 y FC(2)p Fr(\034)2454 2479 y(j)2483 2491 y Fj(l)p Fi(\000)p Fp(1)2589 2454 y Fx(")2635 2413 y Fr(\033)2635 2479 y(j)2664 2491 y Fj(l)p Fi(\000)p Fp(1)-118 2676 y Fy(b)m(y)g(\(3.62\),)g(the)g(ab)s(o)m (v)m(e)g(expression)h(con)m(v)m(erges)g(to)f(0)f(as)g Fx(k)j Fs(!)c(1)p Fy(.)42 b(This)29 b(together)f(with)h(the)f(b)s(ound) h(2)p Fs(jh)p Fv(k)3792 2691 y Fr(j)3821 2700 y Fp(1)3859 2676 y Fs(ijj)p Fx(t)p Fs(j)-118 2796 y Fy(for)j(the)h(last)f(term)g (of)g(\(3.64\))g(pro)m(v)m(es)i(the)f(desired)g(limit)1464 3031 y(lim)1448 3090 y Fr(t)p Ft(!1)1642 2963 y Fs(j)p Fx(X)8 b Fy(\()p Fx(t)p Fy(\))21 b Fs(\000)i Fx(X)8 b Fy(\(0\))p Fs(j)p 1642 3008 591 4 v 1919 3099 a Fx(t)2270 3031 y Fy(=)27 b(0)p Fx(:)-118 3235 y Fh(\003)-118 3518 y Fg(3.4.3)136 b(Some)45 b(more)h(results)f(and)g(applications)-118 3703 y Fy(Eliasson's)39 b(pap)s(er)h([29])g(con)m(tains)g(man)m(y)f (other)h(in)m(teresting)g(results.)65 b(The)41 b(\014rst)f(one)g(that)g (w)m(e)g(state)h(is)e(an)-118 3823 y(immediate)30 b(corollary)h(of)h (theorem)g(3.4.4)-118 4005 y Fv(Corollary)k(3.4.10)49 b FA(F)-7 b(or)34 b(almost)g(every)h Fx(\025)28 b Fs(2)g Fx(\033)t Fy(\()p Fx(H)8 b Fy(\))21 b Fs(\\)i Fy(\()p Fx(\025)2074 4020 y FC(0)2113 4005 y Fx(;)17 b Fy(+)p Fs(1)p Fy(\))p FA(,)34 b(the)h(fol)5 b(lowing)33 b(ine)-5 b(quality)35 b(holds)1625 4184 y Fy(2)p Fx(\032)p Fy(\()p Fx(\025)p Fy(\))p Fx(\032)1907 4143 y Ft(0)1931 4184 y Fy(\()p Fx(\025)p Fy(\))27 b Fs(\025)h Fy(1)p Fx(:)-118 4364 y FA(In)g(p)-5 b(articular,)30 b(the)f(set)g(of)g(al)5 b(l)28 b Fx(\025)g(>)f(\025)1263 4379 y FC(0)1332 4364 y FA(for)h(which)g Fx(\032)p Fy(\()p Fx(\025)p Fy(\))h FA(is)g(neither)g(Diophantine)e(nor)i(r)-5 b(ational)28 b(is)h(of)f(me)-5 b(asur)g(e)-118 4484 y(0.)28 4666 y Fy(The)39 b(reason)f(for)f(this)g(is)g(the)h(follo)m(wing.)55 b(It)38 b(is)f(sho)m(wn)i(in)d([20])h(that)h(2)p Fx(\032)p Fy(\()p Fx(\025)p Fy(\))p Fx(\032)2953 4630 y Ft(0)2976 4666 y Fy(\()p Fx(\025)p Fy(\))e Fs(\025)h Fy(1)g(for)g(almost)e(all)h Fx(\025)-118 4786 y Fy(b)s(elonging)31 b(to)h(the)h(set)1468 4907 y Fs(f)o Fx(\025)28 b Fs(2)g Fx(\033)t Fy(\()p Fx(H)8 b Fy(\);)17 b Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))26 b(=)i(0)p Fs(g)16 b Fx(;)-118 5064 y Fy(where)35 b Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))34 b(is)g(the)h(upp)s(er)g(Ly)m(apuno)m(v)h (exp)s(onen)m(t)f(of)f(\(3.1\).)48 b(If)35 b(no)m(w)g Fx(\032)p Fy(\()p Fx(\025)p Fy(\))f(is)g(Diophan)m(tine,)f(then)i Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))30 b(=)h(0)-118 5184 y(b)m(y)42 b(the)g(\014rst)f(item)f(of)h(theorem)g(3.4.4)g(and)g(if)f Fx(\032)p Fy(\()p Fx(\025)p Fy(\))h(is)g(neither)g(Diophan)m(tine)f (nor)h(rational,)g(then)h(b)m(y)g(the)-118 5304 y(second)35 b(item)e(of)h(the)g(same)g(theorem)g(w)m(e)h(also)e(conclude)i(that)f Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))29 b(=)h(0.)48 b(Hence)35 b Fx(\014)6 b Fy(\()p Fx(\025)p Fy(\))30 b(=)g(0)k(for)f(almost)g(all) -118 5425 y Fx(\025)j Fy(in)f(the)i(sp)s(ectrum)f(and)h(in)e(the)i (domain)d(of)i(application)d(of)j(the)h(theorem)f(\(that)g(is,)g (either)g(in)g(the)g(upp)s(er)-118 5545 y(part)c(of)g(the)h(sp)s (ectrum)g(or)g(close)f(to)g(constan)m(t)i(co)s(e\016cien)m(ts\).)28 5666 y(The)h(follo)m(wing)c(theorem)j(relies)f(on)h(the)g(tec)m (hniques)i(of)d(3.4.4)h(but)g(requires)h(additional)c(w)m(ork.)48 b(It)34 b(deals)-118 5786 y(with)e(the)g(description)g(of)f(the)i(endp) s(oin)m(ts)f(\(b)s(oth)g(collapsed)g(and)g(not\))g(of)f(sp)s(ectral)h (gaps)g(and)h(the)f(existence)-118 5906 y(of)g(non-reducible)g (situations)f(in)h(the)h(domain)e Fx(\025)d(>)f(\025)1919 5921 y FC(0)1959 5906 y Fy(\()p Fs(j)p Fx(Q)p Fs(j)2130 5921 y Fr(r)2167 5906 y Fy(\).)1900 6155 y(90)p eop %%Page: 91 95 91 94 bop -118 219 a Fv(Theorem)37 b(3.4.11)h(\([29]\))48 b FA(F)-7 b(or)34 b Fx(\025)28 b(>)f(\025)1418 234 y FC(0)1457 219 y Fy(\()p Fs(j)p Fx(Q)p Fs(j)1628 234 y Fr(r)1666 219 y Fy(\))35 b FA(the)g(fol)5 b(lowing)33 b(holds:)-33 422 y(\(i\))49 b(The)35 b(matrix)g Fx(A)28 b Fy(=)h Fx(A)p Fy(\()p Fx(\025)p Fy(\))g(=)f(0)35 b FA(if)g Fs(f)p Fx(\025)p Fs(g)g FA(is)g(a)h(c)-5 b(ol)5 b(lapse)-5 b(d)34 b(gap)h(and)f(it)i(is)f(nilp)-5 b(otent)35 b(di\013er)-5 b(ent)35 b(fr)-5 b(om)35 b(zer)-5 b(o)35 b(if)126 542 y Fs(f)p Fx(\025)p Fs(g)f FA(is)h(the)g(endp)-5 b(oint)34 b(of)g(a)h(non-c)-5 b(ol)5 b(lapse)-5 b(d)33 b(gap.)-62 746 y(\(ii\))48 b(F)-7 b(or)31 b(a)i(generic)e(set)i(of)f Fx(Q)1064 710 y Ft(0)1088 746 y Fx(s)g FA(in)g(the)h Fs(j)17 b(\001)f(j)1560 761 y Fr(r)1598 746 y FA(-top)-5 b(olo)g(gy,)32 b(ther)-5 b(e)33 b(exist)f(values)g(of)g Fx(\025)c(>)f(\025)3145 761 y FC(0)3185 746 y Fy(\()p Fs(j)p Fx(Q)p Fs(j)3356 761 y Fr(r)3393 746 y Fy(\))33 b FA(for)f(which)g(the)126 866 y(fundamental)i(matrix)g(is)h(unb)-5 b(ounde)g(d)34 b(and)g Fx(\032)p Fy(\()p Fx(\025)p Fy(\))h FA(is)g(neither)f(Diophantine)g(nor)g(r)-5 b(ational.)28 1094 y Fy(W)d(e)31 b(will)c(come)j(bac)m(k)h(on)f(the)h(second)g(item)e (in)g(section)i(3.5,)f(b)s(ecause)h(it)e(is)h(a)g(result)g(on)g (non-reducibilit)m(y)-8 b(.)28 1215 y(Finally)g(,)41 b(in)g([29],)j(the)e(follo)m(wing)d(result)j(on)f(the)h(nature)g(of)g (the)g(sp)s(ectrum)g(of)f(Sc)m(hr\177)-49 b(odinger)42 b(equation)-118 1335 y(with)32 b(quasi-p)s(erio)s(dic)f(forcing)g(is)h (also)g(sho)m(wn)-118 1563 y Fv(Theorem)37 b(3.4.12)h(\([29]\))48 b FA(F)-7 b(or)34 b Fx(\025)28 b(>)f(\025)1418 1578 y FC(0)1457 1563 y Fy(\()p Fs(j)p Fx(Q)p Fs(j)1628 1578 y Fr(r)1666 1563 y Fy(\))35 b FA(the)g(fol)5 b(lowing)33 b(holds)-33 1767 y(\(i\))49 b(F)-7 b(or)34 b(a)g(generic)g(p)-5 b(otential)35 b Fx(Q)g FA(in)g(the)f Fs(j)22 b(\001)g(j)1652 1782 y Fr(r)1690 1767 y FA(-top)-5 b(olo)g(gy,)34 b Fx(\033)t Fy(\()p Fx(H)8 b Fy(\))22 b Fs(\\)g Fy(\()p Fx(\025)2557 1782 y FC(0)2596 1767 y Fy(\()p Fs(j)p Fx(Q)p Fs(j)2767 1782 y Fr(r)2805 1767 y Fy(\))p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))34 b FA(is)g(a)h(Cantor)g(set.)-62 1970 y(\(ii\))48 b Fx(\033)t Fy(\()p Fx(H)8 b Fy(\))29 b Fs(\\)i Fy(\()p Fx(\025)571 1985 y FC(0)610 1970 y Fy(\()p Fs(j)p Fx(Q)p Fs(j)781 1985 y Fr(r)819 1970 y Fy(\))p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))45 b FA(is)g(pur)-5 b(ely)46 b(absolutely)g(c)-5 b(ontinuous.)77 b(In)45 b(p)-5 b(articular,)48 b(ther)-5 b(e)46 b(ar)-5 b(e)46 b(no)f(p)-5 b(oint)126 2091 y(eigenvalues)33 b(in)i Fy(\()p Fx(\025)851 2106 y FC(0)890 2091 y Fy(\()p Fs(j)p Fx(Q)p Fs(j)1061 2106 y Fr(r)1099 2091 y Fy(\))p Fx(;)17 b Fy(+)p Fs(1)p Fy(\))p FA(.)-118 2319 y Fy(The)27 b(\014rst)g(item)e(follo)m(ws)g(from)g(theorem)h(3.4.4.)40 b(F)-8 b(rom)25 b(this)h(theorem)g(it)f(also)h(follo)m(ws)f(that)h(if)f Fx(\025)i(>)h(\025)3601 2334 y FC(0)3640 2319 y Fy(\()p Fs(j)p Fx(Q)p Fs(j)3811 2334 y Fr(r)3849 2319 y Fy(\))e(no)-118 2439 y(p)s(oin)m(t)31 b(sp)s(ectrum)h(can)h(o)s(ccur)f(\(see)h(also)e 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Fy(Up)23 b(to)g(no)m(w)h(w)m(e)g(ha)m(v)m(e)h(discussed)f(the)g (reducibilit)m(y)d(of)i(Sc)m(hr\177)-49 b(odinger)24 b(equation)e(with)h(quasi-p)s(erio)s(dic)e(p)s(oten)m(tial,)-118 3655 y(and)34 b(the)g(results,)g(in)f(the)h(case)g(of)f(analytic)g(p)s (oten)m(tial)e(and)j(Diophan)m(tine)e(frequencies)j(are)f(quite)g (de\014nitiv)m(e)-118 3776 y(either)22 b(close)h(to)f(constan)m(t)i(co) s(e\016cien)m(ts)g(or)e(with)h(high)e(energies)j(\(they)f(can)g(b)s (oth)g(b)s(e)g(written)f(as)h(p)s(erturbations)-118 3896 y(of)38 b(systems)i(with)d(constan)m(t)j(co)s(e\016cien)m(ts\).)62 b(It)38 b(is)g(therefore)h(natural)e(to)h(lo)s(ok)f(for)h FA(c)-5 b(onverse)44 b Fy(results.)62 b(This)-118 4017 y(means)37 b(lo)s(oking)e(for)i(conditions)g(under)h(whic)m(h)g(there)g (is)f(no)g(reducibilit)m(y)-8 b(,)37 b(and)g(to)g(try)h(to)f (understand)h(the)-118 4137 y(transition)44 b(b)s(et)m(w)m(een)k (reducibilit)m(y)c(and)i(non-reducibilit)m(y)-8 b(.)82 b(The)46 b(results)h(in)e(this)g(direction)g(are)h(not)g(y)m(et)-118 4257 y(complete,)25 b(esp)s(ecially)f(when)i(trying)d(to)i(understand)h (the)f(transition)d(from)i(reducibilit)m(y)f(to)h(non-reducibilit)m(y) -8 b(.)-118 4378 y(Therefore,)37 b(and)f(as)g(a)f(thorough)g (discussion)h(of)f(these)i(results)f(w)m(ould)f(tak)m(e)h(us)h(to)s(o)d (far,)i(w)m(e)h(will)c(giv)m(e)i(only)-118 4498 y(some)d(results,)h (together)g(with)f(some)h(sp)s(eculation.)28 4618 y(W)-8 b(e)33 b(b)s(egin)f(this)g(section)h(with)f(an)h(example)f(of)g(a)g (non-reducible)g(Sc)m(hr\177)-49 b(odinger)33 b(equation.)-118 4878 y Fv(An)k(example)g(with)g(p)s(oin)m(t)f(eigen)m(v)-6 b(alues)-118 5063 y Fy(In)30 b(this)g(subsection)h(w)m(e)g(will)d(giv)m (e)i(an)g(example)f(due)i(to)f(R.)g(Johnson)g(and)g(J.)h(Moser)g(\([47) o(]\))f(exhibiting)f(p)s(oin)m(t)-118 5183 y(eigen)m(v)-5 b(alues.)52 b(Recall)34 b(that)h(a)g(p)s(oin)m(t)g Fx(\025)g Fy(in)f(the)i(sp)s(ectrum)g Fx(\033)t Fy(\()p Fx(H)8 b Fy(\))35 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219 a Fy(The)34 b(construction)e(of)h(this)f(example)g (dep)s(ends)i(on)f(\014nding)f(an)g(o)s(dd)h(quasi-p)s(erio)s(dic)d (function)1565 415 y Fx(f)11 b Fy(\()p Fx(t)p Fy(\))27 b(=)h Fx(F)14 b Fy(\()p Fx(!)2042 430 y FC(1)2081 415 y Fx(t;)j(!)2221 430 y FC(2)2260 415 y Fx(t)p Fy(\))-118 612 y(with)1226 860 y Fx(F)d Fy(\()p Fx(\022)1386 875 y FC(1)1426 860 y Fx(;)j(\022)1515 875 y FC(2)1555 860 y Fy(\))27 b(=)1760 735 y Ft(1)1724 765 y Fq(X)1729 975 y Fr(n)p FC(=1)1884 860 y Fx(a)1935 875 y Fr(n)1999 860 y Fy(sin\()p Fx(j)2197 875 y Fr(n)2244 860 y Fx(\022)2289 875 y FC(1)2351 860 y Fs(\000)c Fx(k)2502 875 y Fr(n)2548 860 y Fx(\022)2593 875 y FC(2)2633 860 y Fy(\))1095 b(\(3.65\))-118 1138 y(for)32 b(whic)m(h)872 1373 y Fx(g)t Fy(\()p Fx(t)p Fy(\))27 b(=)1165 1238 y Fq(Z)1264 1264 y Fr(t)1220 1463 y FC(0)1310 1373 y Fx(f)11 b Fy(\()p Fx(s)p Fy(\))p Fx(ds)27 b Fs(\025)h Fx(c)p Fs(j)p Fx(t)p Fs(j)1853 1332 y FC(1)p Ft(\000)p Fr(\016)2176 1373 y Fy(for)195 b Fs(j)p Fx(t)p Fs(j)27 b(\025)h 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16 R 474 0 V -484 15 R 495 0 V -504 15 R 516 0 V -526 16 R 538 0 V -548 15 R 560 0 V -569 15 R 580 0 V -590 16 R 602 0 V -611 15 R 623 0 V -633 15 R 645 0 V -654 16 R 666 0 V -676 15 R 688 0 V -697 15 R 709 0 V -718 15 R 730 0 V -739 16 R 751 0 V -760 15 R 772 0 V -781 15 R 793 0 V -803 16 R 816 0 V -824 15 R 836 0 V -845 15 R 857 0 V -866 16 R 878 0 V -887 15 R 900 0 V -909 15 R 921 0 V -930 15 R 942 0 V -950 16 R 963 0 V -972 15 R 984 0 V -993 15 R 1006 0 V -1014 16 R 1026 0 V -1035 15 R 1048 0 V -1056 15 R 1068 0 V -1077 16 R 1090 0 V -1098 15 R 1110 0 V -1118 15 R 1131 0 V -1139 16 R 1151 0 V -1160 15 R 1173 0 V -1181 15 R 1194 0 V -1202 15 R 1214 0 V -1222 16 R 1235 0 V -1243 15 R 1256 0 V -1264 15 R 1277 0 V -1285 16 R 1297 0 V -1305 15 R 1318 0 V -1325 15 R 1338 0 V -1346 16 R 1359 0 V -1367 15 R 1380 0 V -1387 15 R 1400 0 V -1408 16 R 1420 0 V -1428 15 R 1441 0 V -1448 15 R 1461 0 V -1469 15 R 1482 0 V -1489 16 R 1502 0 V -1509 15 R 1522 0 V -1530 15 R 1543 0 V -1550 16 R 1563 0 V -1570 15 R 1583 0 V -1590 15 R 1603 0 V -1610 16 R 1623 0 V -1630 15 R 1643 0 V -1650 15 R 1663 0 V -1670 16 R 1684 0 V -1691 15 R 1704 0 V -1711 15 R 1724 0 V -1731 15 R 1744 0 V -1751 16 R 1764 0 V -1770 15 R 1783 0 V -1790 15 R 1803 0 V -1810 16 R 1824 0 V -1830 15 R 1843 0 V -1850 15 R 1863 0 V -1869 16 R 1882 0 V -1889 15 R 1902 0 V -1908 15 R 1922 0 V -1928 16 R 1941 0 V -1948 15 R 1961 0 V -1967 15 R 1980 0 V -1986 15 R 1999 0 V -2005 16 R 2019 0 V -2025 15 R 2038 0 V -2044 15 R 2057 0 V -2063 16 R 2076 0 V -2082 15 R 2096 0 V -2102 15 R 2115 0 V -2121 16 R 2134 0 V -2140 15 R 2153 0 V -2158 15 R 2172 0 V -2178 16 R 2191 0 V -2197 15 R 2210 0 V -2215 15 R 2228 0 V -2234 15 R 2248 0 V -2253 16 R 2266 0 V -2272 15 R 2285 0 V -2290 15 R 2303 0 V -2309 16 R 2323 0 V -2328 15 R 2341 0 V -2346 15 R 2359 0 V -2364 16 R 2377 0 V -2382 15 R 2396 0 V -2402 15 R 2415 0 V -2420 15 R 2433 0 V -2438 16 R 2451 0 V -2456 15 R 2469 0 V -2473 15 R 2487 0 V -2492 16 R 2505 0 V -2510 15 R 2523 0 V -2528 15 R 2541 0 V -2546 16 R 2559 0 V -2563 15 R 2576 0 V -2581 15 R 2595 0 V -2599 16 R 2612 0 V -2617 15 R 2630 0 V -2634 15 R 2647 0 V -2652 15 R 2665 0 V -2669 16 R 2682 0 V -2686 15 R 2699 0 V -2704 15 R 2717 0 V -2721 16 R 2734 0 V -2738 15 R 2751 0 V -2755 15 R 2768 0 V -2772 16 R 2785 0 V -2789 15 R 2802 0 V -2806 15 R 2819 0 V -2823 16 R 2836 0 V -2840 15 R 2853 0 V 998 2484 M 2870 0 V 995 2499 M 2886 0 V 991 2515 M 2903 0 V 987 2530 M 2919 0 V 984 2545 M 2935 0 V 980 2561 M 2952 0 V 977 2576 M 2968 0 V 973 2591 M 2984 0 V 970 2607 M 3000 0 V 967 2622 M 3016 0 V 964 2637 M 3031 0 V 961 2653 M 3047 0 V 957 2668 M 3064 0 V 954 2683 M 3079 0 V 951 2698 M 3095 0 V 949 2714 M 3109 0 V 946 2729 M 3125 0 V 943 2744 M 3140 0 V 940 2760 M 3156 0 V 938 2775 M 3170 0 V 935 2790 M 3186 0 V 933 2806 M 3200 0 V 930 2821 M 3215 0 V 928 2836 M 3229 0 V 925 2852 M 3245 0 V 923 2867 M 3259 0 V 921 2882 M 3273 0 V 919 2897 M 3287 0 V 917 2913 M 3301 0 V 915 2928 M 3315 0 V 913 2943 M 3329 0 V 911 2959 M 3343 0 V 910 2974 M 3356 0 V 908 2989 M 3370 0 V 907 3005 M 3383 0 V 905 3020 M 3396 0 V 904 3035 M 3409 0 V 902 3051 M 3423 0 V 901 3066 M 3435 0 V 900 3081 M 3448 0 V 899 3096 M 3460 0 V 898 3112 M 3473 0 V 897 3127 M 3485 0 V 897 3142 M 3496 0 V 896 3158 M 3509 0 V 896 3173 M 3520 0 V 895 3188 M 3532 0 V 895 3204 M 3543 0 V 895 3219 M 3554 0 V 895 3234 M 3565 0 V 895 3249 M 3576 0 V 895 3265 M 3586 0 V 895 3280 M 3597 0 V 896 3295 M 3606 0 V 896 3311 M 3617 0 V 897 3326 M 3626 0 V currentpoint stroke M 898 3341 M 3635 0 V 899 3357 M 3644 0 V 900 3372 M 3653 0 V 902 3387 M 3661 0 V 903 3403 M 3670 0 V 905 3418 M 3677 0 V 907 3433 M 3685 0 V 909 3448 M 3692 0 V 911 3464 M 3700 0 V 913 3479 M 3707 0 V 915 3494 M 3714 0 V 918 3510 M 3721 0 V 920 3525 M 3728 0 V 923 3540 M 3734 0 V 925 3556 M 3741 0 V 928 3571 M 3747 0 V 931 3586 M 3753 0 V 934 3602 M 3758 0 V 936 3617 M 3765 0 V 940 3632 M 3770 0 V 943 3647 M 3776 0 V 946 3663 M 3781 0 V 949 3678 M 3787 0 V 952 3693 M 3793 0 V 956 3709 M 3797 0 V 959 3724 M 3803 0 V 962 3739 M 3808 0 V 966 3755 M 3813 0 V 970 3770 M 3817 0 V 973 3785 M 3822 0 V 977 3801 M 3827 0 V 981 3816 M 3831 0 V 984 3831 M 3836 0 V 988 3846 M 3841 0 V 992 3862 M 3845 0 V 996 3877 M 3849 0 V -3845 15 R 3853 0 V -3849 16 R 3858 0 V -3854 15 R 3862 0 V -3858 15 R 3866 0 V -3862 16 R 3870 0 V -3866 15 R 3874 0 V -3869 15 R 3877 0 V -3873 16 R 3881 0 V -3877 15 R 3885 0 V -3880 15 R 3888 0 V -3884 15 R 3892 0 V -3888 16 R 3896 0 V -3891 15 R 3899 0 V -3895 15 R 3903 0 V -3898 16 R 3906 0 V -3902 15 R 3910 0 V -3905 15 R 3913 0 V -3909 16 R 3917 0 V -3912 15 R 3920 0 V -3915 15 R 3923 0 V -3919 16 R 3926 0 V -3921 15 R 3929 0 V -3924 15 R 3932 0 V -3927 15 R 3935 0 V -3931 16 R 3939 0 V -3934 15 R 3941 0 V -3936 15 R 3944 0 V -3939 16 R 3947 0 V -3942 15 R 3950 0 V -3945 15 R 3952 0 V -3947 16 R 3955 0 V -3950 15 R 3958 0 V -3953 15 R 3961 0 V -3956 15 R 3963 0 V -3958 16 R 3966 0 V -3961 15 R 3969 0 V -3964 15 R 3971 0 V -3966 16 R 3974 0 V -3969 15 R 3977 0 V -3971 15 R 3978 0 V -3973 16 R 3981 0 V -3976 15 R 3983 0 V -3978 15 R 3986 0 V -3981 16 R 3989 0 V -3983 15 R 3990 0 V -3985 15 R 3993 0 V -3988 15 R 3995 0 V -3989 16 R 3997 0 V -3992 15 R 4000 0 V -3995 15 R 4002 0 V -3996 16 R 4004 0 V -3999 15 R 4006 0 V -4000 15 R 4008 0 V -4003 16 R 4010 0 V -4004 15 R 4012 0 V -4007 15 R 4014 0 V -4008 16 R 4016 0 V -4011 15 R 4019 0 V -4013 15 R 4020 0 V -4015 15 R 4023 0 V -4017 16 R 4024 0 V -4019 15 R 4027 0 V -4021 15 R 4028 0 V -4023 16 R 4031 0 V -4025 15 R 4032 0 V 1.000 UL LT0 2053 280 M -10 15 V -11 16 V -11 15 V -10 15 V -11 16 V -10 15 V -11 15 V -10 15 V -11 16 V -10 15 V -10 15 V -11 16 V -10 15 V -10 15 V -10 16 V -10 15 V -10 15 V -10 16 V -10 15 V -10 15 V -10 15 V -10 16 V -10 15 V -9 15 V -10 16 V -10 15 V -9 15 V -10 16 V -9 15 V -10 15 V -9 16 V -10 15 V -9 15 V -9 15 V -9 16 V -9 15 V -9 15 V -10 16 V -8 15 V -9 15 V -9 16 V -9 15 V -9 15 V -9 15 V -8 16 V -9 15 V -9 15 V -8 16 V -9 15 V -8 15 V -9 16 V -8 15 V -8 15 V -8 16 V -9 15 V -8 15 V -8 15 V -8 16 V -8 15 V -8 15 V -8 16 V -8 15 V -7 15 V -8 16 V -8 15 V -7 15 V -8 16 V -8 15 V -7 15 V -8 15 V -7 16 V -7 15 V -8 15 V -7 16 V -7 15 V -7 15 V -7 16 V -7 15 V -7 15 V -7 16 V -7 15 V -7 15 V -7 15 V -7 16 V -6 15 V -7 15 V -7 16 V -6 15 V -7 15 V -6 16 V -7 15 V -6 15 V -6 16 V -7 15 V -6 15 V -6 15 V -6 16 V -6 15 V -6 15 V -6 16 V -6 15 V -6 15 V -6 16 V -6 15 V -5 15 V -6 16 V -6 15 V -5 15 V -6 15 V -5 16 V -6 15 V -5 15 V -6 16 V -5 15 V -5 15 V -5 16 V -5 15 V -6 15 V -5 15 V -5 16 V -5 15 V -4 15 V -5 16 V -5 15 V -5 15 V -5 16 V -4 15 V -5 15 V -4 16 V -5 15 V -4 15 V -5 15 V -4 16 V -4 15 V -5 15 V -4 16 V -4 15 V -4 15 V -4 16 V -4 15 V -4 15 V -4 16 V -4 15 V -4 15 V -3 15 V -4 16 V -4 15 V -3 15 V -4 16 V -3 15 V -4 15 V -3 16 V -3 15 V -3 15 V -3 16 V -4 15 V -3 15 V -3 15 V -2 16 V -3 15 V -3 15 V -3 16 V -2 15 V -3 15 V -2 16 V -3 15 V -2 15 V -3 16 V -2 15 V -2 15 V -2 15 V -2 16 V -2 15 V -2 15 V -2 16 V -1 15 V -2 15 V -1 16 V -2 15 V -1 15 V -2 16 V -1 15 V -1 15 V -1 15 V -1 16 V -1 15 V 0 15 V -1 16 V 0 15 V -1 15 V 0 16 V 0 15 V 0 15 V 0 15 V 0 16 V 0 15 V 1 15 V 0 16 V 1 15 V 1 15 V 1 16 V 1 15 V 2 15 V 1 16 V 2 15 V 2 15 V 2 15 V 2 16 V 2 15 V 2 15 V 3 16 V 2 15 V 3 15 V 2 16 V 3 15 V 3 15 V 3 16 V 2 15 V 4 15 V 3 15 V 3 16 V 3 15 V 3 15 V 4 16 V 3 15 V 3 15 V 4 16 V 4 15 V 3 15 V 4 16 V 4 15 V 3 15 V 4 15 V 4 16 V 4 15 V 4 15 V 4 16 V 4 15 V 4 15 V 4 16 V 4 15 V 5 15 V 4 16 V 4 15 V 5 15 V 4 15 V 4 16 V 5 15 V 4 15 V 5 16 V 4 15 V 5 15 V 4 16 V 5 15 V 5 15 V 4 16 V 5 15 V 5 15 V 5 15 V 4 16 V 5 15 V 5 15 V 5 16 V 5 15 V 5 15 V 5 16 V 5 15 V 5 15 V 5 15 V 5 16 V 5 15 V 5 15 V 5 16 V 5 15 V 6 15 V 5 16 V 5 15 V 5 15 V 5 16 V 6 15 V 5 15 V 5 15 V 6 16 V 5 15 V 5 15 V 6 16 V 5 15 V 6 15 V 5 16 V 6 15 V 5 15 V 6 16 V 5 15 V 6 15 V 5 15 V 6 16 V 5 15 V 6 15 V 5 16 V 6 15 V 4032 0 V 1.000 UL LT0 2054 280 M 10 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 11 15 V 12 15 V 11 16 V 11 15 V 12 15 V 11 16 V 11 15 V 12 15 V 11 16 V 12 15 V 11 15 V 12 15 V 12 16 V 11 15 V 12 15 V 12 16 V 12 15 V 11 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 13 16 V 12 15 V 12 15 V 12 16 V 13 15 V 12 15 V 12 15 V 13 16 V 12 15 V 13 15 V 12 16 V 13 15 V 12 15 V 13 16 V 12 15 V 13 15 V 12 16 V 13 15 V 13 15 V 12 15 V 13 16 V 13 15 V 13 15 V 12 16 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 12 16 V 13 15 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 14 16 V 13 15 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 14 16 V 13 15 V 13 15 V 13 16 V 13 15 V 14 15 V 13 16 V 13 15 V 13 15 V 13 15 V 14 16 V 13 15 V 13 15 V 13 16 V 14 15 V 13 15 V 13 16 V 13 15 V 14 15 V 13 16 V 13 15 V 13 15 V 14 15 V 13 16 V 13 15 V 13 15 V 14 16 V 13 15 V 13 15 V 13 16 V 14 15 V 13 15 V 13 15 V 13 16 V 13 15 V 14 15 V 13 16 V 13 15 V 13 15 V 13 16 V 13 15 V 14 15 V 13 16 V 13 15 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 15 V 13 16 V 12 15 V 13 15 V 13 16 V 13 15 V 12 15 V 13 16 V 13 15 V 12 15 V 13 16 V 13 15 V 12 15 V 13 15 V 12 16 V 13 15 V 12 15 V 13 16 V 12 15 V 13 15 V 12 16 V 12 15 V 12 15 V 13 16 V 12 15 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 11 15 V 12 15 V 12 16 V 11 15 V 12 15 V 11 15 V 12 16 V 11 15 V 11 15 V 12 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 10 16 V 11 15 V 10 15 V 11 16 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 10 16 V 9 15 V 10 15 V 9 15 V 10 16 V 9 15 V 9 15 V 10 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 8 16 V 9 15 V 9 15 V 9 15 V 8 16 V 9 15 V 9 15 V 8 16 V 9 15 V 8 15 V 9 16 V 8 15 V 8 15 V 9 16 V 8 15 V 8 15 V 9 15 V 8 16 V 8 15 V 8 15 V 9 16 V 8 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 16 V 8 15 V 8 15 V 7 16 V 8 15 V 8 15 V 8 15 V 8 16 V 7 15 V 8 15 V 8 16 V 8 15 V 7 15 V 8 16 V 8 15 V 8 15 V 7 15 V 8 16 V 8 15 V 7 15 V 8 16 V 8 15 V 7 15 V 8 16 V 7 15 V 8 15 V 8 16 V 7 15 V 8 15 V 7 15 V 8 16 V 8 15 V 7 15 V 8 16 V 7 15 V 8 15 V 7 16 V 8 15 V 7 15 V 8 16 V 8 15 V 7 15 V 8 15 V 7 16 V 8 15 V 7 15 V 8 16 V 7 15 V 1.000 UL LT0 3672 280 M -1 0 V 1 15 R 0 16 R 0 15 R 1 15 R -1 0 V 1 16 R -1 0 V 2 15 R -1 0 V 2 15 R -2 0 V 3 15 R -2 0 V 3 16 R -3 0 V 4 15 R -3 0 V 4 15 R -4 0 V 6 16 R -5 0 V 6 15 R -5 0 V 7 15 R -6 0 V 8 16 R -7 0 V 9 15 R -8 0 V 10 15 R -9 0 V 11 16 R -10 0 V 12 15 R -11 0 V 14 15 R -12 0 V 14 15 R -13 0 V 16 16 R -15 0 V 18 15 R -16 0 V 19 15 R -18 0 V 21 16 R -19 0 V 22 15 R -20 0 V 24 15 R -23 0 V 26 16 R -24 0 V 28 15 R -26 0 V 29 15 R -27 0 V 31 16 R -29 0 V 33 15 R -31 0 V 35 15 R -33 0 V 37 15 R -35 0 V 40 16 R -38 0 V 42 15 R -39 0 V 44 15 R -42 0 V 46 16 R -44 0 V 49 15 R -46 0 V 51 15 R -48 0 V 53 16 R -51 0 V 56 15 R -53 0 V 58 15 R -55 0 V 60 15 R -58 0 V 64 16 R -61 0 V 66 15 R -63 0 V 69 15 R -66 0 V 72 16 R -69 0 V 75 15 R -72 0 V 78 15 R -75 0 V 81 16 R -77 0 V 83 15 R -80 0 V 87 15 R -84 0 V 90 16 R -86 0 V 92 15 R -89 0 V 96 15 R -92 0 V 99 15 R -96 0 V 103 16 R -99 0 V 106 15 R -102 0 V 109 15 R -105 0 V 112 16 R -109 0 V 116 15 R -112 0 V 120 15 R -116 0 V 123 16 R -119 0 V 127 15 R -123 0 V 130 15 R -125 0 V 133 16 R -129 0 V 137 15 R -133 0 V 141 15 R -137 0 V 145 15 R -140 0 V 148 16 R -144 0 V 152 15 R -147 0 V 156 15 R -152 0 V 160 16 R -155 0 V 164 15 R -160 0 V 168 15 R -163 0 V 172 16 R -167 0 V 176 15 R -171 0 V 180 15 R -176 0 V 185 16 R -180 0 V 189 15 R -184 0 V 193 15 R -188 0 V 197 15 R -192 0 V 201 16 R -195 0 V 205 15 R -200 0 V 209 15 R -204 0 V 214 16 R -209 0 V 218 15 R -212 0 V 222 15 R -217 0 V 227 16 R -221 0 V 231 15 R -226 0 V 236 15 R -230 0 V 240 16 R -235 0 V 245 15 R -239 0 V 249 15 R -243 0 V 253 15 R -248 0 V 258 16 R -252 0 V 263 15 R -257 0 V 267 15 R -261 0 V 272 16 R -266 0 V 276 15 R -270 0 V 281 15 R -275 0 V 286 16 R -280 0 V 291 15 R -285 0 V 296 15 R -290 0 V 300 16 R -294 0 V 305 15 R -298 0 V 310 15 R -304 0 V 315 15 R -309 0 V 320 16 R -313 0 V 324 15 R -318 0 V 329 15 R -322 0 V 334 16 R -328 0 V 339 15 R -332 0 V 344 15 R -338 0 V 349 16 R -342 0 V 354 15 R -347 0 V 358 15 R -351 0 V 363 15 R -357 0 V 369 16 R -362 0 V 374 15 R -367 0 V 379 15 R -372 0 V 383 16 R -376 0 V 388 15 R -381 0 V 393 15 R -386 0 V 398 16 R -391 0 V 404 15 R -397 0 V 409 15 R -402 0 V 414 16 R -407 0 V 419 15 R -411 0 V 423 15 R -416 0 V 429 15 R -422 0 V 434 16 R -427 0 V 439 15 R -431 0 V 444 15 R -437 0 V 449 16 R -441 0 V 454 15 R -447 0 V 459 15 R -451 0 V 464 16 R -457 0 V 470 15 R -462 0 V 474 15 R -467 0 V 480 16 R -472 0 V 485 15 R -477 0 V 489 15 R -482 0 V 495 15 R -487 0 V 500 16 R -492 0 V 505 15 R -497 0 V 510 15 R -502 0 V 515 16 R -507 0 V 519 15 R -511 0 V 524 15 R -517 0 V 530 16 R -522 0 V 535 15 R -526 0 V 539 15 R -531 0 V 544 16 R -536 0 V 550 15 R -542 0 V 555 15 R -547 0 V 560 15 R -552 0 V 565 16 R -557 0 V 570 15 R -561 0 V 574 15 R -566 0 V 579 16 R -571 0 V 585 15 R -576 0 V 589 15 R -581 0 V 594 16 R -585 0 V 598 15 R -590 0 V 603 15 R -594 0 V 608 16 R -600 0 V 613 15 R -604 0 V 617 15 R -608 0 V 621 15 R -613 0 V 627 16 R -618 0 V 631 15 R -622 0 V 635 15 R -626 0 V 640 16 R -631 0 V 644 15 R -635 0 V 648 15 R -639 0 V 652 16 R -643 0 V 657 15 R -648 0 V 661 15 R -652 0 V 665 16 R -656 0 V 670 15 R -661 0 V 674 15 R -664 0 V 677 15 R -668 0 V 681 16 R -672 0 V 686 15 R -676 0 V 689 15 R -680 0 V 693 16 R -683 0 V 696 15 R -686 0 V 699 15 R -690 0 V 703 16 R -693 0 V 706 15 R -696 0 V 709 15 R -699 0 V 712 15 R -702 0 V 715 16 R -705 0 V 718 15 R -708 0 V 721 15 R -711 0 V 724 16 R -713 0 V 725 15 R -715 0 V 728 15 R -717 0 V 730 16 R -719 0 V currentpoint stroke M 731 15 R -720 0 V 732 15 R -721 0 V 734 16 R -723 0 V 735 15 R -724 0 V 736 15 R -724 0 V 736 15 R -725 0 V 737 16 R -725 0 V 737 15 R -726 0 V 738 15 R -726 0 V 738 16 R -726 0 V 738 15 R -727 0 V 739 15 R -727 0 V 739 16 R -727 0 V 739 15 R -727 0 V 739 15 R -727 0 V 738 16 R -726 0 V 738 15 R -726 0 V 738 15 R -726 0 V 738 15 R -726 0 V 738 16 R -725 0 V 737 15 R -725 0 V 736 15 R -724 0 V 736 16 R -724 0 V 736 15 R -723 0 V 735 15 R -723 0 V 735 16 R -722 0 V 733 15 R -721 0 V 733 15 R -721 0 V 733 16 R -720 0 V 732 15 R -719 0 V 731 15 R -719 0 V 730 15 R -717 0 V 729 16 R -717 0 V 729 15 R -716 0 V 728 15 R -715 0 V 727 16 R -715 0 V 727 15 R -714 0 V 725 15 R -712 0 V 724 16 R -711 0 V 723 15 R -711 0 V 723 15 R -710 0 V 722 16 R -709 0 V 721 15 R -708 0 V 719 15 R -706 0 V 718 15 R -705 0 V 717 16 R -704 0 V 716 15 R -703 0 V 715 15 R -703 0 V 715 16 R -702 0 V 714 15 R -701 0 V 713 15 R -700 0 V 712 16 R -699 0 V 711 15 R -697 0 V 709 15 R -696 0 V 708 16 R -695 0 V 707 15 R -694 0 V 706 15 R -693 0 V 705 15 R -692 0 V 704 16 R -691 0 V 703 15 R -690 0 V 702 15 R -688 0 V 700 16 R -687 0 V 699 15 R -686 0 V 698 15 R -685 0 V 697 16 R -684 0 V 696 15 R -682 0 V 694 15 R -681 0 V 693 15 R -680 0 V 692 16 R -678 0 V 690 15 R -677 0 V 690 15 R -677 0 V 689 16 R -675 0 V 687 15 R -674 0 V 686 15 R -673 0 V 685 16 R -671 0 V 683 15 R -670 0 V 683 15 R -669 0 V 681 16 R -668 0 V 680 15 R -667 0 V 679 15 R -665 0 V 678 15 R -665 0 V 677 16 R -663 0 V 675 15 R -662 0 V 674 15 R -660 0 V 673 16 R -660 0 V 672 15 R -658 0 V 670 15 R -657 0 V 670 16 R -656 0 V 668 15 R -654 0 V 666 15 R -653 0 V 666 16 R -652 0 V 664 15 R -651 0 V 664 15 R -650 0 V 662 15 R -648 0 V 660 16 R -647 0 V 660 15 R -646 0 V 658 15 R -644 0 V 657 16 R -644 0 V 656 15 R -642 0 V 1.000 UL LT0 3672 280 M 0 15 V 0 16 V 0 15 V 1 15 V 0 16 V 1 15 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 2 16 V 1 15 V 2 15 V 2 16 V 2 15 V 2 15 V 2 16 V 2 15 V 3 15 V 2 15 V 3 16 V 3 15 V 3 15 V 3 16 V 3 15 V 4 15 V 3 16 V 4 15 V 3 15 V 4 16 V 4 15 V 4 15 V 4 15 V 5 16 V 4 15 V 5 15 V 4 16 V 5 15 V 5 15 V 5 16 V 5 15 V 5 15 V 5 15 V 6 16 V 5 15 V 6 15 V 6 16 V 6 15 V 6 15 V 6 16 V 6 15 V 7 15 V 6 16 V 6 15 V 7 15 V 7 15 V 7 16 V 7 15 V 7 15 V 7 16 V 7 15 V 8 15 V 7 16 V 8 15 V 7 15 V 8 16 V 8 15 V 8 15 V 8 15 V 8 16 V 8 15 V 9 15 V 8 16 V 9 15 V 8 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 15 V 9 16 V 10 15 V 9 15 V 10 16 V 9 15 V 10 15 V 10 16 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 10 15 V 10 16 V 11 15 V 10 15 V 11 16 V 10 15 V 11 15 V 11 16 V 11 15 V 11 15 V 10 16 V 11 15 V 12 15 V 11 15 V 11 16 V 11 15 V 11 15 V 12 16 V 11 15 V 12 15 V 11 16 V 12 15 V 11 15 V 12 15 V 12 16 V 12 15 V 12 15 V 11 16 V 12 15 V 12 15 V 12 16 V 13 15 V 12 15 V 12 16 V 12 15 V 12 15 V 13 15 V 12 16 V 12 15 V 13 15 V 12 16 V 13 15 V 12 15 V 13 16 V 13 15 V 12 15 V 13 16 V 13 15 V 12 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 16 V 12 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 16 V 14 15 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 16 V 14 15 V 13 15 V 13 16 V 13 15 V 13 15 V 14 16 V 13 15 V 13 15 V 13 15 V 14 16 V 13 15 V 13 15 V 14 16 V 13 15 V 13 15 V 13 16 V 14 15 V 13 15 V 13 16 V 14 15 V 13 15 V 13 15 V 13 16 V 14 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 16 V 12 15 V 13 15 V 13 16 V 12 15 V 12 15 V 13 16 V 12 15 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 11 16 V 12 15 V 12 15 V 12 15 V 12 16 V 12 15 V 11 15 V 12 16 V 12 15 V 12 15 V 12 16 V 11 15 V 12 15 V 12 16 V 12 15 V 12 15 V 11 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 11 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 11 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 15 V 12 16 V 12 15 V 13 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 13 15 V 12 16 V 12 15 V 12 15 V 13 15 V 12 16 V 12 15 V 12 15 V 13 16 V 12 15 V 12 15 V 13 16 V 12 15 V 12 15 V 13 16 V 12 15 V 13 15 V 12 15 V 12 16 V 13 15 V 12 15 V 13 16 V 12 15 V -642 0 V 1.000 UL LT0 3671 280 M 1 15 V 0 16 V 0 15 V 0 15 V 0 16 V 1 15 V 0 15 V 1 15 V 0 16 V 1 15 V 0 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 2 15 V 1 15 V 1 16 V 2 15 V 1 15 V 2 16 V 2 15 V 1 15 V 2 16 V 2 15 V 2 15 V 2 16 V 2 15 V 2 15 V 2 15 V 2 16 V 3 15 V 2 15 V 2 16 V 3 15 V 3 15 V 2 16 V 3 15 V 3 15 V 2 15 V 3 16 V 3 15 V 3 15 V 3 16 V 3 15 V 3 15 V 4 16 V 3 15 V 3 15 V 4 16 V 3 15 V 4 15 V 3 15 V 4 16 V 4 15 V 4 15 V 3 16 V 4 15 V 4 15 V 4 16 V 4 15 V 5 15 V 4 16 V 4 15 V 4 15 V 5 15 V 4 16 V 5 15 V 4 15 V 5 16 V 4 15 V 5 15 V 5 16 V 5 15 V 4 15 V 5 16 V 5 15 V 5 15 V 5 15 V 6 16 V 5 15 V 5 15 V 5 16 V 6 15 V 5 15 V 6 16 V 5 15 V 6 15 V 5 16 V 6 15 V 6 15 V 5 15 V 6 16 V 6 15 V 6 15 V 6 16 V 6 15 V 6 15 V 6 16 V 6 15 V 6 15 V 6 16 V 7 15 V 6 15 V 6 15 V 7 16 V 6 15 V 7 15 V 6 16 V 7 15 V 6 15 V 7 16 V 7 15 V 7 15 V 6 15 V 7 16 V 7 15 V 7 15 V 7 16 V 7 15 V 7 15 V 7 16 V 7 15 V 7 15 V 7 16 V 8 15 V 7 15 V 7 15 V 7 16 V 8 15 V 7 15 V 8 16 V 7 15 V 8 15 V 7 16 V 8 15 V 7 15 V 8 16 V 8 15 V 7 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 16 V 8 15 V 7 15 V 8 16 V 9 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 15 V 8 16 V 9 15 V 8 15 V 8 16 V 9 15 V 8 15 V 9 16 V 8 15 V 9 15 V 8 16 V 9 15 V 9 15 V 8 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 10 15 V 9 15 V 9 16 V 10 15 V 9 15 V 10 16 V 10 15 V 9 15 V 10 16 V 10 15 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 11 16 V 10 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 12 15 V 11 15 V 12 16 V 11 15 V 12 15 V 12 16 V 11 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 15 V 13 16 V 12 15 V 12 15 V 12 16 V 13 15 V 12 15 V 13 16 V 12 15 V 12 15 V 13 16 V 13 15 V 12 15 V 13 15 V 12 16 V 13 15 V 13 15 V 12 16 V 13 15 V 13 15 V 13 16 V 12 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 15 V 13 16 V 13 15 V 12 15 V 13 16 V 13 15 V 13 15 V 13 16 V 14 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 15 V 13 16 V 13 15 V 14 15 V 13 16 V 13 15 V 13 15 V 13 16 V 14 15 V 13 15 V 13 15 V 14 16 V 13 15 V 13 15 V 14 16 V 13 15 V 13 15 V 14 16 V 13 15 V 14 15 V 13 16 V 13 15 V 14 15 V 13 15 V 14 16 V 13 15 V 14 15 V 13 16 V 14 15 V 13 15 V 14 16 V 14 15 V 13 15 V 14 16 V 13 15 V 14 15 V 14 15 V 13 16 V 14 15 V 14 15 V 13 16 V 14 15 V 1.000 UL LT0 4359 280 M 0 15 R 0 16 R 1 0 V 0 15 R 0 15 R 0 16 R 1 15 R 0 15 R 1 15 R 0 16 R 1 15 R 1 15 R 0 16 R 1 0 V 0 15 R 1 0 V 0 15 R 1 16 R 1 0 V 0 15 R 1 0 V 0 15 R 1 0 V 1 16 R 1 15 R 1 0 V 0 15 R 1 0 V 1 15 R 1 0 V 0 16 R 1 0 V 1 15 R 1 0 V 0 15 R 2 0 V 0 16 R 1 0 V 1 15 R 1 0 V 1 15 R 1 0 V 1 16 R 1 0 V 1 15 R 2 0 V 0 15 R 2 0 V 0 16 R 2 0 V 0 15 R 2 0 V 0 15 R 3 0 V -1 15 R 4 0 V -1 16 R 3 0 V -1 15 R 4 0 V -2 15 R 5 0 V -2 16 R 5 0 V -2 15 R 4 0 V -2 15 R 5 0 V -2 16 R 6 0 V -3 15 R 6 0 V -4 15 R 7 0 V -4 15 R 7 0 V -4 16 R 8 0 V -5 15 R 8 0 V -5 15 R 9 0 V -5 16 R 9 0 V -6 15 R 9 0 V -6 15 R 10 0 V -7 16 R 11 0 V -7 15 R 11 0 V -8 15 R 12 0 V -8 16 R 13 0 V -10 15 R 14 0 V -10 15 R 14 0 V -10 15 R 15 0 V -12 16 R 16 0 V -12 15 R 17 0 V -13 15 R 17 0 V -13 16 R 18 0 V -14 15 R 19 0 V -15 15 R 20 0 V -16 16 R 21 0 V -17 15 R 22 0 V -17 15 R 22 0 V -18 16 R 24 0 V -20 15 R 25 0 V -20 15 R 26 0 V -22 15 R 27 0 V -22 16 R 28 0 V -24 15 R 29 0 V -24 15 R 30 0 V -26 16 R 32 0 V -27 15 R 33 0 V -28 15 R 34 0 V -29 16 R 35 0 V -30 15 R 36 0 V -31 15 R 38 0 V -33 16 R 39 0 V -34 15 R 41 0 V -36 15 R 42 0 V -37 15 R 44 0 V -39 16 R 46 0 V -40 15 R 46 0 V -41 15 R 48 0 V -42 16 R 49 0 V -44 15 R 51 0 V -45 15 R 53 0 V -48 16 R 55 0 V -49 15 R 56 0 V -50 15 R 58 0 V -53 16 R 60 0 V -54 15 R 62 0 V -56 15 R 63 0 V -57 15 R 65 0 V -59 16 R 67 0 V -61 15 R 69 0 V -63 15 R 71 0 V -64 16 R 72 0 V -66 15 R 74 0 V -68 15 R 77 0 V -71 16 R 79 0 V -72 15 R 80 0 V -74 15 R 83 0 V -76 16 R 84 0 V -78 15 R 87 0 V -80 15 R 89 0 V -82 15 R 91 0 V -84 16 R 93 0 V -87 15 R 96 0 V -89 15 R 98 0 V -91 16 R 100 0 V -93 15 R 102 0 V -95 15 R 105 0 V -98 16 R 107 0 V -100 15 R 110 0 V -102 15 R 111 0 V -104 15 R 114 0 V -107 16 R 117 0 V -109 15 R 118 0 V -111 15 R 121 0 V -113 16 R 123 0 V -116 15 R 126 0 V -118 15 R 129 0 V -121 16 R 131 0 V -124 15 R 134 0 V -126 15 R 136 0 V -128 16 R 139 0 V -131 15 R 141 0 V -133 15 R 144 0 V -136 15 R 147 0 V -139 16 R 149 0 V -141 15 R 152 0 V -144 15 R 155 0 V -146 16 R 157 0 V -149 15 R 160 0 V -152 15 R 163 0 V -154 16 R 165 0 V -157 15 R 169 0 V -160 15 R 171 0 V -163 16 R 174 0 V -165 15 R 177 0 V -168 15 R 179 0 V -170 15 R 182 0 V -173 16 R 185 0 V -177 15 R 188 0 V -179 15 R 191 0 V -182 16 R 194 0 V -184 15 R 196 0 V -187 15 R 199 0 V -190 16 R 202 0 V -193 15 R 205 0 V -195 15 R 207 0 V -198 16 R 211 0 V -202 15 R 214 0 V -204 15 R 216 0 V -207 15 R 220 0 V -210 16 R 222 0 V -212 15 R 225 0 V -216 15 R 228 0 V -218 16 R 231 0 V -221 15 R 233 0 V -223 15 R 236 0 V -226 16 R 239 0 V -229 15 R 242 0 V -232 15 R 245 0 V -235 16 R 248 0 V -238 15 R 251 0 V -240 15 R 253 0 V -243 15 R 256 0 V -246 16 R 259 0 V -248 15 R 261 0 V -251 15 R 264 0 V -253 16 R 266 0 V -255 15 R 269 0 V -259 15 R 272 0 V -261 16 R 274 0 V -263 15 R 277 0 V -266 15 R 279 0 V -268 16 R 282 0 V -271 15 R 284 0 V -273 15 R 287 0 V -276 15 R 289 0 V -278 16 R 292 0 V -280 15 R 293 0 V -282 15 R 296 0 V -285 16 R 299 0 V -287 15 R 300 0 V -288 15 R 302 0 V -291 16 R 305 0 V -293 15 R 306 0 V -294 15 R 308 0 V -296 15 R 310 0 V -298 16 R 311 0 V -299 15 R 313 0 V -300 15 R 314 0 V -302 16 R 315 0 V -302 15 R 316 0 V -304 15 R 318 0 V -305 16 R 318 0 V -305 15 R 319 0 V -306 15 R 319 0 V -306 16 R 320 0 V -306 15 R 319 0 V -306 15 R 320 0 V currentpoint stroke M -306 15 R 319 0 V -306 16 R 319 0 V -305 15 R 319 0 V -305 15 R 318 0 V -304 16 R 318 0 V -304 15 R 317 0 V -303 15 R 317 0 V -303 16 R 316 0 V -302 15 R 316 0 V -302 15 R 315 0 V -301 16 R 315 0 V -300 15 R 313 0 V -299 15 R 313 0 V -299 15 R 313 0 V -298 16 R 311 0 V -297 15 R 311 0 V -296 15 R 310 0 V -296 16 R 309 0 V -294 15 R 308 0 V -293 15 R 307 0 V -292 16 R 306 0 V -292 15 R 305 0 V -290 15 R 304 0 V -289 16 R 303 0 V -288 15 R 302 0 V -287 15 R 301 0 V -286 15 R 300 0 V -285 16 R 299 0 V -284 15 R 298 0 V -283 15 R 297 0 V -282 16 R 296 0 V -281 15 R 295 0 V -279 15 R 293 0 V -278 16 R 292 0 V -277 15 R 291 0 V -276 15 R 291 0 V -275 16 R 289 0 V -274 15 R 288 0 V -272 15 R 286 0 V -271 15 R 286 0 V -271 16 R 285 0 V -269 15 R 283 0 V -267 15 R 282 0 V -267 16 R 281 0 V -265 15 R 280 0 V -265 15 R 279 0 V -263 16 R 278 0 V -262 15 R 276 0 V -261 15 R 276 0 V -260 16 R 274 0 V -258 15 R 273 0 V -257 15 R 272 0 V -256 15 R 270 0 V -255 16 R 270 0 V -254 15 R 269 0 V -253 15 R 267 0 V -251 16 R 266 0 V -250 15 R 265 0 V -249 15 R 264 0 V -248 16 R 263 0 V -247 15 R 262 0 V -246 15 R 261 0 V -245 15 R 260 0 V -244 16 R 258 0 V -242 15 R 257 0 V -240 15 R 255 0 V -239 16 R 255 0 V -239 15 R 253 0 V -237 15 R 237 0 V -220 16 R 220 0 V -204 15 R 204 0 V -188 15 R 188 0 V -172 16 R 172 0 V -155 15 R 155 0 V -139 15 R 139 0 V -123 15 R 123 0 V -106 16 R 106 0 V -90 15 R 90 0 V -73 15 R 73 0 V -57 16 R 57 0 V -40 15 R 40 0 V -24 15 R 24 0 V -7 16 R 7 0 V 1.000 UL LT0 4359 280 M 0 15 V 0 16 V 1 15 V 0 15 V 0 16 V 1 15 V 0 15 V 1 15 V 0 16 V 1 15 V 1 15 V 0 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 2 16 V 1 15 V 1 15 V 2 15 V 1 16 V 2 15 V 1 15 V 2 16 V 2 15 V 2 15 V 2 16 V 2 15 V 2 15 V 2 16 V 2 15 V 2 15 V 2 15 V 3 16 V 2 15 V 2 15 V 3 16 V 3 15 V 2 15 V 3 16 V 3 15 V 2 15 V 3 15 V 3 16 V 3 15 V 3 15 V 4 16 V 3 15 V 3 15 V 3 16 V 4 15 V 3 15 V 4 16 V 3 15 V 4 15 V 4 15 V 3 16 V 4 15 V 4 15 V 4 16 V 4 15 V 4 15 V 4 16 V 4 15 V 5 15 V 4 16 V 4 15 V 5 15 V 4 15 V 5 16 V 4 15 V 5 15 V 4 16 V 5 15 V 5 15 V 5 16 V 5 15 V 5 15 V 5 16 V 5 15 V 5 15 V 5 15 V 5 16 V 6 15 V 5 15 V 6 16 V 5 15 V 6 15 V 5 16 V 6 15 V 6 15 V 5 16 V 6 15 V 6 15 V 6 15 V 6 16 V 6 15 V 6 15 V 7 16 V 6 15 V 6 15 V 6 16 V 7 15 V 6 15 V 7 16 V 6 15 V 7 15 V 7 15 V 7 16 V 6 15 V 7 15 V 7 16 V 7 15 V 7 15 V 7 16 V 7 15 V 8 15 V 7 15 V 7 16 V 8 15 V 7 15 V 8 16 V 7 15 V 8 15 V 8 16 V 7 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 15 V 8 16 V 8 15 V 8 15 V 9 16 V 8 15 V 8 15 V 9 16 V 8 15 V 9 15 V 8 16 V 9 15 V 9 15 V 9 15 V 9 16 V 8 15 V 9 15 V 9 16 V 10 15 V 9 15 V 9 16 V 9 15 V 10 15 V 9 16 V 9 15 V 10 15 V 9 15 V 10 16 V 10 15 V 9 15 V 10 16 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 10 16 V 10 15 V 11 15 V 10 15 V 10 16 V 11 15 V 10 15 V 11 16 V 11 15 V 10 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 12 15 V 11 15 V 11 16 V 12 15 V 12 15 V 11 16 V 12 15 V 12 15 V 12 15 V 12 16 V 12 15 V 13 15 V 12 16 V 13 15 V 12 15 V 13 16 V 13 15 V 13 15 V 13 16 V 14 15 V 13 15 V 14 15 V 13 16 V 14 15 V 14 15 V 14 16 V 14 15 V 14 15 V 14 16 V 14 15 V 14 15 V 14 16 V 15 15 V 14 15 V 14 15 V 15 16 V 14 15 V 15 15 V 14 16 V 15 15 V 15 15 V 15 16 V 14 15 V 15 15 V 15 16 V 15 15 V 15 15 V 15 15 V 15 16 V 15 15 V 15 15 V 15 16 V 15 15 V 16 15 V 15 16 V 15 15 V 15 15 V 16 16 V 15 15 V 16 15 V 15 15 V 15 16 V 16 15 V 16 15 V 15 16 V 16 15 V 15 15 V 16 16 V 16 15 V 15 15 V 16 16 V 16 15 V 16 15 V 16 15 V 15 16 V 16 15 V 16 15 V 16 16 V 16 15 V 16 15 V 16 16 V 16 15 V 16 15 V 16 15 V 16 16 V 16 15 V 17 15 V 16 16 V 16 15 V 16 15 V 17 16 V 16 15 V 16 15 V 16 16 V 17 15 V 16 15 V 16 15 V 17 16 V 16 15 V 17 15 V 16 16 V 17 15 V 16 15 V 17 16 V 7 6 V 1.000 UL LT0 4359 280 M 0 15 V 1 16 V 0 15 V 0 15 V 0 16 V 1 15 V 0 15 V 1 15 V 0 16 V 1 15 V 1 15 V 1 16 V 1 15 V 0 15 V 2 16 V 1 15 V 1 15 V 1 16 V 2 15 V 1 15 V 2 15 V 1 16 V 2 15 V 2 15 V 1 16 V 2 15 V 2 15 V 2 16 V 3 15 V 2 15 V 2 16 V 2 15 V 3 15 V 3 15 V 2 16 V 3 15 V 3 15 V 3 16 V 2 15 V 3 15 V 4 16 V 3 15 V 3 15 V 3 15 V 4 16 V 3 15 V 4 15 V 4 16 V 3 15 V 4 15 V 4 16 V 4 15 V 4 15 V 5 16 V 4 15 V 4 15 V 5 15 V 4 16 V 5 15 V 4 15 V 5 16 V 5 15 V 5 15 V 5 16 V 5 15 V 5 15 V 6 16 V 5 15 V 6 15 V 5 15 V 6 16 V 5 15 V 6 15 V 6 16 V 6 15 V 6 15 V 6 16 V 6 15 V 7 15 V 6 16 V 7 15 V 6 15 V 7 15 V 7 16 V 6 15 V 7 15 V 7 16 V 7 15 V 8 15 V 7 16 V 7 15 V 8 15 V 7 16 V 8 15 V 7 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 16 V 8 15 V 9 15 V 8 16 V 8 15 V 9 15 V 8 16 V 9 15 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 10 15 V 9 16 V 10 15 V 9 15 V 10 15 V 10 16 V 9 15 V 10 15 V 10 16 V 10 15 V 11 15 V 10 16 V 10 15 V 10 15 V 11 16 V 10 15 V 11 15 V 11 15 V 10 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 12 15 V 11 15 V 11 16 V 12 15 V 11 15 V 12 15 V 12 16 V 11 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 13 16 V 12 15 V 12 15 V 13 15 V 12 16 V 13 15 V 12 15 V 13 16 V 12 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 16 V 14 15 V 13 15 V 13 16 V 14 15 V 13 15 V 14 16 V 13 15 V 14 15 V 13 15 V 14 16 V 13 15 V 14 15 V 14 16 V 13 15 V 14 15 V 14 16 V 13 15 V 14 15 V 14 15 V 13 16 V 14 15 V 14 15 V 13 16 V 14 15 V 14 15 V 13 16 V 14 15 V 13 15 V 14 16 V 13 15 V 14 15 V 13 15 V 13 16 V 14 15 V 13 15 V 14 16 V 13 15 V 14 15 V 13 16 V 14 15 V 13 15 V 14 16 V 13 15 V 14 15 V 14 15 V 13 16 V 14 15 V 14 15 V 13 16 V 14 15 V 14 15 V 14 16 V 13 15 V 14 15 V 14 16 V 14 15 V 14 15 V 14 15 V 14 16 V 14 15 V 14 15 V 14 16 V 14 15 V 14 15 V 14 16 V 14 15 V 15 15 V 14 16 V 14 15 V 14 15 V 15 15 V 14 16 V 14 15 V 15 15 V 14 16 V 15 15 V 14 15 V 15 16 V 14 15 V 15 15 V 14 16 V 15 15 V 15 15 V 14 15 V 15 16 V 15 15 V 14 15 V 15 16 V 15 15 V 15 15 V 15 16 V 15 15 V 15 15 V 15 15 V 14 16 V 15 15 V 15 15 V 16 16 V 14 14 V 1.000 UL LT0 867 280 M 0 15 R 0 16 R 0 15 R 0 15 R 0 16 R 0 15 R -1 0 V 0 15 R 0 15 R 0 16 R 0 15 R -1 0 V 0 15 R 0 16 R -1 0 V 0 15 R 0 15 R -1 0 V 0 16 R 0 15 R -1 0 V 0 15 R -1 16 R 0 15 R -1 0 V 0 15 R -1 15 R -1 16 R 0 15 R -1 0 V 0 15 R -1 0 V 0 16 R -1 15 R -1 15 R -1 16 R -1 15 R -1 15 R -1 16 R -1 0 V 0 15 R -1 0 V 0 15 R -1 0 V 0 15 R -1 0 V 0 16 R -1 0 V -1 15 R -1 15 R -1 0 V 0 16 R -1 0 V 0 15 R -1 0 V 0 15 R -2 0 V 0 16 R -1 0 V 0 15 R -2 0 V 1 15 R -2 0 V 0 15 R -1 0 V 0 16 R -2 0 V 1 15 R -2 0 V 0 15 R -2 0 V 1 16 R -3 0 V 2 15 R -3 0 V 1 15 R -3 0 V 2 16 R -3 0 V 1 15 R -3 0 V 2 15 R -4 0 V 3 16 R -5 0 V 3 15 R -4 0 V 3 15 R -5 0 V 3 15 R -5 0 V 4 16 R -6 0 V 4 15 R -6 0 V 5 15 R -6 0 V 4 16 R -6 0 V 5 15 R -7 0 V 5 15 R -7 0 V 6 16 R -8 0 V 6 15 R -8 0 V 7 15 R -9 0 V 7 16 R -9 0 V 8 15 R -10 0 V 8 15 R -10 0 V 9 15 R -11 0 V 9 16 R -11 0 V 10 15 R -12 0 V 11 15 R -13 0 V 11 16 R -13 0 V 12 15 R -14 0 V 12 15 R -14 0 V 13 16 R -15 0 V 13 15 R -16 0 V 15 15 R -17 0 V 16 16 R -18 0 V 16 15 R -18 0 V 17 15 R -19 0 V 17 15 R -20 0 V 19 16 R -21 0 V 20 15 R -22 0 V 20 15 R -22 0 V 21 16 R -24 0 V 23 15 R -25 0 V 23 15 R -25 0 V 24 16 R -26 0 V 25 15 R -28 0 V 27 15 R -29 0 V 27 16 R -29 0 V 28 15 R -31 0 V 30 15 R -32 0 V 31 15 R -33 0 V 32 16 R -35 0 V 34 15 R -36 0 V 34 15 R -36 0 V 35 16 R -38 0 V 37 15 R -39 0 V 38 15 R -41 0 V 40 16 R -42 0 V 41 15 R -43 0 V 42 15 R -45 0 V 44 16 R -46 0 V 45 15 R -48 0 V 47 15 R -49 0 V 49 15 R -51 0 V 50 16 R -53 0 V 52 15 R -54 0 V 53 15 R -56 0 V 55 16 R -57 0 V 56 15 R -59 0 V 59 15 R -61 0 V 60 16 R -63 0 V 62 15 R -64 0 V 64 15 R -67 0 V 66 15 R -68 0 V 67 16 R -69 0 V 69 15 R -72 0 V 71 15 R -73 0 V 73 16 R -76 0 V 75 15 R -77 0 V 77 15 R -80 0 V 79 16 R -81 0 V 81 15 R -84 0 V 84 15 R -86 0 V 85 16 R -88 0 V 88 15 R -90 0 V 90 15 R -93 0 V 92 15 R -94 0 V 94 16 R -96 0 V 96 15 R -99 0 V 99 15 R -101 0 V 101 16 R -104 0 V 104 15 R -106 0 V 105 15 R -108 0 V 108 16 R -110 0 V 110 15 R -112 0 V 112 15 R -115 0 V 116 16 R -118 0 V 118 15 R -121 0 V 121 15 R -123 0 V 123 15 R -125 0 V 125 16 R -128 0 V 128 15 R -130 0 V 131 15 R -133 0 V 133 16 R -136 0 V 136 15 R -138 0 V 139 15 R -141 0 V 141 16 R -144 0 V 144 15 R -146 0 V 147 15 R -149 0 V 149 16 R -151 0 V 152 15 R -155 0 V 155 15 R -157 0 V 158 15 R -160 0 V 161 16 R -163 0 V 163 15 R -165 0 V 166 15 R -168 0 V 169 16 R -171 0 V 171 15 R -174 0 V 175 15 R -177 0 V 178 16 R -180 0 V 181 15 R -183 0 V 184 15 R -186 0 V 187 16 R -189 0 V 190 15 R -192 0 V 193 15 R -194 0 V 194 15 R -196 0 V 197 16 R -199 0 V 201 15 R -203 0 V 204 15 R -206 0 V 207 16 R -208 0 V 209 15 R -211 0 V 212 15 R -214 0 V 215 16 R -217 0 V 218 15 R -219 0 V 220 15 R -222 0 V 224 16 R -225 0 V 226 15 R -228 0 V 229 15 R -230 0 V 231 15 R -233 0 V 234 16 R -235 0 V 237 15 R -238 0 V 239 15 R -241 0 V 242 16 R -243 0 V 244 15 R -245 0 V 247 15 R -248 0 V 249 16 R -250 0 V 251 15 R -252 0 V 253 15 R -254 0 V 255 15 R -256 0 V 257 16 R -257 0 V 259 15 R -260 0 V 261 15 R -261 0 V 262 16 R -263 0 V 263 15 R -263 0 V 264 15 R -264 0 V 265 16 R -265 0 V 266 15 R -265 0 V 265 15 R -265 0 V 265 16 R -264 0 V 265 15 R -265 0 V 265 15 R -264 0 V 264 15 R -263 0 V 264 16 R -263 0 V 263 15 R -262 0 V 262 15 R -261 0 V 262 16 R -261 0 V currentpoint stroke M 261 15 R -259 0 V 259 15 R -258 0 V 258 16 R -257 0 V 257 15 R -255 0 V 256 15 R -255 0 V 255 16 R -253 0 V 253 15 R -252 0 V 252 15 R -250 0 V 251 15 R -250 0 V 250 16 R -248 0 V 248 15 R -247 0 V 248 15 R -246 0 V 246 16 R -244 0 V 244 15 R -243 0 V 244 15 R -242 0 V 242 16 R -240 0 V 240 15 R -238 0 V 239 15 R -238 0 V 238 16 R -236 0 V 236 15 R -234 0 V 235 15 R -233 0 V 233 15 R -231 0 V 232 16 R -230 0 V 230 15 R -228 0 V 229 15 R -227 0 V 227 16 R -226 0 V 227 15 R -225 0 V 225 15 R -223 0 V 224 16 R -222 0 V 222 15 R -220 0 V 221 15 R -218 0 V 218 16 R -216 0 V 217 15 R -215 0 V 216 15 R -214 0 V 214 15 R -212 0 V 213 16 R -211 0 V 212 15 R -210 0 V 210 15 R -208 0 V 209 16 R -207 0 V 208 15 R -206 0 V 207 15 R -205 0 V 205 16 R -202 0 V 203 15 R -201 0 V 202 15 R -200 0 V 201 16 R -199 0 V 200 15 R -198 0 V 198 15 R -195 0 V 196 15 R -194 0 V 195 16 R -193 0 V 194 15 R -192 0 V 193 15 R -191 0 V 192 16 R -189 0 V 190 15 R -188 0 V 189 15 R -187 0 V 188 16 R -186 0 V 187 15 R -184 0 V 185 15 R -183 0 V 184 15 R -182 0 V 183 16 R -180 0 V 181 15 R -179 0 V 180 15 R -178 0 V 179 16 R -177 0 V 178 15 R -175 0 V 176 15 R -174 0 V 175 16 R -173 0 V 174 15 R -171 0 V 172 15 R -170 0 V 171 16 R -169 0 V 170 15 R -167 0 V 169 15 R -167 0 V 168 15 R -166 0 V 167 16 R -164 0 V 165 15 R -163 0 V 164 15 R -162 0 V 164 16 R -161 0 V 162 15 R -160 0 V 161 15 R -159 0 V 160 16 R -157 0 V 158 15 R -156 0 V 158 15 R -156 0 V 157 16 R -154 0 V 155 15 R -153 0 V 155 15 R -153 0 V 154 15 R -151 0 V 152 16 R -150 0 V 152 15 R -150 0 V 151 15 R -148 0 V 149 16 R -147 0 V 149 15 R -146 0 V 1.000 UL LT0 867 280 M 0 15 V 0 16 V 0 15 V 0 15 V 0 16 V 0 15 V -1 15 V 0 15 V 0 16 V 0 15 V -1 15 V 0 16 V -1 15 V 0 15 V -1 16 V 0 15 V -1 15 V -1 16 V 0 15 V -1 15 V -1 15 V -1 16 V 0 15 V -1 15 V -1 16 V -1 15 V -1 15 V -1 16 V -1 15 V -1 15 V -1 16 V -1 15 V -1 15 V -1 15 V -1 16 V -2 15 V -1 15 V -1 16 V -1 15 V -1 15 V -2 16 V -1 15 V -1 15 V -2 15 V -1 16 V -1 15 V -2 15 V -1 16 V -1 15 V -2 15 V -1 16 V -2 15 V -1 15 V -1 16 V -2 15 V -1 15 V -2 15 V -1 16 V -2 15 V -1 15 V -2 16 V -1 15 V -2 15 V -1 16 V -2 15 V -1 15 V -2 16 V -1 15 V -2 15 V -1 15 V -2 16 V -1 15 V -1 15 V -2 16 V -1 15 V -2 15 V -1 16 V -2 15 V -1 15 V -1 16 V -2 15 V -1 15 V -2 15 V -1 16 V -1 15 V -2 15 V -1 16 V -1 15 V -2 15 V -1 16 V -1 15 V -1 15 V -2 16 V -1 15 V -1 15 V -1 15 V -1 16 V -1 15 V -2 15 V -1 16 V -1 15 V -1 15 V -1 16 V -1 15 V -1 15 V -1 16 V -1 15 V -1 15 V 0 15 V -1 16 V -1 15 V -1 15 V -1 16 V -1 15 V 0 15 V -1 16 V -1 15 V 0 15 V -1 15 V -1 16 V 0 15 V -1 15 V 0 16 V -1 15 V 0 15 V -1 16 V 0 15 V 0 15 V -1 16 V 0 15 V 0 15 V -1 15 V 0 16 V 0 15 V 0 15 V 0 16 V 0 15 V -1 15 V 0 16 V 0 15 V 0 15 V 1 16 V 0 15 V 0 15 V 0 15 V 0 16 V 0 15 V 1 15 V 0 16 V 0 15 V 1 15 V 0 16 V 0 15 V 1 15 V 0 16 V 1 15 V 0 15 V 1 15 V 1 16 V 0 15 V 1 15 V 1 16 V 0 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 0 15 V 1 16 V 2 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 2 16 V 1 15 V 1 15 V 1 15 V 1 16 V 2 15 V 1 15 V 1 16 V 1 15 V 2 15 V 1 16 V 1 15 V 1 15 V 1 15 V 1 16 V 2 15 V 1 15 V 1 16 V 0 15 V 1 15 V 1 16 V 1 15 V 0 15 V 0 16 V 1 15 V 0 15 V 0 15 V 1 16 V 0 15 V 0 15 V 1 16 V 0 15 V 0 15 V 0 16 V 0 15 V 1 15 V 0 16 V 0 15 V 0 15 V 1 15 V 0 16 V 0 15 V 1 15 V 0 16 V 0 15 V 1 15 V 0 16 V 0 15 V 1 15 V 0 16 V 0 15 V 1 15 V 0 15 V 1 16 V 0 15 V 1 15 V 0 16 V 1 15 V 0 15 V 1 16 V 0 15 V 1 15 V 0 16 V 1 15 V 1 15 V 0 15 V 1 16 V 1 15 V 0 15 V 1 16 V 1 15 V 1 15 V 0 16 V 1 15 V 1 15 V 1 16 V 1 15 V 0 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 2 15 V 1 15 V 1 16 V 1 15 V 1 15 V 2 16 V 1 15 V 1 15 V 1 16 V 1 15 V 2 15 V 1 16 V 1 15 V 2 15 V 1 15 V 1 16 V 2 15 V 1 15 V 1 16 V 2 15 V -146 0 V 1.000 UL LT0 867 280 M 0 15 V 0 16 V 0 15 V 0 15 V 0 16 V -1 15 V 0 15 V 0 15 V 0 16 V -1 15 V 0 15 V -1 16 V 0 15 V -1 15 V 0 16 V -1 15 V 0 15 V -1 16 V -1 15 V 0 15 V -1 15 V -1 16 V -1 15 V -1 15 V 0 16 V -1 15 V -1 15 V -1 16 V -1 15 V -1 15 V -2 16 V -1 15 V -1 15 V -1 15 V -1 16 V -1 15 V -2 15 V -1 16 V -1 15 V -2 15 V -1 16 V -2 15 V -1 15 V -1 15 V -2 16 V -1 15 V -2 15 V -2 16 V -1 15 V -2 15 V -1 16 V -2 15 V -2 15 V -2 16 V -1 15 V -2 15 V -2 15 V -2 16 V -2 15 V -1 15 V -2 16 V -2 15 V -2 15 V -2 16 V -2 15 V -2 15 V -2 16 V -2 15 V -2 15 V -2 15 V -2 16 V -2 15 V -2 15 V -2 16 V -2 15 V -2 15 V -2 16 V -3 15 V -2 15 V -2 16 V -2 15 V -2 15 V -3 15 V -2 16 V -2 15 V -2 15 V -3 16 V -2 15 V -2 15 V -2 16 V -3 15 V -2 15 V -2 16 V -3 15 V -2 15 V -2 15 V -3 16 V -2 15 V -2 15 V -3 16 V -2 15 V -3 15 V -2 16 V -2 15 V -3 15 V -2 16 V -3 15 V -2 15 V -2 15 V -3 16 V -2 15 V -3 15 V -2 16 V -3 15 V -2 15 V -3 16 V -2 15 V -3 15 V -2 15 V -2 16 V -3 15 V -2 15 V -3 16 V -2 15 V -3 15 V -2 16 V -3 15 V -2 15 V -3 16 V -2 15 V -3 15 V -2 15 V -2 16 V -3 15 V -2 15 V -3 16 V -2 15 V -3 15 V -2 16 V -2 15 V -3 15 V -2 16 V -3 15 V -2 15 V -2 15 V -3 16 V -2 15 V -2 15 V -3 16 V -2 15 V -2 15 V -3 16 V -2 15 V -2 15 V -2 16 V -3 15 V -2 15 V -2 15 V -2 16 V -2 15 V -2 15 V -2 16 V -3 15 V -2 15 V -2 16 V -2 15 V -2 15 V -2 16 V -2 15 V -1 15 V -2 15 V -2 16 V -2 15 V -2 15 V -1 16 V -2 15 V -2 15 V -2 16 V -1 15 V -2 15 V -1 16 V -2 15 V -1 15 V -2 15 V -1 16 V -1 15 V -2 15 V -1 16 V -1 15 V -1 15 V -1 16 V -1 15 V -1 15 V -1 15 V 0 16 V -1 15 V 0 15 V -1 16 V 0 15 V 0 15 V 0 16 V 1 15 V 0 15 V 1 16 V 0 15 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 2 15 V 1 15 V 1 16 V 2 15 V 1 15 V 2 16 V 1 15 V 2 15 V 1 15 V 2 16 V 1 15 V 2 15 V 2 16 V 1 15 V 2 15 V 2 16 V 2 15 V 1 15 V 2 16 V 2 15 V 2 15 V 2 15 V 2 16 V 2 15 V 2 15 V 1 16 V 2 15 V 2 15 V 2 16 V 2 15 V 3 15 V 2 16 V 2 15 V 2 15 V 2 15 V 2 16 V 2 15 V 2 15 V 2 16 V 2 15 V 2 15 V 3 16 V 2 15 V 2 15 V 2 16 V 2 15 V 3 15 V 2 15 V 2 16 V 2 15 V 2 15 V 3 16 V 2 15 V 2 15 V 2 16 V 3 15 V 2 15 V 2 15 V 3 16 V 2 15 V 2 15 V 2 16 V 3 15 V 2 15 V 2 16 V 3 15 V 2 15 V 2 16 V 3 15 V 2 15 V 2 15 V 3 16 V 2 15 V 2 15 V 3 16 V 2 15 V 2 15 V 3 16 V 2 15 V 2 15 V 3 16 V 2 15 V 2 15 V 3 15 V 2 16 V 2 15 V 3 15 V 2 16 V 3 15 V 1.000 UL LT0 4633 280 M 0 15 R 0 16 R 0 15 R 0 15 R 1 16 R 0 15 R 0 15 R 1 15 R 0 16 R 1 15 R 1 15 R 0 16 R 1 15 R 1 15 R 1 16 R 1 15 R 1 15 R 1 16 R 2 15 R 1 15 R 1 15 R 2 16 R 1 15 R 2 15 R 1 16 R 2 15 R 2 15 R 1 16 R 2 15 R 2 15 R 2 16 R 2 15 R 2 15 R 3 15 R 2 16 R 2 15 R 3 15 R 2 16 R 3 15 R -1 0 V 3 15 R 3 16 R 3 15 R -1 0 V 3 15 R 3 15 R 3 16 R 3 15 R 3 15 R 3 16 R 3 15 R 4 15 R -1 0 V 4 16 R 3 15 R 4 15 R 3 16 R 4 15 R 4 15 R -1 0 V 4 15 R 4 16 R 4 15 R 4 15 R -1 0 V 5 16 R -1 0 V 5 15 R -1 0 V 5 15 R -1 0 V 5 16 R 5 15 R -1 0 V 5 15 R -1 0 V 5 16 R -1 0 V 6 15 R -1 0 V 5 15 R -1 0 V 6 15 R -1 0 V 6 16 R -2 0 V 6 15 R -1 0 V 6 15 R -1 0 V 6 16 R -1 0 V 6 15 R -2 0 V 7 15 R -2 0 V 7 16 R -2 0 V 7 15 R -2 0 V 7 15 R -2 0 V 8 16 R -2 0 V 7 15 R -2 0 V 8 15 R -3 0 V 8 15 R -2 0 V 8 16 R -3 0 V 8 15 R -3 0 V 9 15 R -3 0 V 9 16 R -3 0 V 9 15 R -4 0 V 10 15 R -4 0 V 10 16 R -4 0 V 10 15 R -4 0 V 10 15 R -4 0 V 10 16 R -4 0 V 10 15 R -4 0 V 11 15 R -5 0 V 11 15 R -5 0 V 12 16 R -5 0 V 11 15 R -5 0 V 12 15 R -6 0 V 13 16 R -6 0 V 13 15 R -7 0 V 14 15 R -7 0 V 14 16 R -7 0 V 14 15 R -8 0 V 15 15 R -8 0 V 15 16 R -8 0 V 15 15 R -8 0 V 16 15 R -9 0 V 16 15 R -9 0 V 17 16 R -10 0 V 17 15 R -10 0 V 18 15 R -10 0 V 18 16 R -11 0 V 18 15 R -11 0 V 19 15 R -11 0 V 19 16 R -12 0 V 20 15 R -12 0 V 21 15 R -14 0 V 22 15 R -14 0 V 22 16 R -14 0 V 22 15 R -14 0 V 23 15 R -15 0 V 23 16 R -15 0 V 24 15 R -16 0 V 25 15 R -17 0 V 26 16 R -18 0 V 27 15 R -19 0 V 27 15 R -18 0 V 28 16 R -20 0 V 29 15 R -20 0 V 29 15 R -21 0 V 30 15 R -21 0 V 31 16 R -23 0 V 32 15 R -23 0 V 33 15 R -24 0 V 33 16 R -24 0 V 34 15 R -25 0 V 35 15 R -26 0 V 35 16 R -26 0 V 36 15 R -27 0 V 37 15 R -28 0 V 39 16 R -29 0 V 39 15 R -30 0 V 40 15 R -31 0 V 41 15 R -31 0 V 42 16 R -33 0 V 43 15 R -33 0 V 44 15 R -34 0 V 45 16 R -35 0 V 46 15 R -37 0 V 47 15 R -37 0 V 48 16 R -38 0 V 49 15 R -39 0 V 50 15 R -40 0 V 52 16 R -41 0 V 52 15 R -42 0 V 53 15 R -43 0 V 55 15 R -44 0 V 55 16 R -45 0 V 57 15 R -46 0 V 58 15 R -48 0 V 59 16 R -48 0 V 60 15 R -49 0 V 61 15 R -51 0 V 63 16 R -52 0 V 64 15 R -53 0 V 65 15 R -54 0 V 66 16 R -55 0 V 68 15 R -57 0 V 69 15 R -58 0 V 70 15 R -58 0 V 71 16 R -60 0 V 73 15 R -62 0 V 74 15 R -62 0 V 75 16 R -64 0 V 77 15 R -65 0 V 77 15 R -65 0 V 78 16 R -67 0 V 80 15 R -68 0 V 81 15 R -69 0 V 83 16 R -71 0 V 84 15 R -72 0 V 85 15 R -73 0 V 86 15 R -74 0 V 88 16 R -76 0 V 89 15 R -77 0 V 90 15 R -77 0 V 91 16 R -79 0 V 92 15 R -79 0 V 93 15 R -81 0 V 95 16 R -82 0 V 95 15 R -82 0 V 96 15 R -83 0 V 97 15 R -85 0 V 99 16 R -85 0 V 99 15 R -86 0 V 99 15 R -86 0 V 100 16 R -87 0 V 101 15 R -87 0 V 101 15 R -88 0 V 102 16 R -88 0 V 102 15 R -88 0 V 102 15 R -88 0 V 101 16 R -87 0 V 101 15 R -87 0 V 101 15 R -87 0 V 101 15 R -87 0 V 101 16 R -86 0 V 100 15 R -86 0 V 100 15 R -85 0 V 99 16 R -85 0 V 99 15 R -84 0 V 99 15 R -84 0 V 98 16 R -83 0 V 97 15 R -82 0 V 96 15 R -81 0 V 96 16 R -82 0 V 96 15 R -80 0 V 95 15 R -80 0 V 94 15 R -79 0 V 94 16 R -79 0 V 93 15 R -78 0 V 93 15 R -78 0 V 92 16 R -76 0 V 91 15 R -76 0 V 91 15 R -75 0 V 90 16 R -75 0 V 89 15 R -73 0 V currentpoint stroke M 88 15 R -73 0 V 88 16 R -72 0 V 87 15 R -71 0 V 86 15 R -71 0 V 86 15 R -70 0 V 86 16 R -70 0 V 85 15 R -69 0 V 84 15 R -68 0 V 83 16 R -67 0 V 82 15 R -66 0 V 82 15 R -66 0 V 81 16 R -65 0 V 81 15 R -65 0 V 80 15 R -64 0 V 80 16 R -64 0 V 79 15 R -63 0 V 79 15 R -62 0 V 77 15 R -61 0 V 77 16 R -61 0 V 77 15 R -60 0 V 75 15 R -59 0 V 75 16 R -59 0 V 75 15 R -58 0 V 74 15 R -58 0 V 74 16 R -57 0 V 73 15 R -57 0 V 73 15 R -56 0 V 72 16 R -55 0 V 71 15 R -55 0 V 71 15 R -54 0 V 70 15 R -53 0 V 69 16 R -52 0 V 69 15 R -53 0 V 66 15 R -49 0 V 49 16 R -32 0 V 32 15 R -15 0 V 1.000 UL LT0 4633 280 M 0 15 V 0 16 V 0 15 V 0 15 V 1 16 V 0 15 V 0 15 V 1 15 V 0 16 V 1 15 V 1 15 V 0 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 2 15 V 1 15 V 1 15 V 2 16 V 1 15 V 2 15 V 1 16 V 2 15 V 2 15 V 1 16 V 2 15 V 2 15 V 2 16 V 2 15 V 2 15 V 3 15 V 2 16 V 2 15 V 3 15 V 2 16 V 3 15 V 2 15 V 3 16 V 3 15 V 2 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 16 V 3 15 V 4 15 V 3 16 V 3 15 V 4 15 V 3 16 V 4 15 V 4 15 V 3 15 V 4 16 V 4 15 V 4 15 V 4 16 V 4 15 V 4 15 V 4 16 V 5 15 V 4 15 V 4 16 V 5 15 V 4 15 V 5 15 V 5 16 V 4 15 V 5 15 V 5 16 V 5 15 V 5 15 V 5 16 V 5 15 V 5 15 V 6 16 V 5 15 V 6 15 V 5 15 V 6 16 V 5 15 V 6 15 V 6 16 V 6 15 V 6 15 V 6 16 V 6 15 V 6 15 V 6 16 V 6 15 V 7 15 V 6 15 V 7 16 V 6 15 V 7 15 V 7 16 V 7 15 V 7 15 V 7 16 V 7 15 V 7 15 V 7 16 V 7 15 V 8 15 V 7 15 V 8 16 V 7 15 V 8 15 V 8 16 V 7 15 V 8 15 V 8 16 V 8 15 V 9 15 V 8 15 V 8 16 V 8 15 V 9 15 V 8 16 V 9 15 V 9 15 V 9 16 V 9 15 V 8 15 V 10 16 V 9 15 V 9 15 V 9 15 V 10 16 V 9 15 V 10 15 V 9 16 V 10 15 V 10 15 V 9 16 V 10 15 V 10 15 V 11 16 V 10 15 V 10 15 V 10 15 V 11 16 V 10 15 V 11 15 V 11 16 V 11 15 V 10 15 V 11 16 V 11 15 V 11 15 V 12 16 V 11 15 V 11 15 V 12 15 V 11 16 V 12 15 V 12 15 V 11 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 13 15 V 12 15 V 12 15 V 13 16 V 13 15 V 12 15 V 13 16 V 13 15 V 12 15 V 13 16 V 13 15 V 13 15 V 14 16 V 13 15 V 13 15 V 13 15 V 14 16 V 13 15 V 13 15 V 14 16 V 13 15 V 14 15 V 14 16 V 13 15 V 14 15 V 14 15 V 14 16 V 14 15 V 13 15 V 14 16 V 14 15 V 14 15 V 14 16 V 14 15 V 14 15 V 13 16 V 14 15 V 14 15 V 14 15 V 14 16 V 14 15 V 14 15 V 14 16 V 14 15 V 15 15 V 14 16 V 14 15 V 14 15 V 15 16 V 14 15 V 15 15 V 14 15 V 15 16 V 14 15 V 15 15 V 14 16 V 15 15 V 15 15 V 15 16 V 14 15 V 15 15 V 15 16 V 15 15 V 15 15 V 15 15 V 16 16 V 15 15 V 15 15 V 15 16 V 15 15 V 16 15 V 15 16 V 16 15 V 15 15 V 16 16 V 15 15 V 16 15 V 15 15 V 16 16 V 16 15 V 15 15 V 16 16 V 16 15 V 16 15 V 16 16 V 16 15 V 16 15 V 16 16 V 16 15 V 16 15 V 16 15 V 16 16 V 17 15 V 13 13 V 1.000 UL LT0 4633 280 M 0 15 V 0 16 V 0 15 V 0 15 V 1 16 V 0 15 V 0 15 V 1 15 V 0 16 V 1 15 V 1 15 V 0 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 2 15 V 1 15 V 1 15 V 2 16 V 1 15 V 2 15 V 1 16 V 2 15 V 2 15 V 1 16 V 2 15 V 2 15 V 2 16 V 2 15 V 2 15 V 3 15 V 2 16 V 2 15 V 3 15 V 2 16 V 2 15 V 3 15 V 3 16 V 2 15 V 3 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 16 V 3 15 V 3 15 V 4 16 V 3 15 V 4 15 V 3 16 V 4 15 V 3 15 V 4 15 V 4 16 V 4 15 V 3 15 V 4 16 V 4 15 V 4 15 V 5 16 V 4 15 V 4 15 V 4 16 V 5 15 V 4 15 V 5 15 V 4 16 V 5 15 V 5 15 V 5 16 V 4 15 V 5 15 V 5 16 V 5 15 V 5 15 V 6 16 V 5 15 V 5 15 V 6 15 V 5 16 V 5 15 V 6 15 V 6 16 V 5 15 V 6 15 V 6 16 V 6 15 V 6 15 V 6 16 V 6 15 V 6 15 V 6 15 V 7 16 V 6 15 V 6 15 V 7 16 V 6 15 V 7 15 V 7 16 V 6 15 V 7 15 V 7 16 V 7 15 V 7 15 V 7 15 V 7 16 V 7 15 V 8 15 V 7 16 V 7 15 V 8 15 V 7 16 V 8 15 V 7 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 16 V 8 15 V 9 15 V 8 16 V 9 15 V 8 15 V 9 15 V 8 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 10 16 V 9 15 V 9 15 V 10 15 V 9 16 V 10 15 V 10 15 V 10 16 V 9 15 V 10 15 V 10 16 V 10 15 V 10 15 V 11 16 V 10 15 V 10 15 V 11 15 V 10 16 V 11 15 V 10 15 V 11 16 V 11 15 V 10 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 12 15 V 11 16 V 11 15 V 12 15 V 11 16 V 12 15 V 12 15 V 11 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 15 V 12 16 V 12 15 V 13 15 V 12 16 V 13 15 V 12 15 V 13 16 V 13 15 V 13 15 V 12 15 V 14 16 V 13 15 V 13 15 V 13 16 V 14 15 V 13 15 V 14 16 V 14 15 V 14 15 V 14 16 V 14 15 V 14 15 V 14 15 V 15 16 V 14 15 V 15 15 V 14 16 V 15 15 V 15 15 V 15 16 V 15 15 V 15 15 V 14 16 V 16 15 V 15 15 V 15 15 V 15 16 V 15 15 V 15 15 V 16 16 V 15 15 V 16 15 V 15 16 V 16 15 V 15 15 V 16 16 V 16 15 V 15 15 V 16 15 V 16 16 V 16 15 V 16 15 V 16 16 V 16 15 V 16 15 V 16 16 V 16 15 V 16 15 V 16 16 V 16 15 V 17 15 V 16 15 V 16 16 V 17 15 V 16 15 V 16 16 V 17 15 V 16 15 V 17 16 V 16 15 V 17 15 V 17 16 V 16 15 V 17 15 V 17 15 V 17 16 V 16 15 V 17 15 V 17 16 V 17 15 V 15 13 V 1.000 UL LT0 3115 280 M 0 15 R 0 16 R 1 0 V 0 15 R 0 15 R 1 16 R 1 15 R 0 15 R 1 0 V 0 15 R 1 0 V 1 16 R 1 15 R 1 15 R 1 0 V 1 16 R 2 15 R 1 15 R 1 0 V 1 16 R 1 0 V 2 15 R 2 15 R 2 16 R 3 15 R 2 15 R 1 0 V 2 15 R 3 16 R 3 15 R 3 15 R 3 16 R 1 0 V 3 15 R 3 15 R 1 0 V 3 16 R 4 15 R 4 15 R 4 16 R 4 15 R 4 15 R 4 15 R 1 0 V 4 16 R 5 15 R 4 15 R 5 16 R 5 15 R 5 15 R 5 16 R 1 0 V 5 15 R 5 15 R 6 15 R 5 16 R 6 15 R 6 15 R 6 16 R 6 15 R 6 15 R 6 16 R 6 15 R 1 0 V 6 15 R 6 16 R 7 15 R 6 15 R 1 0 V 6 15 R 1 0 V 6 16 R 7 15 R 7 15 R 7 16 R 1 0 V 6 15 R 1 0 V 7 15 R 7 16 R 1 0 V 6 15 R 1 0 V 7 15 R 1 0 V 7 16 R 7 15 R 1 0 V 7 15 R 1 0 V 7 15 R 1 0 V 7 16 R 1 0 V 7 15 R 1 0 V 7 15 R 1 0 V 7 16 R 1 0 V 7 15 R 1 0 V 7 15 R 2 0 V 6 16 R 2 0 V 7 15 R 1 0 V 7 15 R 2 0 V 6 16 R 2 0 V 7 15 R 2 0 V 7 15 R 2 0 V 6 15 R 2 0 V 7 16 R 2 0 V 6 15 R 3 0 V 6 15 R 3 0 V 6 16 R 3 0 V 6 15 R 3 0 V 6 15 R 3 0 V 6 16 R 3 0 V 6 15 R 3 0 V 6 15 R 4 0 V 5 16 R 4 0 V 5 15 R 4 0 V 5 15 R 5 0 V 4 15 R 5 0 V 4 16 R 6 0 V 4 15 R 5 0 V 4 15 R 6 0 V 3 16 R 6 0 V 4 15 R 6 0 V 3 15 R 7 0 V 2 16 R 7 0 V 3 15 R 7 0 V 2 15 R 8 0 V 2 16 R 8 0 V 1 15 R 9 0 V 1 15 R 9 0 V 0 15 R 9 0 V 1 16 R 9 0 V 0 15 R 10 0 V 0 15 R 10 0 V -1 16 R 12 0 V -2 15 R 12 0 V -2 15 R 12 0 V -3 16 R 13 0 V -3 15 R 13 0 V -3 15 R 13 0 V -4 15 R 15 0 V -5 16 R 15 0 V -5 15 R 15 0 V -6 15 R 16 0 V -6 16 R 17 0 V -7 15 R 17 0 V -7 15 R 17 0 V -8 16 R 19 0 V -9 15 R 19 0 V -9 15 R 20 0 V -10 16 R 20 0 V -11 15 R 22 0 V -12 15 R 22 0 V -12 15 R 23 0 V -14 16 R 24 0 V -14 15 R 25 0 V -15 15 R 25 0 V -15 16 R 26 0 V -17 15 R 27 0 V -17 15 R 28 0 V -18 16 R 29 0 V -19 15 R 29 0 V -20 15 R 31 0 V -21 16 R 31 0 V -21 15 R 32 0 V -23 15 R 34 0 V -24 15 R 35 0 V -25 16 R 35 0 V -25 15 R 36 0 V -27 15 R 38 0 V -28 16 R 38 0 V -28 15 R 39 0 V -30 15 R 41 0 V -31 16 R 42 0 V -32 15 R 43 0 V -34 15 R 44 0 V -34 16 R 45 0 V -35 15 R 46 0 V -37 15 R 48 0 V -38 15 R 49 0 V -40 16 R 51 0 V -41 15 R 51 0 V -41 15 R 52 0 V -43 16 R 54 0 V -44 15 R 55 0 V -46 15 R 57 0 V -47 16 R 58 0 V -48 15 R 59 0 V -50 15 R 61 0 V -51 16 R 62 0 V -53 15 R 64 0 V -54 15 R 65 0 V -55 15 R 66 0 V -57 16 R 68 0 V -58 15 R 69 0 V -60 15 R 71 0 V -61 16 R 72 0 V -63 15 R 74 0 V -64 15 R 75 0 V -65 16 R 76 0 V -67 15 R 78 0 V -68 15 R 79 0 V -69 16 R 80 0 V -71 15 R 82 0 V -72 15 R 83 0 V -74 15 R 85 0 V -75 16 R 86 0 V -76 15 R 87 0 V -78 15 R 89 0 V -79 16 R 90 0 V -80 15 R 91 0 V -81 15 R 92 0 V -83 16 R 94 0 V -84 15 R 95 0 V -85 15 R 96 0 V -86 15 R 97 0 V -87 16 R 98 0 V -88 15 R 99 0 V -89 15 R 100 0 V -90 16 R 101 0 V -91 15 R 102 0 V -91 15 R 102 0 V -92 16 R 102 0 V -91 15 R 102 0 V -92 15 R 102 0 V -91 16 R 102 0 V -91 15 R 101 0 V -90 15 R 100 0 V -89 15 R 100 0 V -89 16 R 99 0 V -89 15 R 99 0 V -88 15 R 98 0 V -87 16 R 98 0 V -87 15 R 97 0 V -85 15 R 95 0 V -84 16 R 94 0 V -83 15 R 94 0 V -83 15 R 93 0 V -82 16 R 92 0 V -81 15 R 91 0 V -80 15 R 90 0 V -79 15 R 90 0 V -79 16 R 89 0 V -78 15 R 88 0 V -76 15 R 86 0 V -75 16 R 86 0 V currentpoint stroke M -75 15 R 85 0 V -74 15 R 84 0 V -73 16 R 83 0 V -72 15 R 83 0 V -71 15 R 81 0 V -70 16 R 80 0 V -69 15 R 80 0 V -69 15 R 79 0 V -68 15 R 78 0 V -67 16 R 78 0 V -66 15 R 76 0 V -65 15 R 75 0 V -64 16 R 74 0 V -63 15 R 74 0 V -63 15 R 73 0 V -62 16 R 72 0 V -60 15 R 71 0 V -60 15 R 70 0 V -59 16 R 70 0 V -59 15 R 69 0 V -58 15 R 68 0 V -57 15 R 68 0 V -56 16 R 66 0 V -55 15 R 65 0 V -54 15 R 65 0 V -54 16 R 64 0 V -53 15 R 64 0 V -53 15 R 63 0 V -52 16 R 62 0 V -50 15 R 61 0 V -50 15 R 60 0 V -49 16 R 60 0 V -49 15 R 59 0 V -48 15 R 59 0 V -48 15 R 58 0 V -47 16 R 58 0 V -46 15 R 56 0 V -45 15 R 55 0 V -44 16 R 55 0 V -44 15 R 54 0 V -43 15 R 54 0 V -43 16 R 53 0 V -42 15 R 53 0 V -42 15 R 52 0 V -41 15 R 52 0 V -41 16 R 51 0 V -39 15 R 50 0 V -39 15 R 49 0 V -38 16 R 49 0 V -38 15 R 48 0 V -37 15 R 48 0 V -37 16 R 47 0 V -36 15 R 47 0 V -36 15 R 46 0 V -35 16 R 46 0 V -35 15 R 45 0 V -34 15 R 45 0 V -34 15 R 45 0 V -33 16 R 43 0 V -32 15 R 43 0 V -32 15 R 42 0 V -31 16 R 42 0 V -31 15 R 41 0 V -30 15 R 41 0 V -30 16 R 41 0 V -30 15 R 40 0 V -29 15 R 40 0 V -29 16 R 39 0 V -28 15 R 39 0 V -28 15 R 39 0 V -28 15 R 38 0 V -27 16 R 38 0 V -27 15 R 37 0 V -26 15 R 37 0 V -25 16 R 36 0 V -25 15 R 35 0 V 1.000 UL LT0 3115 280 M 0 15 V 0 16 V 1 15 V 0 15 V 1 16 V 1 15 V 0 15 V 1 15 V 2 16 V 1 15 V 1 15 V 2 16 V 2 15 V 1 15 V 2 16 V 3 15 V 2 15 V 2 16 V 3 15 V 2 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 16 V 4 15 V 3 15 V 4 16 V 4 15 V 4 15 V 4 16 V 4 15 V 4 15 V 4 15 V 5 16 V 5 15 V 4 15 V 5 16 V 5 15 V 5 15 V 5 16 V 6 15 V 5 15 V 6 15 V 5 16 V 6 15 V 6 15 V 6 16 V 6 15 V 6 15 V 6 16 V 6 15 V 7 15 V 6 16 V 7 15 V 6 15 V 7 15 V 7 16 V 7 15 V 7 15 V 7 16 V 7 15 V 8 15 V 7 16 V 7 15 V 8 15 V 8 16 V 7 15 V 8 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 16 V 9 15 V 8 15 V 8 16 V 9 15 V 9 15 V 8 15 V 9 16 V 8 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 15 V 9 16 V 10 15 V 9 15 V 9 16 V 10 15 V 9 15 V 9 16 V 10 15 V 9 15 V 10 16 V 9 15 V 10 15 V 9 15 V 10 16 V 9 15 V 10 15 V 9 16 V 10 15 V 10 15 V 9 16 V 10 15 V 10 15 V 9 15 V 10 16 V 10 15 V 9 15 V 10 16 V 10 15 V 10 15 V 9 16 V 10 15 V 10 15 V 10 16 V 9 15 V 10 15 V 10 15 V 9 16 V 10 15 V 10 15 V 10 16 V 9 15 V 10 15 V 10 16 V 10 15 V 9 15 V 10 16 V 10 15 V 9 15 V 10 15 V 10 16 V 10 15 V 9 15 V 10 16 V 10 15 V 9 15 V 10 16 V 10 15 V 9 15 V 10 16 V 10 15 V 9 15 V 10 15 V 9 16 V 10 15 V 10 15 V 9 16 V 10 15 V 9 15 V 10 16 V 10 15 V 9 15 V 10 16 V 9 15 V 10 15 V 10 15 V 9 16 V 10 15 V 9 15 V 10 16 V 9 15 V 10 15 V 10 16 V 9 15 V 10 15 V 10 16 V 9 15 V 10 15 V 9 15 V 10 16 V 10 15 V 9 15 V 10 16 V 10 15 V 10 15 V 9 16 V 10 15 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 10 16 V 10 15 V 11 15 V 10 16 V 11 15 V 10 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 10 15 V 11 15 V 11 16 V 11 15 V 12 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 11 15 V 12 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 12 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 12 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 12 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 12 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 12 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 12 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 12 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 12 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 12 16 V 11 15 V 35 0 V 1.000 UL LT0 3115 280 M 0 15 V 1 16 V 0 15 V 0 15 V 1 16 V 1 15 V 1 15 V 1 15 V 1 16 V 1 15 V 2 15 V 1 16 V 2 15 V 2 15 V 2 16 V 2 15 V 2 15 V 2 16 V 3 15 V 3 15 V 2 15 V 3 16 V 3 15 V 3 15 V 4 16 V 3 15 V 4 15 V 3 16 V 4 15 V 4 15 V 4 16 V 4 15 V 4 15 V 5 15 V 4 16 V 5 15 V 4 15 V 5 16 V 5 15 V 5 15 V 6 16 V 5 15 V 5 15 V 6 15 V 5 16 V 6 15 V 6 15 V 6 16 V 6 15 V 6 15 V 6 16 V 7 15 V 6 15 V 6 16 V 7 15 V 7 15 V 7 15 V 6 16 V 7 15 V 7 15 V 8 16 V 7 15 V 7 15 V 8 16 V 7 15 V 8 15 V 7 16 V 8 15 V 8 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 16 V 8 15 V 9 15 V 8 16 V 8 15 V 9 15 V 8 16 V 9 15 V 9 15 V 8 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 10 15 V 9 16 V 9 15 V 10 15 V 9 15 V 10 16 V 9 15 V 10 15 V 9 16 V 10 15 V 10 15 V 9 16 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 9 15 V 10 16 V 10 15 V 10 15 V 11 16 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 11 15 V 10 16 V 10 15 V 10 15 V 11 16 V 10 15 V 10 15 V 11 16 V 10 15 V 11 15 V 10 16 V 11 15 V 10 15 V 11 15 V 10 16 V 11 15 V 10 15 V 11 16 V 10 15 V 11 15 V 11 16 V 10 15 V 11 15 V 10 16 V 11 15 V 11 15 V 11 15 V 10 16 V 11 15 V 11 15 V 10 16 V 11 15 V 11 15 V 11 16 V 11 15 V 10 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 10 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 10 16 V 11 15 V 10 15 V 11 16 V 10 15 V 10 15 V 11 15 V 10 16 V 10 15 V 10 15 V 11 16 V 10 15 V 10 15 V 10 16 V 11 15 V 10 15 V 10 16 V 10 15 V 10 15 V 11 15 V 10 16 V 10 15 V 10 15 V 11 16 V 10 15 V 10 15 V 10 16 V 11 15 V 10 15 V 10 16 V 11 15 V 10 15 V 10 15 V 11 16 V 10 15 V 10 15 V 10 16 V 11 15 V 10 15 V 10 16 V 11 15 V 10 15 V 11 16 V 10 15 V 10 15 V 11 15 V 10 16 V 10 15 V 11 15 V 10 16 V 11 15 V 10 15 V 10 16 V 11 15 V 10 15 V 11 16 V 10 15 V 11 15 V 10 15 V 11 16 V 10 15 V 10 15 V 11 16 V 10 15 V 11 15 V 10 16 V 11 15 V 10 15 V 11 15 V 10 16 V 11 15 V 10 15 V 11 16 V 10 15 V 11 15 V 10 16 V 11 15 V 10 15 V 11 16 V 10 15 V 11 15 V 11 15 V 10 16 V 11 15 V 10 15 V 11 16 V 10 15 V 11 15 V 11 16 V 10 15 V 11 15 V 10 16 V 11 15 V 11 15 V 10 15 V 11 16 V 10 15 V 11 15 V 11 16 V 10 15 V 1.000 UL LT0 2730 280 M 0 15 R 1 16 R -1 0 V 1 15 R 1 15 R 1 16 R 1 15 R 1 15 R 2 15 R 1 16 R 2 15 R 2 15 R 3 16 R 2 15 R 3 15 R 3 16 R 3 15 R 3 15 R 4 16 R -1 0 V 4 15 R 4 15 R 4 15 R 4 16 R 4 15 R 5 15 R 5 16 R -1 0 V 5 15 R 6 15 R -1 0 V 6 16 R -1 0 V 6 15 R 6 15 R -1 0 V 6 16 R 6 15 R 6 15 R 6 15 R 6 16 R 7 15 R 6 15 R 7 16 R 7 15 R 7 15 R 7 16 R 7 15 R 7 15 R 8 15 R 7 16 R 8 15 R 8 15 R 8 16 R 8 15 R 8 15 R 8 16 R 9 15 R 8 15 R 9 16 R 8 15 R 9 15 R 9 15 R 9 16 R 9 15 R 9 15 R 10 16 R -1 0 V 10 15 R 9 15 R 10 16 R 9 15 R 10 15 R 10 16 R -1 0 V 10 15 R 10 15 R 10 15 R 10 16 R 10 15 R 10 15 R 11 16 R -1 0 V 11 15 R 10 15 R 11 16 R -1 0 V 11 15 R 11 15 R -1 0 V 11 16 R 11 15 R -1 0 V 11 15 R 11 15 R 11 16 R -1 0 V 12 15 R -1 0 V 12 15 R -1 0 V 12 16 R -1 0 V 11 15 R 11 15 R 12 16 R -1 0 V 12 15 R -1 0 V 12 15 R -1 0 V 12 16 R -1 0 V 12 15 R -1 0 V 12 15 R -1 0 V 13 15 R -2 0 V 13 16 R -1 0 V 12 15 R -1 0 V 13 15 R -2 0 V 13 16 R -1 0 V 12 15 R -1 0 V 13 15 R -2 0 V 13 16 R -1 0 V 13 15 R -2 0 V 13 15 R -2 0 V 14 16 R -2 0 V 14 15 R -3 0 V 14 15 R -2 0 V 14 15 R -3 0 V 14 16 R -2 0 V 14 15 R -3 0 V 15 15 R -3 0 V 15 16 R -4 0 V 15 15 R -3 0 V 15 15 R -3 0 V 15 16 R -4 0 V 16 15 R -4 0 V 15 15 R -4 0 V 16 15 R -4 0 V 16 16 R -4 0 V 16 15 R -5 0 V 17 15 R -5 0 V 17 16 R -6 0 V 18 15 R -6 0 V 17 15 R -5 0 V 17 16 R -6 0 V 18 15 R -6 0 V 18 15 R -7 0 V 19 16 R -7 0 V 19 15 R -7 0 V 19 15 R -8 0 V 20 15 R -8 0 V 20 16 R -9 0 V 21 15 R -9 0 V 21 15 R -10 0 V 22 16 R -10 0 V 22 15 R -11 0 V 23 15 R -11 0 V 23 16 R -12 0 V 24 15 R -12 0 V 24 15 R -13 0 V 25 16 R -13 0 V 25 15 R -14 0 V 26 15 R -14 0 V 26 15 R -15 0 V 27 16 R -16 0 V 28 15 R -16 0 V 28 15 R -17 0 V 29 16 R -18 0 V 29 15 R -17 0 V 29 15 R -18 0 V 30 16 R -19 0 V 31 15 R -20 0 V 32 15 R -20 0 V 32 16 R -21 0 V 33 15 R -22 0 V 34 15 R -23 0 V 35 15 R -24 0 V 36 16 R -25 0 V 37 15 R -26 0 V 37 15 R -26 0 V 38 16 R -27 0 V 39 15 R -28 0 V 40 15 R -29 0 V 41 16 R -30 0 V 41 15 R -30 0 V 42 15 R -32 0 V 44 16 R -33 0 V 45 15 R -34 0 V 45 15 R -34 0 V 46 15 R -36 0 V 48 16 R -37 0 V 48 15 R -37 0 V 49 15 R -39 0 V 51 16 R -40 0 V 51 15 R -41 0 V 53 15 R -42 0 V 53 16 R -43 0 V 55 15 R -45 0 V 56 15 R -45 0 V 57 16 R -47 0 V 58 15 R -48 0 V 60 15 R -49 0 V 60 15 R -50 0 V 61 16 R -51 0 V 62 15 R -52 0 V 64 15 R -53 0 V 64 16 R -54 0 V 65 15 R -55 0 V 66 15 R -56 0 V 67 16 R -57 0 V 68 15 R -58 0 V 69 15 R -59 0 V 70 15 R -60 0 V 71 16 R -61 0 V 72 15 R -62 0 V 73 15 R -63 0 V 73 16 R -63 0 V 74 15 R -64 0 V 74 15 R -63 0 V 74 16 R -64 0 V 74 15 R -64 0 V 74 15 R -64 0 V 74 16 R -63 0 V 73 15 R -63 0 V 73 15 R -63 0 V 72 15 R -61 0 V 71 16 R -61 0 V 71 15 R -61 0 V 70 15 R -59 0 V 69 16 R -59 0 V 69 15 R -59 0 V 68 15 R -58 0 V 68 16 R -57 0 V 66 15 R -56 0 V 65 15 R -55 0 V 65 16 R -55 0 V 64 15 R -54 0 V 64 15 R -53 0 V 62 15 R -52 0 V 61 16 R -51 0 V 61 15 R -51 0 V 60 15 R -50 0 V 59 16 R -49 0 V 58 15 R -48 0 V 58 15 R -48 0 V 57 16 R -47 0 V 56 15 R -46 0 V 55 15 R -45 0 V 54 16 R -44 0 V 53 15 R -44 0 V 54 15 R -44 0 V 53 15 R -43 0 V 52 16 R -42 0 V 51 15 R -41 0 V 50 15 R -41 0 V currentpoint stroke M 50 16 R -40 0 V 49 15 R -39 0 V 48 15 R -38 0 V 47 16 R -38 0 V 47 15 R -37 0 V 46 15 R -36 0 V 45 16 R -36 0 V 45 15 R -35 0 V 44 15 R -35 0 V 44 15 R -34 0 V 43 16 R -34 0 V 43 15 R -33 0 V 42 15 R -33 0 V 42 16 R -32 0 V 41 15 R -32 0 V 41 15 R -31 0 V 39 16 R -30 0 V 39 15 R -29 0 V 38 15 R -29 0 V 38 16 R -28 0 V 37 15 R -28 0 V 37 15 R -28 0 V 37 15 R -27 0 V 36 16 R -27 0 V 36 15 R -26 0 V 35 15 R -26 0 V 34 16 R -25 0 V 34 15 R -24 0 V 33 15 R -24 0 V 33 16 R -24 0 V 33 15 R -24 0 V 33 15 R -23 0 V 32 15 R -23 0 V 31 16 R -22 0 V 31 15 R -22 0 V 31 15 R -21 0 V 30 16 R -21 0 V 30 15 R -21 0 V 30 15 R -21 0 V 30 16 R -20 0 V 28 15 R -19 0 V 28 15 R -19 0 V 28 16 R -19 0 V 28 15 R -19 0 V 28 15 R -19 0 V 28 15 R -18 0 V 27 16 R -18 0 V 26 15 R -17 0 V 26 15 R -17 0 V 26 16 R -17 0 V 26 15 R -17 0 V 26 15 R -17 0 V 26 16 R -16 0 V 24 15 R -15 0 V 24 15 R -15 0 V 24 16 R -15 0 V 24 15 R -15 0 V 24 15 R -15 0 V 24 15 R -15 0 V 24 16 R -15 0 V 23 15 R -14 0 V 23 15 R -14 0 V 23 16 R -13 0 V 22 15 R -13 0 V 1.000 UL LT0 2730 280 M 0 15 V 1 16 V 0 15 V 1 15 V 1 16 V 1 15 V 1 15 V 2 15 V 1 16 V 2 15 V 2 15 V 3 16 V 2 15 V 3 15 V 3 16 V 3 15 V 3 15 V 4 16 V 3 15 V 4 15 V 4 15 V 4 16 V 4 15 V 5 15 V 5 16 V 4 15 V 6 15 V 5 16 V 5 15 V 6 15 V 5 16 V 6 15 V 6 15 V 6 15 V 6 16 V 7 15 V 6 15 V 7 16 V 7 15 V 7 15 V 7 16 V 7 15 V 7 15 V 8 15 V 7 16 V 8 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 16 V 9 15 V 8 15 V 9 16 V 8 15 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 10 16 V 9 15 V 9 15 V 10 16 V 9 15 V 10 15 V 10 16 V 9 15 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 11 16 V 10 15 V 10 15 V 11 16 V 10 15 V 11 15 V 10 16 V 11 15 V 10 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 10 15 V 11 15 V 12 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 12 15 V 11 16 V 11 15 V 12 15 V 11 16 V 11 15 V 12 15 V 11 16 V 12 15 V 11 15 V 12 16 V 12 15 V 11 15 V 12 15 V 11 16 V 12 15 V 12 15 V 12 16 V 11 15 V 12 15 V 12 16 V 12 15 V 11 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 11 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 11 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 15 V 12 16 V 12 15 V 11 15 V 12 16 V 12 15 V 12 15 V 12 16 V 11 15 V 12 15 V 12 16 V 12 15 V 11 15 V 12 15 V 12 16 V 11 15 V 12 15 V 12 16 V 11 15 V 12 15 V 11 16 V 12 15 V 11 15 V 12 16 V 11 15 V 12 15 V 11 15 V 11 16 V 11 15 V 12 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 10 16 V 11 15 V 10 15 V 11 16 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 9 15 V 10 16 V 10 15 V 9 15 V 10 16 V 10 15 V 9 15 V 10 16 V 9 15 V 9 15 V 10 16 V 9 15 V 10 15 V 9 15 V 9 16 V 10 15 V 9 15 V 9 16 V 9 15 V 10 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 10 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 8 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 8 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 15 V 8 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 8 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 15 V 9 16 V 8 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 8 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 15 V 9 16 V 8 15 V 9 15 V 9 16 V 9 15 V -13 0 V 1.000 UL LT0 2730 280 M 0 15 V 0 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 2 15 V 1 16 V 2 15 V 2 15 V 3 16 V 2 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 16 V 4 15 V 4 15 V 4 15 V 4 16 V 4 15 V 5 15 V 4 16 V 5 15 V 5 15 V 5 16 V 6 15 V 5 15 V 6 16 V 6 15 V 6 15 V 6 15 V 6 16 V 7 15 V 6 15 V 7 16 V 7 15 V 7 15 V 7 16 V 7 15 V 7 15 V 8 15 V 7 16 V 8 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 16 V 9 15 V 8 15 V 9 16 V 8 15 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 10 15 V 9 15 V 10 16 V 9 15 V 10 15 V 9 16 V 10 15 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 10 16 V 11 15 V 10 15 V 10 16 V 11 15 V 10 15 V 11 16 V 10 15 V 11 15 V 11 15 V 10 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 15 V 12 16 V 11 15 V 11 15 V 12 16 V 11 15 V 11 15 V 12 16 V 11 15 V 11 15 V 12 16 V 11 15 V 12 15 V 11 15 V 12 16 V 11 15 V 12 15 V 11 16 V 12 15 V 12 15 V 11 16 V 12 15 V 11 15 V 12 15 V 12 16 V 11 15 V 12 15 V 11 16 V 12 15 V 12 15 V 11 16 V 12 15 V 11 15 V 12 16 V 12 15 V 11 15 V 12 15 V 11 16 V 12 15 V 11 15 V 12 16 V 11 15 V 12 15 V 11 16 V 12 15 V 11 15 V 12 16 V 11 15 V 12 15 V 11 15 V 11 16 V 12 15 V 11 15 V 11 16 V 12 15 V 11 15 V 11 16 V 11 15 V 12 15 V 11 16 V 11 15 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 10 15 V 11 16 V 11 15 V 11 15 V 10 15 V 11 16 V 11 15 V 10 15 V 11 16 V 10 15 V 11 15 V 10 16 V 10 15 V 11 15 V 10 16 V 10 15 V 11 15 V 10 15 V 10 16 V 10 15 V 11 15 V 10 16 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 10 16 V 10 15 V 11 15 V 10 16 V 10 15 V 10 15 V 11 16 V 10 15 V 10 15 V 11 15 V 10 16 V 10 15 V 11 15 V 10 16 V 10 15 V 10 15 V 11 16 V 10 15 V 10 15 V 10 16 V 10 15 V 11 15 V 10 15 V 10 16 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 10 16 V 9 15 V 10 15 V 10 15 V 10 16 V 10 15 V 9 15 V 10 16 V 10 15 V 10 15 V 9 16 V 10 15 V 10 15 V 9 16 V 10 15 V 9 15 V 10 15 V 9 16 V 10 15 V 9 15 V 10 16 V 9 15 V 10 15 V 9 16 V 10 15 V 9 15 V 10 16 V 9 15 V 9 15 V 10 15 V 9 16 V 10 15 V 9 15 V 9 16 V 10 15 V 9 15 V 9 16 V 9 15 V 10 15 V 9 15 V 9 16 V 9 15 V 10 15 V 9 16 V 9 15 V 9 15 V 10 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 10 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 10 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 10 16 V 9 15 V 1.000 UL LT0 4739 280 M 0 15 R 0 16 R 0 15 R 0 15 R 0 16 R 1 0 V 0 15 R 0 15 R 1 15 R 0 16 R 1 15 R 1 15 R 0 16 R 1 15 R 1 15 R 1 16 R 1 15 R 1 15 R 1 16 R 1 15 R 1 15 R 2 15 R 1 16 R 1 15 R 2 15 R 1 16 R 1 0 V 1 15 R 2 15 R 2 16 R 1 15 R 2 15 R 2 16 R 2 15 R 2 15 R 2 15 R 2 16 R 1 0 V 2 15 R 2 15 R 2 16 R 3 15 R 2 15 R 3 16 R 3 15 R 2 15 R 3 15 R 3 16 R 3 15 R 3 15 R 3 16 R 3 15 R 3 15 R 3 16 R 3 15 R 4 15 R 3 16 R 3 15 R 1 0 V 3 15 R 4 15 R 3 16 R 4 15 R 4 15 R 4 16 R 3 15 R 4 15 R 4 16 R 5 15 R 4 15 R 4 16 R 4 15 R 5 15 R 4 15 R 4 16 R 5 15 R 5 15 R 4 16 R 5 15 R 5 15 R 5 16 R 4 15 R 1 0 V 4 15 R 1 0 V 5 16 R 5 15 R 5 15 R 5 15 R 5 16 R 6 15 R 5 15 R 6 16 R 5 15 R 6 15 R 6 16 R 5 15 R 6 15 R 6 16 R 6 15 R 6 15 R 6 15 R 6 16 R 1 0 V 5 15 R 1 0 V 6 15 R 6 16 R 6 15 R 1 0 V 6 15 R 6 16 R 1 0 V 6 15 R 1 0 V 6 15 R 7 16 R 6 15 R 1 0 V 6 15 R 1 0 V 6 15 R 1 0 V 6 16 R 1 0 V 6 15 R 1 0 V 6 15 R 1 0 V 7 16 R 7 15 R 1 0 V 6 15 R 1 0 V 7 16 R 1 0 V 6 15 R 1 0 V 7 15 R 1 0 V 6 15 R 1 0 V 7 16 R 1 0 V 7 15 R 1 0 V 6 15 R 2 0 V 6 16 R 2 0 V 6 15 R 2 0 V 6 15 R 2 0 V 6 16 R 2 0 V 6 15 R 2 0 V 7 15 R 2 0 V 6 16 R 2 0 V 6 15 R 3 0 V 6 15 R 2 0 V 6 15 R 3 0 V 6 16 R 3 0 V 5 15 R 3 0 V 6 15 R 3 0 V 6 16 R 3 0 V 6 15 R 3 0 V 6 15 R 3 0 V 6 16 R 4 0 V 5 15 R 4 0 V 5 15 R 4 0 V 5 16 R 5 0 V 4 15 R 5 0 V 5 15 R 5 0 V 4 15 R 5 0 V 4 16 R 6 0 V 4 15 R 6 0 V 4 15 R 6 0 V 3 16 R 7 0 V 3 15 R 7 0 V 3 15 R 7 0 V 3 16 R 7 0 V 3 15 R 7 0 V 3 15 R 8 0 V 2 16 R 8 0 V 2 15 R 9 0 V 1 15 R 10 0 V 1 15 R 9 0 V 1 16 R 10 0 V 1 15 R 10 0 V 0 15 R 11 0 V 0 16 R 11 0 V 0 15 R 11 0 V -1 15 R 13 0 V -2 16 R 13 0 V -2 15 R 13 0 V -2 15 R 14 0 V -3 16 R 14 0 V -2 15 R 14 0 V -3 15 R 15 0 V -4 15 R 16 0 V -4 16 R 16 0 V -5 15 R 17 0 V -5 15 R 17 0 V -6 16 R 18 0 V -6 15 R 18 0 V -6 15 R 19 0 V -7 16 R 19 0 V -7 15 R 20 0 V -8 15 R 20 0 V -8 16 R 21 0 V -9 15 R 22 0 V -10 15 R 23 0 V -10 15 R 23 0 V -11 16 R 24 0 V -11 15 R 24 0 V -12 15 R 25 0 V -12 16 R 25 0 V -12 15 R 25 0 V -13 15 R 27 0 V -14 16 R 27 0 V -14 15 R 28 0 V -15 15 R 28 0 V -14 15 R 28 0 V -15 16 R 29 0 V -16 15 R 30 0 V -16 15 R 30 0 V -17 16 R 30 0 V -16 15 R 30 0 V -16 15 R 30 0 V -16 16 R 30 0 V -16 15 R 30 0 V -16 15 R 30 0 V -16 16 R 30 0 V -15 15 R 30 0 V -16 15 R 30 0 V -15 15 R 29 0 V -15 16 R 29 0 V -14 15 R 29 0 V -14 15 R 28 0 V -13 16 R 27 0 V -12 15 R 27 0 V -12 15 R 27 0 V -12 16 R 26 0 V -11 15 R 26 0 V -11 15 R 26 0 V -11 16 R 26 0 V -10 15 R 24 0 V -9 15 R 24 0 V -8 15 R 23 0 V -8 16 R 24 0 V -8 15 R 23 0 V -8 15 R 23 0 V -7 16 R 22 0 V -6 15 R 21 0 V -5 15 R 21 0 V -6 16 R 21 0 V -5 15 R 21 0 V -5 15 R 20 0 V -4 16 R 20 0 V -4 15 R 20 0 V -4 15 R 19 0 V -2 15 R 18 0 V -2 16 R 18 0 V -2 15 R 18 0 V -2 15 R 18 0 V -1 16 R 17 0 V -1 15 R 17 0 V 0 15 R 16 0 V 0 16 R 16 0 V 1 15 R 15 0 V 1 15 R 16 0 V 1 16 R 15 0 V 1 15 R 15 0 V 2 15 R 15 0 V 2 15 R 14 0 V 3 16 R 14 0 V 3 15 R 13 0 V currentpoint stroke M 3 15 R 14 0 V 3 16 R 13 0 V 4 15 R 13 0 V 4 15 R 13 0 V 4 16 R 12 0 V 5 15 R 12 0 V 5 15 R 12 0 V 6 16 R 11 0 V 6 15 R 11 0 V 1.000 UL LT0 4739 280 M 0 15 V 0 16 V 0 15 V 0 15 V 0 16 V 1 15 V 0 15 V 1 15 V 0 16 V 1 15 V 1 15 V 0 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 2 15 V 1 16 V 1 15 V 2 15 V 1 16 V 2 15 V 2 15 V 2 16 V 1 15 V 2 15 V 2 16 V 2 15 V 2 15 V 2 15 V 2 16 V 3 15 V 2 15 V 2 16 V 3 15 V 2 15 V 3 16 V 3 15 V 2 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 16 V 3 15 V 4 15 V 3 16 V 3 15 V 4 15 V 4 15 V 3 16 V 4 15 V 4 15 V 4 16 V 3 15 V 4 15 V 4 16 V 5 15 V 4 15 V 4 16 V 4 15 V 5 15 V 4 15 V 4 16 V 5 15 V 5 15 V 4 16 V 5 15 V 5 15 V 5 16 V 4 15 V 5 15 V 6 16 V 5 15 V 5 15 V 5 15 V 5 16 V 6 15 V 5 15 V 6 16 V 5 15 V 6 15 V 6 16 V 5 15 V 6 15 V 6 16 V 6 15 V 6 15 V 6 15 V 6 16 V 6 15 V 7 15 V 6 16 V 6 15 V 7 15 V 6 16 V 7 15 V 7 15 V 7 16 V 6 15 V 7 15 V 7 15 V 7 16 V 7 15 V 7 15 V 8 16 V 7 15 V 7 15 V 8 16 V 7 15 V 8 15 V 7 15 V 8 16 V 8 15 V 7 15 V 8 16 V 8 15 V 8 15 V 8 16 V 8 15 V 9 15 V 8 16 V 8 15 V 9 15 V 8 15 V 9 16 V 8 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 9 15 V 9 16 V 9 15 V 10 15 V 9 15 V 9 16 V 10 15 V 10 15 V 9 16 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 11 15 V 10 16 V 11 15 V 10 15 V 11 16 V 11 15 V 10 15 V 11 16 V 11 15 V 11 15 V 11 16 V 12 15 V 11 15 V 11 15 V 12 16 V 11 15 V 12 15 V 11 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 13 15 V 12 16 V 13 15 V 12 15 V 13 16 V 13 15 V 12 15 V 13 16 V 13 15 V 13 15 V 14 15 V 13 16 V 13 15 V 14 15 V 13 16 V 14 15 V 14 15 V 14 16 V 14 15 V 14 15 V 14 16 V 15 15 V 14 15 V 15 15 V 14 16 V 15 15 V 15 15 V 15 16 V 15 15 V 15 15 V 15 16 V 15 15 V 15 15 V 15 16 V 16 15 V 15 15 V 16 15 V 15 16 V 16 15 V 15 15 V 16 16 V 16 15 V 16 15 V 15 16 V 16 15 V 16 15 V 16 16 V 16 15 V 16 15 V 17 15 V 16 16 V 16 15 V 16 15 V 17 16 V 16 15 V 17 15 V 16 16 V 17 15 V 16 15 V 17 16 V 16 15 V 17 15 V 17 15 V 17 16 V 17 15 V 16 15 V 17 16 V 17 15 V 17 15 V 17 16 V 17 15 V 17 15 V 18 16 V 17 15 V 14 12 V 1.000 UL LT0 4739 280 M 0 15 V 0 16 V 0 15 V 0 15 V 1 16 V 0 15 V 0 15 V 1 15 V 0 16 V 1 15 V 1 15 V 0 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 2 15 V 1 16 V 1 15 V 2 15 V 2 16 V 1 15 V 2 15 V 2 16 V 1 15 V 2 15 V 2 16 V 2 15 V 2 15 V 2 15 V 3 16 V 2 15 V 2 15 V 2 16 V 3 15 V 2 15 V 3 16 V 3 15 V 2 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 16 V 3 15 V 4 15 V 3 16 V 4 15 V 