%!PS-Adobe-2.0 %%Creator: dvips(k) 5.991 Copyright 2011 Radical Eye Software %%Title: arXiv:1208.2324v5 [hep-th] 13 Mar 2013 %%CreationDate: Mon Mar 18 05:17:46 2019 %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: EUFM10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z %+ NSR_superstring_measures_in_genus_5.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2019.03.18:0517 %%BeginProcSet: texc.pro 0 0 %! /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 /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}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: texps.pro 0 0 %! 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TeXDict begin %%EndSetup %%Page: 1 1 gsave %matrix defaultmatrix setmatrix 90 rotate /stampsize 20 def /Times-Roman findfont stampsize scalefont setfont currentfont /FontBBox get aload pop /pdf@top exch 1000 div stampsize mul def pop /pdf@bottom exch 1000 div stampsize mul def pop 252 -32 moveto currentpoint /pdf@lly exch pdf@bottom add def /pdf@llx exch 2 sub def 0.5 setgray (arXiv:1208.2324v5 [hep-th] 13 Mar 2013) show currentpoint /pdf@ury exch pdf@top add def /pdf@urx exch 2 add def /pdfmark where{pop}{userdict /pdfmark /cleartomark load put}ifelse [ /H /I /Border [0 0 1] /BS <</S/D/D[2 6]/W 1>> /Color [0 1 1] /Action << /Subtype /URI /URI (http://arxiv.org/abs/1208.2324v5)>> /Subtype /Link /Rect[pdf@llx pdf@lly pdf@urx pdf@ury] /ANN pdfmark grestore TeXDict begin HPSdict begin 1 0 bop 0 0 a SDict begin [/Producer (dvips + Distiller)/Title ()/Subject ()/Creator (LaTeX with hyperref package)/Author ()/Keywords () /DOCINFO pdfmark end 0 0 a 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 432 133 a SDict begin H.S end 432 133 a 432 133 a SDict begin H.R end 432 133 a 432 133 a SDict begin [/View [/XYZ H.V]/Dest (page.1) cvn /DEST pdfmark end 432 133 a 440 448 a SDict begin [/Count 0/Dest (section.1) cvn/Title (1 Introduction) /OUT pdfmark end 440 448 a 440 448 a SDict begin [/Count 0/Dest (section.2) cvn/Title (2 Definitions: the modular forms from OPSMY and Grushevsky) /OUT pdfmark end 440 448 a 440 448 a SDict begin [/Count 3/Dest (section.3) cvn/Title (3 Degeneration) /OUT pdfmark end 440 448 a 440 448 a SDict begin [/Count 2/Dest (subsection.3.1) cvn/Title (3.1 The expansion of G5\(5\)) /OUT pdfmark end 440 448 a 440 448 a SDict begin [/Count 0/Dest (subsubsection.3.1.1) cvn/Title (3.1.1 Expanding P\(V\)12) /OUT pdfmark end 440 448 a 440 448 a 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a(4)2701 3075 y(2)2739 3125 y Fz(\))p 3076 3155 4 117 v 666 3158 2412 4 v 665 3275 4 117 v 833 3245 a Ft(#)882 3257 y Fy(4)p 1085 3275 V 1298 3245 a Fi(Z)e