3 15 V 4 15 V 3 16 V 4 15 V 4 15 V 4 16 V 3 15 V 4 15 V 4 16 V 5 15 V 4 15 V 4 16 V 4 15 V 5 15 V 4 15 V 4 16 V 5 15 V 5 15 V 4 16 V 5 15 V 5 15 V 5 16 V 5 15 V 5 15 V 5 16 V 5 15 V 5 15 V 5 15 V 5 16 V 6 15 V 5 15 V 6 16 V 5 15 V 6 15 V 6 16 V 5 15 V 6 15 V 6 16 V 6 15 V 6 15 V 6 15 V 7 16 V 6 15 V 6 15 V 6 16 V 7 15 V 6 15 V 7 16 V 7 15 V 6 15 V 7 16 V 7 15 V 7 15 V 7 15 V 7 16 V 7 15 V 7 15 V 7 16 V 8 15 V 7 15 V 8 16 V 7 15 V 8 15 V 7 15 V 8 16 V 8 15 V 8 15 V 8 16 V 8 15 V 8 15 V 8 16 V 8 15 V 9 15 V 8 16 V 9 15 V 8 15 V 9 15 V 9 16 V 8 15 V 9 15 V 9 16 V 9 15 V 9 15 V 10 16 V 9 15 V 9 15 V 10 16 V 9 15 V 10 15 V 9 15 V 10 16 V 10 15 V 10 15 V 10 16 V 10 15 V 10 15 V 10 16 V 10 15 V 11 15 V 10 16 V 11 15 V 11 15 V 10 15 V 11 16 V 11 15 V 11 15 V 11 16 V 11 15 V 12 15 V 11 16 V 11 15 V 12 15 V 11 16 V 12 15 V 12 15 V 12 15 V 12 16 V 12 15 V 12 15 V 12 16 V 12 15 V 13 15 V 12 16 V 13 15 V 12 15 V 13 16 V 13 15 V 13 15 V 13 15 V 13 16 V 13 15 V 13 15 V 13 16 V 13 15 V 14 15 V 13 16 V 14 15 V 13 15 V 14 15 V 14 16 V 14 15 V 14 15 V 13 16 V 14 15 V 14 15 V 14 16 V 14 15 V 14 15 V 14 16 V 15 15 V 14 15 V 14 15 V 14 16 V 15 15 V 14 15 V 14 16 V 15 15 V 15 15 V 14 16 V 15 15 V 15 15 V 15 16 V 14 15 V 15 15 V 15 15 V 16 16 V 15 15 V 15 15 V 15 16 V 15 15 V 16 15 V 15 16 V 16 15 V 15 15 V 16 16 V 16 15 V 15 15 V 16 15 V 16 16 V 16 15 V 16 15 V 16 16 V 16 15 V 16 15 V 16 16 V 16 15 V 17 15 V 16 16 V 16 15 V 17 15 V 16 15 V 17 16 V 16 15 V 17 15 V 16 16 V 17 15 V 17 15 V 16 16 V 17 15 V 17 15 V 17 16 V 17 15 V 3 3 V 1.000 UL LT0 1329 280 M 0 15 R 0 16 R -1 15 R 1 0 V -1 15 R 0 16 R -1 15 R -1 15 R -1 15 R 1 0 V -2 16 R 1 0 V -2 15 R 1 0 V -2 15 R 1 0 V -2 16 R -1 15 R -2 15 R -1 16 R -2 15 R -2 15 R -2 16 R 1 0 V -3 15 R 1 0 V -3 15 R 1 0 V -3 15 R -2 16 R -2 15 R -3 15 R -2 16 R -3 15 R -3 15 R 1 0 V -3 16 R -3 15 R -3 15 R -3 16 R -3 15 R -3 15 R -3 15 R -3 16 R -4 15 R 1 0 V -4 15 R -3 16 R -4 15 R -3 15 R -4 16 R -3 15 R -4 15 R -4 15 R 1 0 V -4 16 R -4 15 R -4 15 R -4 16 R -4 15 R 1 0 V -5 15 R 1 0 V -4 16 R -4 15 R -4 15 R -4 16 R -4 15 R -5 15 R 1 0 V -5 15 R 1 0 V -5 16 R 1 0 V -5 15 R -4 15 R -4 16 R -4 15 R -4 15 R -5 16 R 1 0 V -5 15 R -4 15 R -4 16 R -4 15 R -5 15 R 1 0 V -5 15 R -4 16 R -4 15 R -4 15 R -5 16 R 1 0 V -5 15 R -4 15 R -4 16 R -4 15 R -5 15 R 1 0 V -5 16 R -4 15 R -4 15 R -4 15 R -4 16 R -5 15 R 1 0 V -5 15 R 1 0 V -5 16 R -4 15 R -4 15 R -4 16 R -4 15 R -4 15 R -4 16 R -4 15 R -4 15 R -4 15 R -4 16 R -4 15 R 1 0 V -5 15 R 1 0 V -5 16 R 1 0 V -5 15 R 1 0 V -4 15 R -4 16 R -4 15 R 1 0 V -5 15 R 1 0 V -5 16 R 1 0 V -4 15 R 1 0 V -5 15 R 1 0 V -5 15 R 1 0 V -4 16 R 1 0 V -5 15 R 1 0 V -5 15 R 2 0 V -5 16 R 1 0 V -5 15 R 2 0 V -5 15 R 1 0 V -5 16 R 2 0 V -5 15 R 2 0 V -6 15 R 2 0 V -5 15 R 2 0 V -5 16 R 2 0 V -6 15 R 3 0 V -6 15 R 2 0 V -5 16 R 2 0 V -6 15 R 3 0 V -6 15 R 3 0 V -6 16 R 3 0 V -6 15 R 3 0 V -6 15 R 3 0 V -6 16 R 3 0 V -6 15 R 4 0 V -7 15 R 4 0 V -7 15 R 4 0 V -7 16 R 4 0 V -7 15 R 5 0 V -8 15 R 5 0 V -8 16 R 6 0 V -9 15 R 6 0 V -9 15 R 7 0 V -9 16 R 6 0 V -9 15 R 7 0 V -10 15 R 8 0 V -10 16 R 7 0 V -10 15 R 8 0 V -10 15 R 8 0 V -11 15 R 9 0 V -11 16 R 9 0 V -12 15 R 10 0 V -12 15 R 10 0 V -12 16 R 10 0 V -13 15 R 12 0 V -14 15 R 12 0 V -14 16 R 12 0 V -14 15 R 13 0 V -15 15 R 13 0 V -15 16 R 14 0 V -16 15 R 14 0 V -16 15 R 15 0 V -17 15 R 16 0 V -18 16 R 17 0 V -19 15 R 17 0 V -19 15 R 18 0 V -19 16 R 18 0 V -20 15 R 19 0 V -20 15 R 20 0 V -22 16 R 21 0 V -23 15 R 22 0 V -23 15 R 22 0 V -23 16 R 23 0 V -25 15 R 24 0 V -25 15 R 25 0 V -26 15 R 26 0 V -27 16 R 26 0 V -28 15 R 28 0 V -29 15 R 29 0 V -30 16 R 30 0 V -31 15 R 31 0 V -31 15 R 31 0 V -32 16 R 32 0 V -33 15 R 33 0 V -34 15 R 35 0 V -36 16 R 36 0 V -36 15 R 36 0 V -37 15 R 38 0 V -38 15 R 38 0 V -39 16 R 40 0 V -40 15 R 40 0 V -41 15 R 42 0 V -42 16 R 43 0 V -43 15 R 44 0 V -44 15 R 44 0 V -44 16 R 45 0 V -46 15 R 47 0 V -47 15 R 48 0 V -48 15 R 49 0 V -48 16 R 49 0 V -49 15 R 50 0 V -50 15 R 51 0 V -50 16 R 51 0 V -51 15 R 52 0 V -51 15 R 52 0 V -51 16 R 52 0 V -51 15 R 52 0 V -51 15 R 52 0 V -51 16 R 51 0 V -50 15 R 51 0 V -49 15 R 50 0 V -49 15 R 50 0 V -48 16 R 49 0 V -48 15 R 49 0 V -47 15 R 48 0 V -46 16 R 47 0 V -46 15 R 47 0 V -45 15 R 46 0 V -44 16 R 45 0 V -43 15 R 44 0 V -42 15 R 43 0 V -41 16 R 42 0 V -40 15 R 41 0 V -39 15 R 41 0 V -39 15 R 40 0 V -38 16 R 40 0 V -37 15 R 38 0 V -36 15 R 37 0 V -35 16 R 37 0 V -35 15 R 37 0 V -34 15 R 35 0 V -33 16 R 35 0 V -33 15 R 35 0 V -32 15 R 33 0 V -31 16 R 33 0 V -30 15 R 32 0 V -29 15 R 31 0 V -29 15 R 31 0 V -28 16 R 30 0 V -28 15 R 30 0 V -27 15 R 29 0 V -26 16 R 28 0 V -25 15 R 27 0 V currentpoint stroke M -25 15 R 28 0 V -25 16 R 27 0 V -24 15 R 26 0 V -23 15 R 25 0 V -22 16 R 25 0 V -22 15 R 24 0 V -21 15 R 24 0 V -21 15 R 23 0 V -20 16 R 23 0 V -20 15 R 22 0 V -19 15 R 22 0 V -19 16 R 21 0 V -18 15 R 21 0 V -18 15 R 21 0 V -18 16 R 21 0 V -17 15 R 19 0 V -16 15 R 19 0 V -16 16 R 19 0 V -16 15 R 19 0 V -15 15 R 18 0 V -15 15 R 18 0 V -15 16 R 18 0 V -14 15 R 17 0 V -14 15 R 17 0 V -13 16 R 16 0 V -13 15 R 16 0 V -13 15 R 16 0 V -12 16 R 15 0 V -12 15 R 15 0 V -11 15 R 14 0 V -11 15 R 15 0 V -11 16 R 14 0 V -11 15 R 14 0 V -10 15 R 13 0 V -9 16 R 13 0 V -10 15 R 13 0 V -9 15 R 12 0 V -9 16 R 13 0 V -9 15 R 12 0 V -8 15 R 11 0 V -8 16 R 12 0 V -8 15 R 11 0 V -7 15 R 11 0 V -7 15 R 10 0 V -7 16 R 11 0 V -7 15 R 10 0 V -6 15 R 10 0 V -6 16 R 10 0 V -7 15 R 10 0 V -6 15 R 10 0 V -6 16 R 9 0 V -5 15 R 9 0 V -5 15 R 9 0 V -5 16 R 8 0 V -5 15 R 9 0 V -5 15 R 9 0 V -5 15 R 8 0 V -4 16 R 8 0 V -4 15 R 8 0 V -4 15 R 8 0 V -4 16 R 7 0 V -3 15 R 7 0 V 1.000 UL LT0 1329 280 M 0 15 V 0 16 V -1 15 V 0 15 V 0 16 V -1 15 V -1 15 V -1 15 V -1 16 V -1 15 V -1 15 V -1 16 V -1 15 V -2 15 V -1 16 V -2 15 V -2 15 V -2 16 V -2 15 V -2 15 V -2 15 V -2 16 V -2 15 V -3 15 V -2 16 V -3 15 V -3 15 V -2 16 V -3 15 V -3 15 V -3 16 V -3 15 V -3 15 V -3 15 V -3 16 V -4 15 V -3 15 V -3 16 V -4 15 V -3 15 V -4 16 V -3 15 V -4 15 V -4 15 V -3 16 V -4 15 V -4 15 V -4 16 V -4 15 V -4 15 V -3 16 V -4 15 V -4 15 V -4 16 V -4 15 V -5 15 V -4 15 V -4 16 V -4 15 V -4 15 V -4 16 V -4 15 V -4 15 V -5 16 V -4 15 V -4 15 V -4 16 V -4 15 V -5 15 V -4 15 V -4 16 V -4 15 V -4 15 V -5 16 V -4 15 V -4 15 V -4 16 V -4 15 V -5 15 V -4 16 V -4 15 V -4 15 V -4 15 V -4 16 V -5 15 V -4 15 V -4 16 V -4 15 V -4 15 V -4 16 V -4 15 V -4 15 V -4 16 V -4 15 V -4 15 V -4 15 V -4 16 V -4 15 V -4 15 V -4 16 V -4 15 V -3 15 V -4 16 V -4 15 V -4 15 V -4 16 V -3 15 V -4 15 V -4 15 V -3 16 V -4 15 V -4 15 V -3 16 V -4 15 V -3 15 V -4 16 V -3 15 V -4 15 V -3 15 V -3 16 V -4 15 V -3 15 V -3 16 V -4 15 V -3 15 V -3 16 V -3 15 V -3 15 V -3 16 V -3 15 V -3 15 V -3 15 V -3 16 V -3 15 V -3 15 V -3 16 V -3 15 V -3 15 V -2 16 V -3 15 V -3 15 V -2 16 V -3 15 V -2 15 V -3 15 V -2 16 V -3 15 V -2 15 V -2 16 V -3 15 V -2 15 V -2 16 V -2 15 V -2 15 V -2 16 V -2 15 V -2 15 V -2 15 V -2 16 V -2 15 V -2 15 V -1 16 V -2 15 V -1 15 V -2 16 V -2 15 V -1 15 V -1 16 V -2 15 V -1 15 V -1 15 V -1 16 V -2 15 V -1 15 V -1 16 V -1 15 V 0 15 V -1 16 V -1 15 V -1 15 V -1 16 V 0 15 V -1 15 V 0 15 V -1 16 V 0 15 V -1 15 V 0 16 V 0 15 V 0 15 V 0 16 V -1 15 V 0 15 V 0 15 V 1 16 V 0 15 V 0 15 V 1 16 V 0 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 2 15 V 1 15 V 2 16 V 1 15 V 2 15 V 2 16 V 1 15 V 2 15 V 2 16 V 2 15 V 2 15 V 2 16 V 2 15 V 2 15 V 2 15 V 2 16 V 3 15 V 2 15 V 2 16 V 2 15 V 3 15 V 2 16 V 2 15 V 3 15 V 2 16 V 3 15 V 3 15 V 2 15 V 3 16 V 2 15 V 3 15 V 3 16 V 3 15 V 2 15 V 3 16 V 3 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 16 V 4 15 V 3 15 V 3 16 V 3 15 V 4 15 V 3 15 V 3 16 V 4 15 V 3 15 V 4 16 V 3 15 V 3 15 V 4 16 V 3 15 V 4 15 V 3 15 V 4 16 V 3 15 V 4 15 V 4 16 V 3 15 V 4 15 V 3 16 V 4 15 V 4 15 V 3 16 V 4 15 V 4 15 V 4 15 V 3 16 V 4 15 V 4 15 V 4 16 V 3 15 V 4 15 V 4 16 V 4 15 V 4 15 V 4 16 V 3 15 V 4 15 V 4 15 V 4 16 V 4 15 V 4 15 V 4 16 V 4 15 V 7 0 V 1.000 UL LT0 1329 280 M 0 15 V 0 16 V 0 15 V -1 15 V 0 16 V -1 15 V -1 15 V 0 15 V -1 16 V -1 15 V -1 15 V -2 16 V -1 15 V -2 15 V -1 16 V -2 15 V -2 15 V -1 16 V -2 15 V -2 15 V -3 15 V -2 16 V -2 15 V -3 15 V -2 16 V -3 15 V -2 15 V -3 16 V -3 15 V -3 15 V -3 16 V -3 15 V -3 15 V -3 15 V -3 16 V -3 15 V -4 15 V -3 16 V -4 15 V -3 15 V -4 16 V -3 15 V -4 15 V -3 15 V -4 16 V -4 15 V -4 15 V -4 16 V -3 15 V -4 15 V -4 16 V -4 15 V -4 15 V -4 16 V -4 15 V -4 15 V -4 15 V -4 16 V -5 15 V -4 15 V -4 16 V -4 15 V -4 15 V -4 16 V -5 15 V -4 15 V -4 16 V -4 15 V -4 15 V -5 15 V -4 16 V -4 15 V -4 15 V -4 16 V -5 15 V -4 15 V -4 16 V -4 15 V -4 15 V -5 16 V -4 15 V -4 15 V -4 15 V -4 16 V -4 15 V -4 15 V -5 16 V -4 15 V -4 15 V -4 16 V -4 15 V -4 15 V -4 16 V -4 15 V -4 15 V -4 15 V -4 16 V -3 15 V -4 15 V -4 16 V -4 15 V -4 15 V -4 16 V -3 15 V -4 15 V -4 16 V -3 15 V -4 15 V -4 15 V -3 16 V -4 15 V -3 15 V -4 16 V -3 15 V -4 15 V -3 16 V -3 15 V -4 15 V -3 15 V -3 16 V -3 15 V -4 15 V -3 16 V -3 15 V -3 15 V -3 16 V -3 15 V -3 15 V -3 16 V -2 15 V -3 15 V -3 15 V -3 16 V -2 15 V -3 15 V -2 16 V -3 15 V -2 15 V -3 16 V -2 15 V -2 15 V -3 16 V -2 15 V -2 15 V -2 15 V -2 16 V -2 15 V -2 15 V -2 16 V -1 15 V -2 15 V -2 16 V -1 15 V -2 15 V -1 16 V -2 15 V -1 15 V -1 15 V -1 16 V -2 15 V -1 15 V -1 16 V -1 15 V 0 15 V -1 16 V -1 15 V -1 15 V 0 16 V -1 15 V 0 15 V 0 15 V -1 16 V 0 15 V 0 15 V 0 16 V 0 15 V 0 15 V 0 16 V 0 15 V 1 15 V 0 16 V 0 15 V 1 15 V 0 15 V 1 16 V 0 15 V 1 15 V 1 16 V 1 15 V 0 15 V 1 16 V 1 15 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 0 16 V 1 15 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 1 15 V 1 16 V 1 15 V 2 15 V 1 15 V 2 16 V 1 15 V 1 15 V 2 16 V 2 15 V 1 15 V 2 16 V 2 15 V 1 15 V 2 16 V 2 15 V 2 15 V 2 15 V 2 16 V 2 15 V 2 15 V 2 16 V 2 15 V 3 15 V 2 16 V 2 15 V 2 15 V 3 16 V 2 15 V 3 15 V 2 15 V 3 16 V 2 15 V 3 15 V 2 16 V 3 15 V 3 15 V 3 16 V 2 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 16 V 3 15 V 3 15 V 3 16 V 3 15 V 3 15 V 4 15 V 3 16 V 3 15 V 3 15 V 4 16 V 3 15 V 3 15 V 4 16 V 3 15 V 3 15 V 4 16 V 3 15 V 4 15 V 3 15 V 4 16 V 3 15 V 4 15 V 4 16 V 3 15 V 4 15 V 3 16 V 4 15 V 4 15 V 3 16 V 4 15 V 4 15 V 3 15 V 4 16 V 4 15 V 4 15 V 3 16 V 4 15 V 1.000 UL LT0 4131 280 M -1 0 V 1 15 R -1 0 V 1 16 R 0 15 R 0 15 R 1 16 R -1 0 V 1 15 R 1 15 R -1 0 V 1 15 R 1 16 R 1 15 R -1 0 V 1 15 R 1 16 R 1 15 R 1 15 R 1 16 R 2 15 R -1 0 V 2 15 R 1 16 R 2 15 R 1 15 R 2 15 R 2 16 R -1 0 V 2 15 R 2 15 R 2 16 R 2 15 R 2 15 R 2 16 R 3 15 R -1 0 V 3 15 R 2 16 R 3 15 R 2 15 R 3 15 R 3 16 R 2 15 R 3 15 R 3 16 R 3 15 R 3 15 R 4 16 R -1 0 V 4 15 R 3 15 R 4 15 R -1 0 V 4 16 R 4 15 R 3 15 R 4 16 R 4 15 R 4 15 R 4 16 R 4 15 R 4 15 R 4 16 R 4 15 R 5 15 R 4 15 R 5 16 R 4 15 R 5 15 R 5 16 R 5 15 R -1 0 V 6 15 R -1 0 V 6 16 R -1 0 V 6 15 R -1 0 V 6 15 R 5 16 R 5 15 R 6 15 R 5 15 R 6 16 R 5 15 R 6 15 R 6 16 R 6 15 R -1 0 V 6 15 R 6 16 R 7 15 R -1 0 V 7 15 R 6 16 R 6 15 R 7 15 R 6 15 R 7 16 R -1 0 V 7 15 R 7 15 R 7 16 R 6 15 R 7 15 R 7 16 R 7 15 R 8 15 R -1 0 V 8 16 R 7 15 R 7 15 R 8 15 R 7 16 R 8 15 R 7 15 R 8 16 R 8 15 R 8 15 R 8 16 R -1 0 V 9 15 R -1 0 V 9 15 R -1 0 V 9 16 R 8 15 R 8 15 R 9 15 R -1 0 V 9 16 R 9 15 R -1 0 V 9 15 R 9 16 R -1 0 V 9 15 R 9 15 R -1 0 V 10 16 R -1 0 V 10 15 R -1 0 V 10 15 R -1 0 V 10 15 R -1 0 V 10 16 R -1 0 V 10 15 R -1 0 V 10 15 R -1 0 V 10 16 R -1 0 V 11 15 R -2 0 V 11 15 R -1 0 V 10 16 R -1 0 V 11 15 R -2 0 V 11 15 R -1 0 V 11 16 R -2 0 V 12 15 R -2 0 V 11 15 R -2 0 V 12 15 R -2 0 V 12 16 R -2 0 V 12 15 R -2 0 V 12 15 R -3 0 V 13 16 R -3 0 V 13 15 R -3 0 V 13 15 R -3 0 V 13 16 R -3 0 V 13 15 R -3 0 V 14 15 R -4 0 V 14 16 R -4 0 V 14 15 R -4 0 V 15 15 R -4 0 V 14 15 R -4 0 V 15 16 R -5 0 V 15 15 R -4 0 V 15 15 R -5 0 V 16 16 R -6 0 V 17 15 R -6 0 V 16 15 R -6 0 V 17 16 R -6 0 V 17 15 R -7 0 V 18 15 R -7 0 V 18 16 R -7 0 V 18 15 R -8 0 V 19 15 R -8 0 V 19 15 R -8 0 V 20 16 R -9 0 V 20 15 R -10 0 V 21 15 R -10 0 V 21 16 R -10 0 V 22 15 R -11 0 V 22 15 R -11 0 V 23 16 R -12 0 V 23 15 R -12 0 V 24 15 R -13 0 V 24 16 R -13 0 V 25 15 R -14 0 V 26 15 R -15 0 V 27 15 R -16 0 V 27 16 R -15 0 V 27 15 R -16 0 V 28 15 R -17 0 V 29 16 R -18 0 V 30 15 R -18 0 V 30 15 R -19 0 V 31 16 R -20 0 V 32 15 R -20 0 V 32 15 R -21 0 V 34 16 R -22 0 V 34 15 R -23 0 V 35 15 R -23 0 V 36 15 R -24 0 V 36 16 R -25 0 V 37 15 R -25 0 V 38 15 R -26 0 V 39 16 R -28 0 V 40 15 R -28 0 V 41 15 R -29 0 V 41 16 R -29 0 V 42 15 R -30 0 V 43 15 R -31 0 V 44 15 R -32 0 V 44 16 R -32 0 V 45 15 R -33 0 V 46 15 R -34 0 V 47 16 R -34 0 V 47 15 R -35 0 V 47 15 R -34 0 V 47 16 R -34 0 V 47 15 R -35 0 V 47 15 R -34 0 V 47 16 R -34 0 V 47 15 R -33 0 V 45 15 R -32 0 V 45 15 R -32 0 V 45 16 R -32 0 V 45 15 R -31 0 V 43 15 R -30 0 V 43 16 R -30 0 V 43 15 R -29 0 V 42 15 R -29 0 V 42 16 R -28 0 V 41 15 R -28 0 V 41 15 R -27 0 V 40 16 R -26 0 V 39 15 R -26 0 V 39 15 R -25 0 V 38 15 R -24 0 V 38 16 R -25 0 V 38 15 R -24 0 V 37 15 R -23 0 V 36 16 R -22 0 V 36 15 R -22 0 V 35 15 R -21 0 V 34 16 R -21 0 V 35 15 R -21 0 V 34 15 R -20 0 V 34 16 R -20 0 V 33 15 R -19 0 V 33 15 R -19 0 V 32 15 R -18 0 V 32 16 R -18 0 V 32 15 R -18 0 V 31 15 R -17 0 V 31 16 R -17 0 V 31 15 R -16 0 V 29 15 R -15 0 V 29 16 R -15 0 V 29 15 R -15 0 V 29 15 R -15 0 V 29 16 R -15 0 V 28 15 R -13 0 V 27 15 R -13 0 V currentpoint stroke M 27 15 R -13 0 V 27 16 R -13 0 V 27 15 R -12 0 V 26 15 R -12 0 V 26 16 R -12 0 V 26 15 R -11 0 V 25 15 R -11 0 V 25 16 R -11 0 V 26 15 R -11 0 V 25 15 R -11 0 V 25 16 R -10 0 V 24 15 R -10 0 V 24 15 R -9 0 V 23 15 R -9 0 V 24 16 R -9 0 V 23 15 R -9 0 V 23 15 R -8 0 V 23 16 R -9 0 V 23 15 R -8 0 V 22 15 R -8 0 V 23 16 R -8 0 V 22 15 R -7 0 V 22 15 R -8 0 V 22 15 R -7 0 V 22 16 R -8 0 V 22 15 R -7 0 V 22 15 R -7 0 V 21 16 R -6 0 V 21 15 R -7 0 V 21 15 R -6 0 V 21 16 R -6 0 V 20 15 R -5 0 V 20 15 R 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b(\(for)g(a)-118 4304 y(discussion)g(of)f (this)g(kind)g(of)h(con)m(trol)e(of)i(the)g(m)m(ultiplicites)c(see)k (next)h(c)m(hapter\),)g(whic)m(h)f(allo)m(ws)e(the)i(metho)s(d)-118 4425 y(\(of)e(KAM-t)m(yp)s(e\))i(to)f(succeed.)50 b(Results)34 b(of)f(this)h(kind,)g(but)h(with)e(more)h(restrictiv)m(e)g(h)m(yp)s (othesis)h(where)g(\014rst)-118 4545 y(obtained)24 b(in)g([83)o(],)i ([16])f(and)f([32])g(\(where)i(the)f(con)m(tin)m(uous)g(case)g(of)g (Mathieu's)f(equation)h(with)f(quasi-p)s(erio)s(dic)-118 4665 y(forcing)31 b(is)h(treated\).)28 4786 y(Finally)27 b(note)i(that,)h(due)g(to)f(Eliasson's)f(result)i(3.4.11,)f(in)f(the)i (KAM)f(zone)h(there)g(is)f(generically)e(a)i(dense)-118 4906 y(set)i(of)f(non-reducible)g(systems)h(\(from)e(those)i(with)f (rotation)f(n)m(um)m(b)s(er)i(neither)f(Diophan)m(tine)f(nor)h (rational\).)1900 6155 y(95)p eop %%Page: 96 100 96 99 bop -118 883 a Fz(Chapter)78 b(4)-118 1298 y(Reducibilit)-6 b(y)76 b(in)h(higher)g(dimensions)-118 1750 y Fy(In)48 b(this)f(c)m(hapter)h(w)m(e)g(discuss)h(sev)m(eral)f(results)g (concerning)f(the)h(reducibilit)m(y)e(of)h(linear)e(equations)j(with) -118 1871 y(quasi-p)s(erio)s(dic)30 b(co)s(e\016cien)m(ts)k(in)e (dimension)f(greater)i(than)f(t)m(w)m(o.)28 1991 y(First)42 b(of)f(all)f(w)m(e)k(presen)m(t)g(some)e(reducibilit)m(y)e(theorems)j (that)f(hold)f(in)h(arbitrary)f(groups)i(and)f(whic)m(h)-118 2111 y(are)e(pro)m(v)m(ed)i(b)m(y)f(means)g(of)f(KAM)g(tec)m(hniques,)k (together)d(with)f(their)g(connection)g(with)g(previous)h(theory)-8 b(,)-118 2232 y(esp)s(ecially)29 b(the)i(reducibilit)m(y)e(theorem)h (under)h(the)g(full)d(sp)s(ectrum)j(assumption)e(that)h(w)m(e)i(sa)m(w) f(in)f(the)g(second)-118 2352 y(c)m(hapter.)44 b(W)-8 b(e)33 b(presen)m(t)i(sev)m(eral)e(signi\014can)m(t)f(impro)m(v)m(emen) m(ts)h(of)f(these)i(results.)28 2473 y(In)25 b(the)g(second)g(part)f (of)g(the)h(c)m(hapter)g(w)m(e)h(study)f(the)g(problem)e(of)h (reducibilit)m(y)e(in)i(the)h(con)m(text)g(of)f(compact)-118 2593 y(groups)j(\(eg.)42 b Fx(S)6 b(O)s Fy(\(3\))25 b(or)i Fx(S)6 b(U)k Fy(\(2\)\).)42 b(F)-8 b(or)26 b(this)g(case)i(man)m(y)f (of)g(the)g(tec)m(hniques)i(of)d(the)i(previous)f(section)g(and)g(the) -118 2713 y(previous)i(c)m(hapter)i(are)e(used.)43 b(It)29 b(is)g(also)f(necessary)k(to)d(consider)g(some)g(geometrical)e (de\014nitions)h(in)h(order)g(to)-118 2834 y(b)s(e)35 b(able)f(to)h(prop)s(erly)f(form)m(ulate)g(the)h(results)g(for)g (compact)f(groups.)51 b(W)-8 b(e)36 b(will)c(consider)k(with)e(more)g (detail)-118 2954 y(the)g(case)h(of)e Fx(S)6 b(O)s Fy(\(3)p Fx(;)17 b Fw(R)t Fy(\),)40 b(for)33 b(whic)m(h)i(the)f(main)e(steps)k (of)d(the)h(pro)s(of)f(will)f(b)s(e)i(outlined,)f(but)h(without)g (getting)-118 3074 y(in)m(to)e(an)m(y)h(details.)42 b(Finally)30 b(some)j(con)m(v)m(erse)i(results)e(for)f(the)h(compact)f(case)h(are)g (presen)m(ted.)-118 3406 y Fu(4.1)161 b(Reducibilit)l(y)77 b(for)g(general)f(linear)i(di\013eren)l(tial)f(equations)250 3588 y(with)53 b(quasi-p)t(erio)t(dic)i(co)t(e\016cien)l(ts)-118 3807 y Fy(Consider)33 b(a)f(linear)f(system)j(of)e(di\013eren)m(tial)f (equations)h(with)h(quasi-p)s(erio)s(dic)d(co)s(e\016cien)m(ts:)1613 4137 y Fx(x)1668 4096 y Ft(0)1692 4137 y Fy(\()p Fx(t)p Fy(\))d(=)h Fx(A)p Fy(\()p Fx(t)p Fy(\))p Fx(x)p Fy(\()p Fx(t)p Fy(\))1531 b(\(4.1\))-118 4347 y(where)35 b Fx(x)c Fs(2)g Fw(R)413 4310 y Fr(n)500 4347 y Fy(and)k Fx(t)30 b Fs(7!)g Fx(A)g Fs(2)h Fx(g)i Fs(\032)e Fx(g)t(l)r Fy(\()p Fx(n;)17 b Fw(R)5 b Fy(\))39 b(\(or)34 b(in)f Fx(g)t(l)r Fy(\()p Fx(n;)17 b Fw(C)j Fy(\)\))40 b(is)33 b(quasi-p)s(erio)s(dic)f (with)i(frequency)i Fx(!)e Fs(2)c Fw(R)3969 4310 y Fr(d)-118 4467 y Fy(and)j Fx(g)i Fy(is)d(a)h(Lie)f(sub-algebra)g(of)g Fx(g)t(l)r Fy(\()p Fx(n;)17 b Fw(R)t Fy(\).)49 b(Therefore,)34 b(and)f(as)g(customary)-8 b(,)32 b(w)m(e)i(can)f(lift)d(the)j(ab)s(o)m (v)m(e)h(system)-118 4587 y(to)e(a)g(system)i(in)e Fw(R)586 4551 y Fr(n)661 4587 y Fs(\002)22 b Fw(T)823 4551 y Fr(d)900 4587 y Fy(\(resp.)44 b Fw(C)1248 4551 y Fr(n)1323 4587 y Fs(\002)23 b Fw(T)1486 4551 y Fr(d)1530 4587 y Fy(\))32 b(as)h(follo)m(ws)1443 4797 y Fx(x)1498 4756 y Ft(0)1549 4797 y Fy(=)1678 4771 y(~)1653 4797 y Fx(A)p Fy(\()p Fx(\022)s Fy(\))p Fx(x;)212 b(\022)2192 4756 y Ft(0)2243 4797 y Fy(=)28 b Fx(!)t(;)1376 b Fy(\(4.2\))-118 5006 y(where)38 b Fx(\022)h Fs(2)e Fw(T)418 4970 y Fr(d)499 5006 y Fy(are)g(the)h(angular)e(v)-5 b(ariables.)57 b(If)37 b(w)m(e)h(w)m(an)m(t)h(the)e(group)h(structure)g(giv)m(en)g(b)m(y)g (matrices)f Fx(A)g Fy(to)-118 5126 y(b)s(ecome)32 b(clearer,)h(w)m(e)g (can)g(also)f(consider)h(the)g(corresp)s(onding)f(matrix)f(equation) 1412 5336 y Fx(X)1501 5295 y Ft(0)1552 5336 y Fy(=)1681 5311 y(~)1655 5336 y Fx(A)q Fy(\()p Fx(\022)s Fy(\))p Fx(X)r(;)212 b(\022)2223 5295 y Ft(0)2274 5336 y Fy(=)27 b Fx(!)t(;)1346 b Fy(\(4.3\))-118 5545 y(where)42 b Fx(X)50 b Fs(2)43 b Fx(G)p Fy(,)g(the)f(group)f(generated)h(b)m(y)g(the)f (in\014nitesimal)d(algebra)i Fx(g)t Fy(.)69 b(This)41 b(de\014nes)i(a)e(linear)e(sk)m(ew-)-118 5666 y(pro)s(duct)33 b(\015o)m(w)g(on)f Fx(G)23 b Fs(\002)f Fw(T)848 5629 y Fr(d)925 5666 y Fy(\(for)31 b(the)i(de\014nition)f(see)i(the)f (second)h(c)m(hapter\).)28 5786 y(W)-8 b(e)41 b(ha)m(v)m(e)g(already)f (met)f(with)h(some)g(conditions)f(under)i(whic)m(h)g(a)e(general)h (equation)g(suc)m(h)h(as)g(\(4.1\))e(is)-118 5906 y(reducible)k(to)h (constan)m(ts)h(co)s(e\016cien)m(ts)g(b)m(y)f(means)g(of)f(a)h(quasi-p) s(erio)s(dic)e(transformation)f(with)j(frequency)1900 6155 y(96)p eop %%Page: 97 101 97 100 bop -118 219 a Fx(!)t(=)p Fy(2.)56 b(Essen)m(tially)-8 b(,)38 b(these)h(conditions)d(w)m(ere)j(the)f FA(ful)5 b(l)39 b(sp)-5 b(e)g(ctrum)44 b Fy(\(see)39 b(c)m(hapter)f(2\),)g(whic) m(h)g(amoun)m(ts)f(to)g(the)-118 339 y(normal)d(h)m(yp)s(erb)s(olicit)m (y)g(of)i(the)g(trivial)d(in)m(v)-5 b(arian)m(t)34 b(torus)i(giv)m(en)g (b)m(y)h(the)f(zero)g(section)g Fs(f)p Fy(0)p Fs(g)24 b(\002)g Fw(T)3496 303 y Fr(d)3540 339 y Fy(.)53 b(Moreo)m(v)m(er)-118 459 y(the)41 b(corresp)s(onding)g(in)m(v)-5 b(arian)m(t)40 b(manifolds)e(need)k(to)f(b)s(e)g(one-dimensional)d(and)j(the)g (external)g(frequencies)-118 580 y(need)d(to)e(satisfy)h(Diophan)m (tine)e(conditions)h(in)g(order)h(to)f(guaran)m(tee)h(that)g(the)g (system)h(can)f(b)s(e)g(reduced)h(to)-118 700 y(constan)m(t)e(co)s (e\016cien)m(ts.)55 b(If)35 b(the)h(latter)f(conditions)g(are)h(not)f (satis\014ed,)i(then)f(w)m(e)h(can)f(split)f(the)h(system)g(in)m(to) -118 820 y(t)m(w)m(o)d(b)s(o)m(xes)h(whic)m(h)f(corresp)s(ond)g(to)g (the)g(stable)f(and)h(the)g(unstable)f(pro)5 b(jections.)28 941 y(Historically)-8 b(,)24 b(the)i(\014rst)g(results)g(on)f (reducibilit)m(y)f(are)i(co)m(v)m(ered)h(b)m(y)g(the)e(ab)s(o)m(v)m(e)i (theory)-8 b(,)27 b(and)f(they)g(deal)f(with)-118 1061 y(p)s(erturbations)f(of)h(h)m(yp)s(erb)s(olic)f(systems.)42 b(Bibliograph)m(y)23 b(for)h(these)i(results)g(can)f(b)s(e)g(found)g (in)f(the)h(monograph)-118 1182 y(b)m(y)31 b(Bogoljub)s(o)m(v,)g 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b(omplex)36 b(domain)h Fs(j)p FA(Im)c Fx(\022)s Fs(j)g Fx(<)f(\032)p FA(,)38 b(with)126 2320 y Fx(\032)28 b(>)f Fy(0)p FA(.)-62 2517 y(\(ii\))48 b(The)f(fr)-5 b(e)g(quency)47 b(ve)-5 b(ctor)47 b Fx(!)k FA(satis\014es)46 b(that)i(ther)-5 b(e)47 b(exist)h(c)-5 b(onstants)46 b Fx(K)r(;)17 b(\033)55 b(>)c Fy(0)c FA(such)g(that)h(for)f(every)126 2637 y Fv(k)28 b Fs(2)g Fw(Z)376 2601 y Fr(d)414 2637 y FA(,)1332 2800 y Fs(jh)p Fv(k)p Fx(;)17 b(!)t Fs(ij)25 b(\025)1810 2733 y Fx(K)p 1774 2778 162 4 v 1774 2869 a Fs(j)p Fv(k)p Fs(j)1889 2840 y Fr(\033)1946 2800 y Fx(;)251 b FA(for)199 b Fv(k)27 b Fs(6)p Fy(=)h(0)p Fx(:)-92 3059 y FA(\(iii\))48 b(The)34 b(eigenvalues)g(of)g(the)h(unp)-5 b(erturb)g(e)g(d)35 b(matrix)g Fx(A)g FA(have)f(distinct)h(r)-5 b(e)g(al)34 b(p)-5 b(arts.)-118 3242 y(Then,)34 b(ther)-5 b(e)35 b(exists)f(an)h Fx(")861 3257 y FC(0)927 3242 y Fx(>)28 b Fy(0)35 b FA(such)f(that)i(for)e Fs(j)p Fx(")p Fs(j)27 b Fx(<)g(")1969 3257 y FC(0)2043 3242 y FA(the)35 b(ab)-5 b(ove)34 b(system)h(r)-5 b(e)g(duc)g(es)35 b(to)g(the)g(form) 1496 3438 y Fx(y)1548 3397 y Ft(0)1598 3438 y Fy(=)27 b Fx(A)1774 3453 y FC(0)1814 3438 y Fx(y)t(;)215 b(\022)2156 3397 y Ft(0)2207 3438 y Fy(=)28 b Fx(!)t(;)-118 3635 y FA(with)35 b Fx(A)167 3650 y FC(0)241 3635 y FA(a)g(c)-5 b(onstant)34 b(matrix,)h(by)g(me)-5 b(ans)34 b(of)g(the)h(non-singular) f(change)g(of)g(variable)1730 3831 y Fx(x)28 b Fy(=)f Fx(P)14 b Fy(\()p Fx(\022)s Fy(\))p Fx(y)-118 4028 y FA(with)35 b Fx(P)41 b Fy(:)27 b Fw(T)315 3992 y Fr(d)387 4028 y Fs(!)g Fx(GL)p Fy(\()p Fx(n;)17 b Fw(R)5 b Fy(\))41 b FA(r)-5 b(e)g(al)35 b(analytic)f(and)h(analytic)-5 b(al)5 b(ly)34 b(invertible)g(in)h(the)g(domain)f Fs(j)p FA(Im)f Fx(\022)s Fs(j)28 b(\024)g Fx(\032=)p Fy(2)p FA(.)28 4229 y Fy(Note)40 b(that)e(this)h(theorem)g(is)g(almost)e(iden) m(tical)h(to)h(\(2.5.1\))f(from)g(c)m(hapter)i(2)f(if)f(w)m(e)j(tak)m (e)e(in)m(to)g(accoun)m(t)-118 4350 y(that)c(the)g(full)e(sp)s(ectrum)i 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y(w)m(e)e(do)e(not)h(kno)m(w)g(fore-hand)g(\(there)g(is)f(no)h (indep)s(enden)m(t)g(c)m(haracterization)f(of)g(the)h(imaginary)d (parts)j(of)f(the)-118 5313 y(eigen)m(v)-5 b(alues,)40 b(although)e(a)g(rotation)f(n)m(um)m(b)s(er)i(can)g(b)s(e)f(de\014ned)i (for)e(Hamiltonian)d(linear)i(equations)i(whic)m(h)-118 5433 y(still)e(k)m(eeps)k(some)e(prop)s(erties,)i(see)g([48)o(])f(and)f ([67]\),)i(is)d(to)h(assume)h(that)f(w)m(e)h(ha)m(v)m(e)h(enough)f (parameters)f(to)-118 5553 y(con)m(trol)26 b(these)i(problems.)41 b(This)27 b(is)f(a)h(classical)e(idea)h(\(see,)j(for)e(instance)g ([62]\))f(and)h(the)h(idea)e(is)g(the)h(follo)m(wing.)-118 5674 y(Assume)33 b(that)g(w)m(e)g(split)f(the)h(p)s(erturb)s(ed)g (system)g(\(4.4\))f(in)g(the)h(follo)m(wing)d(w)m(a)m(y)1088 5870 y Fx(x)1143 5829 y Ft(0)1194 5870 y Fy(=)e Fx(A)1371 5885 y FC(0)1410 5870 y Fx(x)23 b Fy(+)f(\()p Fx("Q)p Fy(\()p Fx(\022)s(;)17 b(\030)5 b Fy(\))21 b(+)h Fx(\030)5 b Fy(\)\))16 b Fx(x;)212 b(\022)2564 5829 y Ft(0)2615 5870 y 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FA(wher)-5 b(e)-33 4541 y(\(i\))49 b Fx(Q)35 b FA(is)g(r)-5 b(e)g(al)34 b(analytic)h(in)f Fy(\()p Fx(\022)s(;)17 b(")p Fy(\))34 b FA(in)h(a)g(c)-5 b(omplex)33 b(neighb)-5 b(orho)g(o)g(d)33 b(of)i Fw(T)2594 4505 y Fr(d)2660 4541 y Fs(\002)23 b(f)p Fy(0)p Fs(g)p FA(.)-62 4745 y(\(ii\))48 b(The)34 b(fr)-5 b(e)g(quency)35 b(ve)-5 b(ctor)34 b Fx(!)39 b FA(is)34 b(Diophantine.)-118 4948 y(De\014ne)e(now)h(as)g Fs(R)p Fy(\()p Fx(")p Fy(;)17 b Fx(Q)p Fy(\))33 b FA(the)g(set)h(of)f(matric)-5 b(es)33 b Fx(A)g FA(with)g Fs(j)p Fx(A)p Fs(j)27 b(\024)i Fy(1)k FA(for)g(which)f(the)i(ab)-5 b(ove)32 b(system)h(is)g(r)-5 b(e)g(ducible,)-118 5069 y(and)37 b(on)h(this)g(set)g(take)g(the)g (normalize)-5 b(d)37 b(L)-5 b(eb)g(esgue)37 b(me)-5 b(asur)g(e.)54 b(Then,)38 b(if)g Fx(")g FA(is)g(smal)5 b(l)37 b(enough,)h(the)g(fol)5 b(lowing)-118 5189 y(b)-5 b(ound)34 b(holds)1403 5329 y Fx(meas)17 b Fy(\()q Fs(R)p Fy(\()p Fx(";)g(Q)p Fy(\)\))27 b Fs(\025)2301 5261 y Fy(1)p 2193 5306 265 4 v 2193 5397 a(1)22 b(+)g Fx(")2408 5368 y Fr(\013)2467 5329 y Fx(;)-118 5536 y FA(for)37 b(a)g(suitable)g(c)-5 b(onstant)36 b Fx(\013)i FA(dep)-5 b(ending)36 b(on)g Fx(Q)i FA(and)e(the)i (Diophantine)d(c)-5 b(ondition)36 b(on)h(the)g(fr)-5 b(e)g(quency)37 b(ve)-5 b(ctor)-118 5657 y Fx(!)t FA(.)1900 6155 y Fy(98)p eop %%Page: 99 103 99 102 bop 709 1855 a currentpoint currentpoint translate 0.82678 0.82678 scale neg exch neg exch translate 709 1855 a @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 3600 @rwi @setspecial %%BeginDocument: generic.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: generic.eps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Sat Jun 9 17:44:07 2001 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 1.000 UL LTb 238 168 M 6724 0 V 0 4704 V -6724 0 V 238 168 L 1.000 UL LT0 1650 2990 Pnt 1706 2755 Pnt 1897 2285 Pnt 1964 2097 Pnt 1975 2097 Pnt 1986 2050 Pnt 2098 1767 Pnt 2098 2097 Pnt 2110 2050 Pnt 2132 1626 Pnt 2132 1673 Pnt 2154 1579 Pnt 2199 1438 Pnt 2222 3037 Pnt 2233 1485 Pnt 2244 1720 Pnt 2266 1203 Pnt 2266 1626 Pnt 2266 2849 Pnt 2278 1344 Pnt 2278 1391 Pnt 2278 1579 Pnt 2278 2802 Pnt 2300 1156 Pnt 2311 1062 Pnt 2311 1297 Pnt 2311 1485 Pnt 2322 1062 Pnt 2334 1156 Pnt 2345 1156 Pnt 2356 874 Pnt 2356 968 Pnt 2356 1156 Pnt 2356 1438 Pnt 2367 827 Pnt 2367 1109 Pnt 2367 1344 Pnt 2378 780 Pnt 2378 874 Pnt 2378 1015 Pnt 2378 1297 Pnt 2390 827 Pnt 2390 968 Pnt 2390 1250 Pnt 2401 780 Pnt 2401 921 Pnt 2401 968 Pnt 2401 1015 Pnt 2401 1250 Pnt 2412 638 Pnt 2412 732 Pnt 2412 874 Pnt 2412 921 Pnt 2412 968 Pnt 2412 1344 Pnt 2423 544 Pnt 2423 591 Pnt 2423 685 Pnt 2423 827 Pnt 2423 874 Pnt 2423 921 Pnt 2423 1156 Pnt 2423 1297 Pnt 2423 2473 Pnt 2435 497 Pnt 2435 544 Pnt 2435 591 Pnt 2435 638 Pnt 2435 780 Pnt 2435 827 Pnt 2435 874 Pnt 2435 1203 Pnt 2446 497 Pnt 2446 544 Pnt 2446 591 Pnt 2446 732 Pnt 2446 780 Pnt 2446 827 Pnt 2446 1062 Pnt 2446 1109 Pnt 2446 1250 Pnt 2446 2285 Pnt 2457 356 Pnt 2457 403 Pnt 2457 544 Pnt 2457 638 Pnt 2457 685 Pnt 2457 732 Pnt 2457 780 Pnt 2457 1015 Pnt 2457 1062 Pnt 2457 1156 Pnt 2457 1203 Pnt 2468 262 Pnt 2468 356 Pnt 2468 403 Pnt 2468 450 Pnt 2468 497 Pnt 2468 591 Pnt 2468 638 Pnt 2468 685 Pnt 2468 732 Pnt 2468 968 Pnt 2468 2426 Pnt 2479 168 Pnt 2479 215 Pnt 2479 262 Pnt 2479 356 Pnt 2479 403 Pnt 2479 450 Pnt 2479 544 Pnt 2479 591 Pnt 2479 638 Pnt 2479 685 Pnt 2479 732 Pnt 2479 968 Pnt 2479 1062 Pnt 2479 2191 Pnt 2491 168 Pnt 2491 215 Pnt 2491 262 Pnt 2491 309 Pnt 2491 356 Pnt 2491 450 Pnt 2491 497 Pnt 2491 591 Pnt 2491 638 Pnt 2491 685 Pnt 2491 921 Pnt 2491 1062 Pnt 2491 1109 Pnt 2502 168 Pnt 2502 215 Pnt 2502 262 Pnt 2502 309 Pnt 2502 403 Pnt 2502 450 Pnt 2502 497 Pnt 2502 544 Pnt 2502 591 Pnt 2502 638 Pnt 2502 874 Pnt 2502 921 Pnt 2502 1109 Pnt 2502 2332 Pnt 2513 168 Pnt 2513 215 Pnt 2513 309 Pnt 2513 356 Pnt 2513 403 Pnt 2513 450 Pnt 2513 497 Pnt 2513 544 Pnt 2513 591 Pnt 2513 780 Pnt 2513 827 Pnt 2513 874 Pnt 2513 968 Pnt 2513 2050 Pnt 2524 168 Pnt 2524 215 Pnt 2524 262 Pnt 2524 309 Pnt 2524 403 Pnt 2524 497 Pnt 2524 544 Pnt 2524 732 Pnt 2524 780 Pnt 2524 827 Pnt 2524 921 Pnt 2524 1015 Pnt 2524 2285 Pnt 2535 168 Pnt 2535 215 Pnt 2535 262 Pnt 2535 309 Pnt 2535 356 Pnt 2535 403 Pnt 2535 450 Pnt 2535 497 Pnt 2535 685 Pnt 2535 732 Pnt 2535 780 Pnt 2535 874 Pnt 2535 921 Pnt 2535 1015 Pnt 2535 2003 Pnt 2547 168 Pnt 2547 215 Pnt 2547 262 Pnt 2547 309 Pnt 2547 356 Pnt 2547 403 Pnt 2547 450 Pnt 2547 638 Pnt 2547 685 Pnt 2547 732 Pnt 2547 827 Pnt 2547 874 Pnt 2547 968 Pnt 2547 1908 Pnt 2558 168 Pnt 2558 215 Pnt 2558 262 Pnt 2558 309 Pnt 2558 356 Pnt 2558 544 Pnt 2558 591 Pnt 2558 638 Pnt 2558 685 Pnt 2558 780 Pnt 2558 827 Pnt 2558 921 Pnt 2558 1861 Pnt 2558 2191 Pnt 2569 168 Pnt 2569 215 Pnt 2569 262 Pnt 2569 309 Pnt 2569 497 Pnt 2569 544 Pnt 2569 591 Pnt 2569 638 Pnt 2569 732 Pnt 2569 780 Pnt 2569 874 Pnt 2569 921 Pnt 2569 1814 Pnt 2580 168 Pnt 2580 215 Pnt 2580 262 Pnt 2580 403 Pnt 2580 450 Pnt 2580 497 Pnt 2580 591 Pnt 2580 685 Pnt 2580 732 Pnt 2580 827 Pnt 2580 874 Pnt 2580 1767 Pnt 2591 168 Pnt 2591 262 Pnt 2591 309 Pnt 2591 356 Pnt 2591 403 Pnt 2591 450 Pnt 2591 497 Pnt 2591 544 Pnt 2591 638 Pnt 2591 685 Pnt 2591 732 Pnt 2591 1720 Pnt 2603 168 Pnt 2603 215 Pnt 2603 262 Pnt 2603 309 Pnt 2603 356 Pnt 2603 403 Pnt 2603 450 Pnt 2603 497 Pnt 2603 591 Pnt 2603 638 Pnt 2603 685 Pnt 2603 780 Pnt 2603 827 Pnt 2603 1673 Pnt 2614 168 Pnt 2614 215 Pnt 2614 262 Pnt 2614 309 Pnt 2614 356 Pnt 2614 403 Pnt 2614 450 Pnt 2614 544 Pnt 2614 591 Pnt 2614 638 Pnt 2614 732 Pnt 2614 780 Pnt 2614 1861 Pnt 2614 2050 Pnt 2625 168 Pnt 2625 215 Pnt 2625 262 Pnt 2625 309 Pnt 2625 356 Pnt 2625 403 Pnt 2625 450 Pnt 2625 497 Pnt 2625 544 Pnt 2625 591 Pnt 2625 685 Pnt 2625 732 Pnt 2625 1673 Pnt 2636 168 Pnt 2636 215 Pnt 2636 262 Pnt 2636 309 Pnt 2636 403 Pnt 2636 450 Pnt 2636 497 Pnt 2636 544 Pnt 2636 638 Pnt 2636 685 Pnt 2636 1532 Pnt 2647 168 Pnt 2647 215 Pnt 2647 262 Pnt 2647 309 Pnt 2647 356 Pnt 2647 403 Pnt 2647 450 Pnt 2647 497 Pnt 2647 591 Pnt 2647 638 Pnt 2647 685 Pnt 2647 1485 Pnt 2647 1579 Pnt 2647 1908 Pnt 2647 1956 Pnt 2659 168 Pnt 2659 215 Pnt 2659 262 Pnt 2659 309 Pnt 2659 356 Pnt 2659 403 Pnt 2659 450 Pnt 2659 544 Pnt 2659 591 Pnt 2659 638 Pnt 2659 1532 Pnt 2670 168 Pnt 2670 215 Pnt 2670 262 Pnt 2670 309 Pnt 2670 356 Pnt 2670 403 Pnt 2670 450 Pnt 2670 497 Pnt 2670 591 Pnt 2670 1485 Pnt 2681 168 Pnt 2681 215 Pnt 2681 262 Pnt 2681 309 Pnt 2681 403 Pnt 2681 450 Pnt 2681 497 Pnt 2681 544 Pnt 2681 1297 Pnt 2681 1438 Pnt 2681 1626 Pnt 2681 1673 Pnt 2681 1814 Pnt 2692 168 Pnt 2692 215 Pnt 2692 262 Pnt 2692 309 Pnt 2692 356 Pnt 2692 403 Pnt 2692 450 Pnt 2692 497 Pnt 2692 544 Pnt 2692 1250 Pnt 2692 1297 Pnt 2692 1391 Pnt 2692 1767 Pnt 2703 168 Pnt 2703 215 Pnt 2703 262 Pnt 2703 309 Pnt 2703 356 Pnt 2703 403 Pnt 2703 450 Pnt 2703 497 Pnt 2703 1203 Pnt 2703 1250 Pnt 2703 1344 Pnt 2715 168 Pnt 2715 215 Pnt 2715 262 Pnt 2715 309 Pnt 2715 356 Pnt 2715 403 Pnt 2715 450 Pnt 2715 1109 Pnt 2715 1203 Pnt 2715 1297 Pnt 2715 1532 Pnt 2726 168 Pnt 2726 215 Pnt 2726 262 Pnt 2726 309 Pnt 2726 356 Pnt 2726 403 Pnt 2726 1015 Pnt 2726 1062 Pnt 2726 1156 Pnt 2726 1250 Pnt 2726 1485 Pnt 2737 168 Pnt 2737 215 Pnt 2737 262 Pnt 2737 309 Pnt 2737 356 Pnt 2737 1062 Pnt 2737 1156 Pnt 2737 1203 Pnt 2737 1485 Pnt 2737 1626 Pnt 2737 1673 Pnt 2748 168 Pnt 2748 215 Pnt 2748 262 Pnt 2748 309 Pnt 2748 874 Pnt 2748 921 Pnt 2748 1015 Pnt 2748 1109 Pnt 2748 1438 Pnt 2760 168 Pnt 2760 215 Pnt 2760 262 Pnt 2760 780 Pnt 2760 827 Pnt 2760 921 Pnt 2760 968 Pnt 2760 1062 Pnt 2760 1109 Pnt 2771 168 Pnt 2771 215 Pnt 2771 638 Pnt 2771 685 Pnt 2771 732 Pnt 2771 780 Pnt 2771 874 Pnt 2771 1015 Pnt 2771 1062 Pnt 2771 1297 Pnt 2771 1579 Pnt 2782 168 Pnt 2782 497 Pnt 2782 544 Pnt 2782 591 Pnt 2782 638 Pnt 2782 685 Pnt 2782 780 Pnt 2782 827 Pnt 2782 921 Pnt 2782 968 Pnt 2782 1015 Pnt 2782 1250 Pnt 2782 1485 Pnt 2793 168 Pnt 2793 215 Pnt 2793 262 Pnt 2793 309 Pnt 2793 356 Pnt 2793 403 Pnt 2793 450 Pnt 2793 497 Pnt 2793 544 Pnt 2793 591 Pnt 2793 685 Pnt 2793 732 Pnt 2793 780 Pnt 2793 874 Pnt 2793 921 Pnt 2793 968 Pnt 2793 1203 Pnt 2793 1250 Pnt 2793 1532 Pnt 2804 168 Pnt 2804 215 Pnt 2804 262 Pnt 2804 309 Pnt 2804 356 Pnt 2804 403 Pnt 2804 450 Pnt 2804 497 Pnt 2804 544 Pnt 2804 591 Pnt 2804 638 Pnt 2804 685 Pnt 2804 827 Pnt 2804 874 Pnt 2804 921 Pnt 2804 1156 Pnt 2804 1203 Pnt 2804 1250 Pnt 2804 1438 Pnt 2816 168 Pnt 2816 215 Pnt 2816 262 Pnt 2816 309 Pnt 2816 356 Pnt 2816 403 Pnt 2816 450 Pnt 2816 497 Pnt 2816 544 Pnt 2816 591 Pnt 2816 638 Pnt 2816 732 Pnt 2816 780 Pnt 2816 827 Pnt 2816 874 Pnt 2816 1109 Pnt 2816 1156 Pnt 2816 1203 Pnt 2816 1438 Pnt 2827 168 Pnt 2827 215 Pnt 2827 262 Pnt 2827 309 Pnt 2827 356 Pnt 2827 403 Pnt 2827 450 Pnt 2827 497 Pnt 2827 544 Pnt 2827 685 Pnt 2827 732 Pnt 2827 780 Pnt 2827 827 Pnt 2827 1062 Pnt 2827 1109 Pnt 2827 1156 Pnt 2827 1344 Pnt 2827 1438 Pnt 2838 168 Pnt 2838 215 Pnt 2838 262 Pnt 2838 309 Pnt 2838 356 Pnt 2838 403 Pnt 2838 450 Pnt 2838 497 Pnt 2838 591 Pnt 2838 638 Pnt 2838 685 Pnt 2838 732 Pnt 2838 1015 Pnt 2838 1062 Pnt 2838 1109 Pnt 2838 1297 Pnt 2849 168 Pnt 2849 215 Pnt 2849 262 Pnt 2849 309 Pnt 2849 356 Pnt 2849 403 Pnt 2849 450 Pnt 2849 497 Pnt 2849 544 Pnt 2849 591 Pnt 2849 638 Pnt 2849 685 Pnt 2849 921 Pnt 2849 968 Pnt 2849 1015 Pnt 2849 1297 Pnt 2849 1344 Pnt 2849 4402 Pnt 2860 168 Pnt 2860 215 Pnt 2860 262 Pnt 2860 309 Pnt 2860 356 Pnt 2860 450 Pnt 2860 497 Pnt 2860 544 Pnt 2860 591 Pnt 2860 638 Pnt 2860 874 Pnt 2860 921 Pnt 2860 968 Pnt 2860 1062 Pnt 2860 1250 Pnt 2872 168 Pnt 2872 215 Pnt 2872 262 Pnt 2872 356 Pnt 2872 403 Pnt 2872 450 Pnt 2872 497 Pnt 2872 544 Pnt 2872 591 Pnt 2872 827 Pnt 2872 874 Pnt 2872 921 Pnt 2872 1015 Pnt 2872 1203 Pnt 2883 168 Pnt 2883 215 Pnt 2883 262 Pnt 2883 309 Pnt 2883 356 Pnt 2883 403 Pnt 2883 450 Pnt 2883 497 Pnt 2883 544 Pnt 2883 780 Pnt 2883 827 Pnt 2883 874 Pnt 2883 968 Pnt 2883 1156 Pnt 2894 168 Pnt 2894 215 Pnt 2894 262 Pnt 2894 309 Pnt 2894 356 Pnt 2894 403 Pnt 2894 450 Pnt 2894 497 Pnt 2894 685 Pnt 2894 732 Pnt 2894 827 Pnt 2894 921 Pnt 2894 1109 Pnt 2894 1156 Pnt 2894 1203 Pnt 2894 1250 Pnt 2905 168 Pnt 2905 215 Pnt 2905 262 Pnt 2905 309 Pnt 2905 356 Pnt 2905 403 Pnt 2905 450 Pnt 2905 638 Pnt 2905 685 Pnt 2905 780 Pnt 2905 874 Pnt 2905 1062 Pnt 2905 1109 Pnt 2905 1203 Pnt 2916 168 Pnt 2916 215 Pnt 2916 262 Pnt 2916 309 Pnt 2916 356 Pnt 2916 403 Pnt 2916 544 Pnt 2916 591 Pnt 2916 638 Pnt 2916 685 Pnt 2916 732 Pnt 2916 827 Pnt 2916 1015 Pnt 2916 1062 Pnt 2928 168 Pnt 2928 215 Pnt 2928 262 Pnt 2928 309 Pnt 2928 450 Pnt 2928 497 Pnt 2928 544 Pnt 2928 591 Pnt 2928 638 Pnt 2928 685 Pnt 2928 780 Pnt 2928 968 Pnt 2928 1015 Pnt 2928 1109 Pnt 2928 1156 Pnt 2939 168 Pnt 2939 215 Pnt 2939 262 Pnt 2939 356 Pnt 2939 403 Pnt 2939 450 Pnt 2939 497 Pnt 2939 544 Pnt 2939 591 Pnt 2939 638 Pnt 2939 732 Pnt 2939 968 Pnt 2939 1062 Pnt 2939 1109 Pnt 2939 3649 Pnt 2950 168 Pnt 2950 215 Pnt 2950 262 Pnt 2950 309 Pnt 2950 356 Pnt 2950 403 Pnt 2950 450 Pnt 2950 497 Pnt 2950 544 Pnt 2950 591 Pnt 2950 685 Pnt 2950 732 Pnt 2950 921 Pnt 2950 968 Pnt 2950 4166 Pnt 2961 168 Pnt 2961 215 Pnt 2961 262 Pnt 2961 309 Pnt 2961 356 Pnt 2961 403 Pnt 2961 450 Pnt 2961 497 Pnt 2961 544 Pnt 2961 591 Pnt 2961 638 Pnt 2961 685 Pnt 2961 874 Pnt 2961 921 Pnt 2961 1015 Pnt 2961 1062 Pnt 2972 168 Pnt 2972 215 Pnt 2972 262 Pnt 2972 309 Pnt 2972 356 Pnt 2972 403 Pnt 2972 450 Pnt 2972 497 Pnt 2972 544 Pnt 2972 591 Pnt 2972 638 Pnt 2972 827 Pnt 2972 874 Pnt 2972 968 Pnt 2972 1015 Pnt 2972 3884 Pnt 2984 168 Pnt 2984 215 Pnt 2984 262 Pnt 2984 309 Pnt 2984 356 Pnt 2984 403 Pnt 2984 450 Pnt 2984 497 Pnt 2984 544 Pnt 2984 591 Pnt 2984 780 Pnt 2984 827 Pnt 2984 921 Pnt 2995 168 Pnt 2995 215 Pnt 2995 262 Pnt 2995 309 Pnt 2995 356 Pnt 2995 403 Pnt 2995 450 Pnt 2995 497 Pnt 2995 544 Pnt 2995 732 Pnt 2995 780 Pnt 2995 827 Pnt 2995 874 Pnt 2995 968 Pnt 3006 168 Pnt 3006 215 Pnt 3006 262 Pnt 3006 309 Pnt 3006 356 Pnt 3006 403 Pnt 3006 450 Pnt 3006 497 Pnt 3006 638 Pnt 3006 685 Pnt 3006 732 Pnt 3006 780 Pnt 3006 874 Pnt 3006 921 Pnt 3017 168 Pnt 3017 215 Pnt 3017 262 Pnt 3017 309 Pnt 3017 356 Pnt 3017 403 Pnt 3017 450 Pnt 3017 591 Pnt 3017 638 Pnt 3017 685 Pnt 3017 732 Pnt 3017 827 Pnt 3017 874 Pnt 3028 168 Pnt 3028 215 Pnt 3028 262 Pnt 3028 309 Pnt 3028 356 Pnt 3028 403 Pnt 3028 544 Pnt 3028 591 Pnt 3028 638 Pnt 3028 685 Pnt 3028 780 Pnt 3028 874 Pnt 3040 168 Pnt 3040 215 Pnt 3040 262 Pnt 3040 309 Pnt 3040 356 Pnt 3040 497 Pnt 3040 544 Pnt 3040 591 Pnt 3040 638 Pnt 3040 732 Pnt 3040 780 Pnt 3040 827 Pnt 3051 168 Pnt 3051 215 Pnt 3051 262 Pnt 3051 309 Pnt 3051 403 Pnt 3051 450 Pnt 3051 497 Pnt 3051 544 Pnt 3051 591 Pnt 3051 638 Pnt 3051 685 Pnt 3051 732 Pnt 3051 780 Pnt 3062 168 Pnt 3062 215 Pnt 3062 309 Pnt 3062 356 Pnt 3062 403 Pnt 3062 450 Pnt 3062 497 Pnt 3062 544 Pnt 3062 591 Pnt 3062 638 Pnt 3062 685 Pnt 3062 732 Pnt 3062 780 Pnt 3062 3226 Pnt 3073 168 Pnt 3073 215 Pnt 3073 262 Pnt 3073 309 Pnt 3073 356 Pnt 3073 403 Pnt 3073 450 Pnt 3073 497 Pnt 3073 544 Pnt 3073 638 Pnt 3073 732 Pnt 3084 168 Pnt 3084 215 Pnt 3084 262 Pnt 3084 309 Pnt 3084 356 Pnt 3084 403 Pnt 3084 450 Pnt 3084 497 Pnt 3084 591 Pnt 3084 638 Pnt 3084 685 Pnt 3096 168 Pnt 3096 215 Pnt 3096 262 Pnt 3096 309 Pnt 3096 356 Pnt 3096 403 Pnt 3096 450 Pnt 3096 544 Pnt 3096 591 Pnt 3096 638 Pnt 3096 685 Pnt 3107 168 Pnt 3107 215 Pnt 3107 262 Pnt 3107 309 Pnt 3107 356 Pnt 3107 403 Pnt 3107 497 Pnt 3107 544 Pnt 3107 591 Pnt 3107 638 Pnt 3107 3132 Pnt 3118 168 Pnt 3118 215 Pnt 3118 262 Pnt 3118 309 Pnt 3118 356 Pnt 3118 450 Pnt 3118 497 Pnt 3118 591 Pnt 3129 168 Pnt 3129 215 Pnt 3129 262 Pnt 3129 309 Pnt 3129 356 Pnt 3129 403 Pnt 3129 450 Pnt 3129 497 Pnt 3129 544 Pnt 3129 591 Pnt 3129 2990 Pnt 3141 168 Pnt 3141 215 Pnt 3141 262 Pnt 3141 309 Pnt 3141 356 Pnt 3141 403 Pnt 3141 450 Pnt 3141 497 Pnt 3141 544 Pnt 3152 168 Pnt 3152 215 Pnt 3152 262 Pnt 3152 309 Pnt 3152 356 Pnt 3152 403 Pnt 3152 450 Pnt 3152 497 Pnt 3152 544 Pnt 3163 168 Pnt 3163 215 Pnt 3163 262 Pnt 3163 309 Pnt 3163 356 Pnt 3163 403 Pnt 3163 450 Pnt 3163 497 Pnt 3163 2802 Pnt 3174 168 Pnt 3174 215 Pnt 3174 262 Pnt 3174 309 Pnt 3174 356 Pnt 3174 403 Pnt 3174 450 Pnt 3185 168 Pnt 3185 215 Pnt 3185 262 Pnt 3185 309 Pnt 3185 356 Pnt 3185 403 Pnt 3185 450 Pnt 3197 168 Pnt 3197 215 Pnt 3197 262 Pnt 3197 309 Pnt 3197 356 Pnt 3197 403 Pnt 3197 2849 Pnt 3208 168 Pnt 3208 215 Pnt 3208 262 Pnt 3208 309 Pnt 3208 356 Pnt 3208 2614 Pnt 3219 168 Pnt 3219 215 Pnt 3219 262 Pnt 3219 309 Pnt 3219 356 Pnt 3219 2661 Pnt 3230 168 Pnt 3230 215 Pnt 3230 262 Pnt 3230 309 Pnt 3230 2614 Pnt 3241 168 Pnt 3241 215 Pnt 3241 262 Pnt 3241 309 Pnt 3241 2661 Pnt 3253 168 Pnt 3253 215 Pnt 3253 262 Pnt 3253 2426 Pnt 3264 168 Pnt 3264 215 Pnt 3264 2473 Pnt 3275 168 Pnt 3275 215 Pnt 3275 2426 Pnt 3275 2567 Pnt 3275 3037 Pnt 3275 3084 Pnt 3286 168 Pnt 3297 168 Pnt 3297 2191 Pnt 3309 168 Pnt 3309 215 Pnt 3309 2943 Pnt 3309 2990 Pnt 3309 3320 Pnt 3320 168 Pnt 3320 215 Pnt 3320 262 Pnt 3320 309 Pnt 3320 2003 Pnt 3320 2191 Pnt 3331 168 Pnt 3331 215 Pnt 3331 262 Pnt 3331 309 Pnt 3331 356 Pnt 3331 1956 Pnt 3331 2003 Pnt 3331 2144 Pnt 3331 2332 Pnt 3342 168 Pnt 3342 215 Pnt 3342 262 Pnt 3342 309 Pnt 3342 356 Pnt 3342 403 Pnt 3342 450 Pnt 3342 1861 Pnt 3342 1908 Pnt 3342 2097 Pnt 3342 2285 Pnt 3353 168 Pnt 3353 215 Pnt 3353 262 Pnt 3353 309 Pnt 3353 356 Pnt 3353 403 Pnt 3353 450 Pnt 3353 497 Pnt 3353 544 Pnt 3353 1767 Pnt 3353 2003 Pnt 3353 2238 Pnt 3365 168 Pnt 3365 215 Pnt 3365 262 Pnt 3365 309 Pnt 3365 356 Pnt 3365 403 Pnt 3365 450 Pnt 3365 497 Pnt 3365 544 Pnt 3365 591 Pnt 