Fs(\010)18 b Ft(A)1517 3209 y Fy(+)1517 3267 y(15)p 1797 3275 V 1848 3245 a Fz(\()1890 3212 y Fy(1)p 1891 3226 34 4 v 1891 3274 a(4)1934 3195 y(12)2004 3245 y Ft(;)c Fs(\000)2116 3212 y Fy(3)p 2115 3226 V 2115 3274 a(4)2158 3195 y(4)2196 3245 y Fz(\))p Ft(;)g Fz(\()2307 3212 y Fy(1)p 2307 3226 V 2307 3274 a(2)2350 3195 y(8)2387 3245 y Ft(;)g Fs(\000)2499 3212 y Fy(1)p 2499 3226 V 2499 3274 a(2)2542 3195 y(8)2579 3245 y Fz(\))p Ft(;)g Fz(\()2690 3212 y Fy(3)p 2691 3226 V 2691 3274 a(4)2734 3195 y(4)2771 3245 y Ft(;)g Fs(\000)2883 3212 y Fy(1)p 2882 3226 V 2882 3274 a(4)2925 3195 y(12)2996 3245 y Fz(\))p 3076 3275 4 117 v 666 3278 2412 4 v 665 3395 4 117 v 833 3365 a Ft(#)882 3377 y Fy(5)p 1085 3395 V 1226 3365 a Fz(\()p Ft(D)1327 3377 y Fy(8)1383 3365 y Fs(\010)k Ft(D)1535 3377 y Fy(8)1572 3365 y Fz(\))1604 3323 y Fy(+)p 1797 3395 V 2264 3365 a Fz(\()2306 3332 y Fy(1)p 2306 3346 34 4 v 2306 3394 a(2)2349 3316 y(8)2386 3365 y Ft(;)c Fz(0)2465 3335 y Fy(7)2502 3365 y Ft(;)g Fz(1\))p 3076 3395 4 117 v 666 3398 2412 4 v 665 3498 4 100 v 833 3468 a Ft(#)882 3480 y Fy(6)p 1085 3498 V 1293 3468 a Ft(E)1354 3480 y Fy(8)1410 3468 y Fs(\010)k Ft(E)1554 3480 y Fy(8)p 1797 3498 V 2424 3468 a Fz(-)p 3076 3498 V 666 3501 2412 4 v 665 3618 4 117 v 833 3588 a Ft(#)882 3600 y Fy(7)p 1085 3618 V 1373 3588 a Ft(D)1444 3553 y Fy(+)1442 3611 y(16)p 1797 3618 V 2344 3588 a Fz(\()2386 3556 y Fy(1)p 2386 3570 34 4 v 2386 3617 a(2)2430 3539 y(16)2500 3588 y Fz(\))p 3076 3618 4 117 v 666 3622 2412 4 v 440 3837 a(The)26 b(lattices)f(denoted)h(b)n(y)g(\003)1379 3807 y Fy(+)1461 3837 y Fz(are)f(obtained)g(b)n(y)h(taking)f(the)i(union)e (\003)16 b Fs([)g Fz(\()p Ft(v)2895 3849 y Fy(1)2949 3837 y Fz(+)g(\003\))g Fs([)g Ft(:)e(:)g(:)440 3937 y Fz(of)33 b(the)g(lattice)e(with)h(itself,)h(shifted)f(b)n(y)h(the)g (gluing)d(v)n(ectors)h Ft(v)2497 3949 y Fp(i)2525 3937 y Fz(.)53 b(F)-7 b(or)32 b(example,)f Ft(E)3170 3949 y Fy(8)3240 3937 y Fz(=)440 4048 y Ft(D)511 4013 y Fy(+)509 4070 y(8)589 4048 y Fz(=)23 b Ft(D)746 4060 y Fy(8)801 4048 y Fs([)c Fz(\(\()949 4016 y Fy(1)p 950 4030 34 4 v 950 4077 a(2)993 3999 y(8)1030 4048 y Fz(\))g(+)f Ft(D)1233 4060 y Fy(8)1270 4048 y Fz(\).)440 4239 y(The)32 b(Grushevsky)e (ansatz,)j(from)d([)1581 4239 y SDict begin H.S end 1581 4239 a Fz(32)1665 4239 y SDict begin 12 H.L end 1665 4239 a 1665 4239 a SDict begin [/Subtype /Link/Dest (cite.grushevsky) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1665 4239 a -1 w Fz(],)j(is)e(instead)f (built)h(using)f(Riemann)g(theta)i(func-)440 4389 y(tions,)d(de\014ned) g(as)g(follo)n(ws)d(for)j(a)g(theta)g(c)n(haracteristic)c Ft(m)h Fz(=)2453 4272 y Fl(\024)2500 