3365 638 Pnt 3365 685 Pnt 3365 1626 Pnt 3365 1673 Pnt 3365 1956 Pnt 3365 2144 Pnt 3365 2191 Pnt 3376 168 Pnt 3376 215 Pnt 3376 262 Pnt 3376 309 Pnt 3376 356 Pnt 3376 403 Pnt 3376 450 Pnt 3376 497 Pnt 3376 544 Pnt 3376 591 Pnt 3376 638 Pnt 3376 685 Pnt 3376 732 Pnt 3376 780 Pnt 3376 827 Pnt 3376 874 Pnt 3376 1438 Pnt 3376 1532 Pnt 3376 1626 Pnt 3376 1861 Pnt 3376 1908 Pnt 3376 2097 Pnt 3387 168 Pnt 3387 215 Pnt 3387 262 Pnt 3387 309 Pnt 3387 356 Pnt 3387 403 Pnt 3387 450 Pnt 3387 497 Pnt 3387 591 Pnt 3387 638 Pnt 3387 685 Pnt 3387 732 Pnt 3387 780 Pnt 3387 827 Pnt 3387 874 Pnt 3387 921 Pnt 3387 1109 Pnt 3387 1297 Pnt 3387 1344 Pnt 3387 1391 Pnt 3387 1485 Pnt 3387 1532 Pnt 3387 1767 Pnt 3387 1814 Pnt 3398 168 Pnt 3398 215 Pnt 3398 262 Pnt 3398 309 Pnt 3398 356 Pnt 3398 403 Pnt 3398 450 Pnt 3398 497 Pnt 3398 544 Pnt 3398 591 Pnt 3398 638 Pnt 3398 685 Pnt 3398 732 Pnt 3398 780 Pnt 3398 1297 Pnt 3398 1344 Pnt 3398 1391 Pnt 3398 1673 Pnt 3398 1720 Pnt 3398 1767 Pnt 3409 168 Pnt 3409 215 Pnt 3409 262 Pnt 3409 309 Pnt 3409 356 Pnt 3409 403 Pnt 3409 450 Pnt 3409 497 Pnt 3409 544 Pnt 3409 591 Pnt 3409 638 Pnt 3409 685 Pnt 3409 732 Pnt 3409 780 Pnt 3409 827 Pnt 3409 874 Pnt 3409 921 Pnt 3409 968 Pnt 3409 1015 Pnt 3409 1062 Pnt 3409 1109 Pnt 3409 1156 Pnt 3409 1203 Pnt 3409 1250 Pnt 3409 1579 Pnt 3409 1626 Pnt 3409 1673 Pnt 3409 1908 Pnt 3409 2661 Pnt 3421 168 Pnt 3421 215 Pnt 3421 262 Pnt 3421 309 Pnt 3421 356 Pnt 3421 403 Pnt 3421 450 Pnt 3421 497 Pnt 3421 544 Pnt 3421 591 Pnt 3421 638 Pnt 3421 685 Pnt 3421 732 Pnt 3421 780 Pnt 3421 827 Pnt 3421 874 Pnt 3421 921 Pnt 3421 1438 Pnt 3421 1485 Pnt 3421 1532 Pnt 3421 1579 Pnt 3421 1861 Pnt 3421 1956 Pnt 3421 3132 Pnt 3432 168 Pnt 3432 215 Pnt 3432 262 Pnt 3432 309 Pnt 3432 356 Pnt 3432 1297 Pnt 3432 1344 Pnt 3432 1438 Pnt 3432 1485 Pnt 3432 1814 Pnt 3432 1861 Pnt 3432 1908 Pnt 3443 168 Pnt 3443 215 Pnt 3443 262 Pnt 3443 309 Pnt 3443 356 Pnt 3443 403 Pnt 3443 450 Pnt 3443 497 Pnt 3443 544 Pnt 3443 591 Pnt 3443 638 Pnt 3443 685 Pnt 3443 732 Pnt 3443 780 Pnt 3443 827 Pnt 3443 874 Pnt 3443 921 Pnt 3443 968 Pnt 3443 1015 Pnt 3443 1062 Pnt 3443 1109 Pnt 3443 1156 Pnt 3443 1203 Pnt 3443 1297 Pnt 3443 1391 Pnt 3443 1720 Pnt 3443 1814 Pnt 3443 1861 Pnt 3454 168 Pnt 3454 215 Pnt 3454 262 Pnt 3454 309 Pnt 3454 356 Pnt 3454 403 Pnt 3454 450 Pnt 3454 497 Pnt 3454 544 Pnt 3454 591 Pnt 3454 638 Pnt 3454 685 Pnt 3454 732 Pnt 3454 780 Pnt 3454 827 Pnt 3454 874 Pnt 3454 921 Pnt 3454 968 Pnt 3454 1015 Pnt 3454 1062 Pnt 3454 1109 Pnt 3454 1156 Pnt 3454 1203 Pnt 3454 1250 Pnt 3454 1297 Pnt 3454 1626 Pnt 3454 1673 Pnt 3454 2520 Pnt 3454 2567 Pnt 3466 168 Pnt 3466 215 Pnt 3466 262 Pnt 3466 309 Pnt 3466 356 Pnt 3466 403 Pnt 3466 450 Pnt 3466 497 Pnt 3466 544 Pnt 3466 591 Pnt 3466 638 Pnt 3466 685 Pnt 3466 732 Pnt 3466 780 Pnt 3466 827 Pnt 3466 874 Pnt 3466 921 Pnt 3466 968 Pnt 3466 1015 Pnt 3466 1062 Pnt 3466 1109 Pnt 3466 1156 Pnt 3466 1579 Pnt 3466 1673 Pnt 3477 168 Pnt 3477 215 Pnt 3477 262 Pnt 3477 309 Pnt 3477 356 Pnt 3477 403 Pnt 3477 450 Pnt 3477 497 Pnt 3477 544 Pnt 3477 591 Pnt 3477 638 Pnt 3477 685 Pnt 3477 732 Pnt 3477 780 Pnt 3477 827 Pnt 3477 874 Pnt 3477 921 Pnt 3477 968 Pnt 3477 1015 Pnt 3477 1062 Pnt 3477 1485 Pnt 3477 1532 Pnt 3477 1626 Pnt 3477 1673 Pnt 3477 2379 Pnt 3477 2990 Pnt 3488 168 Pnt 3488 215 Pnt 3488 262 Pnt 3488 309 Pnt 3488 356 Pnt 3488 403 Pnt 3488 450 Pnt 3488 497 Pnt 3488 544 Pnt 3488 591 Pnt 3488 638 Pnt 3488 685 Pnt 3488 732 Pnt 3488 780 Pnt 3488 827 Pnt 3488 874 Pnt 3488 1391 Pnt 3488 1438 Pnt 3488 1532 Pnt 3488 1626 Pnt 3488 2849 Pnt 3499 168 Pnt 3499 215 Pnt 3499 262 Pnt 3499 309 Pnt 3499 356 Pnt 3499 403 Pnt 3499 450 Pnt 3499 497 Pnt 3499 544 Pnt 3499 591 Pnt 3499 638 Pnt 3499 685 Pnt 3499 732 Pnt 3499 1250 Pnt 3499 1297 Pnt 3499 1344 Pnt 3499 1485 Pnt 3499 2896 Pnt 3499 2943 Pnt 3510 168 Pnt 3510 215 Pnt 3510 262 Pnt 3510 309 Pnt 3510 356 Pnt 3510 403 Pnt 3510 450 Pnt 3510 497 Pnt 3510 544 Pnt 3510 591 Pnt 3510 1109 Pnt 3510 1156 Pnt 3510 1203 Pnt 3510 1250 Pnt 3510 1391 Pnt 3510 1485 Pnt 3510 2332 Pnt 3522 168 Pnt 3522 215 Pnt 3522 262 Pnt 3522 309 Pnt 3522 356 Pnt 3522 403 Pnt 3522 450 Pnt 3522 921 Pnt 3522 968 Pnt 3522 1015 Pnt 3522 1062 Pnt 3522 1109 Pnt 3522 1156 Pnt 3522 1297 Pnt 3522 1344 Pnt 3522 1438 Pnt 3522 2238 Pnt 3533 168 Pnt 3533 215 Pnt 3533 262 Pnt 3533 309 Pnt 3533 591 Pnt 3533 638 Pnt 3533 685 Pnt 3533 732 Pnt 3533 780 Pnt 3533 827 Pnt 3533 874 Pnt 3533 921 Pnt 3533 968 Pnt 3533 1015 Pnt 3533 1062 Pnt 3533 1203 Pnt 3533 1250 Pnt 3533 1344 Pnt 3533 2191 Pnt 3533 2755 Pnt 3544 168 Pnt 3544 215 Pnt 3544 309 Pnt 3544 356 Pnt 3544 403 Pnt 3544 450 Pnt 3544 497 Pnt 3544 544 Pnt 3544 591 Pnt 3544 638 Pnt 3544 685 Pnt 3544 732 Pnt 3544 780 Pnt 3544 827 Pnt 3544 874 Pnt 3544 921 Pnt 3544 968 Pnt 3544 1062 Pnt 3544 1109 Pnt 3544 1156 Pnt 3544 1203 Pnt 3544 1250 Pnt 3544 1297 Pnt 3544 2144 Pnt 3544 2285 Pnt 3555 168 Pnt 3555 215 Pnt 3555 262 Pnt 3555 309 Pnt 3555 356 Pnt 3555 403 Pnt 3555 450 Pnt 3555 497 Pnt 3555 544 Pnt 3555 591 Pnt 3555 638 Pnt 3555 685 Pnt 3555 732 Pnt 3555 780 Pnt 3555 827 Pnt 3555 968 Pnt 3555 1015 Pnt 3555 1062 Pnt 3555 1109 Pnt 3555 1203 Pnt 3555 1250 Pnt 3555 2097 Pnt 3555 2238 Pnt 3555 2802 Pnt 3566 168 Pnt 3566 215 Pnt 3566 262 Pnt 3566 309 Pnt 3566 356 Pnt 3566 403 Pnt 3566 450 Pnt 3566 497 Pnt 3566 544 Pnt 3566 591 Pnt 3566 638 Pnt 3566 685 Pnt 3566 827 Pnt 3566 874 Pnt 3566 921 Pnt 3566 968 Pnt 3566 1015 Pnt 3566 1109 Pnt 3566 1156 Pnt 3566 2050 Pnt 3578 168 Pnt 3578 215 Pnt 3578 262 Pnt 3578 309 Pnt 3578 356 Pnt 3578 403 Pnt 3578 450 Pnt 3578 497 Pnt 3578 544 Pnt 3578 591 Pnt 3578 638 Pnt 3578 685 Pnt 3578 732 Pnt 3578 780 Pnt 3578 827 Pnt 3578 874 Pnt 3578 921 Pnt 3578 1015 Pnt 3578 1062 Pnt 3578 1109 Pnt 3578 2003 Pnt 3578 2097 Pnt 3578 2708 Pnt 3589 168 Pnt 3589 215 Pnt 3589 262 Pnt 3589 309 Pnt 3589 356 Pnt 3589 403 Pnt 3589 450 Pnt 3589 497 Pnt 3589 544 Pnt 3589 591 Pnt 3589 638 Pnt 3589 685 Pnt 3589 732 Pnt 3589 780 Pnt 3589 827 Pnt 3589 921 Pnt 3589 968 Pnt 3589 1015 Pnt 3589 1956 Pnt 3589 2050 Pnt 3589 2144 Pnt 3600 168 Pnt 3600 215 Pnt 3600 262 Pnt 3600 309 Pnt 3600 356 Pnt 3600 403 Pnt 3600 450 Pnt 3600 497 Pnt 3600 544 Pnt 3600 591 Pnt 3600 638 Pnt 3600 685 Pnt 3600 732 Pnt 3600 780 Pnt 3600 827 Pnt 3600 874 Pnt 3600 921 Pnt 3600 968 Pnt 3600 1861 Pnt 3600 1908 Pnt 3600 2003 Pnt 3600 2097 Pnt 3600 2708 Pnt 3611 168 Pnt 3611 215 Pnt 3611 262 Pnt 3611 309 Pnt 3611 356 Pnt 3611 403 Pnt 3611 450 Pnt 3611 497 Pnt 3611 544 Pnt 3611 591 Pnt 3611 638 Pnt 3611 685 Pnt 3611 732 Pnt 3611 780 Pnt 3611 827 Pnt 3611 874 Pnt 3611 1861 Pnt 3611 1956 Pnt 3611 2050 Pnt 3622 168 Pnt 3622 215 Pnt 3622 262 Pnt 3622 309 Pnt 3622 356 Pnt 3622 403 Pnt 3622 450 Pnt 3622 497 Pnt 3622 544 Pnt 3622 591 Pnt 3622 638 Pnt 3622 685 Pnt 3622 732 Pnt 3622 780 Pnt 3622 1767 Pnt 3622 1814 Pnt 3622 1908 Pnt 3622 2003 Pnt 3634 168 Pnt 3634 215 Pnt 3634 262 Pnt 3634 309 Pnt 3634 356 Pnt 3634 403 Pnt 3634 450 Pnt 3634 497 Pnt 3634 544 Pnt 3634 591 Pnt 3634 638 Pnt 3634 685 Pnt 3634 732 Pnt 3634 1673 Pnt 3634 1861 Pnt 3634 1956 Pnt 3634 2614 Pnt 3645 168 Pnt 3645 215 Pnt 3645 262 Pnt 3645 309 Pnt 3645 356 Pnt 3645 403 Pnt 3645 450 Pnt 3645 497 Pnt 3645 544 Pnt 3645 591 Pnt 3645 638 Pnt 3645 1626 Pnt 3645 1673 Pnt 3645 1814 Pnt 3645 1908 Pnt 3645 1956 Pnt 3656 168 Pnt 3656 215 Pnt 3656 262 Pnt 3656 309 Pnt 3656 356 Pnt 3656 403 Pnt 3656 450 Pnt 3656 497 Pnt 3656 544 Pnt 3656 591 Pnt 3656 1532 Pnt 3656 1579 Pnt 3656 1626 Pnt 3656 1767 Pnt 3656 2426 Pnt 3667 168 Pnt 3667 215 Pnt 3667 262 Pnt 3667 309 Pnt 3667 356 Pnt 3667 403 Pnt 3667 450 Pnt 3667 497 Pnt 3667 1438 Pnt 3667 1485 Pnt 3667 1532 Pnt 3667 1579 Pnt 3667 1720 Pnt 3667 1861 Pnt 3667 2473 Pnt 3678 168 Pnt 3678 215 Pnt 3678 262 Pnt 3678 309 Pnt 3678 356 Pnt 3678 403 Pnt 3678 450 Pnt 3678 1344 Pnt 3678 1391 Pnt 3678 1438 Pnt 3678 1485 Pnt 3678 1673 Pnt 3678 1814 Pnt 3690 168 Pnt 3690 215 Pnt 3690 262 Pnt 3690 309 Pnt 3690 356 Pnt 3690 1250 Pnt 3690 1297 Pnt 3690 1344 Pnt 3690 1391 Pnt 3690 1438 Pnt 3690 1579 Pnt 3690 1626 Pnt 3690 2285 Pnt 3690 2332 Pnt 3690 2426 Pnt 3701 168 Pnt 3701 215 Pnt 3701 262 Pnt 3701 309 Pnt 3701 1109 Pnt 3701 1156 Pnt 3701 1203 Pnt 3701 1250 Pnt 3701 1297 Pnt 3701 1344 Pnt 3701 1391 Pnt 3701 1532 Pnt 3701 1579 Pnt 3712 168 Pnt 3712 215 Pnt 3712 874 Pnt 3712 921 Pnt 3712 968 Pnt 3712 1015 Pnt 3712 1062 Pnt 3712 1109 Pnt 3712 1156 Pnt 3712 1203 Pnt 3712 1250 Pnt 3712 1297 Pnt 3712 1485 Pnt 3712 1532 Pnt 3712 2238 Pnt 3723 168 Pnt 3723 309 Pnt 3723 356 Pnt 3723 403 Pnt 3723 450 Pnt 3723 497 Pnt 3723 544 Pnt 3723 591 Pnt 3723 638 Pnt 3723 685 Pnt 3723 732 Pnt 3723 780 Pnt 3723 827 Pnt 3723 874 Pnt 3723 921 Pnt 3723 968 Pnt 3723 1015 Pnt 3723 1062 Pnt 3723 1109 Pnt 3723 1156 Pnt 3723 1203 Pnt 3723 1391 Pnt 3723 1438 Pnt 3723 1485 Pnt 3723 1673 Pnt 3723 2238 Pnt 3734 168 Pnt 3734 215 Pnt 3734 262 Pnt 3734 309 Pnt 3734 356 Pnt 3734 403 Pnt 3734 450 Pnt 3734 497 Pnt 3734 544 Pnt 3734 591 Pnt 3734 638 Pnt 3734 685 Pnt 3734 732 Pnt 3734 780 Pnt 3734 827 Pnt 3734 874 Pnt 3734 921 Pnt 3734 968 Pnt 3734 1015 Pnt 3734 1062 Pnt 3734 1109 Pnt 3734 1156 Pnt 3734 1344 Pnt 3734 1391 Pnt 3734 1532 Pnt 3734 1626 Pnt 3734 2144 Pnt 3734 2379 Pnt 3746 168 Pnt 3746 215 Pnt 3746 262 Pnt 3746 309 Pnt 3746 356 Pnt 3746 403 Pnt 3746 450 Pnt 3746 497 Pnt 3746 544 Pnt 3746 591 Pnt 3746 638 Pnt 3746 685 Pnt 3746 732 Pnt 3746 780 Pnt 3746 827 Pnt 3746 874 Pnt 3746 921 Pnt 3746 968 Pnt 3746 1015 Pnt 3746 1062 Pnt 3746 1250 Pnt 3746 1297 Pnt 3746 1344 Pnt 3746 1485 Pnt 3746 1579 Pnt 3746 2285 Pnt 3757 168 Pnt 3757 215 Pnt 3757 262 Pnt 3757 309 Pnt 3757 356 Pnt 3757 403 Pnt 3757 450 Pnt 3757 497 Pnt 3757 544 Pnt 3757 591 Pnt 3757 638 Pnt 3757 685 Pnt 3757 732 Pnt 3757 780 Pnt 3757 827 Pnt 3757 874 Pnt 3757 921 Pnt 3757 968 Pnt 3757 1156 Pnt 3757 1203 Pnt 3757 1250 Pnt 3757 1297 Pnt 3757 1438 Pnt 3757 1532 Pnt 3757 2097 Pnt 3757 2144 Pnt 3757 2238 Pnt 3757 2332 Pnt 3768 168 Pnt 3768 215 Pnt 3768 262 Pnt 3768 309 Pnt 3768 356 Pnt 3768 403 Pnt 3768 450 Pnt 3768 497 Pnt 3768 544 Pnt 3768 591 Pnt 3768 638 Pnt 3768 685 Pnt 3768 732 Pnt 3768 780 Pnt 3768 827 Pnt 3768 874 Pnt 3768 1062 Pnt 3768 1109 Pnt 3768 1156 Pnt 3768 1203 Pnt 3768 1391 Pnt 3768 1485 Pnt 3768 2097 Pnt 3768 2285 Pnt 3779 168 Pnt 3779 215 Pnt 3779 262 Pnt 3779 309 Pnt 3779 356 Pnt 3779 403 Pnt 3779 450 Pnt 3779 497 Pnt 3779 544 Pnt 3779 591 Pnt 3779 638 Pnt 3779 685 Pnt 3779 732 Pnt 3779 780 Pnt 3779 968 Pnt 3779 1015 Pnt 3779 1062 Pnt 3779 1109 Pnt 3779 1156 Pnt 3779 1344 Pnt 3779 1391 Pnt 3779 1438 Pnt 3779 2003 Pnt 3791 168 Pnt 3791 215 Pnt 3791 262 Pnt 3791 309 Pnt 3791 356 Pnt 3791 403 Pnt 3791 450 Pnt 3791 497 Pnt 3791 544 Pnt 3791 591 Pnt 3791 638 Pnt 3791 827 Pnt 3791 874 Pnt 3791 921 Pnt 3791 968 Pnt 3791 1015 Pnt 3791 1062 Pnt 3791 1109 Pnt 3791 1250 Pnt 3791 1297 Pnt 3791 1344 Pnt 3791 1391 Pnt 3791 1956 Pnt 3791 2003 Pnt 3791 2144 Pnt 3802 168 Pnt 3802 215 Pnt 3802 262 Pnt 3802 309 Pnt 3802 356 Pnt 3802 403 Pnt 3802 450 Pnt 3802 497 Pnt 3802 544 Pnt 3802 685 Pnt 3802 732 Pnt 3802 780 Pnt 3802 827 Pnt 3802 874 Pnt 3802 921 Pnt 3802 968 Pnt 3802 1015 Pnt 3802 1203 Pnt 3802 1250 Pnt 3802 1297 Pnt 3802 1344 Pnt 3802 1956 Pnt 3802 2003 Pnt 3813 168 Pnt 3813 215 Pnt 3813 262 Pnt 3813 309 Pnt 3813 356 Pnt 3813 403 Pnt 3813 450 Pnt 3813 544 Pnt 3813 591 Pnt 3813 638 Pnt 3813 685 Pnt 3813 732 Pnt 3813 780 Pnt 3813 827 Pnt 3813 874 Pnt 3813 921 Pnt 3813 968 Pnt 3813 1109 Pnt 3813 1156 Pnt 3813 1203 Pnt 3813 1250 Pnt 3813 1297 Pnt 3813 2097 Pnt 3824 168 Pnt 3824 215 Pnt 3824 262 Pnt 3824 309 Pnt 3824 356 Pnt 3824 403 Pnt 3824 450 Pnt 3824 497 Pnt 3824 544 Pnt 3824 591 Pnt 3824 638 Pnt 3824 685 Pnt 3824 732 Pnt 3824 780 Pnt 3824 827 Pnt 3824 874 Pnt 3824 1062 Pnt 3824 1109 Pnt 3824 1156 Pnt 3824 1203 Pnt 3824 1250 Pnt 3824 2050 Pnt 3824 2144 Pnt 3835 168 Pnt 3835 215 Pnt 3835 262 Pnt 3835 309 Pnt 3835 356 Pnt 3835 403 Pnt 3835 450 Pnt 3835 497 Pnt 3835 544 Pnt 3835 591 Pnt 3835 638 Pnt 3835 685 Pnt 3835 732 Pnt 3835 780 Pnt 3835 968 Pnt 3835 1015 Pnt 3835 1062 Pnt 3835 1109 Pnt 3835 1156 Pnt 3835 1203 Pnt 3835 1814 Pnt 3835 1908 Pnt 3835 2050 Pnt 3847 168 Pnt 3847 215 Pnt 3847 262 Pnt 3847 309 Pnt 3847 356 Pnt 3847 403 Pnt 3847 450 Pnt 3847 497 Pnt 3847 544 Pnt 3847 591 Pnt 3847 638 Pnt 3847 685 Pnt 3847 732 Pnt 3847 874 Pnt 3847 921 Pnt 3847 968 Pnt 3847 1015 Pnt 3847 1062 Pnt 3847 1109 Pnt 3847 1156 Pnt 3847 1767 Pnt 3847 1814 Pnt 3847 1861 Pnt 3847 2003 Pnt 3858 168 Pnt 3858 215 Pnt 3858 262 Pnt 3858 309 Pnt 3858 356 Pnt 3858 403 Pnt 3858 450 Pnt 3858 497 Pnt 3858 544 Pnt 3858 591 Pnt 3858 638 Pnt 3858 780 Pnt 3858 827 Pnt 3858 874 Pnt 3858 921 Pnt 3858 968 Pnt 3858 1015 Pnt 3858 1062 Pnt 3858 1109 Pnt 3858 1720 Pnt 3858 1767 Pnt 3858 1814 Pnt 3858 1956 Pnt 3869 168 Pnt 3869 215 Pnt 3869 262 Pnt 3869 309 Pnt 3869 356 Pnt 3869 403 Pnt 3869 450 Pnt 3869 497 Pnt 3869 544 Pnt 3869 638 Pnt 3869 685 Pnt 3869 732 Pnt 3869 780 Pnt 3869 827 Pnt 3869 874 Pnt 3869 921 Pnt 3869 968 Pnt 3869 1015 Pnt 3869 1062 Pnt 3869 1673 Pnt 3869 1720 Pnt 3869 1956 Pnt 3869 2050 Pnt 3880 168 Pnt 3880 215 Pnt 3880 262 Pnt 3880 309 Pnt 3880 356 Pnt 3880 403 Pnt 3880 450 Pnt 3880 544 Pnt 3880 591 Pnt 3880 638 Pnt 3880 685 Pnt 3880 732 Pnt 3880 780 Pnt 3880 827 Pnt 3880 874 Pnt 3880 921 Pnt 3880 968 Pnt 3880 1626 Pnt 3880 1673 Pnt 3880 1767 Pnt 3880 1908 Pnt 3880 2003 Pnt 3891 168 Pnt 3891 215 Pnt 3891 262 Pnt 3891 309 Pnt 3891 356 Pnt 3891 403 Pnt 3891 450 Pnt 3891 497 Pnt 3891 544 Pnt 3891 591 Pnt 3891 638 Pnt 3891 685 Pnt 3891 732 Pnt 3891 780 Pnt 3891 827 Pnt 3891 874 Pnt 3891 921 Pnt 3891 1579 Pnt 3891 1720 Pnt 3891 1861 Pnt 3891 2003 Pnt 3903 168 Pnt 3903 215 Pnt 3903 262 Pnt 3903 309 Pnt 3903 356 Pnt 3903 403 Pnt 3903 450 Pnt 3903 497 Pnt 3903 544 Pnt 3903 591 Pnt 3903 638 Pnt 3903 685 Pnt 3903 732 Pnt 3903 780 Pnt 3903 827 Pnt 3903 874 Pnt 3903 1532 Pnt 3903 1626 Pnt 3903 1673 Pnt 3903 1861 Pnt 3914 168 Pnt 3914 215 Pnt 3914 262 Pnt 3914 309 Pnt 3914 356 Pnt 3914 403 Pnt 3914 450 Pnt 3914 497 Pnt 3914 544 Pnt 3914 591 Pnt 3914 638 Pnt 3914 685 Pnt 3914 732 Pnt 3914 780 Pnt 3914 827 Pnt 3914 1485 Pnt 3914 1579 Pnt 3914 1626 Pnt 3914 1814 Pnt 3914 1908 Pnt 3925 168 Pnt 3925 215 Pnt 3925 262 Pnt 3925 309 Pnt 3925 356 Pnt 3925 403 Pnt 3925 450 Pnt 3925 497 Pnt 3925 544 Pnt 3925 591 Pnt 3925 638 Pnt 3925 685 Pnt 3925 732 Pnt 3925 780 Pnt 3925 1438 Pnt 3925 1532 Pnt 3925 1579 Pnt 3925 1626 Pnt 3925 1767 Pnt 3925 1908 Pnt 3936 168 Pnt 3936 215 Pnt 3936 262 Pnt 3936 309 Pnt 3936 356 Pnt 3936 403 Pnt 3936 450 Pnt 3936 497 Pnt 3936 544 Pnt 3936 591 Pnt 3936 638 Pnt 3936 685 Pnt 3936 1344 Pnt 3936 1391 Pnt 3936 1485 Pnt 3936 1579 Pnt 3936 1767 Pnt 3936 1861 Pnt 3947 168 Pnt 3947 215 Pnt 3947 262 Pnt 3947 309 Pnt 3947 356 Pnt 3947 403 Pnt 3947 450 Pnt 3947 497 Pnt 3947 544 Pnt 3947 591 Pnt 3947 638 Pnt 3947 1297 Pnt 3947 1344 Pnt 3947 1438 Pnt 3947 1532 Pnt 3947 1720 Pnt 3959 168 Pnt 3959 215 Pnt 3959 262 Pnt 3959 309 Pnt 3959 356 Pnt 3959 403 Pnt 3959 450 Pnt 3959 497 Pnt 3959 544 Pnt 3959 591 Pnt 3959 1250 Pnt 3959 1297 Pnt 3959 1344 Pnt 3959 1391 Pnt 3959 1485 Pnt 3959 1673 Pnt 3959 1720 Pnt 3970 168 Pnt 3970 215 Pnt 3970 262 Pnt 3970 309 Pnt 3970 356 Pnt 3970 403 Pnt 3970 450 Pnt 3970 497 Pnt 3970 544 Pnt 3970 1156 Pnt 3970 1203 Pnt 3970 1250 Pnt 3970 1297 Pnt 3970 1344 Pnt 3970 1438 Pnt 3970 1626 Pnt 3970 1673 Pnt 3970 1767 Pnt 3981 168 Pnt 3981 215 Pnt 3981 262 Pnt 3981 309 Pnt 3981 356 Pnt 3981 403 Pnt 3981 450 Pnt 3981 1062 Pnt 3981 1109 Pnt 3981 1156 Pnt 3981 1250 Pnt 3981 1297 Pnt 3981 1391 Pnt 3981 1438 Pnt 3981 1626 Pnt 3981 1767 Pnt 3992 168 Pnt 3992 215 Pnt 3992 262 Pnt 3992 309 Pnt 3992 356 Pnt 3992 403 Pnt 3992 968 Pnt 3992 1015 Pnt 3992 1062 Pnt 3992 1109 Pnt 3992 1203 Pnt 3992 1250 Pnt 3992 1344 Pnt 3992 1391 Pnt 3992 1626 Pnt 4003 168 Pnt 4003 215 Pnt 4003 262 Pnt 4003 309 Pnt 4003 356 Pnt 4003 874 Pnt 4003 921 Pnt 4003 968 Pnt 4003 1015 Pnt 4003 1062 Pnt 4003 1109 Pnt 4003 1156 Pnt 4003 1203 Pnt 4003 1297 Pnt 4003 1344 Pnt 4003 1532 Pnt 4003 1579 Pnt 4003 1673 Pnt 4015 168 Pnt 4015 215 Pnt 4015 262 Pnt 4015 309 Pnt 4015 732 Pnt 4015 780 Pnt 4015 827 Pnt 4015 874 Pnt 4015 921 Pnt 4015 968 Pnt 4015 1062 Pnt 4015 1109 Pnt 4015 1156 Pnt 4015 1250 Pnt 4015 1297 Pnt 4015 1485 Pnt 4026 168 Pnt 4026 215 Pnt 4026 544 Pnt 4026 591 Pnt 4026 638 Pnt 4026 685 Pnt 4026 732 Pnt 4026 780 Pnt 4026 827 Pnt 4026 874 Pnt 4026 921 Pnt 4026 968 Pnt 4026 1015 Pnt 4026 1062 Pnt 4026 1109 Pnt 4026 1203 Pnt 4026 1250 Pnt 4026 1438 Pnt 4026 1485 Pnt 4026 1532 Pnt 4026 1626 Pnt 4026 1673 Pnt 4037 168 Pnt 4037 262 Pnt 4037 309 Pnt 4037 356 Pnt 4037 403 Pnt 4037 450 Pnt 4037 497 Pnt 4037 544 Pnt 4037 591 Pnt 4037 638 Pnt 4037 685 Pnt 4037 732 Pnt 4037 780 Pnt 4037 827 Pnt 4037 921 Pnt 4037 968 Pnt 4037 1015 Pnt 4037 1062 Pnt 4037 1156 Pnt 4037 1203 Pnt 4037 1438 Pnt 4037 1485 Pnt 4037 1579 Pnt 4037 1626 Pnt 4048 168 Pnt 4048 215 Pnt 4048 262 Pnt 4048 309 Pnt 4048 356 Pnt 4048 403 Pnt 4048 450 Pnt 4048 497 Pnt 4048 544 Pnt 4048 591 Pnt 4048 638 Pnt 4048 685 Pnt 4048 732 Pnt 4048 780 Pnt 4048 827 Pnt 4048 874 Pnt 4048 921 Pnt 4048 968 Pnt 4048 1062 Pnt 4048 1109 Pnt 4048 1156 Pnt 4048 1391 Pnt 4048 1438 Pnt 4059 168 Pnt 4059 215 Pnt 4059 262 Pnt 4059 309 Pnt 4059 356 Pnt 4059 403 Pnt 4059 450 Pnt 4059 497 Pnt 4059 544 Pnt 4059 591 Pnt 4059 638 Pnt 4059 685 Pnt 4059 732 Pnt 4059 780 Pnt 4059 827 Pnt 4059 874 Pnt 4059 921 Pnt 4059 1015 Pnt 4059 1062 Pnt 4059 1109 Pnt 4059 1344 Pnt 4059 1391 Pnt 4059 1532 Pnt 4059 1579 Pnt 4071 168 Pnt 4071 215 Pnt 4071 262 Pnt 4071 309 Pnt 4071 356 Pnt 4071 403 Pnt 4071 450 Pnt 4071 497 Pnt 4071 544 Pnt 4071 591 Pnt 4071 638 Pnt 4071 685 Pnt 4071 732 Pnt 4071 780 Pnt 4071 827 Pnt 4071 874 Pnt 4071 968 Pnt 4071 1015 Pnt 4071 1062 Pnt 4071 1297 Pnt 4071 1344 Pnt 4071 1391 Pnt 4071 1485 Pnt 4071 1532 Pnt 4082 168 Pnt 4082 215 Pnt 4082 262 Pnt 4082 309 Pnt 4082 356 Pnt 4082 403 Pnt 4082 450 Pnt 4082 497 Pnt 4082 544 Pnt 4082 591 Pnt 4082 638 Pnt 4082 685 Pnt 4082 732 Pnt 4082 780 Pnt 4082 827 Pnt 4082 874 Pnt 4082 921 Pnt 4082 968 Pnt 4082 1015 Pnt 4082 1062 Pnt 4082 1250 Pnt 4082 1297 Pnt 4082 1344 Pnt 4082 1532 Pnt 4093 168 Pnt 4093 215 Pnt 4093 262 Pnt 4093 309 Pnt 4093 356 Pnt 4093 403 Pnt 4093 450 Pnt 4093 497 Pnt 4093 544 Pnt 4093 591 Pnt 4093 638 Pnt 4093 685 Pnt 4093 732 Pnt 4093 827 Pnt 4093 874 Pnt 4093 921 Pnt 4093 968 Pnt 4093 1015 Pnt 4093 1203 Pnt 4093 1250 Pnt 4093 1297 Pnt 4093 1438 Pnt 4093 1485 Pnt 4104 168 Pnt 4104 215 Pnt 4104 262 Pnt 4104 309 Pnt 4104 356 Pnt 4104 403 Pnt 4104 450 Pnt 4104 497 Pnt 4104 544 Pnt 4104 591 Pnt 4104 638 Pnt 4104 685 Pnt 4104 732 Pnt 4104 780 Pnt 4104 827 Pnt 4104 874 Pnt 4104 921 Pnt 4104 968 Pnt 4104 1156 Pnt 4104 1203 Pnt 4104 1250 Pnt 4104 1297 Pnt 4104 1391 Pnt 4104 1438 Pnt 4116 168 Pnt 4116 215 Pnt 4116 262 Pnt 4116 309 Pnt 4116 356 Pnt 4116 403 Pnt 4116 450 Pnt 4116 497 Pnt 4116 544 Pnt 4116 591 Pnt 4116 638 Pnt 4116 685 Pnt 4116 732 Pnt 4116 780 Pnt 4116 827 Pnt 4116 874 Pnt 4116 1109 Pnt 4116 1156 Pnt 4116 1203 Pnt 4116 1250 Pnt 4116 1391 Pnt 4116 1438 Pnt 4127 168 Pnt 4127 215 Pnt 4127 262 Pnt 4127 309 Pnt 4127 356 Pnt 4127 403 Pnt 4127 450 Pnt 4127 497 Pnt 4127 544 Pnt 4127 591 Pnt 4127 638 Pnt 4127 685 Pnt 4127 732 Pnt 4127 780 Pnt 4127 827 Pnt 4127 1062 Pnt 4127 1109 Pnt 4127 1156 Pnt 4127 1203 Pnt 4127 1344 Pnt 4127 1391 Pnt 4138 168 Pnt 4138 215 Pnt 4138 262 Pnt 4138 309 Pnt 4138 356 Pnt 4138 403 Pnt 4138 450 Pnt 4138 497 Pnt 4138 544 Pnt 4138 591 Pnt 4138 638 Pnt 4138 685 Pnt 4138 732 Pnt 4138 780 Pnt 4138 1015 Pnt 4138 1062 Pnt 4138 1109 Pnt 4138 1156 Pnt 4138 1203 Pnt 4138 1297 Pnt 4138 1391 Pnt 4149 168 Pnt 4149 215 Pnt 4149 262 Pnt 4149 309 Pnt 4149 356 Pnt 4149 403 Pnt 4149 450 Pnt 4149 497 Pnt 4149 544 Pnt 4149 591 Pnt 4149 638 Pnt 4149 685 Pnt 4149 732 Pnt 4149 968 Pnt 4149 1015 Pnt 4149 1062 Pnt 4149 1109 Pnt 4149 1156 Pnt 4149 1250 Pnt 4149 1297 Pnt 4160 168 Pnt 4160 215 Pnt 4160 262 Pnt 4160 309 Pnt 4160 356 Pnt 4160 403 Pnt 4160 450 Pnt 4160 497 Pnt 4160 544 Pnt 4160 591 Pnt 4160 638 Pnt 4160 685 Pnt 4160 921 Pnt 4160 968 Pnt 4160 1015 Pnt 4160 1062 Pnt 4160 1109 Pnt 4160 1250 Pnt 4160 1297 Pnt 4172 168 Pnt 4172 215 Pnt 4172 262 Pnt 4172 309 Pnt 4172 356 Pnt 4172 403 Pnt 4172 450 Pnt 4172 497 Pnt 4172 544 Pnt 4172 591 Pnt 4172 638 Pnt 4172 827 Pnt 4172 874 Pnt 4172 921 Pnt 4172 968 Pnt 4172 1015 Pnt 4172 1062 Pnt 4172 1203 Pnt 4172 1297 Pnt 4183 168 Pnt 4183 215 Pnt 4183 262 Pnt 4183 309 Pnt 4183 356 Pnt 4183 403 Pnt 4183 450 Pnt 4183 497 Pnt 4183 544 Pnt 4183 591 Pnt 4183 780 Pnt 4183 827 Pnt 4183 874 Pnt 4183 921 Pnt 4183 968 Pnt 4183 1015 Pnt 4183 1062 Pnt 4183 1156 Pnt 4183 1203 Pnt 4183 1250 Pnt 4194 168 Pnt 4194 215 Pnt 4194 262 Pnt 4194 309 Pnt 4194 356 Pnt 4194 403 Pnt 4194 450 Pnt 4194 497 Pnt 4194 685 Pnt 4194 732 Pnt 4194 780 Pnt 4194 827 Pnt 4194 874 Pnt 4194 921 Pnt 4194 968 Pnt 4194 1015 Pnt 4194 1109 Pnt 4194 1156 Pnt 4194 1203 Pnt 4194 1250 Pnt 4205 168 Pnt 4205 215 Pnt 4205 262 Pnt 4205 309 Pnt 4205 356 Pnt 4205 403 Pnt 4205 450 Pnt 4205 638 Pnt 4205 685 Pnt 4205 732 Pnt 4205 780 Pnt 4205 827 Pnt 4205 874 Pnt 4205 921 Pnt 4205 968 Pnt 4205 1109 Pnt 4205 1203 Pnt 4216 168 Pnt 4216 215 Pnt 4216 262 Pnt 4216 309 Pnt 4216 356 Pnt 4216 544 Pnt 4216 591 Pnt 4216 638 Pnt 4216 685 Pnt 4216 732 Pnt 4216 780 Pnt 4216 827 Pnt 4216 874 Pnt 4216 921 Pnt 4216 1062 Pnt 4216 1109 Pnt 4216 1156 Pnt 4228 168 Pnt 4228 215 Pnt 4228 262 Pnt 4228 309 Pnt 4228 403 Pnt 4228 450 Pnt 4228 497 Pnt 4228 544 Pnt 4228 591 Pnt 4228 638 Pnt 4228 685 Pnt 4228 732 Pnt 4228 780 Pnt 4228 827 Pnt 4228 874 Pnt 4228 1015 Pnt 4228 1062 Pnt 4228 1109 Pnt 4228 1156 Pnt 4239 168 Pnt 4239 215 Pnt 4239 309 Pnt 4239 356 Pnt 4239 403 Pnt 4239 450 Pnt 4239 497 Pnt 4239 544 Pnt 4239 591 Pnt 4239 638 Pnt 4239 685 Pnt 4239 732 Pnt 4239 780 Pnt 4239 827 Pnt 4239 874 Pnt 4239 968 Pnt 4239 1015 Pnt 4239 1109 Pnt 4250 168 Pnt 4250 215 Pnt 4250 262 Pnt 4250 309 Pnt 4250 356 Pnt 4250 403 Pnt 4250 450 Pnt 4250 497 Pnt 4250 544 Pnt 4250 591 Pnt 4250 638 Pnt 4250 685 Pnt 4250 732 Pnt 4250 780 Pnt 4250 827 Pnt 4250 921 Pnt 4250 968 Pnt 4250 1015 Pnt 4250 1062 Pnt 4250 1109 Pnt 4261 168 Pnt 4261 215 Pnt 4261 262 Pnt 4261 309 Pnt 4261 356 Pnt 4261 403 Pnt 4261 450 Pnt 4261 497 Pnt 4261 544 Pnt 4261 591 Pnt 4261 638 Pnt 4261 685 Pnt 4261 732 Pnt 4261 780 Pnt 4261 874 Pnt 4261 921 Pnt 4261 968 Pnt 4261 1015 Pnt 4261 1062 Pnt 4272 168 Pnt 4272 215 Pnt 4272 262 Pnt 4272 309 Pnt 4272 356 Pnt 4272 403 Pnt 4272 450 Pnt 4272 497 Pnt 4272 544 Pnt 4272 591 Pnt 4272 638 Pnt 4272 685 Pnt 4272 732 Pnt 4272 874 Pnt 4272 921 Pnt 4272 1015 Pnt 4284 168 Pnt 4284 215 Pnt 4284 262 Pnt 4284 309 Pnt 4284 356 Pnt 4284 403 Pnt 4284 450 Pnt 4284 497 Pnt 4284 544 Pnt 4284 591 Pnt 4284 638 Pnt 4284 685 Pnt 4284 827 Pnt 4284 874 Pnt 4284 921 Pnt 4284 968 Pnt 4284 1015 Pnt 4295 168 Pnt 4295 215 Pnt 4295 262 Pnt 4295 309 Pnt 4295 356 Pnt 4295 403 Pnt 4295 450 Pnt 4295 497 Pnt 4295 544 Pnt 4295 591 Pnt 4295 638 Pnt 4295 780 Pnt 4295 827 Pnt 4295 874 Pnt 4295 921 Pnt 4295 968 Pnt 4306 168 Pnt 4306 215 Pnt 4306 262 Pnt 4306 309 Pnt 4306 356 Pnt 4306 403 Pnt 4306 450 Pnt 4306 497 Pnt 4306 544 Pnt 4306 591 Pnt 4306 732 Pnt 4306 780 Pnt 4306 827 Pnt 4306 874 Pnt 4306 921 Pnt 4317 168 Pnt 4317 215 Pnt 4317 262 Pnt 4317 309 Pnt 4317 356 Pnt 4317 403 Pnt 4317 450 Pnt 4317 497 Pnt 4317 544 Pnt 4317 638 Pnt 4317 685 Pnt 4317 732 Pnt 4317 780 Pnt 4317 874 Pnt 4317 921 Pnt 4328 168 Pnt 4328 215 Pnt 4328 262 Pnt 4328 309 Pnt 4328 356 Pnt 4328 403 Pnt 4328 450 Pnt 4328 497 Pnt 4328 591 Pnt 4328 638 Pnt 4328 685 Pnt 4328 732 Pnt 4328 780 Pnt 4328 827 Pnt 4328 874 Pnt 4340 168 Pnt 4340 215 Pnt 4340 262 Pnt 4340 309 Pnt 4340 356 Pnt 4340 403 Pnt 4340 450 Pnt 4340 544 Pnt 4340 591 Pnt 4340 638 Pnt 4340 685 Pnt 4340 732 Pnt 4340 780 Pnt 4340 827 Pnt 4340 874 Pnt 4351 168 Pnt 4351 215 Pnt 4351 262 Pnt 4351 309 Pnt 4351 356 Pnt 4351 403 Pnt 4351 497 Pnt 4351 544 Pnt 4351 591 Pnt 4351 638 Pnt 4351 685 Pnt 4351 732 Pnt 4351 780 Pnt 4351 827 Pnt 4362 168 Pnt 4362 215 Pnt 4362 262 Pnt 4362 309 Pnt 4362 356 Pnt 4362 403 Pnt 4362 450 Pnt 4362 497 Pnt 4362 544 Pnt 4362 591 Pnt 4362 638 Pnt 4362 732 Pnt 4362 780 Pnt 4373 168 Pnt 4373 215 Pnt 4373 262 Pnt 4373 309 Pnt 4373 356 Pnt 4373 403 Pnt 4373 450 Pnt 4373 497 Pnt 4373 544 Pnt 4373 591 Pnt 4373 638 Pnt 4373 685 Pnt 4373 732 Pnt 4373 780 Pnt 4384 168 Pnt 4384 215 Pnt 4384 262 Pnt 4384 309 Pnt 4384 356 Pnt 4384 403 Pnt 4384 450 Pnt 4384 497 Pnt 4384 544 Pnt 4384 591 Pnt 4384 638 Pnt 4384 685 Pnt 4384 732 Pnt 4396 168 Pnt 4396 215 Pnt 4396 262 Pnt 4396 309 Pnt 4396 356 Pnt 4396 403 Pnt 4396 450 Pnt 4396 497 Pnt 4396 544 Pnt 4396 591 Pnt 4396 638 Pnt 4396 685 Pnt 4396 732 Pnt 4407 168 Pnt 4407 215 Pnt 4407 262 Pnt 4407 309 Pnt 4407 356 Pnt 4407 403 Pnt 4407 450 Pnt 4407 497 Pnt 4407 544 Pnt 4407 591 Pnt 4407 638 Pnt 4407 685 Pnt 4418 168 Pnt 4418 215 Pnt 4418 262 Pnt 4418 309 Pnt 4418 356 Pnt 4418 403 Pnt 4418 450 Pnt 4418 497 Pnt 4418 544 Pnt 4418 591 Pnt 4418 638 Pnt 4429 168 Pnt 4429 215 Pnt 4429 262 Pnt 4429 309 Pnt 4429 356 Pnt 4429 403 Pnt 4429 450 Pnt 4429 497 Pnt 4429 544 Pnt 4429 591 Pnt 4429 638 Pnt 4440 168 Pnt 4440 215 Pnt 4440 262 Pnt 4440 309 Pnt 4440 356 Pnt 4440 403 Pnt 4440 450 Pnt 4440 497 Pnt 4440 544 Pnt 4440 591 Pnt 4452 168 Pnt 4452 215 Pnt 4452 262 Pnt 4452 309 Pnt 4452 356 Pnt 4452 403 Pnt 4452 450 Pnt 4452 497 Pnt 4452 544 Pnt 4452 591 Pnt 4463 168 Pnt 4463 215 Pnt 4463 262 Pnt 4463 309 Pnt 4463 356 Pnt 4463 403 Pnt 4463 450 Pnt 4463 497 Pnt 4463 544 Pnt 4474 168 Pnt 4474 215 Pnt 4474 262 Pnt 4474 309 Pnt 4474 356 Pnt 4474 403 Pnt 4474 450 Pnt 4474 497 Pnt 4485 168 Pnt 4485 215 Pnt 4485 262 Pnt 4485 309 Pnt 4485 356 Pnt 4485 403 Pnt 4485 450 Pnt 4485 497 Pnt 4497 168 Pnt 4497 215 Pnt 4497 262 Pnt 4497 309 Pnt 4497 356 Pnt 4497 403 Pnt 4497 450 Pnt 4508 168 Pnt 4508 215 Pnt 4508 262 Pnt 4508 309 Pnt 4508 356 Pnt 4508 403 Pnt 4508 450 Pnt 4519 168 Pnt 4519 215 Pnt 4519 262 Pnt 4519 309 Pnt 4519 356 Pnt 4519 403 Pnt 4530 168 Pnt 4530 215 Pnt 4530 262 Pnt 4530 309 Pnt 4530 356 Pnt 4541 168 Pnt 4541 215 Pnt 4541 262 Pnt 4541 309 Pnt 4541 356 Pnt 4553 168 Pnt 4553 215 Pnt 4553 262 Pnt 4553 309 Pnt 4564 168 Pnt 4564 215 Pnt 4564 262 Pnt 4564 309 Pnt 4575 168 Pnt 4575 215 Pnt 4575 262 Pnt 4586 168 Pnt 4586 215 Pnt 4597 168 Pnt 4597 215 Pnt 4609 168 Pnt 4620 168 Pnt 4631 168 Pnt 4631 215 Pnt 4642 168 Pnt 4642 215 Pnt 4642 262 Pnt 4653 168 Pnt 4653 215 Pnt 4653 262 Pnt 4653 309 Pnt 4665 168 Pnt 4665 215 Pnt 4665 262 Pnt 4665 309 Pnt 4665 356 Pnt 4676 168 Pnt 4676 215 Pnt 4676 262 Pnt 4676 309 Pnt 4676 356 Pnt 4676 403 Pnt 4676 450 Pnt 4687 168 Pnt 4687 215 Pnt 4687 262 Pnt 4687 309 Pnt 4687 356 Pnt 4687 403 Pnt 4687 450 Pnt 4687 497 Pnt 4698 168 Pnt 4698 215 Pnt 4698 262 Pnt 4698 309 Pnt 4698 356 Pnt 4698 403 Pnt 4698 450 Pnt 4698 497 Pnt 4698 544 Pnt 4709 168 Pnt 4709 215 Pnt 4709 262 Pnt 4709 309 Pnt 4709 356 Pnt 4709 403 Pnt 4709 450 Pnt 4709 497 Pnt 4709 544 Pnt 4721 168 Pnt 4721 215 Pnt 4721 262 Pnt 4721 309 Pnt 4721 356 Pnt 4721 403 Pnt 4721 450 Pnt 4721 497 Pnt 4721 544 Pnt 4721 591 Pnt 4732 168 Pnt 4732 215 Pnt 4732 262 Pnt 4732 309 Pnt 4732 356 Pnt 4732 403 Pnt 4732 450 Pnt 4732 497 Pnt 4732 544 Pnt 4732 591 Pnt 4732 638 Pnt 4743 168 Pnt 4743 215 Pnt 4743 262 Pnt 4743 309 Pnt 4743 356 Pnt 4743 403 Pnt 4743 450 Pnt 4743 497 Pnt 4743 544 Pnt 4743 591 Pnt 4743 638 Pnt 4743 685 Pnt 4754 168 Pnt 4754 215 Pnt 4754 262 Pnt 4754 309 Pnt 4754 356 Pnt 4754 403 Pnt 4754 450 Pnt 4754 497 Pnt 4754 544 Pnt 4754 591 Pnt 4754 638 Pnt 4754 685 Pnt 4754 732 Pnt 4765 168 Pnt 4765 215 Pnt 4765 262 Pnt 4765 309 Pnt 4765 356 Pnt 4765 403 Pnt 4765 450 Pnt 4765 497 Pnt 4765 544 Pnt 4765 591 Pnt 4765 638 Pnt 4765 685 Pnt 4765 732 Pnt 4765 780 Pnt 4777 168 Pnt 4777 215 Pnt 4777 262 Pnt 4777 309 Pnt 4777 356 Pnt 4777 403 Pnt 4777 450 Pnt 4777 497 Pnt 4777 544 Pnt 4777 591 Pnt 4777 638 Pnt 4777 685 Pnt 4777 732 Pnt 4777 780 Pnt 4777 827 Pnt 4788 168 Pnt 4788 215 Pnt 4788 262 Pnt 4788 309 Pnt 4788 356 Pnt 4788 403 Pnt 4788 450 Pnt 4788 497 Pnt 4788 544 Pnt 4788 591 Pnt 4788 685 Pnt 4788 732 Pnt 4788 780 Pnt 4788 827 Pnt 4788 874 Pnt 4799 168 Pnt 4799 215 Pnt 4799 262 Pnt 4799 309 Pnt 4799 356 Pnt 4799 403 Pnt 4799 450 Pnt 4799 497 Pnt 4799 544 Pnt 4799 591 Pnt 4799 638 Pnt 4799 685 Pnt 4799 732 Pnt 4799 780 Pnt 4799 827 Pnt 4799 874 Pnt 4810 168 Pnt 4810 215 Pnt 4810 262 Pnt 4810 309 Pnt 4810 356 Pnt 4810 403 Pnt 4810 450 Pnt 4810 497 Pnt 4810 544 Pnt 4810 591 Pnt 4810 638 Pnt 4810 685 Pnt 4810 732 Pnt 4810 780 Pnt 4810 827 Pnt 4810 874 Pnt 4810 921 Pnt 4822 168 Pnt 4822 215 Pnt 4822 262 Pnt 4822 309 Pnt 4822 356 Pnt 4822 403 Pnt 4822 450 Pnt 4822 497 Pnt 4822 544 Pnt 4822 591 Pnt 4822 638 Pnt 4822 685 Pnt 4822 732 Pnt 4822 780 Pnt 4822 874 Pnt 4822 921 Pnt 4822 968 Pnt 4833 168 Pnt 4833 215 Pnt 4833 262 Pnt 4833 309 Pnt 4833 356 Pnt 4833 403 Pnt 4833 450 Pnt 4833 497 Pnt 4833 544 Pnt 4833 591 Pnt 4833 638 Pnt 4833 685 Pnt 4833 732 Pnt 4833 780 Pnt 4833 827 Pnt 4833 921 Pnt 4833 968 Pnt 4833 1015 Pnt 4844 168 Pnt 4844 215 Pnt 4844 262 Pnt 4844 309 Pnt 4844 356 Pnt 4844 403 Pnt 4844 450 Pnt 4844 497 Pnt 4844 544 Pnt 4844 591 Pnt 4844 638 Pnt 4844 685 Pnt 4844 732 Pnt 4844 780 Pnt 4844 827 Pnt 4844 874 Pnt 4844 968 Pnt 4844 1015 Pnt 4855 168 Pnt 4855 215 Pnt 4855 262 Pnt 4855 309 Pnt 4855 356 Pnt 4855 403 Pnt 4855 450 Pnt 4855 497 Pnt 4855 638 Pnt 4855 685 Pnt 4855 732 Pnt 4855 780 Pnt 4855 827 Pnt 4855 874 Pnt 4855 921 Pnt 4855 1015 Pnt 4855 1062 Pnt 4866 168 Pnt 4866 215 Pnt 4866 262 Pnt 4866 309 Pnt 4866 356 Pnt 4866 403 Pnt 4866 450 Pnt 4866 497 Pnt 4866 544 Pnt 4866 591 Pnt 4866 638 Pnt 4866 780 Pnt 4866 827 Pnt 4866 874 Pnt 4866 921 Pnt 4866 968 Pnt 4866 1015 Pnt 4866 1062 Pnt 4866 1109 Pnt 4878 168 Pnt 4878 215 Pnt 4878 262 Pnt 4878 309 Pnt 4878 356 Pnt 4878 403 Pnt 4878 450 Pnt 4878 497 Pnt 4878 544 Pnt 4878 591 Pnt 4878 638 Pnt 4878 685 Pnt 4878 732 Pnt 4878 827 Pnt 4878 874 Pnt 4878 921 Pnt 4878 968 Pnt 4878 1015 Pnt 4878 1062 Pnt 4878 1109 Pnt 4878 1156 Pnt 4889 168 Pnt 4889 215 Pnt 4889 262 Pnt 4889 309 Pnt 4889 356 Pnt 4889 403 Pnt 4889 450 Pnt 4889 497 Pnt 4889 544 Pnt 4889 591 Pnt 4889 638 Pnt 4889 685 Pnt 4889 732 Pnt 4889 780 Pnt 4889 921 Pnt 4889 968 Pnt 4889 1015 Pnt 4889 1062 Pnt 4889 1109 Pnt 4889 1156 Pnt 4900 168 Pnt 4900 215 Pnt 4900 262 Pnt 4900 309 Pnt 4900 356 Pnt 4900 403 Pnt 4900 450 Pnt 4900 497 Pnt 4900 544 Pnt 4900 591 Pnt 4900 638 Pnt 4900 685 Pnt 4900 732 Pnt 4900 780 Pnt 4900 827 Pnt 4900 968 Pnt 4900 1015 Pnt 4900 1062 Pnt 4900 1109 Pnt 4900 1156 Pnt 4900 1203 Pnt 4911 168 Pnt 4911 215 Pnt 4911 262 Pnt 4911 309 Pnt 4911 356 Pnt 4911 403 Pnt 4911 450 Pnt 4911 497 Pnt 4911 544 Pnt 4911 591 Pnt 4911 638 Pnt 4911 685 Pnt 4911 732 Pnt 4911 780 Pnt 4911 827 Pnt 4911 874 Pnt 4911 921 Pnt 4911 1015 Pnt 4911 1062 Pnt 4911 1109 Pnt 4911 1203 Pnt 4911 1250 Pnt 4922 168 Pnt 4922 215 Pnt 4922 262 Pnt 4922 309 Pnt 4922 356 Pnt 4922 403 Pnt 4922 450 Pnt 4922 497 Pnt 4922 544 Pnt 4922 591 Pnt 4922 638 Pnt 4922 685 Pnt 4922 732 Pnt 4922 780 Pnt 4922 827 Pnt 4922 874 Pnt 4922 921 Pnt 4922 968 Pnt 4922 1062 Pnt 4922 1109 Pnt 4922 1156 Pnt 4922 1250 Pnt 4934 168 Pnt 4934 215 Pnt 4934 262 Pnt 4934 309 Pnt 4934 356 Pnt 4934 403 Pnt 4934 450 Pnt 4934 497 Pnt 4934 544 Pnt 4934 591 Pnt 4934 638 Pnt 4934 685 Pnt 4934 732 Pnt 4934 780 Pnt 4934 827 Pnt 4934 874 Pnt 4934 921 Pnt 4934 968 Pnt 4934 1015 Pnt 4934 1109 Pnt 4934 1156 Pnt 4934 1203 Pnt 4934 1250 Pnt 4934 1297 Pnt 4945 168 Pnt 4945 215 Pnt 4945 262 Pnt 4945 309 Pnt 4945 356 Pnt 4945 403 Pnt 4945 450 Pnt 4945 497 Pnt 4945 544 Pnt 4945 591 Pnt 4945 638 Pnt 4945 685 Pnt 4945 732 Pnt 4945 780 Pnt 4945 827 Pnt 4945 874 Pnt 4945 921 Pnt 4945 968 Pnt 4945 1015 Pnt 4945 1062 Pnt 4945 1156 Pnt 4945 1203 Pnt 4945 1250 Pnt 4945 1297 Pnt 4945 1344 Pnt 4956 168 Pnt 4956 215 Pnt 4956 262 Pnt 4956 309 Pnt 4956 356 Pnt 4956 403 Pnt 4956 450 Pnt 4956 497 Pnt 4956 544 Pnt 4956 591 Pnt 4956 638 Pnt 4956 685 Pnt 4956 732 Pnt 4956 780 Pnt 4956 827 Pnt 4956 874 Pnt 4956 921 Pnt 4956 968 Pnt 4956 1015 Pnt 4956 1062 Pnt 4956 1109 Pnt 4956 1203 Pnt 4956 1250 Pnt 4956 1344 Pnt 4967 168 Pnt 4967 215 Pnt 4967 262 Pnt 4967 309 Pnt 4967 356 Pnt 4967 403 Pnt 4967 450 Pnt 4967 497 Pnt 4967 544 Pnt 4967 591 Pnt 4967 638 Pnt 4967 685 Pnt 4967 732 Pnt 4967 780 Pnt 4967 827 Pnt 4967 874 Pnt 4967 921 Pnt 4967 968 Pnt 4967 1015 Pnt 4967 1062 Pnt 4967 1109 Pnt 4967 1156 Pnt 4967 1250 Pnt 4967 1297 Pnt 4967 1391 Pnt 4978 168 Pnt 4978 215 Pnt 4978 262 Pnt 4978 309 Pnt 4978 356 Pnt 4978 403 Pnt 4978 450 Pnt 4978 497 Pnt 4978 544 Pnt 4978 591 Pnt 4978 638 Pnt 4978 685 Pnt 4978 732 Pnt 4978 780 Pnt 4978 827 Pnt 4978 874 Pnt 4978 921 Pnt 4978 968 Pnt 4978 1015 Pnt 4978 1062 Pnt 4978 1109 Pnt 4978 1156 Pnt 4978 1203 Pnt 4978 1297 Pnt 4978 1344 Pnt 4978 1391 Pnt 4978 1438 Pnt 4990 168 Pnt 4990 215 Pnt 4990 262 Pnt 4990 309 Pnt 4990 356 Pnt 4990 403 Pnt 4990 450 Pnt 4990 497 Pnt 4990 544 Pnt 4990 591 Pnt 4990 638 Pnt 4990 685 Pnt 4990 732 Pnt 4990 780 Pnt 4990 827 Pnt 4990 874 Pnt 4990 921 Pnt 4990 968 Pnt 4990 1015 Pnt 4990 1062 Pnt 4990 1109 Pnt 4990 1156 Pnt 4990 1203 Pnt 4990 1344 Pnt 4990 1391 Pnt 4990 1438 Pnt 5001 168 Pnt 5001 215 Pnt 5001 262 Pnt 5001 309 Pnt 5001 356 Pnt 5001 874 Pnt 5001 921 Pnt 5001 968 Pnt 5001 1015 Pnt 5001 1062 Pnt 5001 1109 Pnt 5001 1156 Pnt 5001 1203 Pnt 5001 1250 Pnt 5001 1344 Pnt 5001 1391 Pnt 5001 1485 Pnt 5012 168 Pnt 5012 215 Pnt 5012 262 Pnt 5012 968 Pnt 5012 1015 Pnt 5012 1062 Pnt 5012 1109 Pnt 5012 1156 Pnt 5012 1203 Pnt 5012 1250 Pnt 5012 1297 Pnt 5012 1391 Pnt 5012 1438 Pnt 5012 1485 Pnt 5012 1532 Pnt 5023 168 Pnt 5023 215 Pnt 5023 262 Pnt 5023 309 Pnt 5023 356 Pnt 5023 403 Pnt 5023 450 Pnt 5023 497 Pnt 5023 544 Pnt 5023 591 Pnt 5023 638 Pnt 5023 1062 Pnt 5023 1109 Pnt 5023 1156 Pnt 5023 1203 Pnt 5023 1250 Pnt 5023 1297 Pnt 5023 1344 Pnt 5023 1438 Pnt 5023 1485 Pnt 5023 1532 Pnt 5034 168 Pnt 5034 215 Pnt 5034 262 Pnt 5034 309 Pnt 5034 356 Pnt 5034 403 Pnt 5034 450 Pnt 5034 497 Pnt 5034 544 Pnt 5034 591 Pnt 5034 638 Pnt 5034 685 Pnt 5034 732 Pnt 5034 780 Pnt 5034 1109 Pnt 5034 1156 Pnt 5034 1203 Pnt 5034 1250 Pnt 5034 1297 Pnt 5034 1344 Pnt 5034 1391 Pnt 5034 1485 Pnt 5034 1579 Pnt 5046 168 Pnt 5046 215 Pnt 5046 262 Pnt 5046 309 Pnt 5046 356 Pnt 5046 403 Pnt 5046 450 Pnt 5046 497 Pnt 5046 544 Pnt 5046 591 Pnt 5046 638 Pnt 5046 685 Pnt 5046 732 Pnt 5046 780 Pnt 5046 827 Pnt 5046 1203 Pnt 5046 1250 Pnt 5046 1297 Pnt 5046 1344 Pnt 5046 1391 Pnt 5046 1532 Pnt 5046 1626 Pnt 5057 168 Pnt 5057 215 Pnt 5057 262 Pnt 5057 309 Pnt 5057 356 Pnt 5057 403 Pnt 5057 450 Pnt 5057 497 Pnt 5057 544 Pnt 5057 591 Pnt 5057 638 Pnt 5057 685 Pnt 5057 732 Pnt 5057 780 Pnt 5057 827 Pnt 5057 874 Pnt 5057 921 Pnt 5057 1250 Pnt 5057 1297 Pnt 5057 1344 Pnt 5057 1391 Pnt 5057 1438 Pnt 5057 1532 Pnt 5057 1579 Pnt 5057 1626 Pnt 5068 168 Pnt 5068 215 Pnt 5068 262 Pnt 5068 309 Pnt 5068 356 Pnt 5068 403 Pnt 5068 450 Pnt 5068 497 Pnt 5068 544 Pnt 5068 591 Pnt 5068 638 Pnt 5068 685 Pnt 5068 732 Pnt 5068 780 Pnt 5068 827 Pnt 5068 874 Pnt 5068 921 Pnt 5068 968 Pnt 5068 1297 Pnt 5068 1344 Pnt 5068 1391 Pnt 5068 1438 Pnt 5068 1485 Pnt 5068 1579 Pnt 5068 1673 Pnt 5079 168 Pnt 5079 215 Pnt 5079 262 Pnt 5079 309 Pnt 5079 356 Pnt 5079 403 Pnt 5079 450 Pnt 5079 497 Pnt 5079 544 Pnt 5079 591 Pnt 5079 638 Pnt 5079 685 Pnt 5079 732 Pnt 5079 780 Pnt 5079 827 Pnt 5079 874 Pnt 5079 921 Pnt 5079 968 Pnt 5079 1015 Pnt 5079 1062 Pnt 5079 1344 Pnt 5079 1391 Pnt 5079 1438 Pnt 5079 1485 Pnt 5079 1532 Pnt 5079 1626 Pnt 5079 1673 Pnt 5090 168 Pnt 5090 215 Pnt 5090 262 Pnt 5090 309 Pnt 5090 356 Pnt 5090 403 Pnt 5090 450 Pnt 5090 497 Pnt 5090 544 Pnt 5090 591 Pnt 5090 638 Pnt 5090 685 Pnt 5090 732 Pnt 5090 780 Pnt 5090 827 Pnt 5090 874 Pnt 5090 921 Pnt 5090 968 Pnt 5090 1015 Pnt 5090 1062 Pnt 5090 1109 Pnt 5090 1391 Pnt 5090 1438 Pnt 5090 1485 Pnt 5090 1532 Pnt 5090 1626 Pnt 5090 1673 Pnt 5090 1720 Pnt 5102 168 Pnt 5102 215 Pnt 5102 262 Pnt 5102 309 Pnt 5102 356 Pnt 5102 403 Pnt 5102 450 Pnt 5102 497 Pnt 5102 544 Pnt 5102 591 Pnt 5102 638 Pnt 5102 685 Pnt 5102 732 Pnt 5102 780 Pnt 5102 827 Pnt 5102 874 Pnt 5102 921 Pnt 5102 968 Pnt 5102 1015 Pnt 5102 1062 Pnt 5102 1109 Pnt 5102 1156 Pnt 5102 1438 Pnt 5102 1485 Pnt 5102 1532 Pnt 5102 1579 Pnt 5102 1673 Pnt 5102 1767 Pnt 5113 168 Pnt 5113 215 Pnt 5113 262 Pnt 5113 309 Pnt 5113 356 Pnt 5113 403 Pnt 5113 450 Pnt 5113 497 Pnt 5113 544 Pnt 5113 591 Pnt 5113 638 Pnt 5113 685 Pnt 5113 732 Pnt 5113 780 Pnt 5113 827 Pnt 5113 874 Pnt 5113 921 Pnt 5113 968 Pnt 5113 1015 Pnt 5113 1062 Pnt 5113 1109 Pnt 5113 1156 Pnt 5113 1203 Pnt 5113 1485 Pnt 5113 1532 Pnt 5113 1579 Pnt 5113 1626 Pnt 5113 1720 Pnt 5113 1767 Pnt 5124 168 Pnt 5124 215 Pnt 5124 262 Pnt 5124 309 Pnt 5124 356 Pnt 5124 403 Pnt 5124 450 Pnt 5124 497 Pnt 5124 544 Pnt 5124 591 Pnt 5124 638 Pnt 5124 685 Pnt 5124 732 Pnt 5124 780 Pnt 5124 827 Pnt 5124 874 Pnt 5124 921 Pnt 5124 968 Pnt 5124 1015 Pnt 5124 1062 Pnt 5124 1109 Pnt 5124 1156 Pnt 5124 1203 Pnt 5124 1250 Pnt 5124 1532 Pnt 5124 1579 Pnt 5124 1626 Pnt 5124 1673 Pnt 5124 1767 Pnt 5124 1814 Pnt 5135 168 Pnt 5135 215 Pnt 5135 262 Pnt 5135 309 Pnt 5135 356 Pnt 5135 403 Pnt 5135 450 Pnt 5135 497 Pnt 5135 544 Pnt 5135 591 Pnt 5135 638 Pnt 5135 685 Pnt 5135 732 Pnt 5135 780 Pnt 5135 827 Pnt 5135 874 Pnt 5135 921 Pnt 5135 968 Pnt 5135 1015 Pnt 5135 1109 Pnt 5135 1156 Pnt 5135 1203 Pnt 5135 1250 Pnt 5135 1297 Pnt 5135 1579 Pnt 5135 1626 Pnt 5135 1673 Pnt 5135 1767 Pnt 5135 1861 Pnt 5147 168 Pnt 5147 215 Pnt 5147 262 Pnt 5147 309 Pnt 5147 356 Pnt 5147 403 Pnt 5147 450 Pnt 5147 497 Pnt 5147 544 Pnt 5147 591 Pnt 5147 638 Pnt 5147 685 Pnt 5147 732 Pnt 5147 780 Pnt 5147 827 Pnt 5147 874 Pnt 5147 921 Pnt 5147 968 Pnt 5147 1015 Pnt 5147 1062 Pnt 5147 1109 Pnt 5147 1156 Pnt 5147 1203 Pnt 5147 1250 Pnt 5147 1297 Pnt 5147 1344 Pnt 5147 1626 Pnt 5147 1673 Pnt 5147 1720 Pnt 5147 1814 Pnt 5147 1861 Pnt 5158 168 Pnt 5158 215 Pnt 5158 262 Pnt 5158 309 Pnt 5158 356 Pnt 5158 403 Pnt 5158 450 Pnt 5158 497 Pnt 5158 544 Pnt 5158 591 Pnt 5158 638 Pnt 5158 685 Pnt 5158 732 Pnt 5158 780 Pnt 5158 827 Pnt 5158 874 Pnt 5158 921 Pnt 5158 968 Pnt 5158 1015 Pnt 5158 1062 Pnt 5158 1109 Pnt 5158 1156 Pnt 5158 1250 Pnt 5158 1297 Pnt 5158 1344 Pnt 5158 1391 Pnt 5158 1626 Pnt 5158 1673 Pnt 5158 1720 Pnt 5158 1767 Pnt 5158 1861 Pnt 5158 1908 Pnt 5169 168 Pnt 5169 215 Pnt 5169 262 Pnt 5169 309 Pnt 5169 356 Pnt 5169 403 Pnt 5169 450 Pnt 5169 497 Pnt 5169 544 Pnt 5169 591 Pnt 5169 638 Pnt 5169 685 Pnt 5169 732 Pnt 5169 780 Pnt 5169 827 Pnt 5169 874 Pnt 5169 921 Pnt 5169 968 Pnt 5169 1015 Pnt 5169 1062 Pnt 5169 1109 Pnt 5169 1156 Pnt 5169 1203 Pnt 5169 1297 Pnt 5169 1344 Pnt 5169 1391 Pnt 5169 1438 Pnt 5169 1673 Pnt 5169 1720 Pnt 5169 1767 Pnt 5169 1861 Pnt 5180 168 Pnt 5180 215 Pnt 5180 262 Pnt 5180 309 Pnt 5180 356 Pnt 5180 403 Pnt 5180 450 Pnt 5180 497 Pnt 5180 544 Pnt 5180 591 Pnt 5180 638 Pnt 5180 685 Pnt 5180 732 Pnt 5180 780 Pnt 5180 827 Pnt 5180 874 Pnt 5180 921 Pnt 5180 968 Pnt 5180 1015 Pnt 5180 1062 Pnt 5180 1109 Pnt 5180 1156 Pnt 5180 1203 Pnt 5180 1250 Pnt 5180 1297 Pnt 5180 1344 Pnt 5180 1391 Pnt 5180 1438 Pnt 5180 1485 Pnt 5180 1720 Pnt 5180 1767 Pnt 5180 1814 Pnt 5180 1908 Pnt 5180 1956 Pnt 5191 168 Pnt 5191 215 Pnt 5191 262 Pnt 5191 309 Pnt 5191 356 Pnt 5191 403 Pnt 5191 450 Pnt 5191 497 Pnt 5191 544 Pnt 5191 591 Pnt 5191 638 Pnt 5191 685 Pnt 5191 732 Pnt 5191 780 Pnt 5191 827 Pnt 5191 874 Pnt 5191 921 Pnt 5191 968 Pnt 