4338 y Ft(\017)2497 4438 y(\016)2537 4272 y Fl(\025)2581 4389 y Fz(,)k(here)f(regarded)e (as)i(a)440 4543 y(v)n(ector)d(in)h Fi(C)840 4513 y Fy(2)p Fp(g)918 4543 y Fz(,)440 4775 y Ft(\022)495 4658 y Fl(\024)542 4725 y Ft(\017)539 4824 y(\016)579 4658 y Fl(\025)637 4775 y Fz(\()p Ft(z)t(;)14 b(\034)9 b Fz(\))23 b(:=)979 4696 y Fl(X)960 4875 y Fp(n)p Fx(2)p Fj(Z)1090 4858 y Fk(g)1133 4775 y Fz(exp)1274 4633 y Fl(\()1340 4775 y Ft(\031)s(i)1433 4658 y Fl(\022)1494 4775 y Ft(n)c Fz(+)1656 4719 y(1)p 1656 4756 42 4 v 1656 4832 a(2)1707 4775 y Ft(\017)1741 4658 y Fl(\023)1802 4675 y Fp(t)1845 4775 y Ft(\034)1904 4658 y Fl(\022)1966 4775 y Ft(n)f Fz(+)2127 4719 y(1)p 2127 4756 V 2127 4832 a(2)2178 4775 y Ft(\017)2212 4658 y Fl(\023)2292 4775 y Fz(+)g(2)p Ft(\031)s(i)2510 4658 y Fl(\022)2570 4775 y Ft(n)g Fz(+)2731 4719 y(1)p 2731 4756 V 2731 4832 a(2)2783 4775 y Ft(\017)2817 4658 y Fl(\023)2878 4675 y Fp(t)2921 4658 y Fl(\022)2982 4775 y Ft(z)j Fz(+)3135 4719 y(1)p 3135 4756 V 3135 4832 a(2)3187 4775 y Ft(\016)3227 4658 y Fl(\023)3288 4633 y(\))3369 4775 y Ft(:)3134 4966 y SDict begin H.S end 3134 4966 a 3134 4966 a SDict begin 15 H.A end 3134 4966 a 3134 4966 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.6) cvn /DEST pdfmark end 3134 4966 a Fz(\(2.6\))440 5124 y(Riemann)32 b(theta)j(functions)f(for)f Ft(z)38 b Fz(=)c(0)g(are)f(called)f (Riemann)h(theta)h(constan)n(ts.)56 b(The)440 5223 y(Riemann)27 b(theta)h(constan)n(ts)g(of)g(o)r(dd)h(c)n(haracteristics)24 b(are)k(zero)f(iden)n(tically)d(on)29 b Fs(H)3088 5235 y Fp(g)3126 5223 y Fz(.)40 b(W)-7 b(e)440 5376 y(will)25 b(write)h Ft(\022)848 5388 y Fp(m)934 5376 y Fz(:=)d Ft(\022)1100 5259 y Fl(\024)1147 5325 y Ft(\017)1144 5425 y(\016)1184 5259 y Fl(\025)1242 5376 y Fz(\(0)p Ft(;)14 b(\034)9 b Fz(\).)1851 5662 y(6)p eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 432 133 a SDict begin H.S end 432 133 a 432 133 a SDict begin H.R end 432 133 a 432 133 a SDict begin [/View [/XYZ H.V]/Dest (page.7) cvn /DEST pdfmark end 432 133 a 440 535 a Fz(The)30 b(mo)r(dular)e(forms)g(used)i(in)f([)1491 535 y SDict begin H.S end 1491 535 a Fz(32)1575 535 y SDict begin 12 H.L end 1575 535 a 1575 535 a SDict begin [/Subtype /Link/Dest (cite.grushevsky) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1575 535 a -1 w Fz(])i(are)e(de\014ned)h(as)g(follo)n(ws.)41 b(Let)30 b Ft(V)46 b Fs(\032)27 b Fi(F)2856 492 y Fy(\(2)p Fp(g)r Fy(\))2856 557 y(2)3015 535 y Fz(b)r(e)k(a)e(set)440 635 y(of)f(c)n(haracteristics)23 b(in)k(gen)n(us)g Ft(g)s Fz(.)36 b(Then,)28 b(w)n(e)f(de\014ne)1552 819 y Ft(P)12 b Fz(\()p Ft(V)19 b Fz(\))24 b(:=)1908 740 y Fl(Y)1882 918 y Fp(m)p Fx(2)p Fp(V)2053 819 y Ft(\022)2092 831 y Fp(m)2169 819 y Ft(:)3134 819 y SDict begin H.S end 3134 819 a 3134 819 a SDict begin 15 H.A end 3134 819 a 3134 819 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.7) cvn /DEST pdfmark end 3134 819 a Fz(\(2.7\))440 1099 y(No)n(w,)38 b(de\014ne)e Fs(S)967 1055 y Fy(\()p Fp(g)r Fy(\))961 1108 y Fp(p)1094 1099 y Fz(to)g(b)r(e)g(the)h(set)f(of)g(all) e Ft(p)p Fz(-dimensional)d(linear)i(subspaces)i(of)h Fi(F)3152 1055 y Fy(\(2)p Fp(g)r Fy(\))3152 1121 y(2)3281 1099 y Fz(.)440 1216 y(Then,)28 b(w)n(e)f(de\014ne)h(the)g(Grushevsky)e (forms)g Fs(f)p Ft(G)1972 1173 y Fy(\()p Fp(g)r Fy(\))1972 1225 y Fp(p)2063 1216 y Ft(;)14 b Fz(0)22 b Fs(\024)h Ft(p)g Fs(\024)f Ft(g)k Fs(2)d Fi(Z)p Fs(g)e Fz(as)27 b(follo)n(ws:)1427 1405 y Ft(G)1492 1371 y Fy(\()p Fp(g)r Fy(\))1492 1425 y Fp(p)1606 1405 y Fz(:=)1768 1326 y Fl(X)1716 1526 y Fp(V)15 b Fx(2S)1860 1496 y Ff(\()p Fk(g)q Ff(\))1856 1535 y Fk(p)1953 1405 y Ft(P)d Fz(\()p Ft(V)19 b Fz(\))2149 1371 y Fy(2)2182 1346 y Ff(4)p Fd(\000)p Fk(p)2295 1405 y Ft(:)3134 1405 y SDict begin H.S end 3134 1405 a 3134 1405 a SDict begin 15 H.A end 3134 1405 a 3134 1405 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.8) cvn /DEST pdfmark end 3134 1405 a Fz(\(2.8\))440 1695 y(These)32 b(forms)f(are)h(mo)r(dular)e(with)i(resp)r(ect)h(to)f (\000\(1)p Ft(;)14 b Fz(2\))2225 1707 y Fp(g)2296 1695 y Fz(and)32 b(of)h(w)n(eigh)n(t)e(8.)51 b(Note)33 b(that)440 1822 y(this)19 b(normalization)13 b(di\013ers)19 b(from)e(that)j(in)f ([)1822 1822 y SDict begin H.S end 1822 1822 a Fz(36)1906 1822 y SDict begin 12 H.L end 1906 1822 a 1906 1822 a SDict begin [/Subtype /Link/Dest (cite.latticevsriemann) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1906 1822 a -1 w Fz(])g(b)n(y)g(a)g(factor)g(of)2431 1730 y Fl(\020)2480 1822 y Fz(2)2532 1763 y Fk(p)p Ff(\()p Fk(p)p Fd(\000)p Ff(1\))p 2532 1779 180 3 v 2608 1812 a(2)2739 1760 y Fl(Q)2818 1780 y Fp(p)2818 1847 y(i)p Fy(=1)2943 1755 y Fl(\000)2981 1822 y Fz(2)3023 1792 y Fp(i)3069 1822 y Fs(\000)f Fz(1)3194 1755 y Fl(\001)3232 1730 y(\021)3281 1822 y Fz(,)440 1942 y(tak)n(en)27 b(to)g(b)r(e)h(1)g (for)f Ft(p)c Fz(=)f(0:)586 2185 y Ft(G)651 2150 y Fy(\()p Fp(g)r Fy(\))651 2205 y Fp(p)765 2185 y Fz(=)852 2043 y Fl( )918 2185 y Fz(2)970 2121 y Fk(p)p Ff(\()p Fk(p)p Fd(\000)p Ff(1\))p 970 2137 V 1045 2170 a(2)1214 2077 y Fp(p)1178 2106 y Fl(Y)1177 2283 y Fp(i)p Fy(=1)1298 2117 y Fl(\000)1337 2185 y Fz(2)1379 2150 y Fp(i)1424 2185 y Fs(\000)c Fz(1)1549 2117 y Fl(\001)1587 2043 y(!)1829 2106 y(X)1681 2306 y Fp(e)1712 2314 y Ff(1)1745 2306 y Fp(;:::)o(;e)1875 2314 y Fk(p)1910 2306 y Fx(2)p Fj(F)1990 2275 y Ff(\(2)p