5191 1015 Pnt 5191 1062 Pnt 5191 1109 Pnt 5191 1156 Pnt 5191 1203 Pnt 5191 1250 Pnt 5191 1297 Pnt 5191 1344 Pnt 5191 1391 Pnt 5191 1438 Pnt 5191 1485 Pnt 5191 1532 Pnt 5191 1767 Pnt 5191 1814 Pnt 5191 1861 Pnt 5191 1908 Pnt 5191 2003 Pnt 5203 168 Pnt 5203 215 Pnt 5203 262 Pnt 5203 309 Pnt 5203 356 Pnt 5203 403 Pnt 5203 450 Pnt 5203 497 Pnt 5203 544 Pnt 5203 591 Pnt 5203 638 Pnt 5203 685 Pnt 5203 732 Pnt 5203 780 Pnt 5203 827 Pnt 5203 874 Pnt 5203 921 Pnt 5203 968 Pnt 5203 1015 Pnt 5203 1062 Pnt 5203 1109 Pnt 5203 1156 Pnt 5203 1203 Pnt 5203 1250 Pnt 5203 1297 Pnt 5203 1344 Pnt 5203 1391 Pnt 5203 1438 Pnt 5203 1485 Pnt 5203 1532 Pnt 5203 1767 Pnt 5203 1814 Pnt 5203 1861 Pnt 5203 1956 Pnt 5203 2003 Pnt 5214 168 Pnt 5214 215 Pnt 5214 262 Pnt 5214 309 Pnt 5214 356 Pnt 5214 403 Pnt 5214 450 Pnt 5214 497 Pnt 5214 544 Pnt 5214 591 Pnt 5214 638 Pnt 5214 685 Pnt 5214 732 Pnt 5214 780 Pnt 5214 827 Pnt 5214 874 Pnt 5214 921 Pnt 5214 968 Pnt 5214 1015 Pnt 5214 1062 Pnt 5214 1109 Pnt 5214 1156 Pnt 5214 1203 Pnt 5214 1250 Pnt 5214 1297 Pnt 5214 1344 Pnt 5214 1391 Pnt 5214 1438 Pnt 5214 1485 Pnt 5214 1532 Pnt 5214 1579 Pnt 5214 1814 Pnt 5214 1861 Pnt 5214 1908 Pnt 5214 2003 Pnt 5214 2050 Pnt 5225 168 Pnt 5225 215 Pnt 5225 262 Pnt 5225 309 Pnt 5225 356 Pnt 5225 403 Pnt 5225 450 Pnt 5225 497 Pnt 5225 544 Pnt 5225 591 Pnt 5225 638 Pnt 5225 685 Pnt 5225 732 Pnt 5225 780 Pnt 5225 827 Pnt 5225 874 Pnt 5225 921 Pnt 5225 968 Pnt 5225 1015 Pnt 5225 1062 Pnt 5225 1109 Pnt 5225 1156 Pnt 5225 1203 Pnt 5225 1250 Pnt 5225 1297 Pnt 5225 1344 Pnt 5225 1391 Pnt 5225 1438 Pnt 5225 1485 Pnt 5225 1532 Pnt 5225 1579 Pnt 5225 1626 Pnt 5225 1861 Pnt 5225 1908 Pnt 5225 2003 Pnt 5236 168 Pnt 5236 215 Pnt 5236 262 Pnt 5236 309 Pnt 5236 356 Pnt 5236 403 Pnt 5236 450 Pnt 5236 497 Pnt 5236 544 Pnt 5236 591 Pnt 5236 638 Pnt 5236 685 Pnt 5236 732 Pnt 5236 780 Pnt 5236 827 Pnt 5236 874 Pnt 5236 921 Pnt 5236 968 Pnt 5236 1015 Pnt 5236 1062 Pnt 5236 1109 Pnt 5236 1156 Pnt 5236 1203 Pnt 5236 1250 Pnt 5236 1297 Pnt 5236 1391 Pnt 5236 1438 Pnt 5236 1485 Pnt 5236 1532 Pnt 5236 1579 Pnt 5236 1626 Pnt 5236 1673 Pnt 5236 1908 Pnt 5236 1956 Pnt 5236 2050 Pnt 5236 2097 Pnt 5247 168 Pnt 5247 215 Pnt 5247 262 Pnt 5247 309 Pnt 5247 356 Pnt 5247 403 Pnt 5247 450 Pnt 5247 497 Pnt 5247 544 Pnt 5247 968 Pnt 5247 1015 Pnt 5247 1062 Pnt 5247 1109 Pnt 5247 1156 Pnt 5247 1203 Pnt 5247 1250 Pnt 5247 1297 Pnt 5247 1344 Pnt 5247 1438 Pnt 5247 1485 Pnt 5247 1532 Pnt 5247 1626 Pnt 5247 1673 Pnt 5247 1720 Pnt 5247 1908 Pnt 5247 1956 Pnt 5247 2003 Pnt 5259 168 Pnt 5259 215 Pnt 5259 262 Pnt 5259 309 Pnt 5259 356 Pnt 5259 403 Pnt 5259 1156 Pnt 5259 1203 Pnt 5259 1250 Pnt 5259 1297 Pnt 5259 1344 Pnt 5259 1391 Pnt 5259 1485 Pnt 5259 1532 Pnt 5259 1579 Pnt 5259 1673 Pnt 5259 1720 Pnt 5259 1956 Pnt 5259 2003 Pnt 5259 2097 Pnt 5259 2144 Pnt 5270 168 Pnt 5270 215 Pnt 5270 262 Pnt 5270 309 Pnt 5270 1250 Pnt 5270 1297 Pnt 5270 1344 Pnt 5270 1391 Pnt 5270 1438 Pnt 5270 1485 Pnt 5270 1532 Pnt 5270 1579 Pnt 5270 1626 Pnt 5270 1720 Pnt 5270 1767 Pnt 5270 2003 Pnt 5270 2050 Pnt 5270 2144 Pnt 5270 2191 Pnt 5281 168 Pnt 5281 215 Pnt 5281 309 Pnt 5281 356 Pnt 5281 403 Pnt 5281 450 Pnt 5281 497 Pnt 5281 544 Pnt 5281 591 Pnt 5281 638 Pnt 5281 1297 Pnt 5281 1344 Pnt 5281 1391 Pnt 5281 1438 Pnt 5281 1485 Pnt 5281 1532 Pnt 5281 1579 Pnt 5281 1626 Pnt 5281 1673 Pnt 5281 1767 Pnt 5281 1814 Pnt 5281 2003 Pnt 5281 2050 Pnt 5281 2097 Pnt 5281 2144 Pnt 5292 168 Pnt 5292 215 Pnt 5292 262 Pnt 5292 309 Pnt 5292 356 Pnt 5292 403 Pnt 5292 450 Pnt 5292 497 Pnt 5292 544 Pnt 5292 591 Pnt 5292 638 Pnt 5292 685 Pnt 5292 732 Pnt 5292 780 Pnt 5292 827 Pnt 5292 1391 Pnt 5292 1438 Pnt 5292 1485 Pnt 5292 1532 Pnt 5292 1579 Pnt 5292 1626 Pnt 5292 1673 Pnt 5292 1720 Pnt 5292 1767 Pnt 5292 1814 Pnt 5292 2050 Pnt 5292 2097 Pnt 5292 2191 Pnt 5303 168 Pnt 5303 215 Pnt 5303 262 Pnt 5303 309 Pnt 5303 356 Pnt 5303 403 Pnt 5303 450 Pnt 5303 497 Pnt 5303 544 Pnt 5303 591 Pnt 5303 638 Pnt 5303 685 Pnt 5303 732 Pnt 5303 780 Pnt 5303 827 Pnt 5303 874 Pnt 5303 921 Pnt 5303 1438 Pnt 5303 1485 Pnt 5303 1532 Pnt 5303 1579 Pnt 5303 1626 Pnt 5303 1673 Pnt 5303 1720 Pnt 5303 1767 Pnt 5303 1814 Pnt 5303 1861 Pnt 5303 2097 Pnt 5303 2144 Pnt 5315 168 Pnt 5315 215 Pnt 5315 262 Pnt 5315 309 Pnt 5315 356 Pnt 5315 403 Pnt 5315 450 Pnt 5315 497 Pnt 5315 544 Pnt 5315 591 Pnt 5315 638 Pnt 5315 685 Pnt 5315 732 Pnt 5315 780 Pnt 5315 827 Pnt 5315 874 Pnt 5315 921 Pnt 5315 968 Pnt 5315 1015 Pnt 5315 1532 Pnt 5315 1579 Pnt 5315 1626 Pnt 5315 1720 Pnt 5315 1767 Pnt 5315 1861 Pnt 5315 1908 Pnt 5315 2097 Pnt 5315 2144 Pnt 5315 2238 Pnt 5315 2285 Pnt 5326 168 Pnt 5326 215 Pnt 5326 262 Pnt 5326 309 Pnt 5326 356 Pnt 5326 403 Pnt 5326 450 Pnt 5326 497 Pnt 5326 544 Pnt 5326 591 Pnt 5326 638 Pnt 5326 685 Pnt 5326 732 Pnt 5326 780 Pnt 5326 827 Pnt 5326 874 Pnt 5326 921 Pnt 5326 968 Pnt 5326 1015 Pnt 5326 1062 Pnt 5326 1109 Pnt 5326 1579 Pnt 5326 1626 Pnt 5326 1673 Pnt 5326 1767 Pnt 5326 1814 Pnt 5326 1908 Pnt 5326 1956 Pnt 5326 2144 Pnt 5326 2191 Pnt 5326 2285 Pnt 5337 168 Pnt 5337 215 Pnt 5337 262 Pnt 5337 309 Pnt 5337 356 Pnt 5337 403 Pnt 5337 450 Pnt 5337 497 Pnt 5337 544 Pnt 5337 591 Pnt 5337 638 Pnt 5337 685 Pnt 5337 732 Pnt 5337 780 Pnt 5337 827 Pnt 5337 874 Pnt 5337 921 Pnt 5337 968 Pnt 5337 1015 Pnt 5337 1062 Pnt 5337 1109 Pnt 5337 1156 Pnt 5337 1626 Pnt 5337 1673 Pnt 5337 1720 Pnt 5337 1814 Pnt 5337 1861 Pnt 5337 1908 Pnt 5337 1956 Pnt 5337 2191 Pnt 5337 2238 Pnt 5337 2285 Pnt 5337 2332 Pnt 5348 168 Pnt 5348 215 Pnt 5348 262 Pnt 5348 309 Pnt 5348 356 Pnt 5348 403 Pnt 5348 450 Pnt 5348 497 Pnt 5348 544 Pnt 5348 591 Pnt 5348 638 Pnt 5348 685 Pnt 5348 732 Pnt 5348 780 Pnt 5348 827 Pnt 5348 874 Pnt 5348 921 Pnt 5348 968 Pnt 5348 1015 Pnt 5348 1062 Pnt 5348 1109 Pnt 5348 1156 Pnt 5348 1203 Pnt 5348 1250 Pnt 5348 1673 Pnt 5348 1720 Pnt 5348 1767 Pnt 5348 1861 Pnt 5348 1908 Pnt 5348 1956 Pnt 5348 2003 Pnt 5348 2191 Pnt 5348 2332 Pnt 5348 2379 Pnt 5359 168 Pnt 5359 215 Pnt 5359 262 Pnt 5359 309 Pnt 5359 356 Pnt 5359 403 Pnt 5359 450 Pnt 5359 497 Pnt 5359 544 Pnt 5359 591 Pnt 5359 638 Pnt 5359 685 Pnt 5359 732 Pnt 5359 780 Pnt 5359 827 Pnt 5359 874 Pnt 5359 921 Pnt 5359 968 Pnt 5359 1015 Pnt 5359 1062 Pnt 5359 1109 Pnt 5359 1156 Pnt 5359 1203 Pnt 5359 1250 Pnt 5359 1297 Pnt 5359 1720 Pnt 5359 1767 Pnt 5359 1814 Pnt 5359 1861 Pnt 5359 1908 Pnt 5359 2003 Pnt 5359 2050 Pnt 5359 2238 Pnt 5359 2285 Pnt 5371 168 Pnt 5371 215 Pnt 5371 262 Pnt 5371 309 Pnt 5371 356 Pnt 5371 403 Pnt 5371 450 Pnt 5371 497 Pnt 5371 544 Pnt 5371 591 Pnt 5371 638 Pnt 5371 685 Pnt 5371 732 Pnt 5371 780 Pnt 5371 827 Pnt 5371 874 Pnt 5371 921 Pnt 5371 968 Pnt 5371 1015 Pnt 5371 1062 Pnt 5371 1109 Pnt 5371 1156 Pnt 5371 1203 Pnt 5371 1250 Pnt 5371 1297 Pnt 5371 1344 Pnt 5371 1767 Pnt 5371 1814 Pnt 5371 1861 Pnt 5371 1908 Pnt 5371 1956 Pnt 5371 2050 Pnt 5371 2285 Pnt 5371 2379 Pnt 5382 168 Pnt 5382 215 Pnt 5382 262 Pnt 5382 309 Pnt 5382 356 Pnt 5382 403 Pnt 5382 450 Pnt 5382 497 Pnt 5382 544 Pnt 5382 591 Pnt 5382 638 Pnt 5382 685 Pnt 5382 732 Pnt 5382 780 Pnt 5382 827 Pnt 5382 874 Pnt 5382 921 Pnt 5382 968 Pnt 5382 1015 Pnt 5382 1062 Pnt 5382 1109 Pnt 5382 1156 Pnt 5382 1203 Pnt 5382 1250 Pnt 5382 1297 Pnt 5382 1344 Pnt 5382 1391 Pnt 5382 1814 Pnt 5382 1861 Pnt 5382 1956 Pnt 5382 2003 Pnt 5382 2050 Pnt 5382 2097 Pnt 5382 2285 Pnt 5382 2332 Pnt 5393 168 Pnt 5393 215 Pnt 5393 262 Pnt 5393 309 Pnt 5393 356 Pnt 5393 403 Pnt 5393 450 Pnt 5393 497 Pnt 5393 544 Pnt 5393 591 Pnt 5393 638 Pnt 5393 685 Pnt 5393 732 Pnt 5393 780 Pnt 5393 827 Pnt 5393 874 Pnt 5393 921 Pnt 5393 968 Pnt 5393 1015 Pnt 5393 1062 Pnt 5393 1109 Pnt 5393 1156 Pnt 5393 1203 Pnt 5393 1250 Pnt 5393 1297 Pnt 5393 1344 Pnt 5393 1391 Pnt 5393 1438 Pnt 5393 1485 Pnt 5393 1861 Pnt 5393 1908 Pnt 5393 2003 Pnt 5393 2050 Pnt 5393 2097 Pnt 5393 2144 Pnt 5393 2332 Pnt 5393 2379 Pnt 5393 2426 Pnt 5393 2473 Pnt 5404 168 Pnt 5404 215 Pnt 5404 262 Pnt 5404 309 Pnt 5404 356 Pnt 5404 403 Pnt 5404 450 Pnt 5404 497 Pnt 5404 544 Pnt 5404 591 Pnt 5404 638 Pnt 5404 685 Pnt 5404 732 Pnt 5404 780 Pnt 5404 827 Pnt 5404 874 Pnt 5404 921 Pnt 5404 968 Pnt 5404 1015 Pnt 5404 1062 Pnt 5404 1109 Pnt 5404 1156 Pnt 5404 1203 Pnt 5404 1250 Pnt 5404 1297 Pnt 5404 1344 Pnt 5404 1391 Pnt 5404 1438 Pnt 5404 1485 Pnt 5404 1532 Pnt 5404 1908 Pnt 5404 1956 Pnt 5404 2050 Pnt 5404 2144 Pnt 5404 2379 Pnt 5404 2473 Pnt 5415 168 Pnt 5415 215 Pnt 5415 262 Pnt 5415 309 Pnt 5415 356 Pnt 5415 403 Pnt 5415 450 Pnt 5415 497 Pnt 5415 544 Pnt 5415 591 Pnt 5415 638 Pnt 5415 685 Pnt 5415 732 Pnt 5415 780 Pnt 5415 827 Pnt 5415 874 Pnt 5415 921 Pnt 5415 968 Pnt 5415 1015 Pnt 5415 1062 Pnt 5415 1109 Pnt 5415 1156 Pnt 5415 1203 Pnt 5415 1250 Pnt 5415 1297 Pnt 5415 1344 Pnt 5415 1391 Pnt 5415 1438 Pnt 5415 1485 Pnt 5415 1532 Pnt 5415 1579 Pnt 5415 1956 Pnt 5415 2003 Pnt 5415 2050 Pnt 5415 2097 Pnt 5415 2144 Pnt 5415 2191 Pnt 5415 2379 Pnt 5415 2426 Pnt 5415 2520 Pnt 5427 168 Pnt 5427 215 Pnt 5427 262 Pnt 5427 309 Pnt 5427 356 Pnt 5427 403 Pnt 5427 450 Pnt 5427 497 Pnt 5427 544 Pnt 5427 591 Pnt 5427 638 Pnt 5427 685 Pnt 5427 732 Pnt 5427 780 Pnt 5427 827 Pnt 5427 874 Pnt 5427 921 Pnt 5427 968 Pnt 5427 1015 Pnt 5427 1062 Pnt 5427 1109 Pnt 5427 1156 Pnt 5427 1203 Pnt 5427 1250 Pnt 5427 1297 Pnt 5427 1344 Pnt 5427 1391 Pnt 5427 1438 Pnt 5427 1485 Pnt 5427 1532 Pnt 5427 1579 Pnt 5427 1626 Pnt 5427 1956 Pnt 5427 2003 Pnt 5427 2050 Pnt 5427 2097 Pnt 5427 2144 Pnt 5427 2191 Pnt 5427 2238 Pnt 5427 2426 Pnt 5427 2520 Pnt 5427 2567 Pnt 5438 168 Pnt 5438 215 Pnt 5438 262 Pnt 5438 309 Pnt 5438 356 Pnt 5438 403 Pnt 5438 450 Pnt 5438 497 Pnt 5438 544 Pnt 5438 591 Pnt 5438 638 Pnt 5438 685 Pnt 5438 732 Pnt 5438 780 Pnt 5438 1062 Pnt 5438 1109 Pnt 5438 1156 Pnt 5438 1203 Pnt 5438 1250 Pnt 5438 1297 Pnt 5438 1344 Pnt 5438 1391 Pnt 5438 1438 Pnt 5438 1485 Pnt 5438 1532 Pnt 5438 1579 Pnt 5438 1626 Pnt 5438 2003 Pnt 5438 2050 Pnt 5438 2144 Pnt 5438 2238 Pnt 5438 2426 Pnt 5438 2473 Pnt 5449 168 Pnt 5449 215 Pnt 5449 262 Pnt 5449 309 Pnt 5449 356 Pnt 5449 403 Pnt 5449 450 Pnt 5449 497 Pnt 5449 544 Pnt 5449 591 Pnt 5449 638 Pnt 5449 685 Pnt 5449 732 Pnt 5449 780 Pnt 5449 827 Pnt 5449 874 Pnt 5449 921 Pnt 5449 968 Pnt 5449 1203 Pnt 5449 1250 Pnt 5449 1297 Pnt 5449 1344 Pnt 5449 1391 Pnt 5449 1438 Pnt 5449 1485 Pnt 5449 1532 Pnt 5449 1579 Pnt 5449 1626 Pnt 5449 1673 Pnt 5449 2050 Pnt 5449 2097 Pnt 5449 2144 Pnt 5449 2191 Pnt 5449 2238 Pnt 5449 2285 Pnt 5449 2473 Pnt 5449 2520 Pnt 5449 2567 Pnt 5449 2614 Pnt 5460 168 Pnt 5460 215 Pnt 5460 262 Pnt 5460 309 Pnt 5460 356 Pnt 5460 403 Pnt 5460 450 Pnt 5460 497 Pnt 5460 544 Pnt 5460 591 Pnt 5460 638 Pnt 5460 685 Pnt 5460 732 Pnt 5460 780 Pnt 5460 827 Pnt 5460 874 Pnt 5460 921 Pnt 5460 968 Pnt 5460 1015 Pnt 5460 1062 Pnt 5460 1109 Pnt 5460 1297 Pnt 5460 1344 Pnt 5460 1391 Pnt 5460 1438 Pnt 5460 1485 Pnt 5460 1532 Pnt 5460 1579 Pnt 5460 1626 Pnt 5460 1673 Pnt 5460 1720 Pnt 5460 2097 Pnt 5460 2144 Pnt 5460 2191 Pnt 5460 2238 Pnt 5460 2285 Pnt 5472 168 Pnt 5472 215 Pnt 5472 262 Pnt 5472 309 Pnt 5472 356 Pnt 5472 403 Pnt 5472 450 Pnt 5472 497 Pnt 5472 544 Pnt 5472 591 Pnt 5472 638 Pnt 5472 685 Pnt 5472 732 Pnt 5472 780 Pnt 5472 827 Pnt 5472 874 Pnt 5472 921 Pnt 5472 968 Pnt 5472 1015 Pnt 5472 1062 Pnt 5472 1109 Pnt 5472 1156 Pnt 5472 1203 Pnt 5472 1391 Pnt 5472 1438 Pnt 5472 1485 Pnt 5472 1532 Pnt 5472 1579 Pnt 5472 1626 Pnt 5472 1673 Pnt 5472 1720 Pnt 5472 1767 Pnt 5472 2144 Pnt 5472 2238 Pnt 5472 2332 Pnt 5472 2520 Pnt 5472 2567 Pnt 5472 2614 Pnt 5472 2661 Pnt 5483 168 Pnt 5483 215 Pnt 5483 262 Pnt 5483 309 Pnt 5483 356 Pnt 5483 403 Pnt 5483 450 Pnt 5483 497 Pnt 5483 544 Pnt 5483 591 Pnt 5483 638 Pnt 5483 685 Pnt 5483 732 Pnt 5483 780 Pnt 5483 827 Pnt 5483 874 Pnt 5483 921 Pnt 5483 968 Pnt 5483 1015 Pnt 5483 1062 Pnt 5483 1109 Pnt 5483 1156 Pnt 5483 1203 Pnt 5483 1250 Pnt 5483 1297 Pnt 5483 1485 Pnt 5483 1532 Pnt 5483 1579 Pnt 5483 1626 Pnt 5483 1673 Pnt 5483 1720 Pnt 5483 1767 Pnt 5483 1814 Pnt 5483 2144 Pnt 5483 2191 Pnt 5483 2285 Pnt 5483 2332 Pnt 5483 2379 Pnt 5483 2567 Pnt 5494 168 Pnt 5494 215 Pnt 5494 262 Pnt 5494 309 Pnt 5494 356 Pnt 5494 403 Pnt 5494 450 Pnt 5494 497 Pnt 5494 544 Pnt 5494 591 Pnt 5494 638 Pnt 5494 685 Pnt 5494 732 Pnt 5494 780 Pnt 5494 827 Pnt 5494 874 Pnt 5494 921 Pnt 5494 968 Pnt 5494 1015 Pnt 5494 1062 Pnt 5494 1109 Pnt 5494 1156 Pnt 5494 1203 Pnt 5494 1250 Pnt 5494 1297 Pnt 5494 1344 Pnt 5494 1532 Pnt 5494 1579 Pnt 5494 1626 Pnt 5494 1673 Pnt 5494 1720 Pnt 5494 1767 Pnt 5494 1814 Pnt 5494 1861 Pnt 5494 2191 Pnt 5494 2238 Pnt 5494 2285 Pnt 5494 2332 Pnt 5494 2379 Pnt 5494 2567 Pnt 5494 2614 Pnt 5494 2708 Pnt 5505 168 Pnt 5505 215 Pnt 5505 262 Pnt 5505 309 Pnt 5505 356 Pnt 5505 403 Pnt 5505 450 Pnt 5505 497 Pnt 5505 544 Pnt 5505 591 Pnt 5505 638 Pnt 5505 685 Pnt 5505 732 Pnt 5505 780 Pnt 5505 827 Pnt 5505 874 Pnt 5505 921 Pnt 5505 968 Pnt 5505 1015 Pnt 5505 1062 Pnt 5505 1109 Pnt 5505 1156 Pnt 5505 1203 Pnt 5505 1250 Pnt 5505 1297 Pnt 5505 1344 Pnt 5505 1391 Pnt 5505 1438 Pnt 5505 1579 Pnt 5505 1626 Pnt 5505 1673 Pnt 5505 1720 Pnt 5505 1767 Pnt 5505 1814 Pnt 5505 1861 Pnt 5505 1908 Pnt 5505 2238 Pnt 5505 2332 Pnt 5505 2426 Pnt 5505 2614 Pnt 5505 2708 Pnt 5516 168 Pnt 5516 215 Pnt 5516 262 Pnt 5516 309 Pnt 5516 356 Pnt 5516 403 Pnt 5516 450 Pnt 5516 497 Pnt 5516 544 Pnt 5516 591 Pnt 5516 638 Pnt 5516 685 Pnt 5516 732 Pnt 5516 780 Pnt 5516 827 Pnt 5516 874 Pnt 5516 921 Pnt 5516 968 Pnt 5516 1015 Pnt 5516 1062 Pnt 5516 1109 Pnt 5516 1156 Pnt 5516 1203 Pnt 5516 1250 Pnt 5516 1297 Pnt 5516 1344 Pnt 5516 1391 Pnt 5516 1438 Pnt 5516 1485 Pnt 5516 1673 Pnt 5516 1720 Pnt 5516 1767 Pnt 5516 1814 Pnt 5516 1861 Pnt 5516 1908 Pnt 5516 2285 Pnt 5516 2379 Pnt 5516 2426 Pnt 5516 2661 Pnt 5516 2755 Pnt 5528 168 Pnt 5528 215 Pnt 5528 262 Pnt 5528 309 Pnt 5528 356 Pnt 5528 403 Pnt 5528 450 Pnt 5528 497 Pnt 5528 544 Pnt 5528 591 Pnt 5528 638 Pnt 5528 685 Pnt 5528 732 Pnt 5528 780 Pnt 5528 827 Pnt 5528 874 Pnt 5528 921 Pnt 5528 968 Pnt 5528 1015 Pnt 5528 1062 Pnt 5528 1109 Pnt 5528 1156 Pnt 5528 1203 Pnt 5528 1250 Pnt 5528 1297 Pnt 5528 1344 Pnt 5528 1391 Pnt 5528 1438 Pnt 5528 1485 Pnt 5528 1532 Pnt 5528 1720 Pnt 5528 1767 Pnt 5528 1814 Pnt 5528 1861 Pnt 5528 1908 Pnt 5528 1956 Pnt 5528 2285 Pnt 5528 2332 Pnt 5528 2379 Pnt 5528 2473 Pnt 5528 2661 Pnt 5528 2755 Pnt 5539 168 Pnt 5539 215 Pnt 5539 262 Pnt 5539 309 Pnt 5539 356 Pnt 5539 403 Pnt 5539 450 Pnt 5539 497 Pnt 5539 544 Pnt 5539 591 Pnt 5539 638 Pnt 5539 685 Pnt 5539 732 Pnt 5539 780 Pnt 5539 827 Pnt 5539 874 Pnt 5539 921 Pnt 5539 968 Pnt 5539 1156 Pnt 5539 1203 Pnt 5539 1250 Pnt 5539 1297 Pnt 5539 1344 Pnt 5539 1391 Pnt 5539 1438 Pnt 5539 1485 Pnt 5539 1532 Pnt 5539 1579 Pnt 5539 1767 Pnt 5539 1814 Pnt 5539 1861 Pnt 5539 1908 Pnt 5539 1956 Pnt 5539 2003 Pnt 5539 2332 Pnt 5539 2426 Pnt 5539 2520 Pnt 5550 168 Pnt 5550 215 Pnt 5550 262 Pnt 5550 309 Pnt 5550 356 Pnt 5550 403 Pnt 5550 450 Pnt 5550 497 Pnt 5550 544 Pnt 5550 591 Pnt 5550 638 Pnt 5550 685 Pnt 5550 732 Pnt 5550 780 Pnt 5550 827 Pnt 5550 874 Pnt 5550 921 Pnt 5550 968 Pnt 5550 1015 Pnt 5550 1062 Pnt 5550 1109 Pnt 5550 1156 Pnt 5550 1297 Pnt 5550 1344 Pnt 5550 1391 Pnt 5550 1438 Pnt 5550 1485 Pnt 5550 1532 Pnt 5550 1579 Pnt 5550 1626 Pnt 5550 1673 Pnt 5550 1814 Pnt 5550 1861 Pnt 5550 1908 Pnt 5550 1956 Pnt 5550 2003 Pnt 5550 2050 Pnt 5550 2379 Pnt 5550 2473 Pnt 5550 2520 Pnt 5550 2708 Pnt 5550 2755 Pnt 5550 2802 Pnt 5550 2849 Pnt 5561 168 Pnt 5561 215 Pnt 5561 262 Pnt 5561 309 Pnt 5561 356 Pnt 5561 403 Pnt 5561 450 Pnt 5561 497 Pnt 5561 544 Pnt 5561 591 Pnt 5561 638 Pnt 5561 685 Pnt 5561 732 Pnt 5561 780 Pnt 5561 827 Pnt 5561 874 Pnt 5561 921 Pnt 5561 968 Pnt 5561 1015 Pnt 5561 1062 Pnt 5561 1109 Pnt 5561 1156 Pnt 5561 1203 Pnt 5561 1250 Pnt 5561 1391 Pnt 5561 1438 Pnt 5561 1485 Pnt 5561 1532 Pnt 5561 1579 Pnt 5561 1626 Pnt 5561 1673 Pnt 5561 1720 Pnt 5561 1861 Pnt 5561 1908 Pnt 5561 1956 Pnt 5561 2003 Pnt 5561 2050 Pnt 5561 2097 Pnt 5561 2379 Pnt 5561 2426 Pnt 5561 2473 Pnt 5561 2567 Pnt 5561 2755 Pnt 5572 168 Pnt 5572 215 Pnt 5572 262 Pnt 5572 309 Pnt 5572 356 Pnt 5572 403 Pnt 5572 450 Pnt 5572 497 Pnt 5572 544 Pnt 5572 591 Pnt 5572 638 Pnt 5572 685 Pnt 5572 732 Pnt 5572 780 Pnt 5572 827 Pnt 5572 874 Pnt 5572 921 Pnt 5572 968 Pnt 5572 1015 Pnt 5572 1062 Pnt 5572 1109 Pnt 5572 1156 Pnt 5572 1203 Pnt 5572 1250 Pnt 5572 1297 Pnt 5572 1344 Pnt 5572 1485 Pnt 5572 1532 Pnt 5572 1579 Pnt 5572 1626 Pnt 5572 1673 Pnt 5572 1720 Pnt 5572 1767 Pnt 5572 1908 Pnt 5572 1956 Pnt 5572 2003 Pnt 5572 2050 Pnt 5572 2097 Pnt 5572 2426 Pnt 5572 2520 Pnt 5572 2567 Pnt 5572 2755 Pnt 5572 2802 Pnt 5572 2849 Pnt 5572 2896 Pnt 5584 168 Pnt 5584 215 Pnt 5584 262 Pnt 5584 309 Pnt 5584 356 Pnt 5584 403 Pnt 5584 450 Pnt 5584 497 Pnt 5584 544 Pnt 5584 591 Pnt 5584 638 Pnt 5584 685 Pnt 5584 732 Pnt 5584 780 Pnt 5584 827 Pnt 5584 874 Pnt 5584 921 Pnt 5584 968 Pnt 5584 1015 Pnt 5584 1062 Pnt 5584 1109 Pnt 5584 1156 Pnt 5584 1203 Pnt 5584 1250 Pnt 5584 1297 Pnt 5584 1344 Pnt 5584 1391 Pnt 5584 1438 Pnt 5584 1579 Pnt 5584 1626 Pnt 5584 1673 Pnt 5584 1720 Pnt 5584 1767 Pnt 5584 1814 Pnt 5584 1956 Pnt 5584 2003 Pnt 5584 2050 Pnt 5584 2097 Pnt 5584 2144 Pnt 5584 2473 Pnt 5584 2520 Pnt 5584 2567 Pnt 5584 2614 Pnt 5584 2802 Pnt 5595 168 Pnt 5595 215 Pnt 5595 262 Pnt 5595 309 Pnt 5595 356 Pnt 5595 403 Pnt 5595 450 Pnt 5595 497 Pnt 5595 544 Pnt 5595 591 Pnt 5595 638 Pnt 5595 685 Pnt 5595 732 Pnt 5595 780 Pnt 5595 827 Pnt 5595 874 Pnt 5595 921 Pnt 5595 968 Pnt 5595 1015 Pnt 5595 1062 Pnt 5595 1109 Pnt 5595 1156 Pnt 5595 1203 Pnt 5595 1250 Pnt 5595 1297 Pnt 5595 1344 Pnt 5595 1391 Pnt 5595 1438 Pnt 5595 1485 Pnt 5595 1626 Pnt 5595 1673 Pnt 5595 1720 Pnt 5595 1767 Pnt 5595 1814 Pnt 5595 1861 Pnt 5595 2003 Pnt 5595 2050 Pnt 5595 2097 Pnt 5595 2144 Pnt 5595 2191 Pnt 5595 2473 Pnt 5595 2520 Pnt 5595 2567 Pnt 5595 2661 Pnt 5595 2849 Pnt 5595 2896 Pnt 5606 168 Pnt 5606 215 Pnt 5606 262 Pnt 5606 309 Pnt 5606 356 Pnt 5606 403 Pnt 5606 450 Pnt 5606 497 Pnt 5606 544 Pnt 5606 591 Pnt 5606 638 Pnt 5606 685 Pnt 5606 732 Pnt 5606 780 Pnt 5606 827 Pnt 5606 874 Pnt 5606 921 Pnt 5606 968 Pnt 5606 1015 Pnt 5606 1062 Pnt 5606 1109 Pnt 5606 1156 Pnt 5606 1203 Pnt 5606 1250 Pnt 5606 1297 Pnt 5606 1344 Pnt 5606 1391 Pnt 5606 1438 Pnt 5606 1485 Pnt 5606 1532 Pnt 5606 1579 Pnt 5606 1720 Pnt 5606 1767 Pnt 5606 1814 Pnt 5606 1861 Pnt 5606 1908 Pnt 5606 2050 Pnt 5606 2097 Pnt 5606 2144 Pnt 5606 2191 Pnt 5606 2520 Pnt 5606 2614 Pnt 5606 2661 Pnt 5606 2849 Pnt 5617 168 Pnt 5617 215 Pnt 5617 262 Pnt 5617 309 Pnt 5617 356 Pnt 5617 403 Pnt 5617 450 Pnt 5617 497 Pnt 5617 544 Pnt 5617 591 Pnt 5617 638 Pnt 5617 685 Pnt 5617 732 Pnt 5617 780 Pnt 5617 827 Pnt 5617 874 Pnt 5617 921 Pnt 5617 968 Pnt 5617 1015 Pnt 5617 1062 Pnt 5617 1109 Pnt 5617 1156 Pnt 5617 1203 Pnt 5617 1250 Pnt 5617 1297 Pnt 5617 1344 Pnt 5617 1391 Pnt 5617 1438 Pnt 5617 1485 Pnt 5617 1532 Pnt 5617 1579 Pnt 5617 1626 Pnt 5617 1767 Pnt 5617 1814 Pnt 5617 1861 Pnt 5617 1908 Pnt 5617 2097 Pnt 5617 2144 Pnt 5617 2191 Pnt 5617 2238 Pnt 5617 2567 Pnt 5617 2614 Pnt 5617 2708 Pnt 5617 2896 Pnt 5617 2990 Pnt 5628 168 Pnt 5628 215 Pnt 5628 262 Pnt 5628 309 Pnt 5628 356 Pnt 5628 403 Pnt 5628 450 Pnt 5628 497 Pnt 5628 544 Pnt 5628 591 Pnt 5628 638 Pnt 5628 685 Pnt 5628 732 Pnt 5628 780 Pnt 5628 827 Pnt 5628 874 Pnt 5628 921 Pnt 5628 968 Pnt 5628 1015 Pnt 5628 1062 Pnt 5628 1109 Pnt 5628 1156 Pnt 5628 1203 Pnt 5628 1250 Pnt 5628 1297 Pnt 5628 1344 Pnt 5628 1391 Pnt 5628 1438 Pnt 5628 1485 Pnt 5628 1532 Pnt 5628 1579 Pnt 5628 1626 Pnt 5628 1673 Pnt 5628 1814 Pnt 5628 1861 Pnt 5628 1908 Pnt 5628 1956 Pnt 5628 2097 Pnt 5628 2144 Pnt 5628 2191 Pnt 5628 2238 Pnt 5628 2285 Pnt 5628 2567 Pnt 5628 2614 Pnt 5628 2661 Pnt 5628 2708 Pnt 5628 2896 Pnt 5628 2990 Pnt 5640 168 Pnt 5640 215 Pnt 5640 262 Pnt 5640 309 Pnt 5640 356 Pnt 5640 403 Pnt 5640 450 Pnt 5640 497 Pnt 5640 544 Pnt 5640 591 Pnt 5640 638 Pnt 5640 685 Pnt 5640 732 Pnt 5640 780 Pnt 5640 827 Pnt 5640 874 Pnt 5640 921 Pnt 5640 968 Pnt 5640 1015 Pnt 5640 1062 Pnt 5640 1109 Pnt 5640 1156 Pnt 5640 1203 Pnt 5640 1250 Pnt 5640 1297 Pnt 5640 1344 Pnt 5640 1391 Pnt 5640 1438 Pnt 5640 1485 Pnt 5640 1532 Pnt 5640 1579 Pnt 5640 1626 Pnt 5640 1673 Pnt 5640 1720 Pnt 5640 1861 Pnt 5640 1908 Pnt 5640 1956 Pnt 5640 2003 Pnt 5640 2144 Pnt 5640 2191 Pnt 5640 2238 Pnt 5640 2285 Pnt 5640 2332 Pnt 5640 2614 Pnt 5640 2708 Pnt 5640 2755 Pnt 5640 3037 Pnt 5651 168 Pnt 5651 215 Pnt 5651 262 Pnt 5651 309 Pnt 5651 356 Pnt 5651 403 Pnt 5651 450 Pnt 5651 497 Pnt 5651 544 Pnt 5651 591 Pnt 5651 638 Pnt 5651 685 Pnt 5651 732 Pnt 5651 780 Pnt 5651 827 Pnt 5651 874 Pnt 5651 921 Pnt 5651 1250 Pnt 5651 1297 Pnt 5651 1344 Pnt 5651 1391 Pnt 5651 1438 Pnt 5651 1485 Pnt 5651 1532 Pnt 5651 1579 Pnt 5651 1626 Pnt 5651 1673 Pnt 5651 1720 Pnt 5651 1767 Pnt 5651 1908 Pnt 5651 1956 Pnt 5651 2003 Pnt 5651 2050 Pnt 5651 2191 Pnt 5651 2238 Pnt 5651 2285 Pnt 5651 2332 Pnt 5651 2661 Pnt 5651 2708 Pnt 5651 2943 Pnt 5651 2990 Pnt 5651 3037 Pnt 5662 168 Pnt 5662 215 Pnt 5662 262 Pnt 5662 309 Pnt 5662 356 Pnt 5662 403 Pnt 5662 450 Pnt 5662 497 Pnt 5662 544 Pnt 5662 591 Pnt 5662 638 Pnt 5662 685 Pnt 5662 732 Pnt 5662 1438 Pnt 5662 1485 Pnt 5662 1532 Pnt 5662 1579 Pnt 5662 1626 Pnt 5662 1673 Pnt 5662 1720 Pnt 5662 1767 Pnt 5662 1814 Pnt 5662 1861 Pnt 5662 1956 Pnt 5662 2003 Pnt 5662 2050 Pnt 5662 2097 Pnt 5662 2238 Pnt 5662 2285 Pnt 5662 2332 Pnt 5662 2379 Pnt 5662 2661 Pnt 5662 2755 Pnt 5662 2802 Pnt 5662 2990 Pnt 5662 3084 Pnt 5673 168 Pnt 5673 215 Pnt 5673 262 Pnt 5673 309 Pnt 5673 356 Pnt 5673 403 Pnt 5673 450 Pnt 5673 497 Pnt 5673 544 Pnt 5673 591 Pnt 5673 1532 Pnt 5673 1579 Pnt 5673 1626 Pnt 5673 1673 Pnt 5673 1720 Pnt 5673 1767 Pnt 5673 1814 Pnt 5673 1861 Pnt 5673 1908 Pnt 5673 2003 Pnt 5673 2050 Pnt 5673 2097 Pnt 5673 2144 Pnt 5673 2285 Pnt 5673 2332 Pnt 5673 2379 Pnt 5673 2426 Pnt 5673 2708 Pnt 5673 2755 Pnt 5673 2849 Pnt 5673 3037 Pnt 5673 3084 Pnt 5684 168 Pnt 5684 215 Pnt 5684 262 Pnt 5684 309 Pnt 5684 356 Pnt 5684 403 Pnt 5684 450 Pnt 5684 497 Pnt 5684 1626 Pnt 5684 1673 Pnt 5684 1720 Pnt 5684 1767 Pnt 5684 1814 Pnt 5684 1861 Pnt 5684 1908 Pnt 5684 1956 Pnt 5684 2050 Pnt 5684 2097 Pnt 5684 2144 Pnt 5684 2285 Pnt 5684 2332 Pnt 5684 2379 Pnt 5684 2426 Pnt 5684 2708 Pnt 5684 2755 Pnt 5684 2802 Pnt 5684 2849 Pnt 5684 3037 Pnt 5696 168 Pnt 5696 215 Pnt 5696 262 Pnt 5696 309 Pnt 5696 356 Pnt 5696 403 Pnt 5696 450 Pnt 5696 1720 Pnt 5696 1767 Pnt 5696 1814 Pnt 5696 1861 Pnt 5696 1908 Pnt 5696 1956 Pnt 5696 2003 Pnt 5696 2097 Pnt 5696 2144 Pnt 5696 2191 Pnt 5696 2332 Pnt 5696 2379 Pnt 5696 2426 Pnt 5696 2473 Pnt 5696 2755 Pnt 5696 2896 Pnt 5696 3084 Pnt 5696 3132 Pnt 5707 168 Pnt 5707 215 Pnt 5707 262 Pnt 5707 309 Pnt 5707 356 Pnt 5707 1767 Pnt 5707 1814 Pnt 5707 1861 Pnt 5707 1908 Pnt 5707 1956 Pnt 5707 2003 Pnt 5707 2144 Pnt 5707 2191 Pnt 5707 2238 Pnt 5707 2379 Pnt 5707 2426 Pnt 5707 2473 Pnt 5707 2802 Pnt 5707 2849 Pnt 5707 2896 Pnt 5707 3084 Pnt 5718 168 Pnt 5718 215 Pnt 5718 262 Pnt 5718 309 Pnt 5718 1861 Pnt 5718 1908 Pnt 5718 1956 Pnt 5718 2003 Pnt 5718 2050 Pnt 5718 2191 Pnt 5718 2238 Pnt 5718 2285 Pnt 5718 2426 Pnt 5718 2473 Pnt 5718 2520 Pnt 5718 2802 Pnt 5718 2896 Pnt 5718 2943 Pnt 5718 3132 Pnt 5718 3179 Pnt 5718 3226 Pnt 5729 168 Pnt 5729 215 Pnt 5729 262 Pnt 5729 1908 Pnt 5729 1956 Pnt 5729 2003 Pnt 5729 2050 Pnt 5729 2097 Pnt 5729 2238 Pnt 5729 2285 Pnt 5729 2332 Pnt 5729 2426 Pnt 5729 2473 Pnt 5729 2520 Pnt 5729 2567 Pnt 5729 2849 Pnt 5729 2896 Pnt 5729 3132 Pnt 5740 168 Pnt 5740 1956 Pnt 5740 2003 Pnt 5740 2050 Pnt 5740 2097 Pnt 5740 2144 Pnt 5740 2238 Pnt 5740 2285 Pnt 5740 2332 Pnt 5740 2473 Pnt 5740 2520 Pnt 5740 2567 Pnt 5740 2943 Pnt 5740 2990 Pnt 5740 3179 Pnt 5740 3226 Pnt 5740 3273 Pnt 5752 168 Pnt 5752 215 Pnt 5752 262 Pnt 5752 309 Pnt 5752 356 Pnt 5752 2003 Pnt 5752 2050 Pnt 5752 2097 Pnt 5752 2144 Pnt 5752 2191 Pnt 5752 2285 Pnt 5752 2332 Pnt 5752 2379 Pnt 5752 2520 Pnt 5752 2567 Pnt 5752 2614 Pnt 5752 2896 Pnt 5752 2943 Pnt 5752 3037 Pnt 5752 3179 Pnt 5763 168 Pnt 5763 215 Pnt 5763 262 Pnt 5763 309 Pnt 5763 356 Pnt 5763 403 Pnt 5763 450 Pnt 5763 497 Pnt 5763 544 Pnt 5763 591 Pnt 5763 2050 Pnt 5763 2097 Pnt 5763 2191 Pnt 5763 2238 Pnt 5763 2332 Pnt 5763 2379 Pnt 5763 2426 Pnt 5763 2520 Pnt 5763 2567 Pnt 5763 2614 Pnt 5763 2661 Pnt 5763 2943 Pnt 5763 2990 Pnt 5763 3037 Pnt 5763 3320 Pnt 5774 168 Pnt 5774 215 Pnt 5774 262 Pnt 5774 309 Pnt 5774 356 Pnt 5774 403 Pnt 5774 450 Pnt 5774 497 Pnt 5774 544 Pnt 5774 591 Pnt 5774 638 Pnt 5774 685 Pnt 5774 2097 Pnt 5774 2144 Pnt 5774 2191 Pnt 5774 2238 Pnt 5774 2285 Pnt 5774 2379 Pnt 5774 2426 Pnt 5774 2567 Pnt 5774 2614 Pnt 5774 2661 Pnt 5774 2943 Pnt 5774 3084 Pnt 5774 3226 Pnt 5785 168 Pnt 5785 215 Pnt 5785 262 Pnt 5785 309 Pnt 5785 356 Pnt 5785 403 Pnt 5785 450 Pnt 5785 497 Pnt 5785 544 Pnt 5785 591 Pnt 5785 638 Pnt 5785 685 Pnt 5785 732 Pnt 5785 780 Pnt 5785 827 Pnt 5785 2144 Pnt 5785 2191 Pnt 5785 2238 Pnt 5785 2285 Pnt 5785 2426 Pnt 5785 2473 Pnt 5785 2614 Pnt 5785 2661 Pnt 5785 2708 Pnt 5785 2990 Pnt 5785 3037 Pnt 5785 3084 Pnt 5785 3273 Pnt 5797 168 Pnt 5797 215 Pnt 5797 262 Pnt 5797 309 Pnt 5797 356 Pnt 5797 403 Pnt 5797 450 Pnt 5797 497 Pnt 5797 544 Pnt 5797 591 Pnt 5797 638 Pnt 5797 685 Pnt 5797 732 Pnt 5797 780 Pnt 5797 827 Pnt 5797 874 Pnt 5797 2191 Pnt 5797 2238 Pnt 5797 2285 Pnt 5797 2332 Pnt 5797 2426 Pnt 5797 2473 Pnt 5797 2520 Pnt 5797 2614 Pnt 5797 2661 Pnt 5797 2708 Pnt 5797 2755 Pnt 5797 2990 Pnt 5797 3084 Pnt 5797 3132 Pnt 5797 3273 Pnt 5797 3320 Pnt 5797 3367 Pnt 5808 168 Pnt 5808 215 Pnt 5808 262 Pnt 5808 309 Pnt 5808 356 Pnt 5808 403 Pnt 5808 450 Pnt 5808 497 Pnt 5808 544 Pnt 5808 591 Pnt 5808 638 Pnt 5808 685 Pnt 5808 732 Pnt 5808 780 Pnt 5808 827 Pnt 5808 874 Pnt 5808 921 Pnt 5808 968 Pnt 5808 2238 Pnt 5808 2285 Pnt 5808 2332 Pnt 5808 2379 Pnt 5808 2473 Pnt 5808 2520 Pnt 5808 2661 Pnt 5808 2708 Pnt 5808 2755 Pnt 5808 3037 Pnt 5808 3084 Pnt 5808 3320 Pnt 5808 3414 Pnt 5819 168 Pnt 5819 215 Pnt 5819 262 Pnt 5819 309 Pnt 5819 356 Pnt 5819 403 Pnt 5819 450 Pnt 5819 497 Pnt 5819 544 Pnt 5819 591 Pnt 5819 638 Pnt 5819 685 Pnt 5819 732 Pnt 5819 780 Pnt 5819 827 Pnt 5819 874 Pnt 5819 921 Pnt 5819 968 Pnt 5819 1015 Pnt 5819 1062 Pnt 5819 2285 Pnt 5819 2332 Pnt 5819 2379 Pnt 5819 2426 Pnt 5819 2520 Pnt 5819 2567 Pnt 5819 2708 Pnt 5819 2755 Pnt 5819 2802 Pnt 5819 3132 Pnt 5819 3179 Pnt 5819 3367 Pnt 5819 3414 Pnt 5830 168 Pnt 5830 215 Pnt 5830 262 Pnt 5830 309 Pnt 5830 356 Pnt 5830 403 Pnt 5830 450 Pnt 5830 497 Pnt 5830 544 Pnt 5830 591 Pnt 5830 638 Pnt 5830 685 Pnt 5830 732 Pnt 5830 780 Pnt 5830 827 Pnt 5830 874 Pnt 5830 921 Pnt 5830 968 Pnt 5830 1015 Pnt 5830 1062 Pnt 5830 1109 Pnt 5830 2332 Pnt 5830 2379 Pnt 5830 2426 Pnt 5830 2567 Pnt 5830 2614 Pnt 5830 2708 Pnt 5830 2755 Pnt 5830 2802 Pnt 5830 3084 Pnt 5830 3132 Pnt 5830 3367 Pnt 5830 3461 Pnt 5841 168 Pnt 5841 215 Pnt 5841 262 Pnt 5841 309 Pnt 5841 356 Pnt 5841 403 Pnt 5841 450 Pnt 5841 497 Pnt 5841 544 Pnt 5841 591 Pnt 5841 638 Pnt 5841 685 Pnt 5841 732 Pnt 5841 780 Pnt 5841 827 Pnt 5841 874 Pnt 5841 921 Pnt 5841 968 Pnt 5841 1015 Pnt 5841 1062 Pnt 5841 1109 Pnt 5841 1156 Pnt 5841 2379 Pnt 5841 2426 Pnt 5841 2473 Pnt 5841 2567 Pnt 5841 2614 Pnt 5841 2755 Pnt 5841 2802 Pnt 5841 2849 Pnt 5841 3132 Pnt 5841 3179 Pnt 5841 3226 Pnt 5841 3414 Pnt 5841 3461 Pnt 5853 168 Pnt 5853 215 Pnt 5853 262 Pnt 5853 309 Pnt 5853 356 Pnt 5853 403 Pnt 5853 450 Pnt 5853 497 Pnt 5853 544 Pnt 5853 591 Pnt 5853 638 Pnt 5853 685 Pnt 5853 732 Pnt 5853 780 Pnt 5853 827 Pnt 5853 874 Pnt 5853 921 Pnt 5853 968 Pnt 5853 1015 Pnt 5853 1062 Pnt 5853 1109 Pnt 5853 1156 Pnt 5853 1203 Pnt 5853 1250 Pnt 5853 2426 Pnt 5853 2473 Pnt 5853 2520 Pnt 5853 2614 Pnt 5853 2661 Pnt 5853 2802 Pnt 5853 2849 Pnt 5853 3132 Pnt 5853 3414 Pnt 5853 3508 Pnt 5864 168 Pnt 5864 215 Pnt 5864 262 Pnt 5864 309 Pnt 5864 356 Pnt 5864 403 Pnt 5864 450 Pnt 5864 497 Pnt 5864 544 Pnt 5864 591 Pnt 5864 638 Pnt 5864 685 Pnt 5864 732 Pnt 5864 780 Pnt 5864 827 Pnt 5864 874 Pnt 5864 921 Pnt 5864 968 Pnt 5864 1015 Pnt 5864 1062 Pnt 5864 1109 Pnt 5864 1156 Pnt 5864 1203 Pnt 5864 1250 Pnt 5864 1297 Pnt 5864 2426 Pnt 5864 2473 Pnt 5864 2520 Pnt 5864 2661 Pnt 5864 2708 Pnt 5864 2802 Pnt 5864 2849 Pnt 5864 2896 Pnt 5864 3179 Pnt 5864 3226 Pnt 5864 3273 Pnt 5864 3461 Pnt 5864 3508 Pnt 5875 168 Pnt 5875 215 Pnt 5875 262 Pnt 5875 309 Pnt 5875 356 Pnt 5875 403 Pnt 5875 450 Pnt 5875 497 Pnt 5875 544 Pnt 5875 591 Pnt 5875 638 Pnt 5875 685 Pnt 5875 732 Pnt 5875 780 Pnt 5875 827 Pnt 5875 874 Pnt 5875 921 Pnt 5875 968 Pnt 5875 1015 Pnt 5875 1062 Pnt 5875 1109 Pnt 5875 1156 Pnt 5875 1203 Pnt 5875 1250 Pnt 5875 1297 Pnt 5875 1344 Pnt 5875 2473 Pnt 5875 2520 Pnt 5875 2567 Pnt 5875 2661 Pnt 5875 2708 Pnt 5875 2849 Pnt 5875 2896 Pnt 5875 2943 Pnt 5875 3179 Pnt 5875 3320 Pnt 5875 3461 Pnt 5875 3555 Pnt 5886 168 Pnt 5886 215 Pnt 5886 262 Pnt 5886 309 Pnt 5886 356 Pnt 5886 403 Pnt 5886 450 Pnt 5886 497 Pnt 5886 544 Pnt 5886 591 Pnt 5886 638 Pnt 5886 685 Pnt 5886 732 Pnt 5886 780 Pnt 5886 827 Pnt 5886 874 Pnt 5886 921 Pnt 5886 968 Pnt 5886 1015 Pnt 5886 1062 Pnt 5886 1109 Pnt 5886 1156 Pnt 5886 1203 Pnt 5886 1250 Pnt 5886 1297 Pnt 5886 1344 Pnt 5886 1391 Pnt 5886 2520 Pnt 5886 2567 Pnt 5886 2614 Pnt 5886 2708 Pnt 5886 2755 Pnt 5886 2896 Pnt 5886 2943 Pnt 5886 3226 Pnt 5886 3273 Pnt 5886 3320 Pnt 5886 3508 Pnt 5886 3555 Pnt 5897 168 Pnt 5897 215 Pnt 5897 262 Pnt 5897 309 Pnt 5897 356 Pnt 5897 403 Pnt 5897 450 Pnt 5897 497 Pnt 5897 544 Pnt 5897 591 Pnt 5897 638 Pnt 5897 685 Pnt 5897 732 Pnt 5897 780 Pnt 5897 827 Pnt 5897 874 Pnt 5897 921 Pnt 5897 968 Pnt 5897 1015 Pnt 5897 1062 Pnt 5897 1109 Pnt 5897 1156 Pnt 5897 1203 Pnt 5897 1250 Pnt 5897 1297 Pnt 5897 1344 Pnt 5897 1391 Pnt 5897 1438 Pnt 5897 2567 Pnt 5897 2614 Pnt 5897 2755 Pnt 5897 2802 Pnt 5897 2896 Pnt 5897 2943 Pnt 5897 2990 Pnt 5897 3320 Pnt 5897 3367 Pnt 5897 3508 Pnt 5897 3602 Pnt 5909 168 Pnt 5909 215 Pnt 5909 262 Pnt 5909 309 Pnt 5909 356 Pnt 5909 403 Pnt 5909 450 Pnt 5909 497 Pnt 5909 544 Pnt 5909 591 Pnt 5909 638 Pnt 5909 685 Pnt 5909 732 Pnt 5909 780 Pnt 5909 827 Pnt 5909 874 Pnt 5909 921 Pnt 5909 968 Pnt 5909 1015 Pnt 5909 1062 Pnt 5909 1109 Pnt 5909 1156 Pnt 5909 1203 Pnt 5909 1250 Pnt 5909 1297 Pnt 5909 1344 Pnt 5909 1391 Pnt 5909 1438 Pnt 5909 1485 Pnt 5909 2567 Pnt 5909 2614 Pnt 5909 2661 Pnt 5909 2755 Pnt 5909 2802 Pnt 5909 2943 Pnt 5909 2990 Pnt 5909 3273 Pnt 5909 3320 Pnt 5909 3367 Pnt 5909 3555 Pnt 5909 3602 Pnt 5920 168 Pnt 5920 215 Pnt 5920 262 Pnt 5920 309 Pnt 5920 356 Pnt 5920 403 Pnt 5920 450 Pnt 5920 497 Pnt 5920 544 Pnt 5920 591 Pnt 5920 638 Pnt 5920 685 Pnt 5920 732 Pnt 5920 780 Pnt 5920 827 Pnt 5920 874 Pnt 5920 921 Pnt 5920 968 Pnt 5920 1015 Pnt 5920 1062 Pnt 5920 1109 Pnt 5920 1156 Pnt 5920 1203 Pnt 5920 1250 Pnt 5920 1297 Pnt 5920 1344 Pnt 5920 1391 Pnt 5920 1438 Pnt 5920 1485 Pnt 5920 1532 Pnt 5920 2614 Pnt 5920 2661 Pnt 5920 2708 Pnt 5920 2802 Pnt 5920 2849 Pnt 5920 2943 Pnt 5920 2990 Pnt 5920 3037 Pnt 5920 3367 Pnt 5920 3414 Pnt 5920 3555 Pnt 5931 168 Pnt 5931 215 Pnt 5931 262 Pnt 5931 309 Pnt 5931 356 Pnt 5931 403 Pnt 5931 450 Pnt 5931 497 Pnt 5931 544 Pnt 5931 591 Pnt 5931 638 Pnt 5931 685 Pnt 5931 732 Pnt 5931 780 Pnt 5931 827 Pnt 5931 874 Pnt 5931 921 Pnt 5931 968 Pnt 5931 1015 Pnt 5931 1062 Pnt 5931 1109 Pnt 5931 1156 Pnt 5931 1203 Pnt 5931 1250 Pnt 5931 1297 Pnt 5931 1344 Pnt 5931 1391 Pnt 5931 1438 Pnt 5931 1485 Pnt 5931 1532 Pnt 5931 1579 Pnt 5931 2661 Pnt 5931 2708 Pnt 5931 2849 Pnt 5931 2990 Pnt 5931 3037 Pnt 5931 3084 Pnt 5931 3320 Pnt 5931 3367 Pnt 5931 3602 Pnt 5931 3649 Pnt 5942 168 Pnt 5942 215 Pnt 5942 262 Pnt 5942 309 Pnt 5942 356 Pnt 5942 403 Pnt 5942 450 Pnt 5942 497 Pnt 5942 544 Pnt 5942 591 Pnt 5942 638 Pnt 5942 685 Pnt 5942 732 Pnt 5942 780 Pnt 5942 827 Pnt 5942 874 Pnt 5942 921 Pnt 5942 968 Pnt 5942 1015 Pnt 5942 1062 Pnt 5942 1109 Pnt 5942 1156 Pnt 5942 1203 Pnt 5942 1250 Pnt 5942 1297 Pnt 5942 1344 Pnt 5942 1391 Pnt 5942 1438 Pnt 5942 1485 Pnt 5942 1532 Pnt 5942 1579 Pnt 5942 1626 Pnt 5942 2708 Pnt 5942 2755 Pnt 5942 2849 Pnt 5942 2896 Pnt 5942 3037 Pnt 5942 3084 Pnt 5942 3367 Pnt 5942 3414 Pnt 5942 3461 Pnt 5942 3602 Pnt 5953 168 Pnt 5953 215 Pnt 5953 262 Pnt 5953 309 Pnt 5953 356 Pnt 5953 403 Pnt 5953 450 Pnt 5953 497 Pnt 5953 544 Pnt 5953 591 Pnt 5953 638 Pnt 5953 685 Pnt 5953 732 Pnt 5953 780 Pnt 5953 827 Pnt 5953 874 Pnt 5953 921 Pnt 5953 968 Pnt 5953 1015 Pnt 5953 1062 Pnt 5953 1109 Pnt 5953 1156 Pnt 5953 1250 Pnt 5953 1297 Pnt 5953 1344 Pnt 5953 1391 Pnt 5953 1438 Pnt 5953 1485 Pnt 5953 1532 Pnt 5953 1579 Pnt 5953 1626 Pnt 5953 1673 Pnt 5953 2708 Pnt 5953 2755 Pnt 5953 2802 Pnt 5953 2896 Pnt 5953 2943 Pnt 5953 3037 Pnt 5953 3084 Pnt 5953 3132 Pnt 5953 3367 Pnt 5953 3414 Pnt 5953 3649 Pnt 5953 3696 Pnt 5965 168 Pnt 5965 215 Pnt 5965 262 Pnt 5965 309 Pnt 5965 356 Pnt 5965 403 Pnt 5965 450 Pnt 5965 497 Pnt 5965 544 Pnt 5965 591 Pnt 5965 638 Pnt 5965 685 Pnt 5965 732 Pnt 5965 780 Pnt 5965 827 Pnt 5965 874 Pnt 5965 921 Pnt 5965 968 Pnt 5965 1015 Pnt 5965 1062 Pnt 5965 1109 Pnt 5965 1156 Pnt 5965 1203 Pnt 5965 1250 Pnt 5965 1297 Pnt 5965 1344 Pnt 5965 1391 Pnt 5965 1438 Pnt 5965 1485 Pnt 5965 1532 Pnt 5965 1579 Pnt 5965 1626 Pnt 5965 1673 Pnt 5965 1720 Pnt 5965 2755 Pnt 5965 2802 Pnt 5965 2943 Pnt 5965 3084 Pnt 5965 3132 Pnt 5965 3414 Pnt 5965 3461 Pnt 5965 3508 Pnt 5965 3649 Pnt 5976 168 Pnt 5976 215 Pnt 5976 262 Pnt 5976 309 Pnt 5976 356 Pnt 5976 403 Pnt 5976 450 Pnt 5976 497 Pnt 5976 544 Pnt 5976 591 Pnt 5976 638 Pnt 5976 685 Pnt 5976 732 Pnt 5976 780 Pnt 5976 827 Pnt 5976 874 Pnt 5976 921 Pnt 5976 968 Pnt 5976 1015 Pnt 5976 1062 Pnt 5976 1109 Pnt 5976 1156 Pnt 5976 1203 Pnt 5976 1250 Pnt 5976 1297 Pnt 5976 1391 Pnt 5976 1438 Pnt 5976 1485 Pnt 5976 1532 Pnt 5976 1579 Pnt 5976 1626 Pnt 5976 1673 Pnt 5976 1720 Pnt 5976 1767 Pnt 5976 2802 Pnt 5976 2849 Pnt 5976 2943 Pnt 5976 2990 Pnt 5976 3084 Pnt 5976 3179 Pnt 5976 3414 Pnt 5976 3696 Pnt 5976 3743 Pnt 5987 168 Pnt 5987 215 Pnt 5987 262 Pnt 5987 309 Pnt 5987 356 Pnt 5987 403 Pnt 5987 450 Pnt 5987 497 Pnt 5987 544 Pnt 5987 591 Pnt 5987 638 Pnt 5987 685 Pnt 5987 732 Pnt 5987 780 Pnt 5987 827 Pnt 5987 874 Pnt 5987 921 Pnt 5987 968 Pnt 5987 1015 Pnt 5987 1062 Pnt 5987 1109 Pnt 5987 1156 Pnt 5987 1203 Pnt 5987 1250 Pnt 5987 1297 Pnt 5987 1344 Pnt 5987 1391 Pnt 5987 1438 Pnt 5987 1485 Pnt 5987 1532 Pnt 5987 1579 Pnt 5987 1626 Pnt 5987 1673 Pnt 5987 1720 Pnt 5987 1767 Pnt 5987 2802 Pnt 5987 2849 Pnt 5987 2896 Pnt 5987 2990 Pnt 5987 3132 Pnt 5987 3179 Pnt 5987 3461 Pnt 5987 3508 Pnt 5987 3555 Pnt 5987 3696 Pnt 5998 168 Pnt 5998 215 Pnt 5998 262 Pnt 5998 309 Pnt 5998 356 Pnt 5998 403 Pnt 5998 450 Pnt 5998 497 Pnt 5998 544 Pnt 5998 591 Pnt 5998 638 Pnt 5998 685 Pnt 5998 732 Pnt 5998 780 Pnt 5998 827 Pnt 5998 874 Pnt 5998 921 Pnt 5998 968 Pnt 5998 1015 Pnt 5998 1062 Pnt 5998 1109 Pnt 5998 1156 Pnt 5998 1203 Pnt 5998 1250 Pnt 5998 1297 Pnt 5998 1344 Pnt 5998 1391 Pnt 5998 1438 Pnt 5998 1532 Pnt 5998 1579 Pnt 5998 1626 Pnt 5998 1673 Pnt 5998 1720 Pnt 5998 1767 Pnt 5998 1814 Pnt 5998 2849 Pnt 5998 2896 Pnt 5998 2990 Pnt 5998 3037 Pnt 5998 3132 Pnt 5998 3179 Pnt 5998 3226 Pnt 5998 3461 Pnt 5998 3790 Pnt 6009 168 Pnt 6009 215 Pnt 6009 262 Pnt 6009 309 Pnt 6009 356 Pnt 6009 403 Pnt 6009 450 Pnt 6009 497 Pnt 6009 544 Pnt 6009 591 Pnt 6009 638 Pnt 6009 685 Pnt 6009 732 Pnt 6009 780 Pnt 6009 827 Pnt 6009 874 Pnt 6009 921 Pnt 6009 968 Pnt 6009 1015 Pnt 6009 1062 Pnt 6009 1109 Pnt 6009 1156 Pnt 6009 1203 Pnt 6009 1250 Pnt 6009 1297 Pnt 6009 1344 Pnt 6009 1391 Pnt 6009 1438 Pnt 6009 1485 Pnt 6009 1579 Pnt 6009 1626 Pnt 6009 1673 Pnt 6009 1720 Pnt 6009 1767 Pnt 6009 1814 Pnt 6009 1861 Pnt 6009 2896 Pnt 6009 2943 Pnt 6009 3037 Pnt 6009 3084 Pnt 6009 3179 Pnt 6009 3226 Pnt 6009 3508 Pnt 6009 3555 Pnt 6009 3602 Pnt 6009 3743 Pnt 6021 168 Pnt 6021 215 Pnt 6021 262 Pnt 6021 309 Pnt 6021 356 Pnt 6021 403 Pnt 6021 450 Pnt 6021 497 Pnt 6021 968 Pnt 6021 1015 Pnt 6021 1062 Pnt 6021 1109 Pnt 6021 1156 Pnt 6021 1203 Pnt 6021 1250 Pnt 6021 1297 Pnt 6021 1344 Pnt 6021 1391 Pnt 6021 1438 Pnt 6021 1485 Pnt 6021 1532 Pnt 6021 1626 Pnt 6021 1673 Pnt 6021 1720 Pnt 6021 1767 Pnt 6021 1814 Pnt 6021 1861 Pnt 6021 1908 Pnt 6021 2896 Pnt 6021 2943 Pnt 6021 3084 Pnt 6021 3226 Pnt 6021 3273 Pnt 6021 3837 Pnt 6032 168 Pnt 6032 215 Pnt 6032 262 Pnt 6032 309 Pnt 6032 356 Pnt 6032 403 Pnt 6032 450 Pnt 6032 497 Pnt 6032 544 Pnt 6032 591 Pnt 6032 638 Pnt 6032 685 Pnt 6032 732 Pnt 6032 780 Pnt 6032 827 Pnt 6032 1109 Pnt 6032 1156 Pnt 6032 1203 Pnt 6032 1250 Pnt 6032 1297 Pnt 6032 1344 Pnt 6032 1391 Pnt 6032 1438 Pnt 6032 1485 Pnt 6032 1532 Pnt 6032 1579 Pnt 6032 1673 Pnt 6032 1720 Pnt 6032 1767 Pnt 6032 1814 Pnt 6032 1861 Pnt 6032 1908 Pnt 6032 1956 Pnt 6032 2943 Pnt 6032 2990 Pnt 6032 3084 Pnt 6032 3132 Pnt 6032 3226 Pnt 6032 3273 Pnt 6032 3320 Pnt 6032 3555 Pnt 6032 3602 Pnt 6032 3649 Pnt 6032 3790 Pnt 6043 168 Pnt 6043 215 Pnt 6043 262 Pnt 6043 309 Pnt 6043 356 Pnt 6043 403 Pnt 6043 450 Pnt 6043 497 Pnt 6043 544 Pnt 6043 591 Pnt 6043 638 Pnt 6043 685 Pnt 6043 732 Pnt 6043 780 Pnt 6043 827 Pnt 6043 874 Pnt 6043 921 Pnt 6043 968 Pnt 6043 1250 Pnt 6043 1297 Pnt 6043 1344 Pnt 6043 1391 Pnt 6043 1438 Pnt 6043 1485 Pnt 6043 1532 Pnt 6043 1579 Pnt 6043 1626 Pnt 6043 1720 Pnt 6043 1767 Pnt 6043 1814 Pnt 6043 1861 Pnt 6043 1908 Pnt 6043 1956 Pnt 6043 2943 Pnt 6043 3037 Pnt 6043 3132 Pnt 6043 3273 Pnt 6043 3320 Pnt 6043 3837 Pnt 6043 3884 Pnt 6054 168 Pnt 6054 215 Pnt 6054 262 Pnt 6054 309 Pnt 6054 356 Pnt 6054 403 Pnt 6054 450 Pnt 6054 497 Pnt 6054 544 Pnt 6054 591 Pnt 6054 638 Pnt 6054 685 Pnt 6054 732 Pnt 6054 780 Pnt 6054 827 Pnt 6054 874 Pnt 6054 921 Pnt 6054 968 Pnt 6054 1015 Pnt 6054 1062 Pnt 6054 1344 Pnt 6054 1391 Pnt 6054 1438 Pnt 6054 1485 Pnt 6054 1532 Pnt 6054 1579 Pnt 6054 1626 Pnt 6054 1673 Pnt 6054 1767 Pnt 6054 1814 Pnt 6054 1861 Pnt 6054 1908 Pnt 6054 1956 Pnt 6054 2003 Pnt 6054 2990 Pnt 6054 3037 Pnt 6054 3132 Pnt 6054 3179 Pnt 6054 3273 Pnt 6054 3320 Pnt 6054 3367 Pnt 6054 3602 Pnt 6054 3649 Pnt 6054 3696 Pnt 6065 168 Pnt 6065 215 Pnt 6065 262 Pnt 6065 309 Pnt 6065 356 Pnt 6065 403 Pnt 6065 450 Pnt 6065 497 Pnt 6065 544 Pnt 6065 591 Pnt 6065 638 Pnt 6065 685 Pnt 6065 732 Pnt 6065 780 Pnt 6065 827 Pnt 6065 874 Pnt 6065 921 Pnt 6065 968 Pnt 6065 1015 Pnt 6065 1062 Pnt 6065 1109 Pnt 6065 1156 Pnt 6065 1391 Pnt 6065 1438 Pnt 6065 1485 Pnt 6065 1532 Pnt 6065 1579 Pnt 6065 1626 Pnt 6065 1673 Pnt 6065 1720 Pnt 6065 1814 Pnt 6065 1861 Pnt 6065 1908 Pnt 6065 1956 Pnt 6065 2003 Pnt 6065 2050 Pnt 6065 3037 Pnt 6065 3084 Pnt 6065 3179 Pnt 6065 3320 Pnt 6065 3367 Pnt 6065 3884 Pnt 6077 168 Pnt 6077 215 Pnt 6077 262 Pnt 6077 309 Pnt 6077 356 Pnt 6077 403 Pnt 6077 450 Pnt 6077 497 Pnt 6077 544 Pnt 6077 591 Pnt 6077 638 Pnt 6077 685 Pnt 6077 732 Pnt 6077 780 Pnt 6077 827 Pnt 6077 874 Pnt 6077 921 Pnt 6077 968 Pnt 6077 1015 Pnt 6077 1062 Pnt 6077 1109 Pnt 6077 1156 Pnt 6077 1203 Pnt 6077 1250 Pnt 6077 1485 Pnt 6077 1532 Pnt 6077 1579 Pnt 6077 1626 Pnt 6077 1673 Pnt 6077 1720 Pnt 6077 1767 Pnt 6077 1861 Pnt 6077 1908 Pnt 6077 1956 Pnt 6077 2050 Pnt 6077 2097 Pnt 6077 3037 Pnt 6077 3084 Pnt 6077 3179 Pnt 6077 3226 Pnt 6077 3320 Pnt 6077 3414 Pnt 6077 3649 Pnt 6077 3696 Pnt 6077 3743 Pnt 6088 168 Pnt 6088 215 Pnt 6088 262 Pnt 6088 309 Pnt 6088 356 Pnt 6088 403 Pnt 6088 450 Pnt 6088 497 Pnt 6088 544 Pnt 6088 591 Pnt 6088 638 Pnt 6088 685 Pnt 6088 732 Pnt 6088 780 Pnt 6088 827 Pnt 6088 874 Pnt 6088 921 Pnt 6088 968 Pnt 6088 1015 Pnt 6088 1062 Pnt 6088 1109 Pnt 6088 1156 Pnt 6088 1203 Pnt 6088 1250 Pnt 6088 1297 Pnt 6088 1532 Pnt 6088 1579 Pnt 6088 