Fk(g)q Ff(\))1990 2324 y(2)1666 2372 y Fp(e)1697 2380 y Ff(1)1730 2372 y Fp(;:::)o(;e)1860 2380 y Fk(p)1906 2372 y Fy(lin)p Fp(:)p Fy(ind)p Fp(:)2126 2185 y Ft(P)12 b Fz(\(span)p Fs(f)p Ft(e)2471 2197 y Fy(1)2507 2185 y Ft(;)i(:)g(:)g(:)g(;)g(e)2731 2197 y Fp(p)2769 2185 y Fs(g)p Fz(\))2843 2150 y Fy(2)2876 2125 y Ff(4)p Fd(\000)p Fk(p)3134 2185 y SDict begin H.S end 3134 2185 a 3134 2185 a SDict begin 15 H.A end 3134 2185 a 3134 2185 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.9) cvn /DEST pdfmark end 3134 2185 a Fz(\(2.9\))440 2632 y(F)-7 b(rom)28 b([)683 2632 y SDict begin H.S end 683 2632 a Fz(36)767 2632 y SDict begin 12 H.L end 767 2632 a 767 2632 a SDict begin [/Subtype /Link/Dest (cite.latticevsriemann) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 767 2632 a -1 w Fz(])i(w)n(e)g(ha)n(v)n(e) e(sev)n(eral)f(linear)g(dep)r(endencies)i(b)r(et)n(w)n(een)h(lattice)e (theta)i(series)e(and)440 2731 y(Riemann)18 b(theta)j(constan)n(ts.)34 b(In)20 b(the)h(presen)n(t)f(notation)f(they)h(lo)r(ok)f(as)g(follo)n (ws,)g(for)h Ft(p)j Fs(\024)g Fz(4:)544 3002 y Ft(G)609 2967 y Fy(\()p Fp(g)r Fy(\))609 3022 y Fp(p)723 3002 y Fz(=)854 2894 y Fp(p)811 2923 y Fl(X)811 3102 y Fp(k)q Fy(=0)932 3002 y Fz(\()p Fs(\000)p Fz(1\))1103 2967 y Fp(k)q Fy(+)p Fp(p)1247 3002 y Fs(\001)18 b Fz(2)1340 2938 y Fk(k)q Ff(\()p Fk(k)q Ff(+2\()p Fk(g)q Fd(\000)p Fk(p)p Ff(\)+1\))p 1340 2954 403 3 v 1527 2987 a(2)1775 3002 y Fs(\001)1816 2860 y Fl( )1917 2898 y Fp(k)1883 2923 y Fl(Y)1882 3100 y Fp(i)p Fy(=1)1989 3002 y Fz(\(2)2063 2967 y Fp(i)2109 3002 y Fs(\000)g Fz(1\))2280 2894 y Fp(p)p Fx(\000)p Fp(k)2288 2923 y Fl(Y)2288 3100 y Fp(i)p Fy(=1)2403 3002 y Fz(\(2)2477 2967 y Fp(i)2523 3002 y Fs(\000)g Fz(1\))2680 2860 y Fl(!)2745 2866 y Fx(\000)p Fy(1)2848 3002 y Ft(#)2897 2959 y Fy(\()p Fp(g)r Fy(\))2897 3027 y Fp(k)3092 3002 y SDict begin H.S end 3092 3002 a 3092 3002 a SDict begin 15 H.A end 3092 3002 a 3092 3002 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.10) cvn /DEST pdfmark end 3092 3002 a Fz(\(2.10\))440 3273 y(where)680 3211 y Fl(Q)758 3231 y Fp(k)758 3298 y(i)p Fy(=1)870 3273 y Fz(\(2)944 3243 y Fp(i)990 3273 y Fs(\000)g Fz(1\))28 b(is)e(tak)n(en)h(to)h(b)r(e)g(1)f(for)g Ft(k)f Fz(=)c(0.)440 3464 y(Throughout)k(the)i(pap)r(er)g(w)n(e)f(will)e(denote)1542 3648 y Ft(f)1592 3613 y Fy(\()p Fp(g)r Fy(\))1706 3648 y Fz(:=)d Ft(#)1865 3605 y Fy(\()p Fp(g)r Fy(\))1865 3670 y(5)1974 3648 y Fs(\000)c Ft(G)2122 3613 y Fy(\()p Fp(g)r Fy(\))2122 3668 y Fp(g)3092 3648 y SDict begin H.S end 3092 3648 a 3092 3648 a SDict begin 15 H.A end 3092 3648 a 3092 3648 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11) cvn /DEST pdfmark end 3092 3648 a Fz(\(2.11\))1531 3800 y Ft(J)1585 3765 y Fy(\()p Fp(g)r Fy(\))1699 3800 y Fz(:=)k Ft(#)1858 3756 y Fy(\()p Fp(g)r Fy(\))1858 3822 y(6)1967 3800 y Fs(\000)c Ft(#)2099 3756 y Fy(\()p Fp(g)r Fy(\))2099 3822 y(7)2190 3800 y Ft(:)3092 3800 y SDict begin H.S end 3092 3800 a 3092 3800 a SDict begin 15 H.A end 3092 3800 a 3092 3800 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.12) cvn /DEST pdfmark end 3092 3800 a Fz(\(2.12\))440 3984 y(It)28 b(w)n(as)e(sho)n(wn)h(in)f([)1060 3984 y SDict begin H.S end 1060 3984 a Fz(36)1144 3984 y SDict begin 12 H.L end 1144 3984 a 1144 3984 a SDict begin [/Subtype /Link/Dest (cite.latticevsriemann) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1144 3984 a -1 w Fz(])i(that)f Ft(f)1423 3954 y Fy(\()p Fp(g)r Fy(\))1541 3984 y Fz(v)-5 b(anishes)26 b(iden)n(tically)d(on)k(the)h(Jacobian)c(lo)r(cus)j Fs(J)3139 3996 y Fp(g)3205 3984 y Fz(for)440 4084 y Ft(g)33 b Fs(\024)e Fz(4.)51 b(In)32 b(the)h(presen)n(t)e(pap)r(er)h(w)n(e)g(sho)n(w)g (that)g Ft(f)2084 4054 y Fy(\(5\))2205 4084 y Fz(do)r(es)g(not)h(v)-5 b(anish)31 b(iden)n(tically)d(on)440 4183 y Fs(J)496 4195 y Fy(5)534 4183 y Fz(.)440 4339 y SDict begin H.S end 440 4339 a 440 4339 a SDict begin 12 H.A end 440 4339 a 440 4339 a SDict begin [/View [/XYZ H.V]/Dest (section.3) cvn /DEST pdfmark end 440 4339 a 179 x Fu(3.)47 b(Degeneration)440 4789 y Fz(The)23 b(conjecture)f(whic)n(h)g(w)n(e)g (in)n(v)n(estigate)e(and)i(dispro)n(v)n(e)e(in)i(this)g(section)g(is)f (whether)i Ft(G)3215 4745 y Fy(\(5\))3215 4811 y(5)440 4906 y Fz(and)k Ft(#)650 4863 y Fy(\(5\))650 4928 y(5)767 4906 y Fz(agree)f(on)i(the)g(Jacobian)d(lo)r(cus)h Fs(J)1856 4918 y Fy(5)1894 4906 y Fz(.)440 5114 y(T)-7 b(o)30 b(sho)n(w)f(that)i Ft(f)1006 5084 y 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b(w)n(e)f(consider)e(subspaces)h(con)n(taining)e(only)i(c)n (haracteristics)d(of)k(the)h(form)648 3812 y Ft(m)27 b Fz(=)840 3695 y Fl(\024)884 3761 y Fz(0)85 b Ft(\017)884 3861 y Fs(\003)d Ft(\016)1048 3695 y Fl(\025)1092 3812 y Fz(.)45 b(Th)n(us,)31 b(expanding)d Ft(P)12 b Fz(\()p Ft(V)1943 3824 y Fy(1)1981 3812 y Fz(\))31 b(for)e Ft(V)2221 3824 y Fy(1)2289 3812 y Fz(of)h(this)g(t)n(yp)r(e,)h(using)f(\()3017 3812 y SDict begin H.S end 3017 3812 a Fz(3.8)3123 3758 y SDict begin H.R end 3123 3758 a 3123 3812 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.8) cvn H.B /ANN pdfmark end 3123 3812 a Fz(\))q(,)h(w)n(e)648 3950 y(get)663 4142 y Ft(P)12 b Fz(\()p Ft(V)808 4154 y Fy(1)846 4142 y Fz(\))23 b(=)1023 4063 y Fl(Y)989 4241 y Fp(m)p Fx(2)p Fp(V)1132 4249 y Ff(1)1178 4142 y Ft(\022)1217 4157 y Fp(\031)r Fy(\()p