1626 Pnt 6088 1673 Pnt 6088 1720 Pnt 6088 1767 Pnt 6088 1814 Pnt 6088 1908 Pnt 6088 1956 Pnt 6088 2003 Pnt 6088 2050 Pnt 6088 2097 Pnt 6088 3084 Pnt 6088 3132 Pnt 6088 3226 Pnt 6088 3273 Pnt 6088 3367 Pnt 6088 3414 Pnt 6088 3931 Pnt 6099 168 Pnt 6099 215 Pnt 6099 262 Pnt 6099 309 Pnt 6099 356 Pnt 6099 403 Pnt 6099 450 Pnt 6099 497 Pnt 6099 544 Pnt 6099 591 Pnt 6099 638 Pnt 6099 685 Pnt 6099 732 Pnt 6099 780 Pnt 6099 827 Pnt 6099 874 Pnt 6099 921 Pnt 6099 968 Pnt 6099 1015 Pnt 6099 1062 Pnt 6099 1109 Pnt 6099 1156 Pnt 6099 1203 Pnt 6099 1250 Pnt 6099 1297 Pnt 6099 1344 Pnt 6099 1391 Pnt 6099 1579 Pnt 6099 1626 Pnt 6099 1673 Pnt 6099 1720 Pnt 6099 1767 Pnt 6099 1814 Pnt 6099 1861 Pnt 6099 1956 Pnt 6099 2003 Pnt 6099 2050 Pnt 6099 2097 Pnt 6099 2144 Pnt 6099 3179 Pnt 6099 3273 Pnt 6099 3414 Pnt 6099 3461 Pnt 6099 3696 Pnt 6099 3743 Pnt 6099 3790 Pnt 6110 168 Pnt 6110 215 Pnt 6110 262 Pnt 6110 309 Pnt 6110 356 Pnt 6110 403 Pnt 6110 450 Pnt 6110 497 Pnt 6110 544 Pnt 6110 591 Pnt 6110 638 Pnt 6110 685 Pnt 6110 732 Pnt 6110 780 Pnt 6110 827 Pnt 6110 874 Pnt 6110 921 Pnt 6110 968 Pnt 6110 1015 Pnt 6110 1062 Pnt 6110 1109 Pnt 6110 1156 Pnt 6110 1203 Pnt 6110 1250 Pnt 6110 1297 Pnt 6110 1344 Pnt 6110 1391 Pnt 6110 1438 Pnt 6110 1673 Pnt 6110 1720 Pnt 6110 1767 Pnt 6110 1814 Pnt 6110 1861 Pnt 6110 1908 Pnt 6110 2003 Pnt 6110 2050 Pnt 6110 2097 Pnt 6110 2144 Pnt 6110 2191 Pnt 6110 3132 Pnt 6110 3179 Pnt 6110 3273 Pnt 6110 3320 Pnt 6110 3414 Pnt 6110 3461 Pnt 6110 3978 Pnt 6121 168 Pnt 6121 215 Pnt 6121 262 Pnt 6121 309 Pnt 6121 356 Pnt 6121 403 Pnt 6121 450 Pnt 6121 497 Pnt 6121 544 Pnt 6121 591 Pnt 6121 638 Pnt 6121 685 Pnt 6121 732 Pnt 6121 780 Pnt 6121 827 Pnt 6121 874 Pnt 6121 921 Pnt 6121 968 Pnt 6121 1015 Pnt 6121 1062 Pnt 6121 1109 Pnt 6121 1156 Pnt 6121 1203 Pnt 6121 1250 Pnt 6121 1297 Pnt 6121 1344 Pnt 6121 1391 Pnt 6121 1438 Pnt 6121 1485 Pnt 6121 1720 Pnt 6121 1767 Pnt 6121 1814 Pnt 6121 1861 Pnt 6121 1908 Pnt 6121 1956 Pnt 6121 2003 Pnt 6121 2050 Pnt 6121 2097 Pnt 6121 2191 Pnt 6121 3179 Pnt 6121 3226 Pnt 6121 3320 Pnt 6121 3461 Pnt 6121 3508 Pnt 6121 3743 Pnt 6121 3790 Pnt 6121 3837 Pnt 6133 168 Pnt 6133 215 Pnt 6133 262 Pnt 6133 309 Pnt 6133 356 Pnt 6133 403 Pnt 6133 450 Pnt 6133 497 Pnt 6133 544 Pnt 6133 591 Pnt 6133 638 Pnt 6133 685 Pnt 6133 732 Pnt 6133 780 Pnt 6133 827 Pnt 6133 874 Pnt 6133 921 Pnt 6133 968 Pnt 6133 1015 Pnt 6133 1062 Pnt 6133 1109 Pnt 6133 1156 Pnt 6133 1203 Pnt 6133 1250 Pnt 6133 1297 Pnt 6133 1344 Pnt 6133 1391 Pnt 6133 1438 Pnt 6133 1485 Pnt 6133 1532 Pnt 6133 1767 Pnt 6133 1814 Pnt 6133 1861 Pnt 6133 1908 Pnt 6133 1956 Pnt 6133 2003 Pnt 6133 2050 Pnt 6133 2097 Pnt 6133 2144 Pnt 6133 2191 Pnt 6133 2238 Pnt 6133 3179 Pnt 6133 3226 Pnt 6133 3320 Pnt 6133 3367 Pnt 6133 3461 Pnt 6133 3508 Pnt 6133 3837 Pnt 6133 3884 Pnt 6133 4025 Pnt 6133 4119 Pnt 6144 168 Pnt 6144 215 Pnt 6144 262 Pnt 6144 309 Pnt 6144 356 Pnt 6144 403 Pnt 6144 450 Pnt 6144 497 Pnt 6144 544 Pnt 6144 591 Pnt 6144 638 Pnt 6144 685 Pnt 6144 732 Pnt 6144 780 Pnt 6144 827 Pnt 6144 874 Pnt 6144 921 Pnt 6144 968 Pnt 6144 1015 Pnt 6144 1062 Pnt 6144 1109 Pnt 6144 1156 Pnt 6144 1203 Pnt 6144 1250 Pnt 6144 1297 Pnt 6144 1344 Pnt 6144 1391 Pnt 6144 1438 Pnt 6144 1485 Pnt 6144 1532 Pnt 6144 1579 Pnt 6144 1626 Pnt 6144 1814 Pnt 6144 1861 Pnt 6144 1908 Pnt 6144 1956 Pnt 6144 2003 Pnt 6144 2097 Pnt 6144 2144 Pnt 6144 2191 Pnt 6144 2238 Pnt 6144 2285 Pnt 6144 3226 Pnt 6144 3273 Pnt 6144 3367 Pnt 6144 3508 Pnt 6144 3555 Pnt 6144 3790 Pnt 6144 3837 Pnt 6144 3884 Pnt 6155 168 Pnt 6155 215 Pnt 6155 262 Pnt 6155 309 Pnt 6155 356 Pnt 6155 403 Pnt 6155 450 Pnt 6155 497 Pnt 6155 544 Pnt 6155 591 Pnt 6155 638 Pnt 6155 685 Pnt 6155 732 Pnt 6155 780 Pnt 6155 827 Pnt 6155 874 Pnt 6155 921 Pnt 6155 968 Pnt 6155 1015 Pnt 6155 1062 Pnt 6155 1109 Pnt 6155 1156 Pnt 6155 1203 Pnt 6155 1250 Pnt 6155 1297 Pnt 6155 1344 Pnt 6155 1391 Pnt 6155 1438 Pnt 6155 1485 Pnt 6155 1532 Pnt 6155 1579 Pnt 6155 1626 Pnt 6155 1673 Pnt 6155 1861 Pnt 6155 1908 Pnt 6155 1956 Pnt 6155 2003 Pnt 6155 2050 Pnt 6155 2144 Pnt 6155 2191 Pnt 6155 2238 Pnt 6155 2285 Pnt 6155 3226 Pnt 6155 3273 Pnt 6155 3367 Pnt 6155 3414 Pnt 6155 3508 Pnt 6155 3555 Pnt 6155 3837 Pnt 6155 3884 Pnt 6155 3931 Pnt 6155 4072 Pnt 6155 4166 Pnt 6166 168 Pnt 6166 215 Pnt 6166 262 Pnt 6166 309 Pnt 6166 356 Pnt 6166 403 Pnt 6166 450 Pnt 6166 497 Pnt 6166 544 Pnt 6166 591 Pnt 6166 638 Pnt 6166 685 Pnt 6166 732 Pnt 6166 780 Pnt 6166 827 Pnt 6166 874 Pnt 6166 921 Pnt 6166 968 Pnt 6166 1015 Pnt 6166 1062 Pnt 6166 1109 Pnt 6166 1156 Pnt 6166 1203 Pnt 6166 1250 Pnt 6166 1297 Pnt 6166 1344 Pnt 6166 1391 Pnt 6166 1438 Pnt 6166 1485 Pnt 6166 1532 Pnt 6166 1579 Pnt 6166 1626 Pnt 6166 1673 Pnt 6166 1720 Pnt 6166 1908 Pnt 6166 1956 Pnt 6166 2003 Pnt 6166 2050 Pnt 6166 2097 Pnt 6166 2191 Pnt 6166 2238 Pnt 6166 2285 Pnt 6166 2332 Pnt 6166 3273 Pnt 6166 3320 Pnt 6166 3414 Pnt 6166 3555 Pnt 6166 3602 Pnt 6166 3837 Pnt 6166 3884 Pnt 6166 3931 Pnt 6178 168 Pnt 6178 215 Pnt 6178 262 Pnt 6178 309 Pnt 6178 356 Pnt 6178 403 Pnt 6178 450 Pnt 6178 497 Pnt 6178 544 Pnt 6178 591 Pnt 6178 638 Pnt 6178 685 Pnt 6178 732 Pnt 6178 780 Pnt 6178 827 Pnt 6178 874 Pnt 6178 921 Pnt 6178 968 Pnt 6178 1015 Pnt 6178 1062 Pnt 6178 1109 Pnt 6178 1156 Pnt 6178 1203 Pnt 6178 1250 Pnt 6178 1297 Pnt 6178 1344 Pnt 6178 1391 Pnt 6178 1438 Pnt 6178 1485 Pnt 6178 1532 Pnt 6178 1579 Pnt 6178 1626 Pnt 6178 1673 Pnt 6178 1720 Pnt 6178 1767 Pnt 6178 1956 Pnt 6178 2003 Pnt 6178 2050 Pnt 6178 2097 Pnt 6178 2144 Pnt 6178 2191 Pnt 6178 2238 Pnt 6178 2285 Pnt 6178 2332 Pnt 6178 2379 Pnt 6178 3367 Pnt 6178 3461 Pnt 6178 3555 Pnt 6178 3602 Pnt 6178 3884 Pnt 6178 3931 Pnt 6178 3978 Pnt 6178 4119 Pnt 6178 4213 Pnt 6189 168 Pnt 6189 215 Pnt 6189 262 Pnt 6189 309 Pnt 6189 356 Pnt 6189 403 Pnt 6189 450 Pnt 6189 497 Pnt 6189 544 Pnt 6189 591 Pnt 6189 638 Pnt 6189 685 Pnt 6189 732 Pnt 6189 780 Pnt 6189 827 Pnt 6189 874 Pnt 6189 921 Pnt 6189 968 Pnt 6189 1015 Pnt 6189 1062 Pnt 6189 1109 Pnt 6189 1156 Pnt 6189 1203 Pnt 6189 1250 Pnt 6189 1297 Pnt 6189 1344 Pnt 6189 1391 Pnt 6189 1438 Pnt 6189 1485 Pnt 6189 1532 Pnt 6189 1579 Pnt 6189 1626 Pnt 6189 1673 Pnt 6189 1720 Pnt 6189 1767 Pnt 6189 2003 Pnt 6189 2050 Pnt 6189 2097 Pnt 6189 2144 Pnt 6189 2238 Pnt 6189 2285 Pnt 6189 2332 Pnt 6189 2379 Pnt 6189 3320 Pnt 6189 3367 Pnt 6189 3461 Pnt 6189 3602 Pnt 6189 3649 Pnt 6189 3884 Pnt 6189 3931 Pnt 6189 3978 Pnt 6200 168 Pnt 6200 215 Pnt 6200 262 Pnt 6200 309 Pnt 6200 356 Pnt 6200 403 Pnt 6200 450 Pnt 6200 497 Pnt 6200 544 Pnt 6200 591 Pnt 6200 638 Pnt 6200 685 Pnt 6200 732 Pnt 6200 780 Pnt 6200 827 Pnt 6200 874 Pnt 6200 921 Pnt 6200 968 Pnt 6200 1015 Pnt 6200 1062 Pnt 6200 1109 Pnt 6200 1156 Pnt 6200 1203 Pnt 6200 1250 Pnt 6200 1297 Pnt 6200 1344 Pnt 6200 1391 Pnt 6200 1438 Pnt 6200 1485 Pnt 6200 1532 Pnt 6200 1579 Pnt 6200 1626 Pnt 6200 1673 Pnt 6200 1720 Pnt 6200 1767 Pnt 6200 1814 Pnt 6200 2050 Pnt 6200 2097 Pnt 6200 2144 Pnt 6200 2191 Pnt 6200 2285 Pnt 6200 2332 Pnt 6200 2426 Pnt 6200 3367 Pnt 6200 3414 Pnt 6200 3508 Pnt 6200 3602 Pnt 6200 3649 Pnt 6200 3931 Pnt 6200 3978 Pnt 6200 4025 Pnt 6200 4166 Pnt 6200 4260 Pnt 6211 168 Pnt 6211 215 Pnt 6211 262 Pnt 6211 309 Pnt 6211 356 Pnt 6211 403 Pnt 6211 450 Pnt 6211 497 Pnt 6211 544 Pnt 6211 591 Pnt 6211 638 Pnt 6211 874 Pnt 6211 921 Pnt 6211 968 Pnt 6211 1015 Pnt 6211 1062 Pnt 6211 1109 Pnt 6211 1156 Pnt 6211 1203 Pnt 6211 1250 Pnt 6211 1297 Pnt 6211 1344 Pnt 6211 1391 Pnt 6211 1438 Pnt 6211 1485 Pnt 6211 1532 Pnt 6211 1579 Pnt 6211 1626 Pnt 6211 1673 Pnt 6211 1720 Pnt 6211 1767 Pnt 6211 1814 Pnt 6211 1861 Pnt 6211 2050 Pnt 6211 2097 Pnt 6211 2144 Pnt 6211 2191 Pnt 6211 2238 Pnt 6211 2332 Pnt 6211 2379 Pnt 6211 2426 Pnt 6211 2473 Pnt 6211 3367 Pnt 6211 3414 Pnt 6211 3508 Pnt 6211 3555 Pnt 6211 3649 Pnt 6211 3696 Pnt 6211 3931 Pnt 6211 3978 Pnt 6211 4025 Pnt 6222 168 Pnt 6222 215 Pnt 6222 262 Pnt 6222 309 Pnt 6222 356 Pnt 6222 403 Pnt 6222 497 Pnt 6222 544 Pnt 6222 591 Pnt 6222 638 Pnt 6222 685 Pnt 6222 732 Pnt 6222 1109 Pnt 6222 1156 Pnt 6222 1203 Pnt 6222 1250 Pnt 6222 1297 Pnt 6222 1344 Pnt 6222 1391 Pnt 6222 1438 Pnt 6222 1485 Pnt 6222 1532 Pnt 6222 1579 Pnt 6222 1626 Pnt 6222 1673 Pnt 6222 1720 Pnt 6222 1767 Pnt 6222 1814 Pnt 6222 1861 Pnt 6222 1908 Pnt 6222 2097 Pnt 6222 2144 Pnt 6222 2191 Pnt 6222 2238 Pnt 6222 2285 Pnt 6222 2332 Pnt 6222 2379 Pnt 6222 2426 Pnt 6222 2473 Pnt 6222 3414 Pnt 6222 3461 Pnt 6222 3555 Pnt 6222 3649 Pnt 6222 3696 Pnt 6222 3743 Pnt 6222 3978 Pnt 6222 4025 Pnt 6222 4072 Pnt 6222 4213 Pnt 6234 168 Pnt 6234 215 Pnt 6234 262 Pnt 6234 309 Pnt 6234 356 Pnt 6234 403 Pnt 6234 450 Pnt 6234 497 Pnt 6234 544 Pnt 6234 591 Pnt 6234 638 Pnt 6234 685 Pnt 6234 732 Pnt 6234 780 Pnt 6234 827 Pnt 6234 874 Pnt 6234 921 Pnt 6234 1250 Pnt 6234 1297 Pnt 6234 1344 Pnt 6234 1391 Pnt 6234 1438 Pnt 6234 1485 Pnt 6234 1532 Pnt 6234 1579 Pnt 6234 1626 Pnt 6234 1673 Pnt 6234 1720 Pnt 6234 1767 Pnt 6234 1814 Pnt 6234 1861 Pnt 6234 1908 Pnt 6234 1956 Pnt 6234 2144 Pnt 6234 2191 Pnt 6234 2238 Pnt 6234 2285 Pnt 6234 2379 Pnt 6234 2426 Pnt 6234 2473 Pnt 6234 2520 Pnt 6234 3414 Pnt 6234 3461 Pnt 6234 3555 Pnt 6234 3602 Pnt 6234 3696 Pnt 6234 3743 Pnt 6234 3978 Pnt 6234 4025 Pnt 6234 4072 Pnt 6245 168 Pnt 6245 215 Pnt 6245 262 Pnt 6245 309 Pnt 6245 356 Pnt 6245 403 Pnt 6245 450 Pnt 6245 497 Pnt 6245 544 Pnt 6245 591 Pnt 6245 638 Pnt 6245 685 Pnt 6245 732 Pnt 6245 780 Pnt 6245 827 Pnt 6245 874 Pnt 6245 921 Pnt 6245 968 Pnt 6245 1015 Pnt 6245 1062 Pnt 6245 1344 Pnt 6245 1391 Pnt 6245 1438 Pnt 6245 1485 Pnt 6245 1532 Pnt 6245 1579 Pnt 6245 1626 Pnt 6245 1673 Pnt 6245 1720 Pnt 6245 1767 Pnt 6245 1814 Pnt 6245 1861 Pnt 6245 1908 Pnt 6245 1956 Pnt 6245 2003 Pnt 6245 2191 Pnt 6245 2238 Pnt 6245 2285 Pnt 6245 2332 Pnt 6245 2426 Pnt 6245 2473 Pnt 6245 2520 Pnt 6245 2567 Pnt 6245 3461 Pnt 6245 3508 Pnt 6245 3602 Pnt 6245 3696 Pnt 6245 3743 Pnt 6245 3790 Pnt 6245 4025 Pnt 6245 4072 Pnt 6245 4119 Pnt 6245 4260 Pnt 6256 168 Pnt 6256 215 Pnt 6256 262 Pnt 6256 309 Pnt 6256 356 Pnt 6256 403 Pnt 6256 450 Pnt 6256 497 Pnt 6256 544 Pnt 6256 591 Pnt 6256 638 Pnt 6256 685 Pnt 6256 732 Pnt 6256 780 Pnt 6256 827 Pnt 6256 874 Pnt 6256 921 Pnt 6256 968 Pnt 6256 1015 Pnt 6256 1062 Pnt 6256 1109 Pnt 6256 1156 Pnt 6256 1438 Pnt 6256 1485 Pnt 6256 1532 Pnt 6256 1579 Pnt 6256 1626 Pnt 6256 1673 Pnt 6256 1720 Pnt 6256 1767 Pnt 6256 1814 Pnt 6256 1861 Pnt 6256 1908 Pnt 6256 1956 Pnt 6256 2003 Pnt 6256 2050 Pnt 6256 2238 Pnt 6256 2285 Pnt 6256 2332 Pnt 6256 2379 Pnt 6256 2426 Pnt 6256 2473 Pnt 6256 2520 Pnt 6256 2567 Pnt 6256 3461 Pnt 6256 3508 Pnt 6256 3602 Pnt 6256 3649 Pnt 6256 3743 Pnt 6256 3790 Pnt 6256 4025 Pnt 6256 4072 Pnt 6256 4119 Pnt 6267 168 Pnt 6267 215 Pnt 6267 262 Pnt 6267 309 Pnt 6267 356 Pnt 6267 403 Pnt 6267 450 Pnt 6267 497 Pnt 6267 544 Pnt 6267 591 Pnt 6267 638 Pnt 6267 685 Pnt 6267 732 Pnt 6267 780 Pnt 6267 827 Pnt 6267 874 Pnt 6267 921 Pnt 6267 968 Pnt 6267 1015 Pnt 6267 1062 Pnt 6267 1109 Pnt 6267 1156 Pnt 6267 1203 Pnt 6267 1250 Pnt 6267 1485 Pnt 6267 1532 Pnt 6267 1579 Pnt 6267 1626 Pnt 6267 1673 Pnt 6267 1720 Pnt 6267 1767 Pnt 6267 1814 Pnt 6267 1861 Pnt 6267 1908 Pnt 6267 1956 Pnt 6267 2003 Pnt 6267 2050 Pnt 6267 2238 Pnt 6267 2285 Pnt 6267 2332 Pnt 6267 2379 Pnt 6267 2473 Pnt 6267 2520 Pnt 6267 2567 Pnt 6267 2614 Pnt 6267 3508 Pnt 6267 3555 Pnt 6267 3649 Pnt 6267 3743 Pnt 6267 3790 Pnt 6267 3837 Pnt 6267 4072 Pnt 6267 4119 Pnt 6267 4166 Pnt 6267 4308 Pnt 6278 168 Pnt 6278 215 Pnt 6278 262 Pnt 6278 309 Pnt 6278 356 Pnt 6278 403 Pnt 6278 450 Pnt 6278 497 Pnt 6278 544 Pnt 6278 591 Pnt 6278 638 Pnt 6278 685 Pnt 6278 732 Pnt 6278 780 Pnt 6278 827 Pnt 6278 874 Pnt 6278 921 Pnt 6278 968 Pnt 6278 1015 Pnt 6278 1062 Pnt 6278 1109 Pnt 6278 1156 Pnt 6278 1203 Pnt 6278 1250 Pnt 6278 1297 Pnt 6278 1344 Pnt 6278 1579 Pnt 6278 1626 Pnt 6278 1673 Pnt 6278 1720 Pnt 6278 1767 Pnt 6278 1814 Pnt 6278 1861 Pnt 6278 1908 Pnt 6278 1956 Pnt 6278 2003 Pnt 6278 2050 Pnt 6278 2097 Pnt 6278 2285 Pnt 6278 2332 Pnt 6278 2379 Pnt 6278 2426 Pnt 6278 2520 Pnt 6278 2567 Pnt 6278 2614 Pnt 6278 3508 Pnt 6278 3555 Pnt 6278 3649 Pnt 6278 3696 Pnt 6278 3790 Pnt 6278 3837 Pnt 6278 4072 Pnt 6278 4119 Pnt 6278 4166 Pnt 6290 168 Pnt 6290 215 Pnt 6290 262 Pnt 6290 309 Pnt 6290 356 Pnt 6290 403 Pnt 6290 450 Pnt 6290 497 Pnt 6290 544 Pnt 6290 591 Pnt 6290 638 Pnt 6290 685 Pnt 6290 732 Pnt 6290 780 Pnt 6290 827 Pnt 6290 874 Pnt 6290 921 Pnt 6290 968 Pnt 6290 1015 Pnt 6290 1062 Pnt 6290 1109 Pnt 6290 1156 Pnt 6290 1203 Pnt 6290 1250 Pnt 6290 1297 Pnt 6290 1344 Pnt 6290 1391 Pnt 6290 1626 Pnt 6290 1673 Pnt 6290 1720 Pnt 6290 1767 Pnt 6290 1814 Pnt 6290 1861 Pnt 6290 1908 Pnt 6290 1956 Pnt 6290 2003 Pnt 6290 2050 Pnt 6290 2097 Pnt 6290 2144 Pnt 6290 2332 Pnt 6290 2379 Pnt 6290 2426 Pnt 6290 2473 Pnt 6290 2520 Pnt 6290 2567 Pnt 6290 2661 Pnt 6290 3555 Pnt 6290 3602 Pnt 6290 3696 Pnt 6290 3790 Pnt 6290 3837 Pnt 6290 3884 Pnt 6290 4119 Pnt 6290 4166 Pnt 6290 4213 Pnt 6290 4355 Pnt 6301 168 Pnt 6301 215 Pnt 6301 262 Pnt 6301 309 Pnt 6301 356 Pnt 6301 403 Pnt 6301 450 Pnt 6301 497 Pnt 6301 544 Pnt 6301 591 Pnt 6301 638 Pnt 6301 685 Pnt 6301 732 Pnt 6301 780 Pnt 6301 827 Pnt 6301 874 Pnt 6301 921 Pnt 6301 968 Pnt 6301 1015 Pnt 6301 1062 Pnt 6301 1109 Pnt 6301 1156 Pnt 6301 1203 Pnt 6301 1250 Pnt 6301 1297 Pnt 6301 1344 Pnt 6301 1391 Pnt 6301 1438 Pnt 6301 1673 Pnt 6301 1720 Pnt 6301 1767 Pnt 6301 1814 Pnt 6301 1861 Pnt 6301 1908 Pnt 6301 1956 Pnt 6301 2003 Pnt 6301 2050 Pnt 6301 2097 Pnt 6301 2144 Pnt 6301 2191 Pnt 6301 2379 Pnt 6301 2426 Pnt 6301 2473 Pnt 6301 2567 Pnt 6301 2614 Pnt 6301 2661 Pnt 6301 2708 Pnt 6301 3602 Pnt 6301 3696 Pnt 6301 3743 Pnt 6301 3837 Pnt 6301 3884 Pnt 6301 4119 Pnt 6301 4166 Pnt 6301 4213 Pnt 6312 168 Pnt 6312 215 Pnt 6312 262 Pnt 6312 309 Pnt 6312 356 Pnt 6312 403 Pnt 6312 450 Pnt 6312 497 Pnt 6312 544 Pnt 6312 591 Pnt 6312 638 Pnt 6312 685 Pnt 6312 732 Pnt 6312 780 Pnt 6312 827 Pnt 6312 874 Pnt 6312 921 Pnt 6312 968 Pnt 6312 1015 Pnt 6312 1062 Pnt 6312 1109 Pnt 6312 1156 Pnt 6312 1203 Pnt 6312 1250 Pnt 6312 1297 Pnt 6312 1344 Pnt 6312 1391 Pnt 6312 1438 Pnt 6312 1485 Pnt 6312 1532 Pnt 6312 1767 Pnt 6312 1814 Pnt 6312 1861 Pnt 6312 1908 Pnt 6312 1956 Pnt 6312 2003 Pnt 6312 2050 Pnt 6312 2097 Pnt 6312 2144 Pnt 6312 2191 Pnt 6312 2379 Pnt 6312 2426 Pnt 6312 2473 Pnt 6312 2520 Pnt 6312 2614 Pnt 6312 2661 Pnt 6312 2708 Pnt 6312 3602 Pnt 6312 3649 Pnt 6312 3743 Pnt 6312 3837 Pnt 6312 3884 Pnt 6312 3931 Pnt 6312 4166 Pnt 6312 4213 Pnt 6312 4260 Pnt 6312 4402 Pnt 6312 4449 Pnt 6323 168 Pnt 6323 215 Pnt 6323 262 Pnt 6323 309 Pnt 6323 356 Pnt 6323 403 Pnt 6323 450 Pnt 6323 497 Pnt 6323 544 Pnt 6323 591 Pnt 6323 638 Pnt 6323 685 Pnt 6323 732 Pnt 6323 780 Pnt 6323 827 Pnt 6323 874 Pnt 6323 921 Pnt 6323 968 Pnt 6323 1015 Pnt 6323 1062 Pnt 6323 1109 Pnt 6323 1156 Pnt 6323 1203 Pnt 6323 1250 Pnt 6323 1297 Pnt 6323 1344 Pnt 6323 1391 Pnt 6323 1438 Pnt 6323 1485 Pnt 6323 1532 Pnt 6323 1579 Pnt 6323 1814 Pnt 6323 1861 Pnt 6323 1908 Pnt 6323 1956 Pnt 6323 2003 Pnt 6323 2050 Pnt 6323 2097 Pnt 6323 2144 Pnt 6323 2191 Pnt 6323 2238 Pnt 6323 2426 Pnt 6323 2473 Pnt 6323 2520 Pnt 6323 2567 Pnt 6323 2614 Pnt 6323 2661 Pnt 6323 2708 Pnt 6323 2755 Pnt 6323 3649 Pnt 6323 3743 Pnt 6323 3790 Pnt 6323 3884 Pnt 6323 3931 Pnt 6323 4166 Pnt 6323 4213 Pnt 6323 4260 Pnt 6334 168 Pnt 6334 215 Pnt 6334 262 Pnt 6334 309 Pnt 6334 356 Pnt 6334 403 Pnt 6334 450 Pnt 6334 497 Pnt 6334 544 Pnt 6334 591 Pnt 6334 638 Pnt 6334 685 Pnt 6334 732 Pnt 6334 780 Pnt 6334 827 Pnt 6334 874 Pnt 6334 921 Pnt 6334 968 Pnt 6334 1015 Pnt 6334 1062 Pnt 6334 1109 Pnt 6334 1156 Pnt 6334 1203 Pnt 6334 1250 Pnt 6334 1297 Pnt 6334 1344 Pnt 6334 1391 Pnt 6334 1438 Pnt 6334 1485 Pnt 6334 1532 Pnt 6334 1579 Pnt 6334 1626 Pnt 6334 1861 Pnt 6334 1908 Pnt 6334 1956 Pnt 6334 2003 Pnt 6334 2050 Pnt 6334 2097 Pnt 6334 2144 Pnt 6334 2191 Pnt 6334 2238 Pnt 6334 2285 Pnt 6334 2473 Pnt 6334 2520 Pnt 6334 2567 Pnt 6334 2661 Pnt 6334 2708 Pnt 6334 2755 Pnt 6334 3649 Pnt 6334 3696 Pnt 6334 3790 Pnt 6334 3884 Pnt 6334 3931 Pnt 6334 3978 Pnt 6334 4213 Pnt 6334 4260 Pnt 6334 4449 Pnt 6334 4496 Pnt 6346 168 Pnt 6346 215 Pnt 6346 262 Pnt 6346 309 Pnt 6346 356 Pnt 6346 403 Pnt 6346 450 Pnt 6346 497 Pnt 6346 544 Pnt 6346 591 Pnt 6346 638 Pnt 6346 685 Pnt 6346 732 Pnt 6346 780 Pnt 6346 827 Pnt 6346 874 Pnt 6346 921 Pnt 6346 968 Pnt 6346 1015 Pnt 6346 1062 Pnt 6346 1109 Pnt 6346 1156 Pnt 6346 1203 Pnt 6346 1250 Pnt 6346 1297 Pnt 6346 1344 Pnt 6346 1391 Pnt 6346 1438 Pnt 6346 1485 Pnt 6346 1532 Pnt 6346 1579 Pnt 6346 1626 Pnt 6346 1673 Pnt 6346 1908 Pnt 6346 1956 Pnt 6346 2003 Pnt 6346 2050 Pnt 6346 2097 Pnt 6346 2144 Pnt 6346 2191 Pnt 6346 2238 Pnt 6346 2285 Pnt 6346 2332 Pnt 6346 2473 Pnt 6346 2520 Pnt 6346 2567 Pnt 6346 2614 Pnt 6346 2661 Pnt 6346 2708 Pnt 6346 2802 Pnt 6346 3696 Pnt 6346 3790 Pnt 6346 3837 Pnt 6346 3931 Pnt 6346 3978 Pnt 6346 4213 Pnt 6346 4260 Pnt 6346 4308 Pnt 6357 168 Pnt 6357 215 Pnt 6357 262 Pnt 6357 309 Pnt 6357 356 Pnt 6357 403 Pnt 6357 450 Pnt 6357 497 Pnt 6357 544 Pnt 6357 591 Pnt 6357 638 Pnt 6357 685 Pnt 6357 732 Pnt 6357 780 Pnt 6357 827 Pnt 6357 874 Pnt 6357 921 Pnt 6357 968 Pnt 6357 1015 Pnt 6357 1062 Pnt 6357 1109 Pnt 6357 1156 Pnt 6357 1203 Pnt 6357 1250 Pnt 6357 1297 Pnt 6357 1344 Pnt 6357 1391 Pnt 6357 1438 Pnt 6357 1485 Pnt 6357 1532 Pnt 6357 1579 Pnt 6357 1626 Pnt 6357 1673 Pnt 6357 1720 Pnt 6357 1956 Pnt 6357 2003 Pnt 6357 2050 Pnt 6357 2097 Pnt 6357 2144 Pnt 6357 2191 Pnt 6357 2238 Pnt 6357 2285 Pnt 6357 2332 Pnt 6357 2520 Pnt 6357 2567 Pnt 6357 2614 Pnt 6357 2708 Pnt 6357 2755 Pnt 6357 2802 Pnt 6357 3696 Pnt 6357 3743 Pnt 6357 3837 Pnt 6357 3931 Pnt 6357 3978 Pnt 6357 4025 Pnt 6357 4543 Pnt 6368 168 Pnt 6368 215 Pnt 6368 262 Pnt 6368 309 Pnt 6368 356 Pnt 6368 403 Pnt 6368 450 Pnt 6368 497 Pnt 6368 544 Pnt 6368 591 Pnt 6368 638 Pnt 6368 685 Pnt 6368 732 Pnt 6368 780 Pnt 6368 827 Pnt 6368 874 Pnt 6368 921 Pnt 6368 968 Pnt 6368 1015 Pnt 6368 1062 Pnt 6368 1109 Pnt 6368 1156 Pnt 6368 1203 Pnt 6368 1250 Pnt 6368 1297 Pnt 6368 1344 Pnt 6368 1391 Pnt 6368 1438 Pnt 6368 1485 Pnt 6368 1532 Pnt 6368 1579 Pnt 6368 1626 Pnt 6368 1673 Pnt 6368 1720 Pnt 6368 1767 Pnt 6368 2003 Pnt 6368 2050 Pnt 6368 2097 Pnt 6368 2144 Pnt 6368 2191 Pnt 6368 2238 Pnt 6368 2285 Pnt 6368 2332 Pnt 6368 2379 Pnt 6368 2567 Pnt 6368 2614 Pnt 6368 2661 Pnt 6368 2755 Pnt 6368 2802 Pnt 6368 2849 Pnt 6368 3743 Pnt 6368 3837 Pnt 6368 3884 Pnt 6368 3978 Pnt 6368 4025 Pnt 6368 4260 Pnt 6368 4308 Pnt 6368 4355 Pnt 6368 4496 Pnt 6379 168 Pnt 6379 215 Pnt 6379 262 Pnt 6379 309 Pnt 6379 356 Pnt 6379 403 Pnt 6379 450 Pnt 6379 497 Pnt 6379 544 Pnt 6379 591 Pnt 6379 638 Pnt 6379 685 Pnt 6379 732 Pnt 6379 780 Pnt 6379 827 Pnt 6379 874 Pnt 6379 921 Pnt 6379 968 Pnt 6379 1015 Pnt 6379 1062 Pnt 6379 1109 Pnt 6379 1156 Pnt 6379 1203 Pnt 6379 1250 Pnt 6379 1297 Pnt 6379 1344 Pnt 6379 1391 Pnt 6379 1438 Pnt 6379 1485 Pnt 6379 1532 Pnt 6379 1579 Pnt 6379 1626 Pnt 6379 1673 Pnt 6379 1720 Pnt 6379 1767 Pnt 6379 1814 Pnt 6379 2003 Pnt 6379 2050 Pnt 6379 2097 Pnt 6379 2144 Pnt 6379 2191 Pnt 6379 2238 Pnt 6379 2285 Pnt 6379 2332 Pnt 6379 2379 Pnt 6379 2426 Pnt 6379 2567 Pnt 6379 2614 Pnt 6379 2661 Pnt 6379 2708 Pnt 6379 2755 Pnt 6379 2802 Pnt 6379 2849 Pnt 6379 2896 Pnt 6379 3743 Pnt 6379 3790 Pnt 6379 3884 Pnt 6379 3978 Pnt 6379 4025 Pnt 6379 4072 Pnt 6379 4590 Pnt 6390 168 Pnt 6390 215 Pnt 6390 262 Pnt 6390 309 Pnt 6390 356 Pnt 6390 403 Pnt 6390 450 Pnt 6390 497 Pnt 6390 544 Pnt 6390 591 Pnt 6390 638 Pnt 6390 685 Pnt 6390 732 Pnt 6390 780 Pnt 6390 827 Pnt 6390 874 Pnt 6390 921 Pnt 6390 968 Pnt 6390 1015 Pnt 6390 1062 Pnt 6390 1109 Pnt 6390 1156 Pnt 6390 1203 Pnt 6390 1250 Pnt 6390 1297 Pnt 6390 1344 Pnt 6390 1391 Pnt 6390 1438 Pnt 6390 1485 Pnt 6390 1532 Pnt 6390 1579 Pnt 6390 1626 Pnt 6390 1673 Pnt 6390 1720 Pnt 6390 1767 Pnt 6390 1814 Pnt 6390 1861 Pnt 6390 2050 Pnt 6390 2097 Pnt 6390 2144 Pnt 6390 2191 Pnt 6390 2238 Pnt 6390 2285 Pnt 6390 2332 Pnt 6390 2379 Pnt 6390 2426 Pnt 6390 2614 Pnt 6390 2661 Pnt 6390 2708 Pnt 6390 2802 Pnt 6390 2849 Pnt 6390 2896 Pnt 6390 3790 Pnt 6390 3884 Pnt 6390 3931 Pnt 6390 4025 Pnt 6390 4072 Pnt 6390 4308 Pnt 6390 4355 Pnt 6390 4402 Pnt 6390 4543 Pnt 6402 168 Pnt 6402 215 Pnt 6402 262 Pnt 6402 309 Pnt 6402 356 Pnt 6402 403 Pnt 6402 450 Pnt 6402 497 Pnt 6402 544 Pnt 6402 591 Pnt 6402 638 Pnt 6402 685 Pnt 6402 732 Pnt 6402 780 Pnt 6402 827 Pnt 6402 874 Pnt 6402 921 Pnt 6402 968 Pnt 6402 1015 Pnt 6402 1062 Pnt 6402 1109 Pnt 6402 1156 Pnt 6402 1203 Pnt 6402 1250 Pnt 6402 1297 Pnt 6402 1344 Pnt 6402 1391 Pnt 6402 1438 Pnt 6402 1485 Pnt 6402 1532 Pnt 6402 1579 Pnt 6402 1626 Pnt 6402 1673 Pnt 6402 1720 Pnt 6402 1767 Pnt 6402 1814 Pnt 6402 1861 Pnt 6402 1908 Pnt 6402 2097 Pnt 6402 2144 Pnt 6402 2191 Pnt 6402 2238 Pnt 6402 2285 Pnt 6402 2332 Pnt 6402 2379 Pnt 6402 2426 Pnt 6402 2473 Pnt 6402 2661 Pnt 6402 2708 Pnt 6402 2755 Pnt 6402 2802 Pnt 6402 2849 Pnt 6402 2896 Pnt 6402 2943 Pnt 6402 3790 Pnt 6402 3837 Pnt 6402 3931 Pnt 6402 4025 Pnt 6402 4072 Pnt 6402 4637 Pnt 6413 168 Pnt 6413 215 Pnt 6413 262 Pnt 6413 309 Pnt 6413 356 Pnt 6413 403 Pnt 6413 450 Pnt 6413 497 Pnt 6413 544 Pnt 6413 591 Pnt 6413 638 Pnt 6413 685 Pnt 6413 732 Pnt 6413 780 Pnt 6413 827 Pnt 6413 874 Pnt 6413 921 Pnt 6413 968 Pnt 6413 1015 Pnt 6413 1062 Pnt 6413 1109 Pnt 6413 1156 Pnt 6413 1203 Pnt 6413 1250 Pnt 6413 1297 Pnt 6413 1344 Pnt 6413 1391 Pnt 6413 1438 Pnt 6413 1485 Pnt 6413 1532 Pnt 6413 1579 Pnt 6413 1626 Pnt 6413 1673 Pnt 6413 1720 Pnt 6413 1767 Pnt 6413 1814 Pnt 6413 1861 Pnt 6413 1908 Pnt 6413 1956 Pnt 6413 2144 Pnt 6413 2191 Pnt 6413 2238 Pnt 6413 2285 Pnt 6413 2332 Pnt 6413 2379 Pnt 6413 2426 Pnt 6413 2473 Pnt 6413 2661 Pnt 6413 2708 Pnt 6413 2755 Pnt 6413 2849 Pnt 6413 2896 Pnt 6413 2943 Pnt 6413 3837 Pnt 6413 3931 Pnt 6413 3978 Pnt 6413 4072 Pnt 6413 4119 Pnt 6413 4355 Pnt 6413 4402 Pnt 6413 4449 Pnt 6413 4590 Pnt 6424 168 Pnt 6424 215 Pnt 6424 262 Pnt 6424 309 Pnt 6424 356 Pnt 6424 403 Pnt 6424 450 Pnt 6424 497 Pnt 6424 544 Pnt 6424 591 Pnt 6424 638 Pnt 6424 685 Pnt 6424 732 Pnt 6424 780 Pnt 6424 827 Pnt 6424 874 Pnt 6424 921 Pnt 6424 968 Pnt 6424 1015 Pnt 6424 1062 Pnt 6424 1109 Pnt 6424 1156 Pnt 6424 1203 Pnt 6424 1250 Pnt 6424 1297 Pnt 6424 1344 Pnt 6424 1391 Pnt 6424 1438 Pnt 6424 1485 Pnt 6424 1532 Pnt 6424 1579 Pnt 6424 1626 Pnt 6424 1673 Pnt 6424 1720 Pnt 6424 1767 Pnt 6424 1814 Pnt 6424 1861 Pnt 6424 1908 Pnt 6424 1956 Pnt 6424 2191 Pnt 6424 2238 Pnt 6424 2285 Pnt 6424 2332 Pnt 6424 2379 Pnt 6424 2426 Pnt 6424 2473 Pnt 6424 2520 Pnt 6424 2708 Pnt 6424 2755 Pnt 6424 2802 Pnt 6424 2896 Pnt 6424 2990 Pnt 6424 3837 Pnt 6424 3884 Pnt 6424 3978 Pnt 6424 4072 Pnt 6424 4119 Pnt 6424 4637 Pnt 6424 4684 Pnt 6435 168 Pnt 6435 215 Pnt 6435 262 Pnt 6435 309 Pnt 6435 356 Pnt 6435 403 Pnt 6435 450 Pnt 6435 497 Pnt 6435 544 Pnt 6435 591 Pnt 6435 638 Pnt 6435 685 Pnt 6435 732 Pnt 6435 780 Pnt 6435 827 Pnt 6435 874 Pnt 6435 921 Pnt 6435 968 Pnt 6435 1015 Pnt 6435 1062 Pnt 6435 1109 Pnt 6435 1156 Pnt 6435 1203 Pnt 6435 1250 Pnt 6435 1297 Pnt 6435 1344 Pnt 6435 1391 Pnt 6435 1438 Pnt 6435 1485 Pnt 6435 1532 Pnt 6435 1579 Pnt 6435 1626 Pnt 6435 1720 Pnt 6435 1767 Pnt 6435 1814 Pnt 6435 1861 Pnt 6435 1908 Pnt 6435 1956 Pnt 6435 2003 Pnt 6435 2238 Pnt 6435 2285 Pnt 6435 2332 Pnt 6435 2379 Pnt 6435 2426 Pnt 6435 2473 Pnt 6435 2520 Pnt 6435 2567 Pnt 6435 2708 Pnt 6435 2755 Pnt 6435 2802 Pnt 6435 2849 Pnt 6435 2896 Pnt 6435 2943 Pnt 6435 2990 Pnt 6435 3884 Pnt 6435 3978 Pnt 6435 4025 Pnt 6435 4119 Pnt 6435 4166 Pnt 6435 4402 Pnt 6435 4449 Pnt 6435 4496 Pnt 6435 4637 Pnt 6446 168 Pnt 6446 215 Pnt 6446 262 Pnt 6446 309 Pnt 6446 356 Pnt 6446 403 Pnt 6446 450 Pnt 6446 497 Pnt 6446 544 Pnt 6446 591 Pnt 6446 638 Pnt 6446 685 Pnt 6446 732 Pnt 6446 780 Pnt 6446 827 Pnt 6446 874 Pnt 6446 921 Pnt 6446 968 Pnt 6446 1015 Pnt 6446 1062 Pnt 6446 1109 Pnt 6446 1156 Pnt 6446 1203 Pnt 6446 1250 Pnt 6446 1297 Pnt 6446 1344 Pnt 6446 1391 Pnt 6446 1438 Pnt 6446 1485 Pnt 6446 1532 Pnt 6446 1579 Pnt 6446 1626 Pnt 6446 1673 Pnt 6446 1767 Pnt 6446 1814 Pnt 6446 1861 Pnt 6446 1908 Pnt 6446 1956 Pnt 6446 2003 Pnt 6446 2050 Pnt 6446 2238 Pnt 6446 2285 Pnt 6446 2332 Pnt 6446 2379 Pnt 6446 2426 Pnt 6446 2473 Pnt 6446 2520 Pnt 6446 2567 Pnt 6446 2755 Pnt 6446 2802 Pnt 6446 2849 Pnt 6446 2943 Pnt 6446 2990 Pnt 6446 3037 Pnt 6446 3884 Pnt 6446 3931 Pnt 6446 4025 Pnt 6446 4119 Pnt 6446 4166 Pnt 6446 4684 Pnt 6446 4731 Pnt 6458 168 Pnt 6458 215 Pnt 6458 262 Pnt 6458 309 Pnt 6458 356 Pnt 6458 403 Pnt 6458 450 Pnt 6458 497 Pnt 6458 544 Pnt 6458 591 Pnt 6458 638 Pnt 6458 685 Pnt 6458 732 Pnt 6458 780 Pnt 6458 827 Pnt 6458 874 Pnt 6458 921 Pnt 6458 968 Pnt 6458 1015 Pnt 6458 1062 Pnt 6458 1109 Pnt 6458 1156 Pnt 6458 1203 Pnt 6458 1250 Pnt 6458 1297 Pnt 6458 1344 Pnt 6458 1391 Pnt 6458 1438 Pnt 6458 1485 Pnt 6458 1532 Pnt 6458 1579 Pnt 6458 1626 Pnt 6458 1673 Pnt 6458 1720 Pnt 6458 1767 Pnt 6458 1814 Pnt 6458 1861 Pnt 6458 1908 Pnt 6458 1956 Pnt 6458 2003 Pnt 6458 2050 Pnt 6458 2097 Pnt 6458 2285 Pnt 6458 2332 Pnt 6458 2379 Pnt 6458 2426 Pnt 6458 2473 Pnt 6458 2520 Pnt 6458 2567 Pnt 6458 2614 Pnt 6458 2802 Pnt 6458 2849 Pnt 6458 2896 Pnt 6458 2943 Pnt 6458 2990 Pnt 6458 3037 Pnt 6458 3931 Pnt 6458 4025 Pnt 6458 4072 Pnt 6458 4166 Pnt 6458 4213 Pnt 6458 4449 Pnt 6458 4496 Pnt 6458 4543 Pnt 6458 4684 Pnt 6458 4778 Pnt 6469 168 Pnt 6469 215 Pnt 6469 262 Pnt 6469 309 Pnt 6469 356 Pnt 6469 403 Pnt 6469 450 Pnt 6469 497 Pnt 6469 544 Pnt 6469 591 Pnt 6469 638 Pnt 6469 685 Pnt 6469 732 Pnt 6469 780 Pnt 6469 827 Pnt 6469 874 Pnt 6469 921 Pnt 6469 968 Pnt 6469 1015 Pnt 6469 1062 Pnt 6469 1109 Pnt 6469 1156 Pnt 6469 1203 Pnt 6469 1250 Pnt 6469 1297 Pnt 6469 1344 Pnt 6469 1391 Pnt 6469 1438 Pnt 6469 1485 Pnt 6469 1532 Pnt 6469 1579 Pnt 6469 1626 Pnt 6469 1673 Pnt 6469 1720 Pnt 6469 1767 Pnt 6469 1861 Pnt 6469 1908 Pnt 6469 1956 Pnt 6469 2003 Pnt 6469 2050 Pnt 6469 2097 Pnt 6469 2144 Pnt 6469 2332 Pnt 6469 2379 Pnt 6469 2426 Pnt 6469 2473 Pnt 6469 2520 Pnt 6469 2567 Pnt 6469 2614 Pnt 6469 2661 Pnt 6469 2802 Pnt 6469 2849 Pnt 6469 2896 Pnt 6469 2990 Pnt 6469 3037 Pnt 6469 3084 Pnt 6469 3931 Pnt 6469 3978 Pnt 6469 4072 Pnt 6469 4166 Pnt 6469 4213 Pnt 6469 4731 Pnt 6469 4778 Pnt 6480 168 Pnt 6480 215 Pnt 6480 262 Pnt 6480 309 Pnt 6480 356 Pnt 6480 403 Pnt 6480 450 Pnt 6480 497 Pnt 6480 544 Pnt 6480 591 Pnt 6480 638 Pnt 6480 685 Pnt 6480 732 Pnt 6480 780 Pnt 6480 827 Pnt 6480 874 Pnt 6480 921 Pnt 6480 968 Pnt 6480 1015 Pnt 6480 1062 Pnt 6480 1109 Pnt 6480 1156 Pnt 6480 1203 Pnt 6480 1250 Pnt 6480 1297 Pnt 6480 1344 Pnt 6480 1391 Pnt 6480 1438 Pnt 6480 1485 Pnt 6480 1532 Pnt 6480 1579 Pnt 6480 1626 Pnt 6480 1673 Pnt 6480 1720 Pnt 6480 1767 Pnt 6480 1814 Pnt 6480 1908 Pnt 6480 1956 Pnt 6480 2003 Pnt 6480 2050 Pnt 6480 2097 Pnt 6480 2144 Pnt 6480 2332 Pnt 6480 2379 Pnt 6480 2426 Pnt 6480 2473 Pnt 6480 2520 Pnt 6480 2567 Pnt 6480 2614 Pnt 6480 2661 Pnt 6480 2849 Pnt 6480 2896 Pnt 6480 2943 Pnt 6480 2990 Pnt 6480 3037 Pnt 6480 3084 Pnt 6480 3978 Pnt 6480 4072 Pnt 6480 4119 Pnt 6480 4213 Pnt 6480 4260 Pnt 6480 4496 Pnt 6480 4543 Pnt 6480 4590 Pnt 6480 4731 Pnt 6480 4825 Pnt 6491 168 Pnt 6491 215 Pnt 6491 262 Pnt 6491 309 Pnt 6491 356 Pnt 6491 403 Pnt 6491 450 Pnt 6491 497 Pnt 6491 544 Pnt 6491 591 Pnt 6491 638 Pnt 6491 685 Pnt 6491 732 Pnt 6491 780 Pnt 6491 827 Pnt 6491 874 Pnt 6491 921 Pnt 6491 968 Pnt 6491 1015 Pnt 6491 1062 Pnt 6491 1109 Pnt 6491 1156 Pnt 6491 1203 Pnt 6491 1250 Pnt 6491 1297 Pnt 6491 1344 Pnt 6491 1391 Pnt 6491 1438 Pnt 6491 1485 Pnt 6491 1532 Pnt 6491 1579 Pnt 6491 1626 Pnt 6491 1673 Pnt 6491 1720 Pnt 6491 1767 Pnt 6491 1814 Pnt 6491 1861 Pnt 6491 1956 Pnt 6491 2003 Pnt 6491 2050 Pnt 6491 2097 Pnt 6491 2144 Pnt 6491 2191 Pnt 6491 2379 Pnt 6491 2426 Pnt 6491 2473 Pnt 6491 2520 Pnt 6491 2567 Pnt 6491 2614 Pnt 6491 2661 Pnt 6491 2708 Pnt 6491 2849 Pnt 6491 2896 Pnt 6491 2943 Pnt 6491 3037 Pnt 6491 3084 Pnt 6491 3132 Pnt 6491 3978 Pnt 6491 4025 Pnt 6491 4119 Pnt 6491 4213 Pnt 6491 4260 Pnt 6491 4778 Pnt 6491 4825 Pnt 6503 168 Pnt 6503 215 Pnt 6503 262 Pnt 6503 309 Pnt 6503 356 Pnt 6503 403 Pnt 6503 450 Pnt 6503 497 Pnt 6503 544 Pnt 6503 591 Pnt 6503 638 Pnt 6503 685 Pnt 6503 732 Pnt 6503 780 Pnt 6503 827 Pnt 6503 874 Pnt 6503 921 Pnt 6503 968 Pnt 6503 1015 Pnt 6503 1062 Pnt 6503 1109 Pnt 6503 1156 Pnt 6503 1203 Pnt 6503 1250 Pnt 6503 1297 Pnt 6503 1344 Pnt 6503 1391 Pnt 6503 1438 Pnt 6503 1485 Pnt 6503 1532 Pnt 6503 1579 Pnt 6503 1626 Pnt 6503 1673 Pnt 6503 1720 Pnt 6503 1767 Pnt 6503 1814 Pnt 6503 1861 Pnt 6503 1908 Pnt 6503 2003 Pnt 6503 2050 Pnt 6503 2097 Pnt 6503 2144 Pnt 6503 2191 Pnt 6503 2238 Pnt 6503 2426 Pnt 6503 2473 Pnt 6503 2520 Pnt 6503 2567 Pnt 6503 2614 Pnt 6503 2661 Pnt 6503 2708 Pnt 6503 2896 Pnt 6503 2943 Pnt 6503 2990 Pnt 6503 3037 Pnt 6503 3084 Pnt 6503 3132 Pnt 6503 3179 Pnt 6503 3978 Pnt 6503 4025 Pnt 6503 4119 Pnt 6503 4260 Pnt 6503 4308 Pnt 6503 4543 Pnt 6503 4590 Pnt 6503 4637 Pnt 6503 4778 Pnt 6503 4872 Pnt 6514 168 Pnt 6514 215 Pnt 6514 262 Pnt 6514 309 Pnt 6514 356 Pnt 6514 403 Pnt 6514 450 Pnt 6514 497 Pnt 6514 544 Pnt 6514 591 Pnt 6514 638 Pnt 6514 685 Pnt 6514 1062 Pnt 6514 1109 Pnt 6514 1156 Pnt 6514 1203 Pnt 6514 1250 Pnt 6514 1297 Pnt 6514 1344 Pnt 6514 1391 Pnt 6514 1438 Pnt 6514 1485 Pnt 6514 1532 Pnt 6514 1579 Pnt 6514 1626 Pnt 6514 1673 Pnt 6514 1720 Pnt 6514 1767 Pnt 6514 1814 Pnt 6514 1861 Pnt 6514 1908 Pnt 6514 1956 Pnt 6514 2003 Pnt 6514 2050 Pnt 6514 2097 Pnt 6514 2144 Pnt 6514 2191 Pnt 6514 2238 Pnt 6514 2473 Pnt 6514 2520 Pnt 6514 2567 Pnt 6514 2614 Pnt 6514 2661 Pnt 6514 2708 Pnt 6514 2755 Pnt 6514 2896 Pnt 6514 2943 Pnt 6514 2990 Pnt 6514 3084 Pnt 6514 3132 Pnt 6514 3179 Pnt 6514 4025 Pnt 6514 4072 Pnt 6514 4166 Pnt 6514 4260 Pnt 6514 4308 Pnt 6514 4543 Pnt 6514 4825 Pnt 6514 4872 Pnt 6525 168 Pnt 6525 215 Pnt 6525 262 Pnt 6525 309 Pnt 6525 356 Pnt 6525 403 Pnt 6525 450 Pnt 6525 497 Pnt 6525 1250 Pnt 6525 1297 Pnt 6525 1344 Pnt 6525 1391 Pnt 6525 1438 Pnt 6525 1485 Pnt 6525 1532 Pnt 6525 1579 Pnt 6525 1626 Pnt 6525 1673 Pnt 6525 1720 Pnt 6525 1767 Pnt 6525 1814 Pnt 6525 1861 Pnt 6525 1908 Pnt 6525 1956 Pnt 6525 2003 Pnt 6525 2050 Pnt 6525 2097 Pnt 6525 2144 Pnt 6525 2191 Pnt 6525 2238 Pnt 6525 2285 Pnt 6525 2473 Pnt 6525 2520 Pnt 6525 2567 Pnt 6525 2614 Pnt 6525 2661 Pnt 6525 2708 Pnt 6525 2755 Pnt 6525 2943 Pnt 6525 2990 Pnt 6525 3037 Pnt 6525 3084 Pnt 6525 3132 Pnt 6525 3179 Pnt 6525 3226 Pnt 6525 4025 Pnt 6525 4072 Pnt 6525 4166 Pnt 6525 4308 Pnt 6525 4355 Pnt 6525 4590 Pnt 6525 4637 Pnt 6525 4684 Pnt 6525 4825 Pnt 6536 168 Pnt 6536 215 Pnt 6536 262 Pnt 6536 309 Pnt 6536 356 Pnt 6536 403 Pnt 6536 1344 Pnt 6536 1391 Pnt 6536 1438 Pnt 6536 1485 Pnt 6536 1532 Pnt 6536 1579 Pnt 6536 1626 Pnt 6536 1673 Pnt 6536 1720 Pnt 6536 1767 Pnt 6536 1814 Pnt 6536 1861 Pnt 6536 1908 Pnt 6536 1956 Pnt 6536 2003 Pnt 6536 2050 Pnt 6536 2097 Pnt 6536 2144 Pnt 6536 2191 Pnt 6536 2238 Pnt 6536 2285 Pnt 6536 2332 Pnt 6536 2520 Pnt 6536 2567 Pnt 6536 2614 Pnt 6536 2661 Pnt 6536 2708 Pnt 6536 2755 Pnt 6536 2802 Pnt 6536 2990 Pnt 6536 3037 Pnt 6536 3132 Pnt 6536 3179 Pnt 6536 3226 Pnt 6536 4072 Pnt 6536 4119 Pnt 6536 4213 Pnt 6536 4308 Pnt 6536 4355 Pnt 6536 4590 Pnt 6536 4872 Pnt 6547 168 Pnt 6547 215 Pnt 6547 262 Pnt 6547 309 Pnt 6547 1438 Pnt 6547 1485 Pnt 6547 1532 Pnt 6547 1579 Pnt 6547 1626 Pnt 6547 1673 Pnt 6547 1720 Pnt 6547 1767 Pnt 6547 1814 Pnt 6547 1861 Pnt 6547 1908 Pnt 6547 1956 Pnt 6547 2003 Pnt 6547 2050 Pnt 6547 2144 Pnt 6547 2191 Pnt 6547 2238 Pnt 6547 2285 Pnt 6547 2332 Pnt 6547 2520 Pnt 6547 2567 Pnt 6547 2614 Pnt 6547 2661 Pnt 6547 2708 Pnt 6547 2755 Pnt 6547 2802 Pnt 6547 2849 Pnt 6547 2990 Pnt 6547 3037 Pnt 6547 3084 Pnt 6547 3179 Pnt 6547 3226 Pnt 6547 3273 Pnt 6547 4072 Pnt 6547 4119 Pnt 6547 4213 Pnt 6547 4355 Pnt 6547 4402 Pnt 6547 4637 Pnt 6547 4684 Pnt 6547 4731 Pnt 6559 168 Pnt 6559 215 Pnt 6559 309 Pnt 6559 356 Pnt 6559 403 Pnt 6559 450 Pnt 6559 497 Pnt 6559 544 Pnt 6559 591 Pnt 6559 1532 Pnt 6559 1579 Pnt 6559 1626 Pnt 6559 1673 Pnt 6559 1720 Pnt 6559 1767 Pnt 6559 1814 Pnt 6559 1861 Pnt 6559 1908 Pnt 6559 1956 Pnt 6559 2003 Pnt 6559 2050 Pnt 6559 2097 Pnt 6559 2191 Pnt 6559 2238 Pnt 6559 2285 Pnt 6559 2332 Pnt 6559 2379 Pnt 6559 2567 Pnt 6559 2614 Pnt 6559 2661 Pnt 6559 2708 Pnt 6559 2755 Pnt 6559 2802 Pnt 6559 2849 Pnt 6559 3037 Pnt 6559 3084 Pnt 6559 3179 Pnt 6559 3226 Pnt 6559 3273 Pnt 6559 4119 Pnt 6559 4166 Pnt 6559 4213 Pnt 6559 4260 Pnt 6559 4355 Pnt 6559 4402 Pnt 6559 4637 Pnt 6559 4684 Pnt 6559 4731 Pnt 6570 168 Pnt 6570 215 Pnt 6570 262 Pnt 6570 309 Pnt 6570 356 Pnt 6570 403 Pnt 6570 450 Pnt 6570 497 Pnt 6570 544 Pnt 6570 591 Pnt 6570 638 Pnt 6570 685 Pnt 6570 732 Pnt 6570 780 Pnt 6570 1626 Pnt 6570 1673 Pnt 6570 1720 Pnt 6570 1767 Pnt 6570 1814 Pnt 6570 1861 Pnt 6570 1908 Pnt 6570 1956 Pnt 6570 2003 Pnt 6570 2050 Pnt 6570 2097 Pnt 6570 2144 Pnt 6570 2191 Pnt 6570 2238 Pnt 6570 2285 Pnt 6570 2332 Pnt 6570 2379 Pnt 6570 2426 Pnt 6570 2614 Pnt 6570 2661 Pnt 6570 2708 Pnt 6570 2755 Pnt 6570 2802 Pnt 6570 2849 Pnt 6570 2896 Pnt 6570 3037 Pnt 6570 3084 Pnt 6570 3132 Pnt 6570 3226 Pnt 6570 3273 Pnt 6570 3320 Pnt 6570 4119 Pnt 6570 4166 Pnt 6570 4260 Pnt 6570 4449 Pnt 6570 4684 Pnt 6570 4731 Pnt 6570 4778 Pnt 6581 168 Pnt 6581 215 Pnt 6581 262 Pnt 6581 309 Pnt 6581 356 Pnt 6581 403 Pnt 6581 450 Pnt 6581 497 Pnt 6581 544 Pnt 6581 591 Pnt 6581 638 Pnt 6581 685 Pnt 6581 732 Pnt 6581 780 Pnt 6581 827 Pnt 6581 874 Pnt 6581 921 Pnt 6581 1673 Pnt 6581 1720 Pnt 6581 1767 Pnt 6581 1814 Pnt 6581 1861 Pnt 6581 1908 Pnt 6581 1956 Pnt 6581 2003 Pnt 6581 2050 Pnt 6581 2097 Pnt 6581 2144 Pnt 6581 2191 Pnt 6581 2238 Pnt 6581 2285 Pnt 6581 2332 Pnt 6581 2379 Pnt 6581 2426 Pnt 6581 2614 Pnt 6581 2661 Pnt 6581 2708 Pnt 6581 2755 Pnt 6581 2802 Pnt 6581 2849 Pnt 6581 2896 Pnt 6581 3084 Pnt 6581 3132 Pnt 6581 3179 Pnt 6581 3226 Pnt 6581 3273 Pnt 6581 3320 Pnt 6581 4260 Pnt 6581 4308 Pnt 6581 4402 Pnt 6581 4449 Pnt 6581 4684 Pnt 6581 4731 Pnt 6581 4778 Pnt 6592 168 Pnt 6592 215 Pnt 6592 262 Pnt 6592 309 Pnt 6592 356 Pnt 6592 403 Pnt 6592 450 Pnt 6592 497 Pnt 6592 544 Pnt 6592 591 Pnt 6592 638 Pnt 6592 685 Pnt 6592 732 Pnt 6592 780 Pnt 6592 827 Pnt 6592 874 Pnt 6592 921 Pnt 6592 968 Pnt 6592 1015 Pnt 6592 1720 Pnt 6592 1767 Pnt 6592 1814 Pnt 6592 1861 Pnt 6592 1908 Pnt 6592 1956 Pnt 6592 2003 Pnt 6592 2050 Pnt 6592 2097 Pnt 6592 2144 Pnt 6592 2191 Pnt 6592 2285 Pnt 6592 2332 Pnt 6592 2379 Pnt 6592 2426 Pnt 6592 2473 Pnt 6592 2661 Pnt 6592 2708 Pnt 6592 2755 Pnt 6592 2802 Pnt 6592 2849 Pnt 6592 2896 Pnt 6592 2943 Pnt 6592 3084 Pnt 6592 3132 Pnt 6592 3179 Pnt 6592 3273 Pnt 6592 3320 Pnt 6592 3367 Pnt 6592 4166 Pnt 6592 4213 Pnt 6592 4308 Pnt 6592 4402 Pnt 6592 4449 Pnt 6592 4496 Pnt 6592 4778 Pnt 6592 4825 Pnt 6603 168 Pnt 6603 215 Pnt 6603 262 Pnt 6603 309 Pnt 6603 356 Pnt 6603 403 Pnt 6603 450 Pnt 6603 497 Pnt 6603 544 Pnt 6603 591 Pnt 6603 638 Pnt 6603 685 Pnt 6603 732 Pnt 6603 780 Pnt 6603 827 Pnt 6603 874 Pnt 6603 921 Pnt 6603 968 Pnt 6603 1015 Pnt 6603 1062 Pnt 6603 1767 Pnt 6603 1814 Pnt 6603 1861 Pnt 6603 1908 Pnt 6603 1956 Pnt 6603 2003 Pnt 6603 2050 Pnt 6603 2097 Pnt 6603 2144 Pnt 6603 2191 Pnt 6603 2238 Pnt 6603 2332 Pnt 6603 2379 Pnt 6603 2426 Pnt 6603 2473 Pnt 6603 2520 Pnt 6603 2708 Pnt 6603 2755 Pnt 6603 2802 Pnt 6603 2849 Pnt 6603 2896 Pnt 6603 2943 Pnt 6603 3132 Pnt 6603 3179 Pnt 6603 3226 Pnt 6603 3273 Pnt 6603 3320 Pnt 6603 3367 Pnt 6603 4213 Pnt 6603 4308 Pnt 6603 4355 Pnt 6603 4449 Pnt 6603 4496 Pnt 6603 4731 Pnt 6603 4778 Pnt 6603 4825 Pnt 6615 168 Pnt 6615 215 Pnt 6615 262 Pnt 6615 309 Pnt 6615 356 Pnt 6615 403 Pnt 6615 450 Pnt 6615 497 Pnt 6615 544 Pnt 6615 591 Pnt 6615 638 Pnt 6615 685 Pnt 6615 732 Pnt 6615 780 Pnt 6615 827 Pnt 6615 874 Pnt 6615 921 Pnt 6615 968 Pnt 6615 1015 Pnt 6615 1062 Pnt 6615 1109 Pnt 6615 1156 Pnt 6615 1814 Pnt 6615 1861 Pnt 6615 1908 Pnt 6615 1956 Pnt 6615 2003 Pnt 6615 2050 Pnt 6615 2097 Pnt 6615 2144 Pnt 6615 2191 Pnt 6615 2238 Pnt 6615 2285 Pnt 6615 2332 Pnt 6615 2379 Pnt 6615 2426 Pnt 6615 2473 Pnt 6615 2520 Pnt 6615 2708 Pnt 6615 2755 Pnt 6615 2802 Pnt 6615 2849 Pnt 6615 2896 Pnt 6615 2943 Pnt 6615 2990 Pnt 6615 3132 Pnt 6615 3179 Pnt 6615 3226 Pnt 6615 3320 Pnt 6615 3367 Pnt 6615 3414 Pnt 6615 4213 Pnt 6615 4260 Pnt 6615 4355 Pnt 6615 4449 Pnt 6615 4496 Pnt 6615 4872 Pnt 6626 168 Pnt 6626 215 Pnt 6626 262 Pnt 6626 309 Pnt 6626 356 Pnt 6626 403 Pnt 6626 450 Pnt 6626 497 Pnt 6626 544 Pnt 6626 591 Pnt 6626 638 Pnt 6626 685 Pnt 6626 732 Pnt 6626 780 Pnt 6626 827 Pnt 6626 874 Pnt 6626 921 Pnt 6626 968 Pnt 6626 1015 Pnt 6626 1062 Pnt 6626 1109 Pnt 6626 1156 Pnt 6626 1203 Pnt 6626 1861 Pnt 6626 1908 Pnt 6626 1956 Pnt 6626 2003 Pnt 6626 2050 Pnt 6626 2097 Pnt 6626 2144 Pnt 6626 2191 Pnt 6626 2238 Pnt 6626 2285 Pnt 6626 2379 Pnt 6626 2426 Pnt 6626 2473 Pnt 6626 2520 Pnt 6626 2567 Pnt 6626 2755 Pnt 6626 2802 Pnt 6626 2849 Pnt 6626 2896 Pnt 6626 2943 Pnt 6626 2990 Pnt 6626 3179 Pnt 6626 3226 Pnt 6626 3273 Pnt 6626 3320 Pnt 6626 3367 Pnt 6626 3414 Pnt 6626 4213 Pnt 6626 4260 Pnt 6626 4355 Pnt 6626 4402 Pnt 6626 4496 Pnt 6626 4543 Pnt 6626 4778 Pnt 6626 4825 Pnt 6626 4872 Pnt 6637 168 Pnt 6637 215 Pnt 6637 262 Pnt 6637 309 Pnt 6637 356 Pnt 6637 403 Pnt 6637 450 Pnt 6637 497 Pnt 6637 544 Pnt 6637 591 Pnt 6637 638 Pnt 6637 685 Pnt 6637 732 Pnt 6637 780 Pnt 6637 827 Pnt 6637 874 Pnt 6637 921 Pnt 6637 968 Pnt 6637 1015 Pnt 6637 1062 Pnt 6637 1109 Pnt 6637 1156 Pnt 6637 1203 Pnt 6637 1250 Pnt 6637 1297 Pnt 6637 1908 Pnt 6637 1956 Pnt 6637 2003 Pnt 6637 2050 Pnt 6637 2097 Pnt 6637 2144 Pnt 6637 2191 Pnt 6637 2238 Pnt 6637 2285 Pnt 6637 2332 Pnt 6637 2426 Pnt 6637 2473 Pnt 6637 2520 Pnt 6637 2567 Pnt 6637 2755 Pnt 6637 2802 Pnt 6637 2849 Pnt 6637 2896 Pnt 6637 2943 Pnt 6637 2990 Pnt 6637 3037 Pnt 6637 3179 Pnt 6637 3226 Pnt 6637 3273 Pnt 6637 3367 Pnt 6637 3414 Pnt 6637 3461 Pnt 6637 4260 Pnt 6637 4308 Pnt 6637 4402 Pnt 6637 4496 Pnt 6637 4543 Pnt 6648 168 Pnt 6648 215 Pnt 6648 262 Pnt 6648 309 Pnt 6648 356 Pnt 6648 403 Pnt 6648 450 Pnt 6648 497 Pnt 6648 544 Pnt 6648 591 Pnt 6648 638 Pnt 6648 685 Pnt 6648 732 Pnt 6648 780 Pnt 6648 827 Pnt 6648 874 Pnt 6648 921 Pnt 6648 968 Pnt 6648 1015 Pnt 6648 1062 Pnt 6648 1109 Pnt 6648 1156 Pnt 6648 1203 Pnt 6648 1250 Pnt 6648 1297 Pnt 6648 1344 Pnt 6648 1956 Pnt 6648 2003 Pnt 6648 2050 Pnt 6648 2097 Pnt 6648 2144 Pnt 6648 2191 Pnt 6648 2238 Pnt 6648 2285 Pnt 6648 2332 Pnt 6648 2379 Pnt 6648 2426 Pnt 6648 2473 Pnt 6648 2520 Pnt 6648 2567 Pnt 6648 2614 Pnt 6648 2802 Pnt 6648 2849 Pnt 6648 2896 Pnt 6648 2943 Pnt 6648 2990 Pnt 6648 3037 Pnt 6648 3226 Pnt 6648 3273 Pnt 6648 3320 Pnt 6648 3367 Pnt 6648 3414 Pnt 6648 3461 Pnt 6648 4260 Pnt 6648 4308 Pnt 6648 4402 Pnt 6648 4543 Pnt 6648 4590 Pnt 6648 4825 Pnt 6648 4872 Pnt 6659 168 Pnt 6659 215 Pnt 6659 262 Pnt 6659 309 Pnt 6659 356 Pnt 6659 403 Pnt 6659 450 Pnt 6659 497 Pnt 6659 544 Pnt 6659 591 Pnt 6659 638 Pnt 6659 685 Pnt 6659 732 Pnt 6659 780 Pnt 6659 827 Pnt 6659 874 Pnt 6659 921 Pnt 6659 968 Pnt 6659 1015 Pnt 6659 1062 Pnt 6659 1109 Pnt 6659 1156 Pnt 6659 1203 Pnt 6659 1250 Pnt 6659 1297 Pnt 6659 1344 Pnt 6659 1391 Pnt 6659 2003 Pnt 6659 2050 Pnt 6659 2097 Pnt 6659 2144 Pnt 6659 2191 Pnt 6659 2238 Pnt 6659 2285 Pnt 6659 2332 Pnt 6659 2379 Pnt 6659 2473 Pnt 6659 2520 Pnt 6659 2567 Pnt 6659 2614 Pnt 6659 2661 Pnt 6659 2802 Pnt 6659 2849 Pnt 6659 2896 Pnt 6659 2943 Pnt 6659 2990 Pnt 6659 3037 Pnt 6659 3084 Pnt 6659 3226 Pnt 6659 3273 Pnt 6659 3320 Pnt 6659 3414 Pnt 6659 3461 Pnt 6659 3508 Pnt 6659 4308 Pnt 6659 4355 Pnt 6659 4449 Pnt 6659 4543 Pnt 6659 4590 Pnt 6671 168 Pnt 6671 215 Pnt 6671 262 Pnt 6671 309 Pnt 6671 356 Pnt 6671 403 Pnt 6671 450 Pnt 6671 497 Pnt 6671 544 Pnt 6671 591 Pnt 6671 638 Pnt 6671 685 Pnt 6671 732 Pnt 6671 780 Pnt 6671 827 Pnt 6671 874 Pnt 6671 921 Pnt 6671 968 Pnt 6671 1015 Pnt 6671 1062 Pnt 6671 1109 Pnt 6671 1156 Pnt 6671 1203 Pnt 6671 1250 Pnt 6671 1297 Pnt 6671 1344 Pnt 6671 1391 Pnt 6671 1438 Pnt 6671 2050 Pnt 6671 2097 Pnt 6671 2144 Pnt 6671 2191 Pnt 6671 2238 Pnt 6671 2285 Pnt 6671 2332 Pnt 6671 2379 Pnt 6671 2426 Pnt 6671 2473 Pnt 6671 2520 Pnt 6671 2567 Pnt 6671 2614 Pnt 6671 2661 Pnt 6671 2849 Pnt 6671 2896 Pnt 6671 2943 Pnt 6671 2990 Pnt 6671 3037 Pnt 6671 3084 Pnt 6671 3273 Pnt 6671 3320 Pnt 6671 3414 Pnt 6671 3461 Pnt 6671 3508 Pnt 6671 4308 Pnt 6671 4355 Pnt 6671 4449 Pnt 6671 4590 Pnt 6671 4637 Pnt 6671 4872 Pnt 6682 168 Pnt 6682 215 Pnt 6682 262 Pnt 6682 309 Pnt 6682 356 Pnt 6682 403 Pnt 6682 450 Pnt 6682 497 Pnt 6682 544 Pnt 6682 591 Pnt 6682 638 Pnt 6682 685 Pnt 6682 732 Pnt 6682 780 Pnt 6682 827 Pnt 6682 874 Pnt 6682 921 Pnt 6682 968 Pnt 6682 1015 Pnt 6682 1062 Pnt 6682 1109 Pnt 6682 1156 Pnt 6682 1203 Pnt 6682 1250 Pnt 6682 1297 Pnt 6682 1344 Pnt 6682 1391 Pnt 6682 1438 Pnt 6682 1485 Pnt 6682 2097 Pnt 6682 2144 Pnt 6682 2191 Pnt 6682 2238 Pnt 6682 2285 Pnt 6682 2332 Pnt 6682 2379 Pnt 6682 2426 Pnt 6682 2473 Pnt 6682 2520 Pnt 6682 2567 Pnt 6682 2614 Pnt 6682 2661 Pnt 6682 2708 Pnt 6682 2849 Pnt 6682 2896 Pnt 6682 2943 Pnt 6682 2990 Pnt 6682 3037 Pnt 6682 3084 Pnt 6682 3132 Pnt 6682 3273 Pnt 6682 3320 Pnt 6682 3367 Pnt 6682 3461 Pnt 6682 3508 Pnt 6682 4449 Pnt 6682 4496 Pnt 6682 4590 Pnt 6682 4637 Pnt 6682 4872 Pnt 6693 168 Pnt 6693 215 Pnt 6693 262 Pnt 6693 309 Pnt 6693 356 Pnt 6693 403 Pnt 6693 450 Pnt 6693 497 Pnt 6693 544 Pnt 6693 591 Pnt 6693 638 Pnt 6693 685 Pnt 6693 732 Pnt 6693 780 Pnt 6693 827 Pnt 6693 874 Pnt 6693 921 Pnt 6693 968 Pnt 6693 1015 Pnt 6693 1062 Pnt 6693 1109 Pnt 6693 1156 Pnt 6693 1203 Pnt 6693 1250 Pnt 6693 1297 Pnt 6693 1344 Pnt 6693 1391 Pnt 6693 1438 Pnt 6693 1485 Pnt 6693 1532 Pnt 6693 2144 Pnt 6693 2191 Pnt 6693 2238 Pnt 6693 2285 Pnt 6693 2332 Pnt 6693 2379 Pnt 6693 2426 Pnt 6693 2473 Pnt 6693 2567 Pnt 6693 2614 Pnt 6693 2661 Pnt 6693 2708 Pnt 6693 2896 Pnt 6693 2943 Pnt 6693 2990 Pnt 6693 3037 Pnt 6693 3084 Pnt 6693 3132 Pnt 6693 3320 Pnt 6693 3367 Pnt 6693 3461 Pnt 6693 3508 Pnt 6693 3555 Pnt 6693 4355 Pnt 6693 4402 Pnt 6693 4496 Pnt 6693 4590 Pnt 6693 4684 Pnt 6704 168 Pnt 6704 215 Pnt 6704 262 Pnt 6704 309 Pnt 6704 356 Pnt 6704 403 Pnt 6704 450 Pnt 6704 497 Pnt 6704 544 Pnt 6704 591 Pnt 6704 638 Pnt 6704 685 Pnt 6704 732 Pnt 6704 780 Pnt 6704 827 Pnt 6704 874 Pnt 6704 921 Pnt 6704 968 Pnt 6704 1015 Pnt 6704 1062 Pnt 6704 1109 Pnt 6704 1156 Pnt 6704 1203 Pnt 6704 1250 Pnt 6704 1297 Pnt 6704 1344 Pnt 6704 1391 Pnt 6704 1438 Pnt 6704 1485 Pnt 6704 1532 Pnt 6704 1579 Pnt 6704 2144 Pnt 6704 2191 Pnt 6704 2238 Pnt 6704 2285 Pnt 6704 2332 Pnt 6704 2379 Pnt 6704 2426 Pnt 6704 2473 Pnt 6704 2520 Pnt 6704 2567 Pnt 6704 2614 Pnt 6704 2661 Pnt 6704 2708 Pnt 6704 2755 Pnt 6704 2896 Pnt 6704 2943 Pnt 6704 2990 Pnt 6704 3037 Pnt 6704 3084 Pnt 6704 3132 Pnt 6704 3179 Pnt 6704 3320 Pnt 6704 3367 Pnt 6704 3414 Pnt 6704 3461 Pnt 6704 3508 Pnt 6704 3555 Pnt 6704 4402 Pnt 6704 4496 Pnt 6704 4543 Pnt 6704 4637 Pnt 6704 4684 Pnt 6715 168 Pnt 6715 215 Pnt 6715 262 Pnt 6715 309 Pnt 6715 356 Pnt 6715 403 Pnt 6715 450 Pnt 6715 497 Pnt 6715 544 Pnt 6715 591 Pnt 6715 638 Pnt 6715 685 Pnt 6715 732 Pnt 6715 780 Pnt 6715 827 Pnt 6715 874 Pnt 6715 921 Pnt 6715 968 Pnt 6715 1015 Pnt 6715 1062 Pnt 6715 1109 Pnt 6715 1156 Pnt 6715 1203 Pnt 6715 1250 Pnt 6715 1297 Pnt 6715 1344 Pnt 6715 1391 Pnt 6715 1438 Pnt 6715 1485 Pnt 6715 1532 Pnt 6715 1579 Pnt 6715 1626 Pnt 6715 2191 Pnt 6715 2238 Pnt 6715 2285 Pnt 6715 2332 Pnt 6715 2379 Pnt 6715 2426 Pnt 6715 2473 Pnt 6715 2520 Pnt 6715 2614 Pnt 6715 2661 Pnt 6715 2708 Pnt 6715 2755 Pnt 6715 2943 Pnt 6715 2990 Pnt 6715 3037 Pnt 6715 3084 Pnt 6715 3132 Pnt 6715 3179 Pnt 6715 3367 Pnt 6715 3414 Pnt 6715 3508 Pnt 6715 3555 Pnt 6715 3602 Pnt 6715 4402 Pnt 6715 4449 Pnt 6715 4543 Pnt 6715 4637 Pnt 6715 4684 Pnt 6715 4731 Pnt 6727 168 Pnt 6727 215 Pnt 6727 262 Pnt 6727 309 Pnt 6727 356 Pnt 6727 403 Pnt 6727 450 Pnt 6727 497 Pnt 6727 544 Pnt 6727 591 Pnt 6727 638 Pnt 6727 685 Pnt 6727 732 Pnt 6727 780 Pnt 6727 827 Pnt 6727 874 Pnt 6727 921 Pnt 6727 968 Pnt 6727 1015 Pnt 6727 1062 Pnt 6727 1109 Pnt 6727 1156 Pnt 6727 1203 Pnt 6727 1250 Pnt 6727 1297 Pnt 6727 1344 Pnt 6727 1391 Pnt 6727 1438 Pnt 6727 1485 Pnt 6727 1532 Pnt 6727 1579 Pnt 6727 1626 Pnt 6727 1673 Pnt 6727 2238 Pnt 6727 2285 Pnt 6727 2332 Pnt 6727 2379 Pnt 6727 2426 Pnt 6727 2473 Pnt 6727 2520 Pnt 6727 2567 Pnt 6727 2614 Pnt 6727 2661 Pnt 6727 2708 Pnt 6727 2755 Pnt 6727 2802 Pnt 6727 2990 Pnt 6727 3037 Pnt 6727 3084 Pnt 6727 3132 Pnt 6727 3179 Pnt 6727 3226 Pnt 6727 3367 Pnt 6727 3414 Pnt 6727 3461 Pnt 6727 3508 Pnt 6727 3555 Pnt 6727 3602 Pnt 6727 4402 Pnt 6727 4449 Pnt 6727 4543 Pnt 6727 4590 Pnt 6727 4684 Pnt 6727 4731 Pnt 6738 168 Pnt 6738 215 Pnt 6738 262 Pnt 6738 309 Pnt 6738 356 Pnt 6738 403 Pnt 6738 450 Pnt 6738 497 Pnt 6738 544 Pnt 6738 591 Pnt 6738 638 Pnt 6738 685 Pnt 6738 732 Pnt 6738 780 Pnt 6738 827 Pnt 6738 874 Pnt 6738 921 Pnt 6738 968 Pnt 6738 1015 Pnt 6738 1062 Pnt 6738 1109 Pnt 6738 1156 Pnt 6738 1203 Pnt 6738 1250 Pnt 6738 1297 Pnt 6738 1344 Pnt 6738 1391 Pnt 6738 1438 Pnt 6738 1485 Pnt 6738 1532 Pnt 6738 1579 Pnt 6738 1626 Pnt 6738 1673 Pnt 6738 2238 Pnt 6738 2285 Pnt 6738 2332 Pnt 6738 2379 Pnt 6738 2426 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6850 2661 Pnt 6850 2708 Pnt 6850 2755 Pnt 6850 2802 Pnt 6850 2849 Pnt 6850 2896 Pnt 6850 2943 Pnt 6850 2990 Pnt 6850 3037 Pnt 6850 3226 Pnt 6850 3273 Pnt 6850 3320 Pnt 6850 3367 Pnt 6850 3414 Pnt 6850 3461 Pnt 6850 3602 Pnt 6850 3649 Pnt 6850 3696 Pnt 6850 3743 Pnt 6850 3790 Pnt 6850 3837 Pnt 6850 4637 Pnt 6850 4684 Pnt 6850 4778 Pnt 6861 168 Pnt 6861 215 Pnt 6861 262 Pnt 6861 309 Pnt 6861 356 Pnt 6861 403 Pnt 6861 450 Pnt 6861 497 Pnt 6861 544 Pnt 6861 591 Pnt 6861 638 Pnt 6861 685 Pnt 6861 732 Pnt 6861 780 Pnt 6861 827 Pnt 6861 874 Pnt 6861 921 Pnt 6861 968 Pnt 6861 1015 Pnt 6861 1062 Pnt 6861 1109 Pnt 6861 1156 Pnt 6861 1203 Pnt 6861 1250 Pnt 6861 1297 Pnt 6861 1344 Pnt 6861 1391 Pnt 6861 1438 Pnt 6861 1485 Pnt 6861 1532 Pnt 6861 1579 Pnt 6861 1626 Pnt 6861 1673 Pnt 6861 1720 Pnt 6861 1767 Pnt 6861 1814 Pnt 6861 1861 Pnt 6861 1908 Pnt 6861 1956 Pnt 6861 2003 Pnt 6861 2050 Pnt 6861 2567 Pnt 6861 2614 Pnt 6861 2661 Pnt 6861 2708 Pnt 6861 2755 Pnt 6861 2802 Pnt 6861 2849 Pnt 6861 2943 Pnt 6861 2990 Pnt 6861 3037 Pnt 6861 3084 Pnt 6861 3226 Pnt 6861 3273 Pnt 6861 3320 Pnt 6861 3367 Pnt 6861 3414 Pnt 6861 3461 Pnt 6861 3649 Pnt 6861 3696 Pnt 6861 3790 Pnt 6861 3884 Pnt 6861 4637 Pnt 6861 4684 Pnt 6861 4778 Pnt 6861 4825 Pnt 6872 168 Pnt 6872 215 Pnt 6872 262 Pnt 6872 309 Pnt 6872 356 Pnt 6872 403 Pnt 6872 450 Pnt 6872 497 Pnt 6872 544 Pnt 6872 591 Pnt 6872 638 Pnt 6872 685 Pnt 6872 732 Pnt 6872 780 Pnt 6872 827 Pnt 6872 874 Pnt 6872 921 Pnt 6872 968 Pnt 6872 1015 Pnt 6872 1062 Pnt 6872 1109 Pnt 6872 1156 Pnt 6872 1203 Pnt 6872 1250 Pnt 6872 1297 Pnt 6872 1344 Pnt 6872 1391 Pnt 6872 1438 Pnt 6872 1485 Pnt 6872 1532 Pnt 6872 1579 Pnt 6872 1626 Pnt 6872 1673 Pnt 6872 1720 Pnt 6872 1767 Pnt 6872 1814 Pnt 6872 1861 Pnt 6872 1908 Pnt 6872 1956 Pnt 6872 2003 Pnt 6872 2050 Pnt 6872 2097 Pnt 6872 2614 Pnt 6872 2661 Pnt 6872 2708 Pnt 6872 2755 Pnt 6872 2802 Pnt 6872 2849 Pnt 6872 2896 Pnt 6872 2943 Pnt 6872 2990 Pnt 6872 3037 Pnt 6872 3084 Pnt 6872 3273 Pnt 6872 3320 Pnt 6872 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6884 3743 Pnt 6884 3837 Pnt 6884 3884 Pnt 6884 3931 Pnt 6884 4684 Pnt 6884 4731 Pnt 6884 4825 Pnt 6895 168 Pnt 6895 215 Pnt 6895 262 Pnt 6895 309 Pnt 6895 356 Pnt 6895 403 Pnt 6895 450 Pnt 6895 497 Pnt 6895 544 Pnt 6895 591 Pnt 6895 638 Pnt 6895 685 Pnt 6895 732 Pnt 6895 780 Pnt 6895 827 Pnt 6895 874 Pnt 6895 921 Pnt 6895 968 Pnt 6895 1015 Pnt 6895 1062 Pnt 6895 1109 Pnt 6895 1156 Pnt 6895 1203 Pnt 6895 1250 Pnt 6895 1297 Pnt 6895 1344 Pnt 6895 1391 Pnt 6895 1438 Pnt 6895 1485 Pnt 6895 1532 Pnt 6895 1579 Pnt 6895 1626 Pnt 6895 1720 Pnt 6895 1767 Pnt 6895 1814 Pnt 6895 1861 Pnt 6895 1908 Pnt 6895 1956 Pnt 6895 2003 Pnt 6895 2050 Pnt 6895 2097 Pnt 6895 2144 Pnt 6895 2661 Pnt 6895 2708 Pnt 6895 2755 Pnt 6895 2802 Pnt 6895 2849 Pnt 6895 2896 Pnt 6895 2943 Pnt 6895 2990 Pnt 6895 3037 Pnt 6895 3084 Pnt 6895 3132 Pnt 6895 3320 Pnt 6895 3367 Pnt 6895 3414 Pnt 6895 3461 Pnt 6895 3508 Pnt 6895 3555 Pnt 6895 3696 Pnt 6895 3743 Pnt 6895 3790 Pnt 6895 3837 Pnt 6895 3884 Pnt 6895 3931 Pnt 6895 4684 Pnt 6895 4731 Pnt 6895 4825 Pnt 6895 4872 Pnt 6906 168 Pnt 6906 215 Pnt 6906 262 Pnt 6906 309 Pnt 6906 356 Pnt 6906 403 Pnt 6906 450 Pnt 6906 497 Pnt 6906 544 Pnt 6906 591 Pnt 6906 638 Pnt 6906 685 Pnt 6906 732 Pnt 6906 780 Pnt 6906 827 Pnt 6906 874 Pnt 6906 921 Pnt 6906 968 Pnt 6906 1015 Pnt 6906 1062 Pnt 6906 1109 Pnt 6906 1156 Pnt 6906 1203 Pnt 6906 1250 Pnt 6906 1297 Pnt 6906 1344 Pnt 6906 1391 Pnt 6906 1438 Pnt 6906 1485 Pnt 6906 1532 Pnt 6906 1579 Pnt 6906 1626 Pnt 6906 1673 Pnt 6906 1720 Pnt 6906 1767 Pnt 6906 1814 Pnt 6906 1861 Pnt 6906 1908 Pnt 6906 1956 Pnt 6906 2003 Pnt 6906 2050 Pnt 6906 2097 Pnt 6906 2144 Pnt 6906 2191 Pnt 6906 2661 Pnt 6906 2708 Pnt 6906 2755 Pnt 6906 2802 Pnt 6906 2849 Pnt 6906 2896 Pnt 6906 2943 Pnt 6906 2990 Pnt 6906 3037 Pnt 6906 3084 Pnt 6906 3132 Pnt 6906 3179 Pnt 6906 3320 Pnt 6906 3367 Pnt 6906 3414 Pnt 6906 3461 Pnt 6906 3508 Pnt 6906 3555 Pnt 6906 3696 Pnt 6906 3743 Pnt 6906 3790 Pnt 6906 3837 Pnt 6906 3884 Pnt 6906 3931 Pnt 6906 4731 Pnt 6906 4778 Pnt 6906 4872 Pnt 6917 168 Pnt 6917 215 Pnt 6917 262 Pnt 6917 309 Pnt 6917 356 Pnt 6917 403 Pnt 6917 450 Pnt 6917 497 Pnt 6917 544 Pnt 6917 591 Pnt 6917 638 Pnt 6917 685 Pnt 6917 732 Pnt 6917 780 Pnt 6917 827 Pnt 6917 874 Pnt 6917 921 Pnt 6917 968 Pnt 6917 1015 Pnt 6917 1062 Pnt 6917 1109 Pnt 6917 1156 Pnt 6917 1203 Pnt 6917 1250 Pnt 6917 1297 Pnt 6917 1344 Pnt 6917 1391 Pnt 6917 1438 Pnt 6917 1485 Pnt 6917 1532 Pnt 6917 1579 Pnt 6917 1626 Pnt 6917 1673 Pnt 6917 1720 Pnt 6917 1767 Pnt 6917 1814 Pnt 6917 1861 Pnt 6917 1908 Pnt 6917 1956 Pnt 6917 2003 Pnt 6917 2050 Pnt 6917 2097 Pnt 6917 2144 Pnt 6917 2191 Pnt 6917 2708 Pnt 6917 2755 Pnt 6917 2802 Pnt 6917 2849 Pnt 6917 2896 Pnt 6917 2943 Pnt 6917 2990 Pnt 6917 3037 Pnt 6917 3084 Pnt 6917 3132 Pnt 6917 3179 Pnt 6917 3320 Pnt 6917 3367 Pnt 6917 3414 Pnt 6917 3461 Pnt 6917 3508 Pnt 6917 3555 Pnt 6917 3743 Pnt 6917 3790 Pnt 6917 3884 Pnt 6917 3931 Pnt 6917 3978 Pnt 6917 4731 Pnt 6917 4778 Pnt 6917 4872 Pnt 6928 168 Pnt 6928 215 Pnt 6928 262 Pnt 6928 309 Pnt 6928 356 Pnt 6928 403 Pnt 6928 450 Pnt 6928 497 Pnt 6928 544 Pnt 6928 591 Pnt 6928 638 Pnt 6928 685 Pnt 6928 732 Pnt 6928 780 Pnt 6928 827 Pnt 6928 874 Pnt 6928 921 Pnt 6928 968 Pnt 6928 1015 Pnt 6928 1062 Pnt 6928 1109 Pnt 6928 1156 Pnt 6928 1203 Pnt 6928 1250 Pnt 6928 1297 Pnt 6928 1344 Pnt 6928 1391 Pnt 6928 1438 Pnt 6928 1485 Pnt 6928 1532 Pnt 6928 1579 Pnt 6928 1626 Pnt 6928 1673 Pnt 6928 1720 Pnt 6928 1767 Pnt 6928 1814 Pnt 6928 1861 Pnt 6928 1908 Pnt 6928 1956 Pnt 6928 2003 Pnt 6928 2050 Pnt 6928 2097 Pnt 6928 2144 Pnt 6928 2191 Pnt 6928 2238 Pnt 6928 2708 Pnt 6928 2755 Pnt 6928 2802 Pnt 6928 2849 Pnt 6928 2896 Pnt 6928 2943 Pnt 6928 2990 Pnt 6928 3037 Pnt 6928 3084 Pnt 6928 3132 Pnt 6928 3179 Pnt 6928 3367 Pnt 6928 3414 Pnt 6928 3461 Pnt 6928 3508 Pnt 6928 3555 Pnt 6928 3602 Pnt 6928 3743 Pnt 6928 3790 Pnt 6928 3837 Pnt 6928 3884 Pnt 6928 3931 Pnt 6928 3978 Pnt 6928 4778 Pnt 6928 4825 Pnt 6940 168 Pnt 6940 215 Pnt 6940 262 Pnt 6940 309 Pnt 6940 356 Pnt 6940 403 Pnt 6940 450 Pnt 6940 497 Pnt 6940 544 Pnt 6940 591 Pnt 6940 638 Pnt 6940 685 Pnt 6940 732 Pnt 6940 780 Pnt 6940 827 Pnt 6940 874 Pnt 6940 921 Pnt 6940 968 Pnt 6940 1015 Pnt 6940 1062 Pnt 6940 1109 Pnt 6940 1156 Pnt 6940 1203 Pnt 6940 1250 Pnt 6940 1344 Pnt 6940 1391 Pnt 6940 1438 Pnt 6940 1485 Pnt 6940 1532 Pnt 6940 1579 Pnt 6940 1626 Pnt 6940 1673 Pnt 6940 1720 Pnt 6940 1767 Pnt 6940 1861 Pnt 6940 1908 Pnt 6940 1956 Pnt 6940 2003 Pnt 6940 2050 Pnt 6940 2097 Pnt 6940 2144 Pnt 6940 2191 Pnt 6940 2238 Pnt 6940 2755 Pnt 6940 2802 Pnt 6940 2849 Pnt 6940 2896 Pnt 6940 2943 Pnt 6940 2990 Pnt 6940 3084 Pnt 6940 3132 Pnt 6940 3179 Pnt 6940 3226 Pnt 6940 3367 Pnt 6940 3414 Pnt 6940 3461 Pnt 6940 3508 Pnt 6940 3555 Pnt 6940 3602 Pnt 6940 3743 Pnt 6940 3790 Pnt 6940 3837 Pnt 6940 3931 Pnt 6940 4025 Pnt 6940 4778 Pnt 6940 4825 Pnt 6951 168 Pnt 6951 215 Pnt 6951 262 Pnt 6951 309 Pnt 6951 356 Pnt 6951 403 Pnt 6951 450 Pnt 6951 497 Pnt 6951 544 Pnt 6951 591 Pnt 6951 638 Pnt 6951 685 Pnt 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b(this)g(subsection)h(w)m(e)g(will)c(follo)m(w,)i (again,)h([52)o(],)h(together)f(with)g([51)o(].)43 b(The)30 b(idea)e(of)g(the)i(pro)s(of)e(is)g(similar)e(to)-118 524 y(that)k(of)f(KAM)h(theorem)g(\(see)h([1]\))f(and)g(it)e(consists)j (in)e(the)i(successiv)m(e)h(application)c(of)h(a)h(quadratic)f(metho)s (d)-118 644 y(to)d(o)m(v)m(ercome)i(the)f(small)d(divisors)j(problems)f (that)g(arise)h(in)f(trying)g(to)g(pro)m(v)m(e)i(the)f(con)m(v)m (ergence)j(of)c(the)h(formal)-118 764 y(series)38 b(that)f FA(solve)44 b Fy(the)38 b(conjugating)e(equation.)57 b(A)m(t)38 b(eac)m(h)g(step)g(of)f(the)h(metho)s(d)f(w)m(e)h(m)m(ust)f (solv)m(e)h(a)f(certain)-118 885 y(linear)31 b(homological)d(equation.) 28 1005 y(F)-8 b(or)32 b(a)g(function)g Fx(F)42 b Fy(:)27 b Fw(T)888 969 y Fr(d)960 1005 y Fs(!)g Fw(R)5 b Fy(,)39 b Fx(F)1282 1020 y Fl(k)1362 1005 y Fy(shall)31 b(denote)i(its)f(F)-8 b(ourier)32 b(co)s(e\016cien)m(ts,)h(that)g(is)1316 1259 y Fx(F)1379 1274 y Fl(k)1454 1259 y Fy(=)1655 1192 y(1)p 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Fy(the)h(eigen)m(v)-5 b(alues)33 b(of)f(the)h(matrix)1529 5670 y(\026)1503 5695 y Fx(A)p Fy(.)43 b(Then,)34 b(making)d(the)i(c)m (hange)h(of)e(v)-5 b(ariables)1495 5906 y Fx(x)28 b Fy(=)g(\()p Fx(I)i Fy(+)22 b Fx("P)14 b Fy(\()p Fx(\022)s(;)j(")p Fy(\)\))e Fx(x)2336 5921 y FC(1)2376 5906 y Fx(;)1876 6155 y Fy(101)p eop %%Page: 102 106 102 105 bop -118 219 a Fy(the)33 b(new)g(system)h(b)s(ecomes)1083 439 y Fx(x)1138 454 y FC(1)1178 397 y Ft(0)1229 439 y Fy(=)1332 358 y Fq(\000)1404 413 y Fy(\026)1378 439 y Fx(A)p Fy(\()p Fx(")p Fy(\))22 b(+)g Fx(")1739 397 y FC(2)1778 439 y Fx(Q)1855 454 y FC(2)1895 439 y Fy(\()p Fx(\022)s(;)17 b(")p Fy(\))2109 358 y Fq(\001)2170 439 y Fx(x)2225 454 y FC(1)2265 439 y Fx(;)212 b(\022)2552 397 y Ft(0)2603 439 y Fy(=)28 b Fx(!)t(;)967 b Fy(\(4.12\))-118 674 y(where)34 b(the)f(new)g(p)s(erturbation)f(is)g Fx(Q)1279 689 y FC(2)1346 674 y Fy(=)c(\()p Fx(I)i Fy(+)22 b Fx("P)14 b Fy(\))1820 638 y Ft(\000)p FC(1)1936 649 y Fy(~)1913 674 y Fx(Q)1990 689 y FC(1)2030 674 y Fx(P)g Fy(.)28 794 y(If)27 b(w)m(e)i(apply)e(this)g(pro)s(cedure)h Fx(r)i Fy(times)d(\(p)s(ostp)s(oning)f(the)i(discussion)f(of)g(the)h (resonance)h(condition)c(\(4.11\))-118 915 y(and)33 b(its)f(con)m (trol\))f(w)m(e)j(get)f(an)f(equation)g(whic)m(h)i(lo)s(oks)d(lik)m(e)h (the)h(follo)m(wing)1057 1135 y Fx(x)1112 1150 y Fr(r)1151 1094 y Ft(0)1202 1135 y Fy(=)1305 1054 y Fq(\000)1351 1135 y Fx(A)1424 1150 y Fr(r)1462 1135 y Fy(\()p Fx(")p Fy(\))22 b(+)g Fx(")1750 1094 y FC(2)1785 1070 y Fj(r)1823 1135 y Fx(Q)1900 1150 y Fr(r)1938 1135 y Fy(\()p Fx(\022)s(;)17 b(")p Fy(\))2152 1054 y Fq(\001)2214 1135 y Fx(x)2269 1150 y Fr(r)2308 1135 y Fx(;)211 b(\022)2594 1094 y Ft(0)2645 1135 y Fy(=)28 b Fx(!)t(:)925 b Fy(\(4.13\))-118 1355 y(If)29 b(the)g(norms)g(of)f Fx(A)610 1370 y Fr(n)686 1355 y Fy(and)h Fx(Q)949 1370 y Fr(n)1026 1355 y Fy(do)g(not)f(gro)m(w) i(to)s(o)e(fast,)h(the)h(sc)m(heme)g(will)d(b)s(e)i(con)m(v)m(ergen)m (t)i(to)e(an)g(equation)f(lik)m(e)1720 1575 y Fx(y)1772 1534 y Ft(0)1822 1575 y Fy(=)g Fx(B)5 b Fy(\()p Fx(")p Fy(\))p Fx(y)-118 1795 y Fy(whic)m(h)31 b(is)f(already)g(in)f(Flo)s (quet)h(form.)41 b(Therefore,)32 b(the)f(core)g(of)e(the)i(pro)s(of)f (is)f(the)i(con)m(v)m(ergence)i(of)d(the)h(ab)s(o)m(v)m(e)-118 1915 y(sc)m(heme.)-118 2175 y Fv(The)38 b(resonances)-118 2360 y Fy(Recall)31 b(that)h(at)g(eac)m(h)i(step)f(of)g(the)g(iterativ) m(e)e(pro)s(cess,)j(w)m(e)g(need)f(a)g(Diophan)m(tine)e(condition)g (lik)m(e)1402 2597 y Fs(j)1434 2571 y Fy(\026)1430 2597 y Fx(\025)1487 2612 y Fr(j)1545 2597 y Fs(\000)1649 2571 y Fy(\026)1645 2597 y Fx(\025)1702 2612 y Fr(l)1750 2597 y Fs(\000)22 b Fx(i)p Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(ij)27 b Fx(>)2370 2530 y(c)p 2296 2574 190 4 v 2296 2666 a Fs(j)p Fv(k)p Fs(j)2411 2637 y Fr(\015)2447 2646 y Fp(0)-118 2873 y Fy(in)g(order)g(to)g(b)s(e)h(able)f(to)g(solv)m(e)h(the)g (linear)e(homological)d(equation)k(\(4.10\).)41 b(Note)28 b(that)f(the)h(eigen)m(v)-5 b(alues)27 b Fx(\025)h Fy(are)-118 2993 y(c)m(hanged)j(at)f(eac)m(h)h(step)g(of)f(the)g(pro)s(cess)i(\(b)s (ecause)f(its)f(matrix)e(is)i(c)m(hanged\).)44 b(Therefore)31 b(w)m(e)g(do)f(not)g(kno)m(w)h(in)-118 3114 y(adv)-5 b(ance)33 b(whether)h(this)e(condition)f(will)g(b)s(e)h(ful\014lled)f (or)h(not.)28 3234 y(T)-8 b(o)31 b(deal)f(with)g(this)g(problem)f(a)h (con)m(trol)g(for)g(the)h(v)-5 b(ariation)28 b(of)i(the)h(eigen)m(v)-5 b(alues)30 b(at)g(eac)m(h)i(step)f(is)f(needed,)-118 3355 y(and)c(this)g(con)m(trol)f(is)h(ac)m(hiev)m(ed)h(b)m(y)g(the)f (third)g(condition)e(in)h(theorem)h(4.1.4.)41 b(Indeed,)29 b(assume)d(that)g(condition)-118 3475 y(\(4.11\))36 b(is)g(v)m (eri\014ed)i(at)f(the)g(\014rst)g(step.)57 b(Then)38 b(w)m(e)g(can)f(solv)m(e)h(the)f(linearized)e(homological)e(equation)j (\(4.10\))-118 3595 y(and)e(apply)f(the)h(transformation)d(that)i(it)g (de\014nes)i(as)f(an)f Fx(")p Fy(-p)s(erturbation)f(of)h(the)h(iden)m (tit)m(y)-8 b(.)47 b(The)34 b(new)h(time-)-118 3716 y(free)k(matrix,)g (as)h(already)e(stated,)j(is)1363 3691 y(\026)1337 3716 y Fx(A)p Fy(\()p Fx(")p Fy(\))d(=)h Fx(A)26 b Fy(+)h Fx(")1955 3691 y Fy(\026)1933 3716 y Fx(Q)2010 3731 y FC(1)2049 3716 y Fy(\()p Fx(")p Fy(\),)40 b(and)f(of)g(its)f(eigen)m(v) -5 b(alues)39 b(w)m(e)h(only)f(kno)m(w)h(that)-118 3836 y(the)i(condition)f(\(4.10\))g(is)g(satis\014ed)i(for)e Fx(")i Fy(=)g(0)f(\(when)h(the)f(eigen)m(v)-5 b(alues)42 b(are)g(precisely)g(those)h(of)e Fx(A)p Fy(\).)72 b(T)-8 b(o)-118 3956 y(see)35 b(ho)m(w)g(to)e(con)m(trol)g(the)i(ful\014llmen) m(t)c(of)j(this)f(condition)g(for)g Fx(")d(>)g Fy(0,)k(let)2654 3930 y(\026)2650 3956 y Fx(\025)2707 3971 y Fr(j)2744 3956 y Fy(\()p Fx(")p Fy(\))f(b)s(e)h(the)h(eigen)m(v)-5 b(alues)34 b(of)3847 3931 y(\026)3821 3956 y Fx(A)p Fy(\()p Fx(")p Fy(\))-118 4077 y(for)k Fx(i)i Fy(=)e(1)p Fx(;)17 b(:)g(:)g(:)f(;)h(n)p Fy(.)63 b(As)40 b(w)m(e)g(are)f(assuming)f(the)i (eigen)m(v)-5 b(alues)39 b(to)g(b)s(e)g(di\013eren)m(t,)i(w)m(e)f(can)g (write)e(their)h(T)-8 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FC(\(1\))442 5528 y Fr(i)537 5502 y Fy(,)g(but)h(as)g(at)f(eac)m(h)i(step)f(w)m(e)h(tak)m(e)f(out)f (a)g(Can)m(tor-lik)m(e)g(set)h(the)g(dep)s(endence)i(on)e Fx(")f Fy(cannot)g(b)s(e)-118 5623 y(made)g(di\013eren)m(tiable.)42 b(This)32 b(is)f(wh)m(y)i(the)f(Lipsc)m(hitz)g(condition)e(replaces)j (the)f(condition)e(on)i(the)g(deriv)-5 b(ativ)m(es)-118 5743 y(of)26 b(the)i(eigen)m(v)-5 b(alues.)41 b(Moreo)m(v)m(er,)30 b(as)d(it)f(will)f(the)i(sho)m(w)h(in)e(a)h(momen)m(t,)g(this)g (assumption)f(allo)m(ws)g(us)i(to)e(con)m(trol)-118 5864 y(the)33 b(size)g(\(in)e(measure\))i(of)f(the)h(set)h(of)e Fx(")p Fy('s)h(that)f(w)m(e)h(tak)m(e)h(out.)1876 6155 y(102)p eop %%Page: 103 107 103 106 bop -118 219 a Fv(The)38 b(measure)f(of)h(the)f(resonan)m(t)h (set)-118 403 y Fy(Assuming)24 b(that)h Fx(")f Fy(b)s(elongs)h(to)f(an) h(in)m(terv)-5 b(al)24 b([0)p Fx(;)17 b(")1689 418 y FC(0)1727 403 y Fy(],)27 b(with)d Fx(")2068 418 y FC(0)2132 403 y Fy(small)f(enough,)k(w)m(e)f(w)m(an)m(t)f(to)g(b)s(ound)g(the)g (measure)-118 524 y(of)37 b(the)h(set)g(of)f(v)-5 b(alues)37 b(of)g Fx(")g Fy(to)s(o)g(close)g(to)g(a)g(resonance.)60 b(It)37 b(will)e(b)s(e)j(sho)m(wn)h(that)e(the)h(exp)s(onen)m(tially)e (small)-118 644 y(b)s(ounds)j(on)f(the)g(measure)h(of)e(this)h(set)h (whic)m(h)g(app)s(ear)f(in)f(the)h(theorem)g(can)h(b)s(e)f(ac)m(hiev)m (ed)i(from)c(a)i(careful)-118 764 y(analysis)32 b(of)g(the)h(Diophan)m (tine)e(condition)g(\(4.11\).)28 885 y(On)44 b(one)f(hand,)j(w)m(e)f (can)e(use)i(the)e(Diophan)m(tine)f(condition)g(\(4.11\))g(to)h (describ)s(e)h(the)g(set)g(of)f(resonan)m(t)-118 1005 y(eigen)m(v)-5 b(alues)391 979 y(\026)386 1005 y Fx(\025)p Fy(.)44 b(Indeed,)34 b(the)f(set)g(of)f(resonan)m(t)i Fx(\025)27 b Fy(=)h(\()p Fx(\025)1966 1020 y FC(1)2005 1005 y Fx(;)17 b(:)g(:)g(:)f(;)h(\025)2281 1020 y Fr(n)2328 1005 y Fy(\))32 b(is)g(giv)m(en)h(b)m(y)g(the)g(union)1057 1188 y Fq([)1026 1404 y Ft(j)p Fl(k)p Ft(j6)p FC(=0)1215 1142 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b(The)37 b(w)m(a)m(y)h(used)f(in)f([52)o(])h (to)f(o)m(v)m(ercome)h(this)-118 2089 y(problem)31 b(is)h(to)h(use,)g (at)f(the)h Fx(r)s Fy(-th)f(step,)i(the)f(Diophan)m(tine)e(condition) 1058 2326 y Fs(j)p Fx(\025)1143 2341 y Fr(j)1201 2326 y Fs(\000)23 b Fx(\025)1358 2341 y Fr(l)1406 2326 y Fs(\000)g Fx(i)p Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(ij)26 b Fx(>)2026 2259 y(c)p 1952 2304 V 1952 2395 a Fs(j)p Fv(k)p Fs(j)2067 2366 y Fr(\015)2103 2374 y Fj(r)2152 2326 y Fx(e)2197 2285 y Ft(\000)p Fr(\027)2287 2293 y Fj(r)2322 2285 y Ft(j)p Fl(k)p Ft(j)2435 2326 y Fy(=)i Fx(R)q Fy(\()p Fv(k)p Fx(;)17 b(r)s Fy(\))-118 2602 y(where)40 b(\()p Fx(\015)259 2617 y Fr(r)296 2602 y Fy(\))334 2617 y Fr(r)411 2602 y Fy(is)e(a)g(geometrically)e(increasing)i(sequence)j(\(tak)m(en)f Fx(\015)2440 2617 y Fr(r)2515 2602 y Fy(=)e Fx(\015)2680 2617 y FC(0)2719 2602 y Fx(z)2768 2566 y Fr(r)2807 2602 y Fy(,)i(with)e(1)g Fx(<)g(z)43 b(<)37 b Fy(2\))i(and)f Fx(\027)3873 2617 y Fr(r)3950 2602 y Fy(is)-118 2723 y(precisely)1401 2861 y Fx(\027)1449 2876 y Fr(r)1515 2861 y Fy(=)1750 2793 y Fx(\027)1798 2808 y FC(0)p 1629 2838 331 4 v 1629 2929 a Fy(\()p Fx(r)24 b Fy(+)e(1\))1920 2900 y FC(2)1970 2861 y Fx(;)211 b(\027)2256 2876 y FC(0)2324 2861 y Fy(=)27 b Fx(c:)-118 3085 y Fy(A)m(t)h(eac)m(h)i(step)f(of)e (the)i(inductiv)m(e)f(pro)s(cess)i(the)e(measure)h(of)f(the)g(resonan)m (t)h(set)g(for)f(the)h(eigen)m(v)-5 b(alues)28 b Fx(\025)g Fy(is)g(giv)m(en)-118 3206 y(b)m(y)33 b(the)g(sum)1680 3343 y Fq(X)1679 3566 y Fd(k)p Fi(2)p Fc(Z)1796 3552 y Fj(d)1686 3626 y Fl(k)p Ft(6)p FC(=0)1843 3438 y Fy(2)p Fx(R)q Fy(\()p Fv(k)p Fx(;)17 b(r)s Fy(\))p Fx(;)-118 3826 y Fy(and)33 b(the)g(total)e(measure)i(is)1516 3951 y Fq(X)1526 4161 y Fr(r)r Ft(\025)p FC(0)1706 3951 y Fq(X)1704 4174 y Fd(k)p Fi(2)p Fc(Z)1821 4160 y Fj(d)1711 4234 y Fl(k)p Ft(6)p FC(=0)1868 4046 y Fy(2)2000 3978 y Fx(c)p 1927 4023 190 4 v 1927 4114 a Fs(j)p Fv(k)p Fs(j)2042 4085 y Fr(\015)2078 4093 y Fj(r)2126 4046 y Fx(e)2171 4005 y Ft(\000)p Fr(\027)2261 4013 y Fj(r)2296 4005 y Ft(j)p Fl(k)p Ft(j)3766 4046 y Fy(\(4.15\))-118 4434 y(whic)m(h)g(implies)d(that)i(the)h(con)m(v)m(ergence)j(of)c(the)h (ab)s(o)m(v)m(e)g(series)g(m)m(ust)g(b)s(e)g(examined.)28 4554 y(Let)27 b(us)h(no)m(w)g(explain)e(where)i(the)f(exp)s(onen)m (tially)f(small)f(c)m(haracter)j(of)e(set)i(of)f(resonan)m(t)g(v)-5 b(alues)27 b(of)g Fx(")f Fy(comes)-118 4674 y(from.)42 b(As)31 b(already)g(stated,)h(the)f(eigen)m(v)-5 b(alues)32 b(\(and)f(hence)h(its)f(di\013erences\))h(of)e(the)i(matrix)d Fx(A)j Fy(mo)m(v)m(e)f(at)g(eac)m(h)-118 4795 y(step)e(of)f(the)h (inductiv)m(e)g(pro)s(cess)g(b)m(y)h(an)e(amoun)m(t)g(of)g(order)h Fx(")f Fy(\(this)g(comes)g(from)g(the)h(fact)f(that)g(the)h(a)m(v)m (eraging)-118 4915 y(pro)s(cedure)38 b(do)s(es)f(not)f(c)m(hange)i (previously)f(computed)g(terms,)h(but)f(only)f(mo)s(di\014es)g(those)h (of)g(order)g(greater)-118 5036 y(than)d(the)h(step\).)50 b(Let)34 b(us)h(call)e Fx(I)1096 5051 y Fr(j)t(l)1154 5036 y Fy(\()p Fx(")p Fy(\))h(the)h(in)m(terv)-5 b(al)33 b(\(with)h(diameter)f Fs(O)s Fy(\()p Fx(")p Fy(\)\))h(where)i(the)f (di\013erence)g(b)s(et)m(w)m(een)-118 5156 y(the)29 b(eigen)m(v)-5 b(alues)29 b Fx(\025)604 5171 y Fr(j)670 5156 y Fy(and)g Fx(\025)913 5171 y Fr(l)968 5156 y Fy(mo)m(v)m(es.)43 b(This)29 b(implies)e(that)i(if)f(the)h(di\013erence)h(satis\014es)g(a) f(condition)e(lik)m(e)h(\(4.11\))-118 5276 y(then)j(the)g(v)-5 b(alues)31 b Fs(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)29 b Fy(are)i(outside)g Fx(I)1371 5291 y Fr(j)t(l)1429 5276 y Fy(\()p Fx(")p Fy(\))f(if)g Fs(j)p Fv(k)p Fs(j)d Fx(<)h(N)10 b Fy(\()p Fx(")p Fy(\),)31 b(for)f(a)g(suitable)g(v)-5 b(alue)30 b Fx(N)10 b Fy(\()p Fx(")p Fy(\).)43 b(Therefore,)32 b(in)e(the)-118 5397 y(sum)g(\(4.15\))g(it)g(is)g(enough)h(to)f(start)h (with)f Fs(j)p Fv(k)p Fs(j)d(\025)i Fx(N)10 b Fy(\()p Fx(")p Fy(\).)42 b(This)31 b(in)f(turn)h(implies)d(that)i(if)g(w)m(e)i (are)e(able)g(to)g(pro)m(v)m(e)-118 5517 y(the)41 b(exp)s(onen)m(tial)g (deca)m(y)h(of)f Fx(R)q Fy(\()p Fv(k)p Fx(;)17 b(r)s Fy(\))40 b(with)h Fv(k)p Fy(,)i(then)f(w)m(e)g(will)d(obtain)h (something)g(whic)m(h)h(is)g(exp)s(onen)m(tially)-118 5637 y(small)30 b(with)i Fx(")405 5652 y FC(0)444 5637 y Fy(.)28 5758 y(T)-8 b(o)33 b(b)s(e)h(able)e(to)h(pro)m(v)m(e)i(the)e (latter,)g(it)f(is)h(necessary)i(to)e(resort)g(to)g(v)-5 b(arying)33 b Fx(\027)39 b Fy(and)34 b Fx(\015)k Fy(at)33 b(eac)m(h)h(step)g(in)e(the)-118 5878 y(constan)m(ts)k(of)d(the)i (Diophan)m(tine)e(condition)g(\(4.11\).)47 b(Indeed,)37 b(since)d(the)h(v)-5 b(alue)34 b(exp)q(\()p Fs(\000)p Fx(\027)6 b Fs(j)p Fv(k)p Fs(j)p Fy(\))34 b(will)e(app)s(ear)i(in)1876 6155 y(103)p eop %%Page: 104 108 104 107 bop -118 219 a Fy(the)31 b(denominators)e(of)h(the)h(F)-8 b(ourier)30 b(expansions)h(for)f(solving)f(the)i(homological)c (equation)j(\(4.10\),)g(the)h(v)-5 b(alue)-118 339 y(exp)q(\()p Fx(\027)6 b Fs(j)p Fv(k)p Fs(j)p Fy(\))37 b(will)d(b)s(e)k(m)m (ultiplying)33 b(the)38 b(co)s(e\016cien)m(ts)g(of)f(these)h(series,)h (therefore)e(reducing)g(the)h(width)f(of)f(the)-118 459 y(analyticit)m(y)31 b(strip.)43 b(If)32 b(the)h(v)-5 b(alue)32 b(of)g Fx(\027)38 b Fy(w)m(as)c(k)m(ept)f(\014xed,)h(the)f (functions)f(w)m(ould)h(b)s(e)f(no)g(longer)g(analytic)f(after)-118 580 y(some)26 b(steps)h(and)f(the)g(pro)s(cess)i(w)m(ould)d(ha)m(v)m(e) j(to)d(b)s(e)h(stopp)s(ed.)43 b(As)26 b(it)f(is)g(classical)g(in)g(KAM) h(setup,)j(a)c(v)-5 b(arying)26 b Fx(\027)3978 595 y Fr(r)-118 700 y Fy(at)i(eac)m(h)h(step)g(is)f(c)m(hosen,)j(in)c(suc)m (h)j(a)e(w)m(a)m(y)i(that)e(the)g(reduction)h(on)f(the)g(analyticit)m (y)f(strip)h(b)s(ecomes)h(b)s(ounded.)28 820 y(Whic)m(h)34 b(is)f(the)h(reason)g(for)f(c)m(hanging)g(also)f(at)h(eac)m(h)i(step)f (the)g(exp)s(onen)m(t)h Fx(\015)j Fy(in)33 b(condition)f(\(4.11\))h(?) 46 b(Note)-118 941 y(that)31 b(with)h(the)g(c)m(hoice)g(of)f Fx(\027)928 956 y Fr(r)966 941 y Fy(,)h(the)h(exp)s(onen)m(tial)e(exp)q (\()p Fs(\000)p Fx(\027)2027 956 y Fr(r)2065 941 y Fs(j)p Fv(k)p Fs(j)p Fy(\))g(go)s(es)h(to)g(one)g(as)g Fx(r)e Fs(!)d(1)p Fy(.)43 b(Th)m(us)33 b(the)f(addition)-118 1061 y(of)37 b(some)g(factor)g(in)g(fron)m(t)g(of)g(the)h(exp)s(onen)m (tial)f(is)g(needed)i(to)e(ensure)i(that)e(the)h(sum)f(with)g(resp)s (ect)i(to)e Fx(r)j Fy(is)-118 1182 y(still)31 b(exp)s(onen)m(tially)i (small.)45 b(In)35 b([52)o(])f(the)h(v)-5 b(alue)33 b Fx(\015)1756 1197 y Fr(r)1823 1182 y Fy(=)d Fx(\015)1980 1197 y FC(0)2019 1182 y Fx(z)2068 1145 y Fr(r)2141 1182 y Fy(is)j(used)i(\(although)e(it)g(is)g(not)h(necessary)i(to)e(tak)m(e) -118 1302 y(precisely)f(this)f(sequence\).)46 b(The)34 b(v)-5 b(alue)32 b(of)g Fx(z)37 b Fy(m)m(ust)c(b)s(e)g(tak)m(en)g(b)s (et)m(w)m(een)i(1)d(and)h(2)f(to)g(ensure)i(con)m(v)m(ergence.)28 1422 y(The)g(pro)s(of)d(of)i(theorem)f(4.1.4)g(that)g(w)m(e)i(ha)m(v)m (e)g(just)f(sk)m(etc)m(hed)i(is)d(found)h(in)f(the)h(pap)s(er)f([52].) -118 1651 y Fv(Remark)37 b(4.1.7)49 b FA(In)36 b(this)g(se)-5 b(ction)36 b(we)g(have)g(always)f(assume)-5 b(d)36 b(that)h(the)f(dep) -5 b(endenc)g(e)35 b(on)h(the)g(angles)g Fx(\022)j FA(was)-118 1771 y(analytic)49 b(in)f(a)h(c)-5 b(omplex)48 b(strip.)87 b(Ther)-5 b(e)49 b(ar)-5 b(e,)52 b(however,)f(some)d(r)-5 b(esults)50 b(assuming)e(only)h(some)f(de)-5 b(gr)g(e)g(e)48 b(of)-118 1891 y(smo)-5 b(othness.)73 b(This)44 b(kind)g(of)h(r)-5 b(esults)45 b(ar)-5 b(e)44 b(usual)h(in)g(KAM)g(the)-5 b(ory,)47 b(se)-5 b(e)45 b(for)f(instanc)-5 b(e)44 b([10].)74 b(F)-7 b(or)44 b(\014nitely)-118 2012 y(di\013er)-5 b(entiable)33 b(r)-5 b(esults)36 b(on)e(r)-5 b(e)g(ducibility)35 b(se)-5 b(e)34 b([8])h(and)f(r)-5 b(efer)g(enc)g(es)34 b(ther)-5 b(ein.)-118 2240 y Fv(Remark)37 b(4.1.8)49 b FA(In)f(some)g(c)-5 b(ases)48 b(one)h(do)-5 b(es)48 b(not)h(want)f(to)h(r)-5 b(e)g(duc)g(e)49 b(c)-5 b(ompletely)48 b(a)h(line)-5 b(ar)48 b(quasi-p)-5 b(erio)g(dic)-118 2360 y(system,)40 b(but)g(only)g(up)f(to)h(a)f(smal)5 b(l)39 b(r)-5 b(emainder.)57 b(Then)39 b(we)g(c)-5 b(an)39 b(sp)-5 b(e)g(ak)39 b(of)g(e\013e)-5 b(ctive)38 b(r)-5 b(e)g(ducibility)40 b(or)f(almost)-118 2481 y(r)-5 b(e)g(ducibility)35 b(\(se)-5 b(e)34 b([50])g(and)h([28)o (]\).)-118 2770 y Fg(4.1.2)136 b(Some)67 b(results)h(on)f (non-reducibilit)l(y)-11 b(.)127 b(Non-uniform)67 b(h)l(yp)t(erb)t (olicit)l(y)293 2919 y(and)44 b(ergo)t(dicit)l(y)-118 3104 y Fy(In)35 b(this)g(subsection,)i(w)m(e)g(shall)d(review)i(some)f (results)g(on)g(non-reducibilit)m(y)f(whic)m(h)i(apply)e(to)h (non-compact)-118 3224 y(groups,)f(mainly)e Fx(S)6 b(L)p Fy(\()p Fx(n;)17 b Fw(R)5 b Fy(\),)40 b(the)35 b(subgroup)f(of)f Fx(GL)p Fy(\()p Fx(n;)17 b Fw(R)6 b Fy(\))39 b(whose)c(elemen)m(ts)f (ha)m(v)m(e)i(determinan)m(t)d(one.)48 b(Note)-118 3345 y(that,)32 b(if)g Fx(n)27 b Fy(=)h(2,)k(then)h(w)m(e)h(are,)f(after)f (a)g(c)m(hange)h(of)f(v)-5 b(ariables,)32 b(in)f(the)i(situation)e(of)h (c)m(hapter)i(3.)43 b(Despite)32 b(this)-118 3465 y(fact)k(w)m(e)i 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(enough)h(to)f(guaran)m(tee)g(uniform)-118 4308 y(h)m(yp)s(erb)s (olicit)m(y)32 b(but)h(only)g(that)g(the)g(stable)g(and)g(unstable)g (bundles)h(are)f(measurable)f(\(see)i([44])f(for)f(the)i(case)-118 4428 y(of)f(Sc)m(hr\177)-49 b(odinger)33 b(equation\),)g(since)h(they)g (ma)m(y)f(fail)d(to)j(b)s(e)h(con)m(tin)m(uous.)46 b(This)33 b(situation)e(is)i(referred)h(as)f FA(non-)-118 4548 y(uniform)38 b(hyp)-5 b(erb)g(olicity)46 b Fy(and)37 b(implies)d(that)j(the)h(system)g(is)e(non-reducible)h(\(pro)m(vided)g (that)g(the)g(system)h(is)-118 4669 y(smo)s(oth,)31 b(otherwise)i(see)h (the)e(example)g(in)g([11)o(]\).)44 b(Note,)32 b(\014rst,)h(that)f (this)g(prop)s(ert)m(y)h(do)s(es)g(not)f(o)s(ccur)h(in)e(non-)-118 4789 y(compact)37 b(groups,)j(since)e(p)s(ositiv)m(e)f(Ly)m(apuno)m(v)j (exp)s(onen)m(ts)g(imply)c(that)h(there)i(exist)f(un)m(b)s(ounded)h (orbits)e(in)-118 4910 y(this)32 b(group,)h(whic)m(h)g(is)f(clearly)f (incompatible)f(with)i(the)h(compactness)h(of)e(the)h(group.)28 5030 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Ft(0)2282 5752 y Fy(=)28 b Fx(!)t(;)1876 6155 y Fy(104)p eop %%Page: 105 109 105 108 bop -118 219 a Fy(where)29 b(\()p Fx(X)r(;)17 b(\022)s Fy(\))28 b Fs(2)g Fx(G)13 b Fs(\002)g Fw(T)775 182 y Fr(d)818 219 y Fy(.)42 b(Let)28 b Fx(X)1138 234 y Fr(r)1176 219 y Fy(\()p Fx(\022)1259 234 y FC(2)1299 219 y Fx(;)17 b(:)g(:)g(:)33 b(;)17 b(\022)1580 234 y Fr(d)1620 219 y Fy(\))28 b(b)s(e)g(the)g(fundamen)m(tal)f(matrix)g(for) g(this)h(system)g(ev)-5 b(aluated)28 b(at)-118 339 y(time)22 b(2)p Fx(\031)t(r)s(=!)360 354 y FC(1)421 339 y Fy(and)i(with)f (initial)d(phase)k(for)f(the)g(angle)g Fx(\022)j Fy(b)s(eing)d(\(0)p Fx(;)17 b(\022)2395 354 y FC(2)2434 339 y Fx(;)g(:)g(:)g(:)33 b(;)17 b(\022)2715 354 y Fr(d)2755 339 y Fy(\).)41 b(Let)23 b Fx(\036)28 b Fy(=)f(\()p Fx(\022)3298 354 y FC(2)3338 339 y Fx(;)17 b(:)g(:)g(:)32 b(;)17 b(\022)3618 354 y Fr(d)3659 339 y Fy(\))28 b Fs(2)g Fw(T)3882 303 y Fr(d)p Ft(\000)p FC(1)-118 459 y Fy(and)33 b Fx(\027)h Fy(=)27 b(2)p Fx(\031)t Fy(\()p Fx(!)464 474 y 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y(discrete)35 b(\015o)m(ws.)50 b(Hence)36 b(it)d(can)i(not)f(b)s(e)h(applied)e(to)h(linear)f(sk)m (ew-pro)s(duct)j(\015o)m(ws)g(coming)c(from)i(di\013eren)m(tial)-118 1913 y(equations.)44 b(The)33 b(pro)s(of)f(is)g(a)g(simple)f (application)f(of)i(the)h(follo)m(wing)d(theorem)-118 2142 y Fv(Theorem)37 b(4.1.10)h(\([39]\))48 b FA(L)-5 b(et)932 2362 y Fx(D)1016 2321 y Fr(d)1013 2386 y(h)1085 2362 y Fy(=)1189 2281 y Fq(\010)1247 2362 y Fy(\()p Fx(z)1330 2377 y FC(1)1370 2362 y Fx(;)17 b(:)g(:)g(:)32 b(;)17 b(z)1650 2377 y Fr(d)1691 2362 y Fy(\))27 b Fs(2)h Fw(C)1916 2321 y Fr(d)1963 2362 y Fy(;)44 b Fs(j)p Fx(z)2107 2377 y Fr(i)2135 2362 y Fs(j)28 b(\024)g Fx(r)m(;)44 b(i)28 b Fy(=)g(1)p Fx(;)17 b(:)g(:)g(:)32 b(;)17 b(d)2908 2281 y Fq(\011)-118 2582 y FA(b)-5 b(e)34 b(a)h(p)-5 b(olydisc)34 b(ar)-5 b(ound)35 b Fy(0)27 b Fs(2)h Fw(C)1016 2546 y Fr(d)1098 2582 y FA(for)34 b(some)g(p)-5 b(ositive)35 b Fx(h)27 b(>)h Fy(0)35 b FA(and)940 2802 y Fx(T)1011 2761 y Fr(d)997 2826 y(h)1079 2802 y Fy(=)1182 2721 y Fq(\010)1241 2802 y Fy(\()p Fx(z)1324 2817 y FC(1)1363 2802 y Fx(;)17 b(:)g(:)g(:)33 b(;)17 b(z)1644 2817 y Fr(d)1684 2802 y Fy(\))28 b Fs(2)g Fw(C)1910 2761 y Fr(d)1956 2802 y Fy(;)45 b Fs(j)p Fx(z)2101 2817 y Fr(i)2129 2802 y Fs(j)27 b Fy(=)h Fx(r)m(;)44 b(i)28 b Fy(=)g(1)p Fx(;)17 b(:)g(:)g(:)32 b(;)17 b(d)2900 2721 y Fq(\011)-118 3022 y FA(the)38 b(c)-5 b(orr)g(esp)g(onding)37 b(p)-5 b(olycir)g(cle)38 b(\(which)f(is)h(di\013e)-5 b(omorphic)37 b(to)i Fw(T)2319 2986 y Fr(d)2363 3022 y FA(\).)55 b(Assume)38 b(that)h(the)g(map)e Fx(f)45 b Fy(:)35 b Fx(D)3680 2986 y Fr(d)3677 3048 y(h)3756 3022 y Fs(!)f Fx(D)3974 2986 y Fr(d)3971 3048 y(h)-118 3142 y FA(satis\014es)g(the)h(fol)5 b(lowing)33 b(c)-5 b(onditions)-33 3345 y(\(i\))49 b Fx(f)c FA(is)35 b(holomorphic)e(in)i (a)f(neighb)-5 b(ourho)g(o)g(d)34 b(of)g Fx(D)1912 3309 y Fr(d)1909 3371 y(h)1954 3345 y FA(.)-62 3549 y(\(ii\))48 b Fx(f)11 b Fy(\()p Fx(D)307 3513 y Fr(d)304 3575 y(h)348 3549 y Fy(\))28 b Fs(\032)g Fx(D)603 3513 y Fr(d)600 3575 y(h)680 3549 y FA(and)34 b Fx(f)11 b Fy(\()p Fx(T)1037 3513 y Fr(d)1023 3575 y(h)1077 3549 y Fy(\))27 b Fs(\032)i Fx(T)1319 3513 y Fr(d)1305 3575 y(h)1359 3549 y FA(.)-92 3752 y(\(iii\))48 b Fx(f)11 b Fy(\(0\))27 b(=)g(0)p FA(.)-77 3956 y(\(iv\))48 b Fx(f)d FA(le)-5 b(aves)34 b(invariant)g(the)h(Haar)g (me)-5 b(asur)g(e)35 b(de\014ne)-5 b(d)33 b(on)i Fx(T)2247 3920 y Fr(d)2233 3982 y(h)2287 3956 y FA(.)-118 4159 y(Assume,)27 b(mor)-5 b(e)g(over,)26 b(that)f Fx(A)h FA(is)e(a)h(holomorphic)f(matrix)h(function)f(fr)-5 b(om)25 b(a)g(neighb)-5 b(ourho)g(o)g(d)24 b(of)g Fx(D)3495 4123 y Fr(d)3492 4185 y(h)3562 4159 y FA(to)i Fx(L)p Fy(\()p Fx(n;)17 b Fw(C)j Fy(\))p FA(,)-118 4279 y(and)34 b(let)h Fs(k)22 b(\001)g(k)35 b FA(b)-5 b(e)34 b(some)g(norm.)44 b(Consider)34 b(the)h(skew-pr)-5 b(o)g(duct)34 b(semi-\015ow)g(given)g (on)g Fx(L)p Fy(\()p Fx(n;)17 b Fw(C)k Fy(\))28 b Fs(\002)22 b Fx(T)3572 4243 y Fr(d)3558 4305 y(h)3647 4279 y FA(by)1058 4499 y Fy(\()p Fx(Z)1163 4514 y Fr(r)r FC(+1)1291 4499 y Fx(;)17 b(z)1380 4514 y Fr(r)r FC(+1)1508 4499 y Fy(\))28 b(=)f(\()p Fx(A)p Fy(\()p Fx(z)1871 4514 y Fr(r)1910 4499 y Fy(\))p Fx(Z)2015 4514 y Fr(r)2052 4499 y Fx(;)17 b(f)11 b Fy(\()p Fx(z)2238 4514 y Fr(r)2276 4499 y Fy(\)\))16 b Fx(;)216 b(r)31 b Fs(\025)d Fy(0)-118 4720 y FA(and)34 b(the)h(c)-5 b(orr)g(esp)g(onding)33 b(upp)-5 b(er)35 b(Lyapunov)g(exp)-5 b(onent)557 4982 y Fx(\014)612 4997 y FC(+)672 4982 y Fy(\()p Fx(A;)17 b(f)11 b Fy(\))27 b(=)g(lim)17 b(inf)1074 5042 y Fr(r)r Ft(!)p FC(+)p Ft(1)1341 4847 y Fq(Z)1396 5072 y Fr(z)s Ft(2)p Fr(T)1530 5049 y Fj(d)1520 5096 y(h)1587 4982 y Fy(log)1729 4898 y Fq(\015)1729 4958 y(\015)1785 4982 y Fx(A)1875 4902 y Fq(\000)1920 4982 y Fx(f)1979 4941 y Fr(r)r Ft(\000)p FC(1)2107 4982 y Fy(\()p Fx(z)t Fy(\))2232 4902 y Fq(\001)2300 4982 y Fs(\001)22 b Fx(:)17 b(:)g(:)22 b Fs(\001)g Fx(A)p Fy(\()p Fx(f)11 b Fy(\()p Fx(z)t Fy(\)\))22 b Fs(\001)g Fx(A)p Fy(\()p Fx(z)t Fy(\))3140 4898 y Fq(\015)3140 4958 y(\015)3213 4982 y Fx(dz)t(:)-118 5273 y FA(Then)1470 5394 y Fx(\014)1525 5409 y FC(+)1584 5394 y Fy(\()p Fx(A;)17 b(f)11 b Fy(\))27 b Fs(\025)h Fy(log)17 b(\()p Fx(A)p Fy(\(0\)\))f Fx(:)28 5622 y Fy(More)41 b(results)g(on)f(non-uniformly)e (h)m(yp)s(erb)s(olic)i(co)s(cyles)h(in)f Fx(S)6 b(L)p Fy(\(2)p Fx(;)17 b Fw(R)5 b Fy(\))46 b(w)m(ere)c(obtained)e(b)m(y)h (L.S.)g(Y)-8 b(oung)-118 5742 y(\([86)o(])34 b(and)f([87]\).)46 b(The)34 b(follo)m(wing)c(theorems)k(imply)d(this)i(non-uniform)f(h)m (yp)s(erb)s(olicit)m(y)g(in)h(some)g(cases.)47 b(The)-118 5863 y(\014rst)33 b(one)g(refers)g(to)f(the)h(p)s(ositiv)m(eness)h(on)e (the)h(Ly)m(apuno)m(v)h(exp)s(onen)m(t:)1876 6155 y(106)p eop %%Page: 107 111 107 110 bop -118 219 a Fv(Theorem)37 b(4.1.11)h(\([86]\))48 b FA(L)-5 b(et)34 b Fx(\013)28 b Fs(2)g Fy([0)p Fx(;)17 b Fy(1])33 b FA(b)-5 b(e)34 b(a)f(Brjuno)g(numb)-5 b(er,)33 b(which)g(me)-5 b(ans)33 b(that)h(its)f(c)-5 b(ontinue)g(d)33 b(fr)-5 b(ac-)-118 339 y(tion)35 b(exp)-5 b(ansion)33 b Fy(\()p Fx(q)613 354 y Fr(n)660 339 y Fy(\))698 354 y Fr(n)p Ft(\025)p FC(1)870 339 y FA(satis\014es)h(that)1582 502 y Fq(X)1752 529 y Fy(log)17 b Fx(q)1938 544 y Fr(n)p FC(+1)p 1752 574 323 4 v 1869 665 a Fx(q)1912 680 y Fr(n)2113 597 y Fx(<)27 b Fs(1)-118 859 y FA(and)46 b(let)h Fy(\()p Fx(A)343 874 y Fr(s)380 859 y Fy(\))418 874 y Fr(s)p Ft(2)p FC([0)p Fr(;)p FC(1])679 859 y FA(b)-5 b(e)46 b(a)h(one)g(p)-5 b(ar)g(ameter)46 b(family)h(of)f(maps)h(fr)-5 b(om)46 b Fw(T)2581 823 y FC(1)2671 859 y FA(to)h Fx(S)6 b(L)p Fy(\(2)p Fx(;)17 b Fw(R)5 b Fy(\))p FA(.)87 b(L)-5 b(et,)50 b(for)d Fx(\022)53 b Fs(2)e Fw(T)3943 823 y FC(1)3986 859 y FA(,)-92 954 y Fy(\026)-118 979 y Fx(A)-45 994 y Fr(s)-8 979 y Fy(\()p Fx(\022)s Fy(\))30 b(:)g Fw(P)262 943 y FC(1)333 979 y Fs(!)g Fw(P)522 943 y FC(1)599 979 y FA(b)-5 b(e)36 b(the)g(pr)-5 b(oje)g(ctivization)35 b(of)h Fx(A)1752 994 y Fr(s)1788 979 y Fy(\()p Fx(\022)s Fy(\))30 b Fs(2)h Fx(S)6 b(L)p Fy(\(2)p Fx(;)17 b Fw(R)t Fy(\))42 b FA(and)36 b Fx(\014)6 b Fy(\()p Fx(\022)s(;)17 b(s)p Fy(\))29 b(=)3074 954 y(\026)3048 979 y Fx(A)3121 994 y Fr(s)3158 979 y Fy(\()p Fx(\022)s Fy(\))3282 943 y Ft(\000)p FC(1)3376 979 y Fy(\()p Fx(\031)t(=)p Fy(2\))p FA(.)48 b(Assume)-118 1100 y(that)35 b(the)g(fol)5 b(lowing)33 b(tr)-5 b(ansversality)35 b(c)-5 b(onditions)34 b(hold)27 1292 y Fs(\017)49 b FA(The)34 b(map)g Fx(\022)d Fs(7!)c Fy(\()p Fx(\022)s(;)17 b(\014)6 b Fy(\()p Fx(\022)s(;)17 b(s)p Fy(\)\))34 b FA(is)h(tr)-5 b(ansversal)34 b(to)h Fs(f)p Fx(\022)c Fy(=)c(0)p Fs(g)p FA(.)27 1492 y Fs(\017)49 b Fy(\()p Fx(@)5 b(\014)h(=@)f(s)q Fy(\))p Fx(=)p Fy(\()p Fx(@)g(\014)h(=@)f(\022)t Fy(\))35 b FA(takes)f(distinct)h(values)f(at) h(di\013er)-5 b(ent)35 b(p)-5 b(oints)34 b(in)h Fx(\014)6 b Fy(\()p Fs(\001)p Fx(;)17 b(s)p Fy(\))2954 1456 y Ft(\000)p FC(1)3047 1492 y Fy(\(0\))p FA(.)-118 1684 y(F)-7 b(or)34 b Fx(")110 1699 y FC(0)177 1684 y Fx(>)27 b Fy(0)p FA(,)35 b(let)g Fy(\001\()p Fx(\025)p Fy(\))g FA(b)-5 b(e)34 b(the)h(set)g(of)g(those)f Fx(s)28 b Fs(2)g Fy([0)p Fx(;)17 b Fy(1])34 b FA(such)h(that)g(the)g(c)-5 b(o)g(cycle)34 b(on)h Fw(T)3089 1648 y FC(1)3154 1684 y Fs(\002)22 b Fx(S)6 b(L)p Fy(\(2)p Fx(;)17 b Fw(T)p Fy(\))38 b FA(given)c(by)1443 1811 y Fq(\000)1489 1892 y Fy(2)p Fx(\031)t(\013)q(;)17 b FA(diag)n Fy(\()p Fx(\025;)g(\025)2073 1851 y Ft(\000)p FC(1)2167 1892 y Fy(\))22 b Fs(\016)g Fx(A)2372 1907 y Fr(s)2409 1811 y Fq(\001)-118 2099 y FA(has)34 b(an)h(exp)-5 b(onent)34 b(which)g(is)g(strictly)i(gr)-5 b(e)g(ater)35 b(that)g Fy(\(1)22 b Fs(\000)g Fx(")2085 2114 y FC(0)2124 2099 y Fy(\))17 b(log)g Fx(\025)p FA(.)28 2219 y(Then)34 b(me)-5 b(as)491 2234 y Fr(Leb)607 2219 y Fy(\(\001\()p Fx(\025)p Fy(\)\))27 b Fs(!)g Fy(1)35 b FA(as)g Fx(\025)27 b Fs(!)h(1)p FA(.)28 2433 y Fy(T)-8 b(o)22 b(pro)m(v)m(e)h(non-uniform) d(h)m(yp)s(erb)s(olicit)m(y)-8 b(,)23 b(one)g(m)m(ust)f(sho)m(w)h(that) e(there)i(is)e(some)h(kind)g(of)g(elliptic)d(b)s(eha)m(viour.)