Fp(m)p Fy(\))1391 4142 y Fz(+)18 b(2)p Ft(q)1556 4107 y Fy(1)p Fp(=)p Fy(2)1701 4063 y Fl(X)1674 4241 y Fp(m)p 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Ft(V)1704 543 y Fy(1)1741 531 y Fz(.)38 b(Unless)26 b Ft(m)e Fz(=)e Ft(n)d Fz(+)f(\001,)28 b(eac)n(h)f(term)g Ft(m;)14 b(n)27 b Fz(in)g(the)648 631 y(summation)f(in)j(the)h(third)f (term)g(from)f(\()1964 631 y SDict begin H.S end 1964 631 a Fz(3.10)2112 577 y SDict begin H.R end 2112 577 a 2112 631 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.10) cvn H.B /ANN pdfmark end 2112 631 a Fz(\))i(will)d(b)r(e)j(canceled) e(b)n(y)i(a)f Ft(m)20 b Fz(+)f(\001)p Ft(;)14 b(n)648 731 y Fz(term.)34 b(Com)n(bining)21 b(these)i(facts,)i(w)n(e)f(can)f (rewrite)f(the)j(ab)r(o)n(v)n(e)d(form)n(ula)f(as)i(follo)n(ws:)785 890 y Ft(P)12 b Fz(\()p Ft(V)930 902 y Fy(1)968 890 y Fz(\))23 b(=)1192 811 y Fl(Y)1111 993 y Fp(m)p Fx(2)p Fp(\031)r Fy(\()p Fp(V)1321 1001 y Ff(1)1353 993 y Fy(\))1393 890 y Ft(\022)1434 856 y Fy(2)1432 911 y Fp(m)1513 890 y Fs(\000)18 b Fz(4)p Ft(q)1766 811 y Fl(X)1692 993 y Fp(m)p Fx(2)p Fp(\031)r Fy(\()p Fp(V)1902 1001 y Ff(1)1934 993 y Fy(\))1974 890 y Ft(\022)2015 856 y Fy(2)2013 911 y Fp(m)2076 890 y Fz(\()p Ft(\034)5 b(;)14 b(z)t Fz(\))2346 811 y Fl(Y)2275 993 y Fp(n)p Fx(2)p Fp(\031)r Fy(\()p Fp(V)2467 1001 y Ff(1)2499 993 y Fy(\))2324 1057 y Fp(n)p Fx(6)p Fy(=)p Fp(m)2538 890 y Ft(\022)2579 856 y Fy(2)2577 911 y Fp(n)2641 890 y Fz(+)k Ft(O)r Fz(\()p Ft(q)2861 856 y Fy(2)2899 890 y Fz(\))p Ft(:)3092 890 y SDict begin H.S end 3092 890 a 3092 890 a SDict begin 15 H.A end 3092 890 a 3092 890 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.11) cvn /DEST pdfmark end 3092 890 a Fz(\(3.11\))648 1190 y(Expanding)25 b(the)j(square)e(ro)r(ot)h(then)h(easily)d(yields) 749 1386 y Ft(P)12 b Fz(\()p Ft(V)894 1398 y Fy(1)932 1386 y Fz(\))974 1329 y Ff(1)p 974 1338 29 3 v 974 1372 a(2)1040 1386 y Fz(=)1208 1307 y Fl(Y)1127 1489 y Fp(m)p Fx(2)p Fp(\031)r Fy(\()p Fp(V)1337 1497 y Ff(1)1369 1489 y Fy(\))1409 1386 y Ft(\022)1448 1398 y Fp(m)1530 1386 y Fs(\000)18 b Fz(2)p Ft(q)1782 1307 y Fl(X)1708 1489 y Fp(m)p Fx(2)p Fp(\031)r Fy(\()p Fp(V)1918 1497 y Ff(1)1950 1489 y Fy(\))2000 1330 y Ft(\022)2041 1300 y Fy(2)2039 1350 y Fp(m)2102 1330 y Fz(\()p Ft(\034)5 b(;)14 b(z)t Fz(\))p 2000 1367 287 4 v 2000 1443 a Ft(\022)2041 1419 y Fy(2)2039 1464 y Fp(m)2102 1443 y Fz(\()p Ft(\034)5 b(;)14 b Fz(0\))2382 1307 y Fl(Y)2310 1489 y Fp(n)p Fx(2)p Fp(\031)r Fy(\()p Fp(V)2502 1497 y Ff(1)2535 1489 y Fy(\))2574 1386 y Ft(\022)2613 1398 y Fp(n)2677 1386 y Fz(+)k Ft(O)r Fz(\()p Ft(q)2897 1352 y Fy(2)2935 1386 y Fz(\))p Ft(:)3092 1386 y SDict begin H.S end 