-118 2554 y(In)26 b(Y)-8 b(oung's)26 b(pap)s(er)g(the)h(follo)m(wing)c (notion)i(is)g(used.)43 b(W)-8 b(e)26 b(shall)e(sa)m(y)j(that)f(a)g(co) s(cyle)g(\(2)p Fx(\031)t(\013)q(;)17 b(A)p Fy(\))25 b(of)g Fw(T)3506 2517 y FC(1)3557 2554 y Fs(\002)9 b Fx(S)d(L)p Fy(\(2)p Fx(;)17 b Fw(R)5 b Fy(\))-118 2674 y(has)31 b FA(el)5 b(liptic)33 b(p)-5 b(erio)g(dic)32 b(orbits)39 b Fy(if)29 b Fx(\013)j Fy(is)e(rational)e(of)j(the)g(form)e Fx(p=q)34 b Fy(and)d(there)h(is)e(an)h(in)m(terv)-5 b(al)29 b Fx(J)37 b Fs(\032)28 b Fy([0)p Fx(;)17 b Fy(2)p Fx(\031)t Fy(])30 b(suc)m(h)-118 2794 y(that)i(for)g(ev)m(ery)j Fx(\032)28 b Fs(2)g Fx(J)41 b Fy(there)34 b(is)e(a)g Fx(\022)f Fs(2)d Fw(T)1429 2758 y FC(1)1504 2794 y Fy(for)k(whic)m(h)i (the)f(matrix)1045 3051 y Fx(A)p Fy(\(2)p Fx(\031)1274 2984 y(p)p Fy(\()p Fx(q)25 b Fs(\000)e Fy(1\))p 1274 3028 343 4 v 1421 3119 a Fx(q)1648 3051 y Fy(+)f Fx(\022)s Fy(\))g Fs(\001)g Fx(:)17 b(:)g(:)k Fs(\001)h Fx(A)p Fy(\(2)p Fx(\031)2319 2984 y(p)p 2319 3028 49 4 v 2320 3119 a(q)2400 3051 y Fy(+)g Fx(\022)s Fy(\))g Fs(\001)g Fx(A)p Fy(\()p Fx(\022)s Fy(\))-118 3314 y(has)33 b(eigen)m(v)-5 b(alues)32 b Fx(e)605 3278 y Ft(\006)p FC(2)p Fr(\031)r(i\032)803 3314 y Fy(.)28 3435 y(The)40 b(k)m(ey)h(for)e(pro)m(ving)g(non-uniform) e(h)m(yp)s(erb)s(olicit)m(y)i(is)g(the)h(follo)m(wing)c(theorem,)41 b(whic)m(h,)h(under)e(some)-118 3555 y(h)m(yp)s(othesis)34 b(asserts)g(the)f(existence)h(of)e(co)s(cycles)h(with)f(elliptic)e(p)s (erio)s(dic)h(orbits)-118 3769 y Fv(Theorem)37 b(4.1.12)h(\([86]\))48 b FA(L)-5 b(et)33 b Fy(\()p Fx(A)1268 3784 y Fr(s)1305 3769 y Fy(\))1343 3784 y Fr(s)1412 3769 y FA(satisfy)g(the)g(hyp)-5 b(othesis)32 b(of)g(\(4.1.11\))g(and)g Fx(\013)c Fs(2)g Fy([0)p Fx(;)17 b Fy(1])p FA(.)44 b(Assume)32 b(that,)-118 3889 y(for)37 b(some)f Fy(\()p Fx(\022)s(;)17 b(s)p Fy(\))p FA(,)37 b Fx(\014)6 b Fy(\()p Fx(\022)s(;)17 b(s)p Fy(\))31 b(=)h(0)p FA(.)51 b(L)-5 b(et)37 b Fy(\000\()p Fx(\025)p Fy(\))g FA(b)-5 b(e)37 b(the)g(set)g(of)g Fy(\()p Fx(\013)q(;)17 b(s)p Fy(\))31 b Fs(2)h Fy([0)p Fx(;)17 b Fy(1])23 b Fs(\002)h Fy([0)p Fx(;)17 b Fy(1])37 b FA(for)f(which)h(the)g(c)-5 b(o)g(cycle)36 b(on)-118 4009 y Fw(T)-55 3973 y FC(1)10 4009 y Fs(\002)23 b Fx(S)6 b(L)p Fy(\(2)p Fx(;)17 b Fw(T)p Fy(\))37 b FA(given)d(by)1443 4049 y Fq(\000)1489 4130 y Fy(2)p Fx(\031)t(\013)q(;)17 b FA(diag)n Fy(\()p Fx(\025;)g(\025)2073 4089 y Ft(\000)p FC(1)2167 4130 y Fy(\))22 b Fs(\016)g Fx(A)2372 4145 y Fr(s)2409 4049 y Fq(\001)-118 4316 y FA(has)34 b(el)5 b(liptic)35 b(p)-5 b(erio)g(dic)33 b(orbits.)45 b(Then)34 b(me)-5 b(as)1518 4331 y Fr(Leb)1633 4316 y Fy(\()p 1671 4230 194 4 v(\000\()p Fx(\025)p Fy(\)\))28 b Fs(!)f Fy(1)35 b FA(as)f Fx(\025)28 b Fs(!)f(1)p FA(.)28 4530 y Fy(These)40 b(t)m(w)m(o)e(theorems)g(can)g(b)s(e)g(used)h(in)e (com)m(bination)f(to)i(sho)m(w)h(non-uniform)c(h)m(yp)s(erb)s(olicit)m (y)i(since,)j(if)-118 4650 y Fx(\014)6 b Fy(\()p Fx(\022)s Fy(\))47 b(=)g(0)c(for)h(some)f Fx(\022)s Fy(,)k(the)e(co)s(cycle)f (\(2)p Fx(\031)t(\013)q(;)17 b Fy(diag)n(\()p Fx(\025;)g(\025)2021 4614 y Ft(\000)p FC(1)2115 4650 y Fy(\))22 b Fs(\016)g Fx(A)2320 4665 y Fr(s)2357 4650 y Fy(\))44 b(cannot)g(b)s(e)g (uniformly)e(h)m(yp)s(erb)s(olic)h(if)g Fx(\025)-118 4771 y Fy(b)s(elongs)32 b(to)g(\001\()p Fx(\025)p Fy(\))h(and)f Fx(\025)h Fy(is)f(large)f(enough.)28 4891 y(Another)f(mec)m(hanism)e (for)g(the)h(non-reducibilit)m(y)e(of)i(a)f(certain)h(sk)m(ew-pro)s (duct)h(system)g(is)e(the)i(ergo)s(dicit)m(y)-118 5012 y(of)h(the)h(\015o)m(w)g(on)f Fx(G)20 b Fs(\002)g Fw(T)752 4975 y Fr(d)796 5012 y Fy(,)31 b(since)h(this)f(ergo)s(dicit)m(y)g(is)g (incompatible)d(with)j(reducibilit)m(y)f(\(for)h(examples)g(of)g(this) -118 5132 y(in)i(the)i(compact)f(con)m(text)h(see)g(section)f(4.2.3\).) 48 b(F)-8 b(or)33 b(linear)g(di\013eren)m(tial)f(equations,)j(one)f (has)h(the)f(follo)m(wing)-118 5252 y(theorem,)e(due)i(to)e(M.)h(G.)f (Nerurk)-5 b(ar)-118 5466 y Fv(Theorem)37 b(4.1.13)h(\([66]\))48 b FA(L)-5 b(et)35 b Fx(!)c Fs(2)d Fw(R)1411 5430 y Fr(d)1493 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Fr(d)2241 5881 y Fx(;)17 b Fy(2)p Fx(\031)t(!)t(;)g(G)p Fy(\))33 b FA(in)i Fx(Z)2844 5845 y Ft(1)2837 5906 y Fk(R)2918 5881 y Fy(\()p Fw(T)3019 5845 y Fr(d)3063 5881 y Fx(;)17 b Fy(2)p Fx(\031)t(!)t(;)g(G) p Fy(\))33 b FA(\).)1876 6155 y Fy(107)p eop %%Page: 108 112 108 111 bop -118 219 a Fu(4.2)161 b(Reducibilit)l(y)55 b(in)f(compact)e(groups)-118 438 y Fy(This)23 b(section)g(is)g(dev)m (oted)i(to)e(the)g(results)h(obtained)e(b)m(y)i(R.)f(Krik)m(orian)f(in) g([55],)j([56])e(and)g([54])g(on)g(the)h(reducibil-)-118 558 y(it)m(y)32 b(of)g(linear)f(equations)i(with)f(quasi-p)s(erio)s (dic)e(co)s(e\016cien)m(ts)k(whic)m(h)f(tak)m(e)g(v)-5 b(alues)32 b(on)h(compact)f(groups.)44 b(Our)-118 678 y(atten)m(tion)26 b(will)f(fo)s(cus)i(on)g(the)g(case)h(of)f Fx(S)6 b(O)s Fy(\(3\))25 b(for)i(whic)m(h,)h(together)g(with)e Fx(S)6 b(U)k Fy(\(2\),)28 b(there)g(is)f(a)f(fairly)f(complete)-118 799 y(picture)32 b(of)h(the)g FA(amount)41 b Fy(of)32 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Fs(T)54 b(\032)28 b Fx(g)k Fy(of)c(a)g(maximal)d(torus)k Fx(T)42 b Fs(\032)28 b Fx(G)g Fy(is)h(an)f(Ab)s(elian)f(Lie)-118 459 y(sub-algebra)38 b(whic)m(h)h(is)f(maximal)e(for)i(the)h(inclusion)e(in)h Fx(g)t Fy(.)61 b(Con)m(v)m(ersely)-8 b(,)43 b(ev)m(ery)d(Ab)s(elian)d (Lie)h(sub-algebra)-118 580 y(whic)m(h)f(is)g(maximal,)e Fs(T)61 b(\032)35 b Fx(g)t Fy(,)j(is)e(the)i(Lie)e(algebra)g(of)h(a)f (maximal)e(torus)j Fx(T)49 b Fs(\032)36 b Fx(G)p Fy(.)57 b(W)-8 b(e)37 b(shall)f(call)f FA(maximal)-118 700 y(toric)g (sub-algebr)-5 b(as)39 b Fy(these)34 b(sub-algebras.)28 820 y(There)e(are)e(man)m(y)g(prop)s(erties)h(whic)m(h)f(can)h(help)f (to)g(describ)s(e)h(compact)f(Lie)g(algebras)f(and)i(whic)m(h)f(will)e (b)s(e)-118 941 y(of)k(in)m(terest)h(in)f(the)h(sequel.)45 b(W)-8 b(e)33 b(ha)m(v)m(e)h(already)e(said)g(that)h(for)f(these)i (algebras,)e(the)h(exp)s(onen)m(tial)f(map)g(from)-118 1061 y Fx(g)41 b Fy(to)d Fx(G)h Fy(is)e(surjectiv)m(e.)62 b(Moreo)m(v)m(er,)42 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b Fs(2)i Fx(G)k FA(\(r)-5 b(esp.)53 b Fx(X)41 b Fs(2)33 b Fx(g)t FA(\))k(ther)-5 b(e)38 b(is)f(at)h(le)-5 b(ast)38 b(a)g(maximal)e(torus)i(\(r)-5 b(esp.)53 b(a)38 b(maximal)126 2095 y(toric)d(sub-algebr)-5 b(a\).)-62 2298 y(\(ii\))48 b(If)42 b Fx(T)57 b FA(and)42 b Fx(T)618 2262 y Ft(0)684 2298 y FA(\(r)-5 b(esp.)68 b Fs(T)h FA(and)42 b Fs(T)1393 2262 y Ft(0)1416 2298 y FA(\))h(ar)-5 b(e)42 b(two)h(maximal)f(tori)h(of)f Fx(G)h FA(\(r)-5 b(esp.)69 b(two)42 b(maximal)g(toric)h(sub-)126 2419 y(algebr)-5 b(as)35 b(of)g Fx(g)t FA(\),)g(ther)-5 b(e)36 b(exists)f(an)g(element)g Fx(x)30 b Fs(2)g Fx(G)36 b FA(which)e(c)-5 b(onjugates)35 b Fx(T)50 b FA(and)35 b Fx(T)3191 2382 y Ft(0)3250 2419 y FA(\(r)-5 b(esp.)47 b Fs(T)61 b FA(and)35 b Fs(T)3923 2382 y Ft(0)3946 2419 y FA(\).)126 2539 y(That)f(is,)1052 2659 y Fx(T)1123 2618 y Ft(0)1173 2659 y Fy(=)28 b Fx(x)22 b Fs(\001)g Fx(T)36 b Fs(\001)22 b Fx(x)1602 2618 y Ft(\000)p FC(1)1697 2659 y Fx(;)216 b Fy(\()p Fx(r)s(esp:)199 b Fs(T)2470 2618 y Ft(0)2521 2659 y Fy(=)28 b FA(A)-5 b(d)p Fy(\()p Fx(x)p Fy(\))22 b Fs(\001)g(T)k Fy(\))p Fx(:)-92 2875 y FA(\(iii\))48 b(The)41 b Fy(cen)m(ter)c FA(of)d(a)h(maximal)f (torus)h Fx(T)14 b FA(,)34 b(i.e.)1222 3095 y Fx(C)7 b Fy(\()p Fx(T)14 b Fy(\))27 b(=)h Fs(f)o Fx(x)g Fs(2)h Fx(G)p Fy(;)44 b Fx(xy)31 b Fy(=)d Fx(y)t(x)34 b FA(for)h(al)5 b(l)34 b Fx(y)d Fs(2)d Fx(T)14 b Fs(g)126 3315 y FA(is)34 b Fx(T)14 b FA(.)-77 3519 y(\(iv\))48 b(If)34 b Fx(N)10 b Fy(\()p Fx(T)k Fy(\))35 b FA(is)g(the)g(normalizator)e(of)i(a)g (maximal)e(torus)i Fx(T)14 b FA(,)1356 3739 y Fx(N)c Fy(\()p Fx(T)k Fy(\))28 b(=)f Fs(f)p Fx(x)h Fs(2)g Fx(G)p Fy(;)44 b FA(A)-5 b(d)q Fy(\()p Fx(x)p Fy(\))22 b Fs(\001)g Fx(T)41 b Fs(\032)29 b Fx(T)14 b Fs(g)i Fx(;)126 3959 y FA(then)30 b Fx(T)44 b FA(is)30 b(distinguishe)-5 b(d)30 b(in)g Fx(N)10 b Fy(\()p Fx(T)k Fy(\))30 b FA(and)g(the)h(gr)-5 b(oup)30 b Fx(W)2191 3974 y Fr(T)2274 3959 y Fy(=)d Fx(N)10 b Fy(\()p Fx(T)k Fy(\))p Fx(=T)45 b FA(has)29 b(\014nite)i(c)-5 b(ar)g(dinal.)42 b(A)n(l)5 b(l)30 b(gr)-5 b(oups)126 4079 y Fx(W)218 4094 y Fr(T)308 4079 y FA(of)34 b(this)h(kind)f(ar)-5 b(e)35 b(isomorphic,)e(and)i(this)f(gr)-5 b(oup)35 b(is)g(c)-5 b(al)5 b(le)-5 b(d)34 b(the)42 b Fy(W)-8 b(eyl's)33 b(group)g(of)f Fx(G)p FA(.)28 4307 y Fy(Let)25 b Fx(T)37 b Fy(b)s(e)25 b(a)f(maximal)c(torus)25 b(of)f(a)f(compact)h(semi-simple)d(connected) 26 b(group)e Fx(G)g Fy(and)h Fs(T)49 b Fy(its)24 b(corresp)s(onding) -118 4428 y(maximal)38 b(toric)i(Lie)h(sub-algebra.)68 b(F)-8 b(or)41 b(ev)m(ery)i(elemen)m(t)e Fx(H)49 b Fs(2)43 b(T)26 b Fy(,)43 b(ad\()p Fx(H)8 b Fy(\))42 b Fs(2)h Fx(g)t(l)r Fy(\()p Fx(g)t Fy(\))c(is)i(an)g(an)m(tisymmetric)-118 4548 y(endomorphism)27 b(of)h Fx(g)t(l)r Fy(\()p Fx(g)t Fy(\).)40 b(On)29 b(the)f(other)h(hand,)g(as)g Fs(T)53 b Fy(is)28 b(a)g(maximal)d(toric)i(sub-algebra,)i(it)e(is)h(Ab)s(elian) f(and)-118 4669 y(hence,)35 b(all)d(elemen)m(ts)i(ad\()p Fx(H)8 b Fy(\),)34 b(with)f Fx(H)k Fs(2)30 b(T)c Fy(,)34 b(comm)m(ute.)47 b(This)34 b(altogether)e(implies)g(that)h(these)i (ad\()p Fx(H)8 b Fy(\))33 b(can)-118 4789 y(b)s(e)g(sim)m(ultaneously)e (diagonalized)f(on)j Fw(C)58 b Fy(\(see)34 b([55)o(])f(and)g([71)o (]\).)44 b(W)-8 b(e)33 b(then)g(ha)m(v)m(e)h(the)f(follo)m(wing)-118 5017 y Fv(Theorem)k(4.2.6)h(\([55]\))48 b FA(In)28 b(the)h(ab)-5 b(ove)28 b(situation,)i(ther)-5 b(e)29 b(exists)f(a)h(set)g Fy(\001)g FA(of)f(non-zer)-5 b(o)28 b(r)-5 b(e)g(al)29 b(line)-5 b(ar)28 b(forms)g Fx(\013)-118 5138 y FA(on)k Fs(T)58 b FA(which)32 b(c)-5 b(an)32 b(b)-5 b(e)32 b(written)g(as)h Fy(\001)28 b(=)1380 5112 y(~)1364 5138 y(\001)17 b Fs([)h Fy(\()p Fs(\000)1677 5112 y Fy(~)1661 5138 y(\001\))33 b FA(\(henc)-5 b(e)31 b(they)i(ar)-5 b(e)32 b(invariant)g(by)h(the)f (op)-5 b(er)g(ation)32 b Fx(\013)c Fs(7!)g(\000)p Fx(\013)q FA(\))-118 5258 y(and)34 b(elements)g Fx(X)553 5273 y Fr(\013)630 5258 y Fs(2)28 b Fx(g)39 b FA(such)34 b(that)-33 5461 y(\(i\))49 b Fx(g)31 b Fy(=)c Fs(\010)384 5488 y Fr(\013)p Ft(2)489 5472 y FC(~)476 5488 y(\001)557 5461 y Fy(\()p Fw(R)5 b Fx(X)742 5476 y Fr(\013)819 5461 y Fs(\010)23 b Fw(R)5 b Fx(X)1066 5476 y Ft(\000)p Fr(\013)1176 5461 y Fy(\))p FA(.)-62 5665 y(\(ii\))48 b(F)-7 b(or)34 b(al)5 b(l)34 b Fx(H)h Fs(2)28 b(T)61 b FA(and)34 b Fx(\013)29 b Fs(2)f Fy(\001)1536 5785 y FA(ad)o Fy(\()p Fx(H)8 b Fy(\))22 b Fs(\001)g Fx(X)1953 5800 y Fr(\013)2030 5785 y Fy(=)27 b Fx(i\013)q Fy(\()p Fx(H)8 b Fy(\))p Fx(X)2475 5800 y Ft(\000)p Fr(\013)2579 5785 y Fx(:)1876 6155 y Fy(110)p eop %%Page: 111 115 111 114 bop 28 219 a Fy(The)33 b(elemen)m(ts)g Fx(\013)g Fy(are)f(called)f(the)i FA(r)-5 b(o)g(ots)40 b Fy(of)32 b Fx(g)j Fy(with)d(resp)s(ect)i(to)d Fs(T)26 b Fy(.)43 b(An)33 b(elemen)m(t)f Fx(X)j Fs(2)29 b Fx(g)35 b Fy(it)c(is)h(said)g (to)f(b)s(e)-118 339 y FA(r)-5 b(e)g(gular)45 b Fy(if)34 b(for)g(all)f Fx(\013)f Fs(2)g Fy(\001,)k Fx(\013)q Fy(\()p Fx(X)8 b Fy(\))31 b Fs(6)p Fy(=)g(0)k(\(this)f(means)h(that)g Fx(e)2173 303 y Fr(X)2275 339 y 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Fs(2)28 b Fx(C)2569 2281 y Fr(a)2611 2317 y Fy(\()p Fw(T)2712 2281 y Fr(d)2756 2317 y Fx(;)17 b(S)6 b(O)s Fy(\(3)p Fx(;)17 b Fw(R)s Fy(\)\))39 b FA(and)32 b Fx(B)3515 2332 y Fr(\025)3589 2317 y Fs(2)c Fx(so)p Fy(\(3)p Fx(;)17 b Fw(R)t Fy(\))-118 2437 y FA(such)35 b(that)857 2657 y Fx(@)908 2672 y Fr(!)959 2657 y Fx(Z)1026 2672 y Fr(\025)1071 2657 y Fy(\()p Fx(\022)s Fy(\))23 b Fs(\001)e Fx(Z)1334 2672 y Fr(\025)1379 2657 y Fy(\()p Fx(\022)s Fy(\))1503 2616 y Ft(\000)p FC(1)1620 2657 y Fy(+)h FA(A)-5 b(d)p Fy(\()p Fx(Z)1940 2672 y Fr(\025)1985 2657 y Fy(\()p Fx(\022)s Fy(\)\))22 b Fs(\001)g Fy(\()p Fx(\025A)g Fy(+)g Fx(F)14 b Fy(\()p Fx(\022)s Fy(\)\))28 b(=)f Fx(B)2951 2672 y Fr(\025)2997 2657 y Fx(;)742 b Fy(\(4.22\))-118 2877 y FA(for)35 b(al)5 b(l)34 b Fx(\022)d Fs(2)d Fw(T)410 2841 y Fr(d)454 2877 y FA(,)35 b(and)f Fx(B)782 2892 y Fr(\025)862 2877 y FA(do)-5 b(es)34 b(not)h(dep)-5 b(end)34 b(on)h(time.)-118 3106 y Fv(Remark)i(4.2.10)49 b FA(The)35 b(p)-5 b(erturb)g(ation)35 b Fx(F)50 b FA(may)35 b(dep)-5 b(end)34 b(also)h(on)g(the)h(p)-5 b(ar)g(ameter)35 b Fx(\025)g FA(in)g(a)g(r)-5 b(e)g(al)36 b(analytic)f(way)-118 3226 y(and)i(the)g(r)-5 b(esult)38 b(is)f(stil)5 b(l)37 b(true)h(\(se)-5 b(e)37 b([55]\).)52 b(In)37 b(fact,)g(the)h(pr)-5 b(o)g(of)37 b(is)g(b)-5 b(ase)g(d)36 b(on)h(an)g(in\014nite)g(se)-5 b(quenc)g(e)37 b(of)g(tr)-5 b(ans-)-118 3347 y(formations)34 b(in)g(which,)g(after)h(the)g(\014rst)g(step,)g(the)g(p)-5 b(erturb)g(ation)34 b(term)h(wil)5 b(l)35 b(also)f(dep)-5 b(end)34 b(on)g Fx(\025)p FA(.)28 3575 y Fy(As)27 b(it)e(is)h(stated)h (in)e(the)i(theorem,)g(w)m(e)h(m)m(ust)e FA(solve)33 b Fy(the)26 b(homological)d(equation)j(\(4.22\))f(in)h(order)g(to)g (render)-118 3695 y(equation)32 b(\(4.17\))g(to)g(constan)m(t)i(co)s (e\016cien)m(ts.)44 b(The)34 b(pro)s(of)e(consists)h(in)f(the)h(follo)m (wing)d(steps:)27 3899 y Fs(\017)49 b Fy(First)34 b(of)g(all,)g(a)h (theorem)g(similar)d(to)i(\(4.1.4\))h(is)f(used)i(to)f(pro)m(v)m(e)h (reducibilit)m(y)e(b)m(y)i(means)f(of)f(transfor-)126 4019 y(mations)f(close)i(to)f(the)i(iden)m(tit)m(y)e(for)h(a)f(set)i (of)e Fx(\025)h Fy(of)f(p)s(ositiv)m(e)g(measure.)51 b(Moreo)m(v)m(er,)37 b(as)e Fx(\025)g Fy(is)f(closer)h(to)126 4139 y(zero)f(\(sa)m(y)h(less)f(than)g(a)f(certain)h(constan)m(t)h Fx(\025)1812 4154 y FC(0)1851 4139 y Fy(\),)f(the)h(measure)f(of)f(the) h FA(r)-5 b(e)g(ducible)41 b Fx(\025)p Fy('s)34 b(is)g(exp)s(onen)m (tially)126 4260 y(small)c(in)i Fx(\025)552 4275 y FC(0)591 4260 y Fy(.)27 4463 y Fs(\017)49 b Fy(T)-8 b(o)38 b(pro)m(v)m(e)h(the)f (ab)s(o)m(v)m(e)h(result)f(some)f(resonance)j(regions)d(ha)m(v)m(e)i(b) s(een)g(let)e(aside.)59 b(In)38 b(the)h(second)g(step)126 4584 y(metho)s(dology)27 b(is)i(giv)m(en)g(to)f(skip)h(these)i (resonances)g(b)m(y)e(enlarging)f(the)h(class)g(of)g(transformations)e (\(not)126 4704 y(close)36 b(to)g(the)h(iden)m(tit)m(y\).)55 b(The)37 b(tec)m(hnique)h(has)f(the)f(problem)g(that)g(the)h(dep)s (endence)h(of)e(the)h(ro)s(ots)f(of)126 4824 y(the)e(unp)s(erturb)s(ed) h(matrices)f(on)f(the)i(parameter)e Fx(\025)h Fy(\(whic)m(h)h(giv)m(e)f (rise)g(to)f(small)f(divisor)h(problems)g(in)126 4945 y(solving)e(homological)e(equations)k(\))f(b)s(ecomes)h(v)m(ery)h(bad.) 27 5148 y Fs(\017)49 b Fy(T)-8 b(o)31 b(con)m(trol)f(the)h(dep)s (endence)i(on)e(the)g(parameter)f Fx(\025)h Fy(it)f(is)g(used)i(the)f (notion)f(of)g FA(Pyartli)k(tr)-5 b(ansversality)126 5269 y Fy(together)33 b(with)f(some)g(useful)h(results)g(for)f(the)h (presen)m(t)h(framew)m(ork.)27 5472 y Fs(\017)49 b Fy(Finally)-8 b(,)35 b(using)h(Py)m(artli)g(transv)m(ersalit)m(y)-8 b(,)38 b(it)d(is)h(pro)m(v)m(ed)i(that)f(the)g(inductiv)m(e)f(pro)s (cedure)i(describ)s(ed)f(in)126 5592 y(the)c(second)h(item)d(is)h(con)m (v)m(ergen)m(t.)28 5796 y(W)-8 b(e)33 b(will)d(giv)m(e)j(more)f (details)f(on)i(the)g(ab)s(o)m(v)m(e)g FA(pr)-5 b(o)g(gr)g(am)39 b Fy(in)32 b(the)h(follo)m(wing)d(sections.)1876 6155 y(114)p eop %%Page: 115 119 115 118 bop -118 219 a Fv(Results)36 b(of)i(p)s(ositiv)m(e)e(measure) -118 403 y Fy(Here)i(w)m(e)h(will)c(only)j(presen)m(t)h(the)f(setting)g (necessary)i(for)d(the)h(sequel,)i(b)s(ecause)f(results)f(of)f(this)h (t)m(yp)s(e)g(ha)m(v)m(e)-118 524 y(already)32 b(b)s(een)h(discussed)i (in)c(section)i(4.1.)28 644 y(Assume)46 b(that)f Fx(!)52 b Fs(2)e Fx(D)s(C)7 b Fy(\()p Fx(\015)e(;)17 b(\033)t Fy(\))44 b(and)h Fx(h)1553 659 y FC(1)1642 644 y Fy(=)k Fx(h)g(>)g Fy(0)c(are)g(\014xed)h(and)f(that)g Fx(A)3013 659 y FC(1)3102 644 y Fs(2)k Fx(so)p Fy(\(3)p Fx(;)17 b Fw(R)5 b Fy(\))51 b(and)45 b Fx(F)3861 659 y FC(1)3949 644 y Fs(2)-118 764 y Fx(C)-41 728 y Fr(a)-48 790 y(h)-7 799 y Fp(1)31 764 y Fy(\()p Fw(T)132 728 y Fr(d)176 764 y Fx(;)17 b(so)p Fy(\(3)p Fx(;)g Fw(R)t Fy(\)\))38 b(are)33 b(suc)m(h)h(that)1708 885 y Fs(j)p Fx(F)1799 900 y FC(1)1839 885 y Fs(j)1867 900 y Fr(h)1908 909 y Fp(1)1973 885 y Fx(<)28 b(")2123 900 y FC(1)2162 885 y Fx(;)-118 1055 y Fy(with)k Fx(")150 1070 y FC(1)222 1055 y Fy(small)e(enough.)28 1175 y(If)k Fx(N)5 b(;)17 b(K)41 b Fy(are)34 b(real)f(p)s(ositiv)m(e)h (n)m(um)m(b)s(ers)h(\(think)e(of)h(them)g(as)g FA(lar)-5 b(ge)41 b Fy(n)m(um)m(b)s(ers\),)35 b(and)f Fx(P)44 b Fs(\025)30 b Fy(1)k(is)g(an)g(in)m(teger,)-118 1295 y(w)m(e)g(de\014ne) f(the)g(set)481 1537 y Fx(D)s(S)6 b Fy(\()p Fx(N)f(;)17 b(K)r(;)g(P)d Fy(\))26 b(=)1170 1396 y Fq(\032)1244 1537 y Fx(\013)j Fs(2)f Fw(R)5 b Fx(;)50 b Fs(j)p Fx(\013)22 b Fs(\000)1794 1469 y(h)p Fv(k)p Fx(;)17 b(!)t Fs(i)p 1794 1514 245 4 v 1878 1605 a Fx(P)2049 1537 y Fs(j)27 b(\025)h Fx(K)2299 1496 y Ft(\000)p FC(1)2394 1537 y Fx(;)77 b Fy(for)32 b(all)e(0)e Fx(<)f Fs(j)p Fv(k)p Fs(j)g(\024)h Fx(N)3297 1396 y Fq(\033)3389 1537 y Fx(;)-118 1803 y Fy(and)33 b(w)m(e)i(also)d(de\014ne)j(the)f(set)g Fx(R)q(S)6 b Fy(\()p Fx(N)f(;)17 b(K)r(;)g(P)d Fy(\))31 b(as)j(its)f(complemen)m(tary)f(in)h Fw(R)5 b Fy(.)52 b(When)34 b Fx(P)42 b Fy(=)29 b(1,)34 b(w)m(e)g(will)d(denote)-118 1924 y(these)41 b(sets)g(b)m(y)g Fx(D)s(S)6 b Fy(\()p Fx(N)f(;)17 b(K)7 b Fy(\))39 b(and)h Fx(R)q(S)6 b Fy(\()p Fx(N)f(;)17 b(K)7 b Fy(\))40 b(resp)s(ectiv)m(ely)-8 b(.)66 b(Note)40 b(that,)h(if)e Fx(\034)52 b(>)40 b(\033)k Fy(is)39 b(\014xed,)k(then)e(w)m(e)f(can)-118 2044 y(de\014ne)34 b(the)f(classical)e(set)i(of)f(p)s(oin)m(ts)g(satisfying)g(the)h (Diophan)m(tine)e(condition)515 2307 y Fx(D)s(C)669 2322 y Fr(\034)712 2307 y Fy(\()p Fx(N)5 b(;)17 b(K)7 b Fy(\))27 b(=)1136 2167 y Fq(\032)1210 2307 y Fx(\013)i Fs(2)f Fw(R)5 b Fx(;)50 b Fs(j)p Fx(\013)23 b Fs(\000)f(h)p Fv(k)p Fx(;)17 b(!)t Fs(ij)26 b(\025)2165 2240 y Fx(K)2255 2204 y Ft(\000)p FC(1)p 2165 2284 185 4 v 2179 2375 a Fs(j)p Fv(k)p Fs(j)2294 2347 y Fr(\034)2360 2307 y Fx(;)77 b Fy(for)32 b(all)e(0)e Fx(<)f Fs(j)p Fv(k)p Fs(j)g(\024)i Fx(N)3264 2167 y Fq(\033)3355 2307 y Fx(;)-118 2568 y Fy(and)k(the)g(inclusions)1063 2688 y Fx(D)s(S)6 b Fy(\()p Fx(N)f(;)17 b(K)7 b Fy(\))27 b Fs(\032)h Fx(D)s(C)1792 2703 y Fr(\034)1835 2688 y Fy(\()p Fx(N)5 b(;)17 b(K)7 b Fy(\))27 b Fs(\032)i Fx(D)s(S)6 b Fy(\()p Fx(N)f(;)17 b(K)7 b(N)2754 2647 y Fr(\034)2797 2688 y Fy(\))-118 2858 y(hold.)40 b(The)26 b(sets)g Fx(D)s(S)6 b Fy(\()p Fx(N)f(;)17 b(K)r(;)g(P)d Fy(\))24 b(and)h Fx(R)q(S)6 b Fy(\()p Fx(N)f(;)17 b(K)r(;)g(P)d Fy(\))23 b(are,)k(resp)s(ectiv)m (ely)-8 b(,)28 b(the)d(sets)h(of)f(Diophan)m(tine)f(and)h(nearly)-118 2979 y(resonan)m(t)39 b(real)d(n)m(um)m(b)s(ers)j FA(up)h(to)g(or)-5 b(der)39 b Fx(N)10 b Fy(.)60 b(Moreo)m(v)m(er,)40 b(if)d Fx(K)45 b Fy(is)37 b(c)m(hosen)i(large)e(enough)h(\(of)g(the)g(order)g (of)f(a)-118 3099 y(p)s(o)m(w)m(er)c(of)g Fx(N)43 b Fy(for)32 b(a)g(\014xed)i Fx(N)10 b Fy(\),)33 b(and)f(if)g Fx(\013)c Fs(2)g Fx(R)q(S)6 b Fy(\()p Fx(N)f(;)17 b(K)7 b Fy(\),)33 b(there)g(exists)g(a)f(unique)i(0)27 b Fx(<)h Fs(j)p Fv(k)3192 3114 y FC(0)3231 3099 y Fs(j)f(\024)h Fx(N)43 b Fy(suc)m(h)34 b(that)1230 3308 y Fs(j)p Fx(\013)23 b Fs(\000)f(h)p Fv(k)1540 3323 y FC(0)1579 3308 y Fx(;)17 b(!)t Fs(ij)27 b(\024)h Fx(K)1977 3267 y Ft(\000)p FC(1)2071 3308 y Fs(j)p Fv(k)2158 3323 y FC(0)2197 3308 y Fs(j)2225 3267 y Ft(\000)p Fr(\034)2351 3308 y Fs(\024)g Fx(K)2546 3267 y Ft(\000)p FC(1)2641 3308 y Fx(:)-118 3517 y Fy(The)36 b(reason)g(wh)m(y)g(these)h(Diophan)m(tine)c(and)j(nearly)f(resonan)m (t)h(sets)g(actually)e(describ)s(e)i(the)f(resonances)i(for)-118 3638 y(our)32 b(small)f(divisors)h(problem)f(will)f(b)s(ecome)j(clear)f (in)g(a)g(momen)m(t.)28 3758 y(F)-8 b(or)29 b(the)h(KAM)h(result)e(of)h (reducibilit)m(y)e(in)h(p)s(ositiv)m(e)g(measure)h(w)m(e)h(will)c(in)m (tro)s(duce)j(sequences)j Fx(N)3640 3773 y Fr(r)3708 3758 y Fy(and)d Fx(K)3978 3773 y Fr(r)-118 3879 y Fy(going)i(to)h (in\014nit)m(y)f(\(w)m(e)j(will)c(c)m(ho)s(ose)j Fx(N)1355 3894 y Fr(r)1422 3879 y Fy(=)28 b(constan)m(t)c Fs(\001)e Fy(2)2009 3842 y FC(2)p Fr(r)2115 3879 y Fy(and)34 b Fx(K)2389 3894 y Fr(r)2456 3879 y Fy(=)28 b(constan)m(t)c Fs(\001)e Fx(N)3082 3842 y Fr(\027)3072 3903 y(r)3126 3879 y Fy(,)33 b(with)g Fx(\027)40 b Fy(p)s(ositiv)m(e)33 b(and)-118 3999 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y Fs(\\)23 b Fy(\003)911 5476 y Fr(t)911 5537 y(r)r FC(+1)126 5709 y Fy(\(0)p Fs(!)p Fy(t\):)42 b Fx(\025)28 b Fs(2)g Fy(\003)705 5673 y FC(0)705 5734 y Fr(r)766 5709 y Fs(\\)23 b Fy(\003)923 5673 y Fr(t)923 5734 y(r)r FC(+1)126 5906 y Fy(\(t)p Fs(!)p Fy(0\):)42 b Fx(\025)28 b Fs(2)g Fy(\003)705 5870 y Fr(t)705 5931 y(r)765 5906 y Fs(\\)22 b Fy(\003)921 5870 y FC(0)921 5931 y Fr(r)r FC(+1)1876 6155 y Fy(119)p eop %%Page: 120 124 120 123 bop 413 2302 a currentpoint currentpoint translate 0.5048 0.5048 scale neg exch neg exch translate 413 2302 a @beginspecial 0 @llx 0 @lly 730 @urx 519 @ury 7300 @rwi @setspecial %%BeginDocument: simbol.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: simbol.eps %%Creator: fig2dev Version 3.2 Patchlevel 3a %%CreationDate: Fri Jun 8 12:32:41 2001 %%For: puig@atlas (Joaquim Puig,,,) %%BoundingBox: 0 0 730 519 %%Magnification: 1.0000 %%EndComments /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 519 moveto 0 0 lineto 730 0 lineto 730 519 lineto closepath clip newpath -42.0 546.0 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index show % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % left45 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 32 true [ 32 0 0 -32 0 32 ] {<808080804040404020202020101010100808080804040404 020202020101010180808080404040402020202010101010 080808080404040402020202010101018080808040404040 202020201010101008080808040404040202020201010101 808080804040404020202020101010100808080804040404 0202020201010101>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P4 exch def /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#055 /minus 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /hyphen 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin %%Page: 1 1 10 setmiterlimit 0.06299 0.06299 sc 7.500 slw % Ellipse n 2250 8550 101 101 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr % Ellipse n 900 8550 90 90 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr % Ellipse n 3600 8550 90 90 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P4 [16 0 0 -16 234.00 564.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 4950 8550 90 90 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 8550 3150 90 90 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 9450 3150 90 90 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 10350 3150 90 90 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 11250 3150 90 90 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 12150 3150 90 90 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 6750 4050 90 90 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 5850 4050 90 90 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 4050 4950 90 90 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 2250 6750 90 90 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr % Ellipse n 3150 5850 90 90 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr % Ellipse n 3150 6750 101 101 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr % Ellipse n 4950 4950 101 101 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr % Ellipse n 7650 4050 101 101 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr % Ellipse n 3150 4950 90 90 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P4 [16 0 0 -16 204.00 324.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 1350 6750 90 90 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P4 [16 0 0 -16 84.00 444.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 4950 4050 90 90 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P4 [16 0 0 -16 324.00 264.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 7650 3150 90 90 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P4 [16 0 0 -16 504.00 204.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline [60] 0 sd n 2250 7650 m 2250 450 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 3150 7650 m 3150 450 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 4050 7650 m 4050 450 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 4950 7650 m 4950 450 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 5850 7650 m 5850 450 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 6750 7650 m 6750 450 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 7650 7650 m 7650 450 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 8550 7650 m 8550 450 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 9450 7650 m 9450 450 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 10350 7650 m 10350 450 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 11250 7650 m 11250 450 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1350 450 m 12150 450 l 12150 7650 l 1350 7650 l cp gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1350 6750 m 12150 6750 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1350 5850 m 12150 5850 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1350 4950 m 12150 4950 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1350 4050 m 12150 4050 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1350 3150 m 12150 3150 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1350 2250 m 12150 2250 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1350 1350 m 12150 1350 l gs col0 s gr [] 0 sd % Polyline 15.000 slw gs clippath 7215 8610 m 7215 8490 l 6928 8490 l 7168 8550 l 6928 8610 l cp eoclip n 6300 8550 m 7200 8550 l gs col0 s gr gr % arrowhead n 6928 8610 m 7168 8550 l 6928 8490 l 6928 8610 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 1410 6735 m 1290 6735 l 1290 7022 l 1350 6782 l 1410 7022 l cp eoclip n 1350 7650 m 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(erio)s(dic)f(sc)m(hr\177)-49 b(odinger)82 5436 y(equations)33 b(in)f(the)h(adiabatic)e(case.)44 b FA(Comm.)34 b(Math.)h(Phys.)p Fy(,)d(227\(1\):1{92,)f(2002.)-118 5640 y([31])48 b(A)27 b(Fink.)33 b FA(A)n(lmost)c(p)-5 b(erio)g(dic)28 b(di\013er)-5 b(ential)28 b(e)-5 b(quations)p Fy(.)33 b(Num)m(b)s(er)26 b(377)g(in)g(Lecture)h(Notes)g(in)f(Mathematics.)82 5760 y(Springer,)33 b(1974.)1876 6155 y(124)p eop %%Page: 125 129 125 128 bop -118 219 a Fy([32])48 b(J.)38 b(F)-8 b(r\177)-49 b(ohlic)m(h,)38 b(T.)g(Sp)s(encer,)i(and)e(P)-8 b(.)38 b(Witt)m(w)m(er.)58 b(Lo)s(calization)35 b(for)i(a)g(class)h(of)f (one-dimensional)e(quasi-)82 339 y(p)s(erio)s(dic)c(Sc)m(hr\177)-49 b(odinger)33 b(op)s(erators.)43 b FA(Comm.)34 b(Math.)h(Phys.)p Fy(,)e(132\(1\):5{25,)e(1990.)-118 541 y([33])48 b(D.)e(J.)g(Gilb)s (ert)e(and)j(D.)e(B.)h(P)m(earson.)85 b(On)46 b(sub)s(ordinacy)h(and)f (analysis)g(of)f(the)i(sp)s(ectrum)f(of)g(one-)82 662 y(dimensional)30 b(Sc)m(hr\177)-49 b(odinger)33 b(op)s(erators.)44 b FA(J.)34 b(Math.)i(A)n(nal.)e(Appl.)p Fy(,)e(128\(1\):30{56,)f(1987.) -118 864 y([34])48 b(M.)34 b(Goldstein)d(and)i(W.)g(Sc)m(hlag.)44 b(H\177)-49 b(older)32 b(con)m(tin)m(uit)m(y)h(of)f(the)h(in)m (tegrated)g(densit)m(y)h(of)e(states)i(for)e(quasi-)82 984 y(p)s(erio)s(dic)22 b(Sc)m(hr\177)-49 b(odinger)24 b(equations)g(and)g(a)m(v)m(erages)h(of)e(shifts)h(of)f(subharmonic)g (functions.)29 b FA(A)n(nn.)d(of)g(Math.)82 1105 y(\(2\))p Fy(,)33 b(154\(1\):155{203,)d(2001.)-118 1307 y([35])48 b(G.)32 b(G\023)-49 b(omez,)32 b(J.M.)g(Mondelo,)h(and)f(Sim\023)-49 b(o)30 b(C.)43 b(Re\014ned)33 b(fourier)f(analysis:)42 b(pro)s(cedures,)34 b(error)e(estimates)82 1427 y(and)h(applications.) 41 b FA(Pr)-5 b(eprint)p Fy(,)33 b(2001.)42 b(Accessible)34 b(at)e Fa(http://www.maia.ub.es/ds)q(g/20)q(01)p Fy(.)-118 1630 y([36])48 b(A.)27 b(Gordon.)34 b(On)27 b(the)g(p)s(oin)m(t)f(sp)s (ectrum)h(of)g(the)g(one-dimensional)d(Sc)m(hr\177)-49 b(odinger)27 b(op)s(erator.)33 b FA(R)n(uss.)c(Math.)82 1750 y(Surv.)p Fy(,)k(31:257{258,)e(1976.)-118 1952 y([37])48 b(A.)c(Y.)h(Gordon,)h(S.)e(Jitomirsk)-5 b(a)m(y)m(a,)45 b(Y.)f(Last,)j(and)d(B.)g(Simon.)75 b(Dualit)m(y)43 b(and)h(singular)e (con)m(tin)m(uous)82 2073 y(sp)s(ectrum)33 b(in)f(the)h(almost)e (Mathieu)i(equation.)43 b FA(A)-5 b(cta)35 b(Math.)p Fy(,)e(178\(2\):169{183,)d(1997.)-118 2275 y([38])48 b(M.R.)33 b(Herman.)43 b(Non)33 b(top)s(ological)c(conjugacy)k(of)f(sk) m(ew-pro)s(ducts)j(in)d Fx(su)p Fy(\(2\).)42 b(man)m(uscript.)-118 2477 y([39])48 b(M.R.)37 b(Herman.)54 b(Une)37 b(m)m(\023)-46 b(etho)s(de)36 b(p)s(our)g(minorer)f(les)h(exp)s(osan)m(ts)i(de)f(ly)m (apuno)m(v)g(et)f(quelques)i(exemples)82 2598 y(mon)m(tran)m(t)f(le)f (caract)m(\022)-46 b(ere)38 b(lo)s(cal)c(d'un)k(th)m(\023)-46 b(eor)m(\022)g(eme)37 b(d'arnold)f(et)h(de)h(moser)e(sur)i(le)e(tore)h (de)g(dimension)f(2.)82 2718 y FA(Comment.)e(Math.)h(Helvetici)p Fy(,)e(58:262{288,)d(1981.)-118 2920 y([40])48 b(E.)33 b(Hille.)42 b FA(L)-5 b(e)g(ctur)g(es)35 b(on)f(or)-5 b(dinary)35 b(di\013er)-5 b(ential)34 b(e)-5 b(quations.)42 b Fy(Addison-W)-8 b(esley)g(,)33 b(Reading,)f(Mass.,)i(1969.)-118 3123 y([41])48 b(S.)d(Y.)f(Jitomirsk)-5 b(a)m(y)m(a.)77 b(Metal-insulator)41 b(transition)i(for)h(the)g(almost)f(Mathieu)h(op)s (erator.)77 b FA(A)n(nn.)45 b(of)82 3243 y(Math.)36 b(\(2\))p Fy(,)c(150\(3\):1159{1175,)d(1999.)-118 3445 y([42])48 b(R.)33 b(Johnson.)44 b(Ergo)s(dic)32 b(theory)h(and)f(linear)f (di\013eren)m(tial)g(equations.)44 b FA(J.)35 b(Di\013.)e(Eq.)p Fy(,)g(28:165{194,)e(1978.)-118 3648 y([43])48 b(R.)33 b(Johnson.)44 b(The)33 b(recurren)m(t)h(Hill's)d(equation.)43 b FA(J.)35 b(Di\013.)f(Eq.)p Fy(,)e(46:165{193,)f(1982.)-118 3850 y([44])48 b(R.)43 b(Johnson.)74 b(Ly)m(apuno)m(v)44 b(n)m(um)m(b)s(ers)g(for)e(the)h(almost)e(p)s(erio)s(dic)g(Sc)m(hr\177) -49 b(odinger)43 b(equation.)73 b FA(Il)5 b(linois)43 b(J.)82 3970 y(Math.)p Fy(,)33 b(28\(3\):397{419,)e(1984.)-118 4173 y([45])48 b(R.)36 b(Johnson.)54 b(Exp)s(onen)m(tial)35 b(dic)m(hotom)m(y)-8 b(,)36 b(rotation)f(n)m(um)m(b)s(er)h(and)g (linear)e(di\013eren)m(tial)g(equations)i(with)82 4293 y(b)s(ounded)e(co)s(e\016cien)m(ts.)44 b FA(J.)35 b(Di\013.)f(Eq.)p Fy(,)f(61:54{78,)e(1986.)-118 4495 y([46])48 b(R.)f(Johnson.)85 b(Can)m(tor)46 b(sp)s(ectrum)h(for)f(the)h(quasi-p)s(erio)s(dic)d(Sc)m (hr\177)-49 b(odinger)46 b(equation.)85 b FA(J.)47 b(Di\013.)g(Eq.)p Fy(,)82 4616 y(91:88{110,)31 b(1991.)-118 4818 y([47])48 b(R.)29 b(Johnson)g(and)g(J.)g(Moser.)38 b(The)29 b(rotation)e(n)m(um)m (b)s(er)i(for)g(almost)e(p)s(erio)s(dic)g(p)s(oten)m(tials.)35 b FA(Comm.)30 b(Math.)82 4938 y(Phys.)p Fy(,)j(84:403{438,)e(1982.)-118 5141 y([48])48 b(R.)32 b(Johnson)f(and)h(M.)g(Nerurk)-5 b(ar.)41 b(Exp)s(onen)m(tial)31 b(dic)m(hotom)m(y)g(and)g(rotation)f(n) m(um)m(b)s(er)h(for)g(linear)f(Hamil-)82 5261 y(tonian)i(systems.)45 b FA(J.)35 b(Di\013er)-5 b(ential)33 b(Equations)p Fy(,)g (108\(1\):201{216,)d(1994.)-118 5463 y([49])48 b(R.)e(Johnson)h(and)f (G.R.)g(Sell.)82 b(Smo)s(othness)46 b(of)g(sp)s(ectral)g(subbundles)h (and)f(reducibilit)m(y)f(of)g(quasi-)82 5584 y(p)s(erio)s(dic)31 b(linear)g(di\013eren)m(tial)g(systems.)45 b FA(J.)35 b(Di\013.)f(Eq.)p Fy(,)e(41:262{288,)f(1981.)-118 5786 y([50])95 5761 y(\022)82 5786 y(A.)37 b(Jorba,)h(Ram)-11 b(\023)-38 b(\020rez-Ros)35 b(R.,)j(and)f(J.)g(Villan)m(uev)-5 b(a.)54 b(E\013ectiv)m(e)38 b(reducibilit)m(y)d(of)i(quasi-p)s(erio)s (dic)d(linear)82 5906 y(equations)f(close)g(to)f(constan)m(t)h(co)s (e\016cien)m(ts.)45 b FA(SIAM)35 b(J.)g(Math.)g(A)n(nal.)p Fy(,)d(28\(1\):178{188,)e(1997.)1876 6155 y(125)p eop %%Page: 126 130 126 129 bop -118 219 a Fy([51])95 193 y(\022)82 219 y(A.)32 b(Jorba)g(and)g(C.)h(Sim\023)-49 b(o.)40 b(On)32 b(the)h(reducibilit)m (y)d(of)h(linear)f(di\013eren)m(tial)h(equations)h(with)f(quasip)s (erio)s(dic)82 339 y(co)s(e\016cien)m(ts.)45 b FA(J.)35 b(Di\013er)-5 b(ential)34 b(Equations)p Fy(,)e(98\(1\):111{124,)e (1992.)-118 542 y([52])95 517 y(\022)82 542 y(A.)j(Jorba)f(and)g(C.)h (Sim\023)-49 b(o.)41 b(On)32 b(quasi-p)s(erio)s(dic)f(p)s(erturbations) h(of)f(elliptic)f(equilibrium)f(p)s(oin)m(ts.)43 b FA(SIAM)82 663 y(J.)35 b(Math.)g(A)n(nal.)p Fy(,)d(27\(6\):1704{1737,)e(1996.)-118 866 y([53])48 b(S.)38 b(Kotani.)56 b(Ljapuno)m(v)38 b(indices)f (determine)g(absolutely)g(con)m(tin)m(uous)h(sp)s(ectra)g(of)f (stationary)g(random)82 986 y(one-dimensional)d(Sc)m(hr\177)-49 b(odinger)36 b(op)s(erators.)52 b(In)37 b FA(Sto)-5 b(chastic)37 b(analysis)g(\(Katata/Kyoto,)h(1982\))p Fy(,)e(pages)82 1107 y(225{247.)31 b(North-Holland,)g(Amsterdam,)h(1984.)-118 1310 y([54])48 b(R.)35 b(Krik)m(orian.)48 b Fx(C)776 1274 y FC(0)815 1310 y Fy(-densit)m(\023)-46 b(e)35 b(globale)e(des)j (syst)m(\022)-46 b(emes)36 b(pro)s(duits-crois)m(\023)-46 b(es)34 b(sur)i(le)e(cercle)h(r)m(\023)-46 b(eductibles.)50 b FA(Er-)82 1431 y(go)-5 b(dic)34 b(The)-5 b(ory)35 b(Dynam.)f(Systems) p Fy(,)e(19\(1\):61{100,)f(1999.)-118 1634 y([55])48 b(R.)23 b(Krik)m(orian.)i(R)m(\023)-46 b(eductibilit)m(\023)g(e)20 b(des)j(syst)m(\022)-46 b(emes)24 b(pro)s(duits-crois)m(\023)-46 b(es)22 b(\022)-49 b(a)22 b(v)-5 b(aleurs)22 b(dans)h(des)h(group)s(es) e(compacts.)82 1754 y FA(Ast)n(\023)-47 b(erisque)p Fy(,)32 b(259,)g(1999.)-118 1958 y([56])48 b(R.)34 b(Krik)m(orian.)47 b(R)m(\023)-46 b(eductibilit)m(\023)g(e)32 b(presque)k(partout)e(des)i (\015ots)e(\014br)m(\023)-46 b(es)35 b(quasi-p)m(\023)-46 b(erio)s(diques)34 b(\022)-49 b(a)34 b(v)-5 b(aleurs)34 b(dans)82 2078 y(les)f(group)s(es)g(compacts.)43 b FA(A)n(nn.)35 b(scient.)1621 2053 y(\023)1607 2078 y(Ec.)g(Norm.)f(Sup.)p Fy(,)f(32:187{240,)d(1999.)-118 2282 y([57])48 b(R.)40 b(Krik)m(orian.)64 b(Global)37 b(densit)m(y)k(of)f(reducible)f(quasi-p) s(erio)s(dic)f(co)s(cycles)j(on)f Fw(T)3151 2245 y FC(1)3221 2282 y Fs(\002)28 b Fx(S)6 b(U)k Fy(\(2\).)65 b FA(A)n(nn.)41 b(of)82 2402 y(Math.)36 b(\(2\))p Fy(,)c(154\(2\):269{326,)e(2001.)-118 2605 y([58])48 b(J.)42 b(Lask)-5 b(ar.)70 b(In)m(tro)s(duction)41 b(to)g(frequency)j(map)d(analysis.)69 b(In)42 b FA(Hamiltonian)g (systems)h(with)g(thr)-5 b(e)g(e)43 b(or)82 2726 y(mor)-5 b(e)44 b(de)-5 b(gr)g(e)g(es)44 b(of)g(fr)-5 b(e)g(e)g(dom)44 b(\(S'A)-5 b(gar\023)-50 b(o,)46 b(1995\))p Fy(,)e(pages)g(134{150.)d (Klu)m(w)m(er)j(Acad.)f(Publ.,)j(Dordrec)m(h)m(t,)82 2846 y(1999.)-118 3050 y([59])i(J.)25 b(Lask)-5 b(ar,)27 b(C.)e(F)-8 b(ro)s(esc)m(hl)m(\023)-46 b(e,)27 b(and)e(A.)g(Celletti.)k (The)d(measure)f(of)f(c)m(haos)i(b)m(y)g(the)f(n)m(umerical)f(analysis) g(of)g(the)82 3170 y(fundamen)m(tal)40 b(frequencies.)h(Application)d (to)i(the)g(standard)h(mapping.)64 b FA(Phys.)42 b(D)p Fy(,)e(56\(2-3\):253{269,)82 3290 y(1992.)-118 3494 y([60])48 b(Y.)c(Last.)77 b(A)44 b(relation)e(b)s(et)m(w)m(een)k(a.c.)e(sp)s (ectrum)g(of)f(ergo)s(dic)g(Jacobi)g(matrices)g(and)h(the)g(sp)s(ectra) g(of)82 3614 y(p)s(erio)s(dic)31 b(appro)m(ximan)m(ts.)43 b FA(Comm.)34 b(Math.)h(Phys.)p Fy(,)e(151\(1\):183{192,)d(1993.)-118 3818 y([61])48 b(Y.)42 b(Last.)71 b(Almost)40 b(ev)m(erything)j(ab)s (out)e(the)h(almost)e(Mathieu)i(op)s(erator.)f(I.)71 b(In)42 b FA(XIth)h(International)82 3938 y(Congr)-5 b(ess)31 b(of)g(Mathematic)-5 b(al)31 b(Physics)g(\(Paris,)h(1994\))p Fy(,)d(pages)h(366{372.)d(In)m(ternat.)j(Press,)h(Cam)m(bridge,)82 4058 y(MA,)j(1995.)-118 4262 y([62])48 b(J.)29 b(Moser.)36 b(Con)m(v)m(ergen)m(t)31 b(series)d(expansions)h(for)f(quasi-p)s(erio)s (dic)e(motions.)34 b FA(Math.)d(A)n(nn.)p Fy(,)e(169:136{176,)82 4382 y(1967.)-118 4585 y([63])48 b(J.)c(Moser.)75 b FA(Stable)44 b(and)g(r)-5 b(andom)44 b(motions)g(in)g(dynamic)-5 b(al)43 b(systems)p Fy(.)75 b(Princeton)43 b(Univ)m(ersit)m(y)h(Press,)82 4706 y(Princeton,)25 b(N.)f(J.,)h(1973.)h(With)d(sp)s(ecial)e(emphasis) i(on)g(celestial)e(mec)m(hanics,)26 b(Hermann)c(W)-8 b(eyl)23 b(Lectures,)82 4826 y(the)33 b(Institute)g(for)f(Adv)-5 b(anced)34 b(Study)-8 b(,)34 b(Princeton,)e(N.)h(J,)g(Annals)f(of)g (Mathematics)g(Studies,)h(No.)g(77.)-118 5030 y([64])48 b(J.)38 b(Moser.)58 b(An)37 b(example)g(of)f(sc)m(hr\177)-49 b(odinger)38 b(equation)f(with)g(almost)e(p)s(erio)s(dic)h(p)s(oten)m (tial)f(and)i(no)m(where)82 5150 y(dense)e(sp)s(ectrum.)43 b FA(Comment.)34 b(Math.)h(Helvetici)p Fy(,)e(56:198{224,)d(1981.)-118 5353 y([65])48 b(J.)39 b(Moser)g(and)f(J.)g(P\177)-49 b(osc)m(hel.)61 b(An)39 b(extension)g(of)e(a)h(result)g(b)m(y)h (dinaburg)f(and)g(sinai)f(on)h(quasi-p)s(erio)s(dic)82 5474 y(p)s(oten)m(tials.)43 b FA(Comment.)33 b(Math.)i(Helvetici)p Fy(,)e(59:39{85,)e(1984.)-118 5677 y([66])48 b(M.)e(G.)f(Nerurk)-5 b(ar.)82 b(On)45 b(the)h(construction)g(of)e(smo)s(oth)h(ergo)s(dic)f (sk)m(ew-pro)s(ducts.)84 b FA(Er)-5 b(go)g(dic)46 b(The)-5 b(ory)82 5798 y(Dynam.)34 b(Systems)p Fy(,)f(8\(2\):311{326,)d(1988.) 1876 6155 y(126)p eop %%Page: 127 131 127 130 bop -118 219 a Fy([67])48 b(S.)31 b(No)m(v)m(o,)h(C.)f(N)s (\023)-51 b(u)s(~)g(nez,)30 b(and)h(R.)f(Oba)m(y)m(a.)41 b(Ergo)s(dic)30 b(prop)s(erties)g(and)h(rotation)e(n)m(um)m(b)s(er)i (for)f(linear)f(Hamil-)82 339 y(tonian)j(systems.)45 b FA(J.)35 b(Di\013er)-5 b(ential)33 b(Equations)p Fy(,)g (148\(1\):148{185,)d(1998.)-118 539 y([68])48 b(A.)h(Nunes)h(and)e(X.)h (Y)-8 b(uan.)90 b(A)49 b(note)f(on)h(the)g(reducibilit)m(y)d(of)i (linear)f(di\013eren)m(tial)f(equations)j(with)82 659 y(quasip)s(erio)s(dic)31 b(co)s(e\016cien)m(ts.)45 b FA(Pr)-5 b(eprint)p Fy(,)33 b(2001.)-118 859 y([69])48 b(R.)28 b(Oba)m(y)m(a)h(and)f(M.)g(P)m(aramio.)35 b(Directional)25 b(di\013eren)m(tiabilit)m(y)g(of)i(the)i(rotation)d(n)m(um)m(b)s(er)j (for)e(the)h(almost)82 980 y(p)s(erio)s(dic)j(Sc)m(hr\177)-49 b(odinger)33 b(equation.)43 b FA(Duke)35 b(Math.)g(J.)p Fy(,)e(66:521{552,)d(1992.)-118 1180 y([70])48 b(L.)31 b(P)m(astur)h(and)f(A.)g(Figotin.)38 b FA(Sp)-5 b(e)g(ctr)g(a)33 b(of)g(r)-5 b(andom)32 b(and)h(almost-p)-5 b(erio)g(dic)31 b(op)-5 b(er)g(ators)p Fy(.)41 b(Springer-V)-8 b(erlag,)82 1300 y(Berlin,)32 b(1992.)-118 1500 y([71])48 b(M.)42 b(P)m(ostnik)m(o)m(v.)72 b FA(Lie)43 b(gr)-5 b(oups)43 b(and)f(Lie)h(algebr)-5 b(as)p Fy(.)70 b(\\Mir",)42 b(Mosco)m(w,)k (1986.)69 b(Lectures)43 b(in)e(geometry)-8 b(.)82 1621 y(Semester)34 b(V.)-118 1821 y([72])48 b(A.S.)35 b(Py)m(artli.)45 b(Diophan)m(tine)32 b(appro)m(ximations)g(on)i(submanifolds)e(of)h (euclidean)g(space.)48 b FA(F)-7 b(unkt.)35 b(A)n(nal.)82 1941 y(i.)g(Priloz.)p Fy(,)d(3:303{306,)f(1969.)-118 2141 y([73])48 b(M.)33 b(Reed)h(and)e(B.)h(Simon.)42 b FA(F)-7 b(unctional)33 b(A)n(nalysis)p Fy(,)f(v)m(olume)g(1.)44 b(Academic)32 b(Press,)i(1972.)-118 2341 y([74])48 b(W.)33 b(Rudin.)43 b FA(R)-5 b(e)g(al)35 b(and)f(Complex)f(A)n(nalysis)p Fy(.)43 b(McGra)m(w-Hill)30 b(Bo)s(ok)j(Co.,)g(New)g(Y)-8 b(ork,)33 b(1987.)-118 2541 y([75])48 b(H.)35 b(R)s(\177)-51 b(ussmann.)48 b(On)35 b(optimal)c(estimates)k(for)e(the)i(solutions)f (of)g(linear)f(partial)f(di\013eren)m(tial)g(equations)82 2661 y(of)47 b(\014rst)h(order)g(with)f(constan)m(t)h(co)s(e\016cien)m 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