3092 1386 a 3092 1386 a SDict begin 15 H.A end 3092 1386 a 3092 1386 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.12) cvn /DEST pdfmark end 3092 1386 a Fz(\(3.12\))648 1626 y(Finally)-7 b(,)24 b(w)n(e)j(use)g(the)h(heat)f(equation)f(for)g(the)i(theta)g(functions,) e(where)h Ft(\016)3020 1638 y Fp(ij)3106 1626 y Fz(is)f(the)648 1725 y(Kronec)n(k)n(er)e(delta,)1530 1865 y Ft(@)1579 1835 y Fy(2)1616 1865 y Ft(\022)1655 1877 y Fp(m)p 1505 1902 238 4 v 1505 1978 a Ft(@)5 b(z)1593 1990 y Fp(i)1620 1978 y Ft(@)g(z)1708 1990 y Fp(j)1775 1921 y Fz(=)23 b(2)p Ft(\031)s(i)p Fz(\(1)18 b(+)g Ft(\016)2196 1933 y Fp(ij)2254 1921 y Fz(\))2296 1865 y Ft(@)5 b(\022)2384 1877 y Fp(m)p 2296 1902 151 4 v 2300 1978 a Ft(@)g(\034)2385 1990 y Fp(ij)3092 1921 y SDict begin H.S end 3092 1921 a 3092 1921 a SDict begin 15 H.A end 3092 1921 a 3092 1921 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.13) cvn /DEST pdfmark end 3092 1921 a Fz(\(3.13\))648 2121 y(and)32 b(the)g(fact)h(that)f Ft(\022)1353 2133 y Fp(m)1416 2121 y Fz(\()p Ft(z)t Fz(\))g(is)f(an)h(ev)n(en)g(function)g(of)g Ft(z)j Fz(whenev)n(er)d Ft(m)g Fz(is)f(an)h(ev)n(en)648 2220 y(c)n(haracteristic,)23 b(to)28 b(obtain)807 2407 y Ft(@)856 2376 y Fy(2)893 2407 y Ft(P)12 b Fz(\()p Ft(V)1038 2419 y Fy(1)1075 2407 y Fz(\))1117 2354 y Ff(1)p 1118 2363 29 3 v 1118 2396 a(2)p 807 2444 354 4 v 865 2520 a Ft(@)5 b(z)953 2532 y Fp(i)980 2520 y Ft(@)g(z)1068 2532 y Fp(j)1170 2317 y Fl(\014)1170 2367 y(\014)1170 2417 y(\014)1170 2467 y(\014)1170 2517 y(\014)1198 2571 y Fp(z)r Fy(=0)1343 2463 y Fz(=)23 b Fs(\000)p Fz(8)p Ft(\031)s(i)p Fz(\(1)17 b(+)h Ft(\016)1828 2475 y Fp(ij)1886 2463 y Fz(\))p Ft(q)1972 2296 y Fl(0)1972 2445 y(@)2119 2384 y(X)2045 2566 y Fp(m)p Fx(2)p Fp(\031)r Fy(\()p Fp(V)2255 2574 y Ff(1)2287 2566 y Fy(\))2337 2407 y Ft(@)5 b(\022)2425 2419 y Fp(m)p 2337 2444 151 4 v 2341 2520 a Ft(@)g(\034)2426 2532 y Fp(ij)2534 2384 y Fl(Y)2511 2563 y Fp(n)p Fx(6)p Fy(=)p Fp(m)2676 2463 y Ft(\022)2715 2475 y Fp(n)2760 2296 y Fl(1)2760 2445 y(A)2852 2463 y Fz(+)18 b Ft(O)r Fz(\()p Ft(q)3072 2428 y Fy(2)3110 2463 y Fz(\))p Ft(:)3092 2669 y SDict begin H.S end 3092 2669 a 3092 2669 a SDict begin 15 H.A end 3092 2669 a 3092 2669 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.14) cvn /DEST pdfmark end 3092 2669 a Fz(\(3.14\))648 2818 y(Note)27 b(that)g Ft(P)12 b Fz(\()p Ft(V)1172 2830 y Fy(1)1210 2818 y Fz(\))27 b(is)f(an)g(ev)n(en)h(function)f(of)h Ft(z)j Fz(and)d(th)n(us)g(the)g(o)r(dd)g(partial)d(deriv)-5 b(a-)648 2917 y(tiv)n(es)26 b(v)-5 b(anish)26 b(\(up)i(to)g Ft(O)r Fz(\()p Ft(q)1489 2887 y Fy(2)1527 2917 y Fz(\)\).)440 2944 y SDict begin H.S end 440 2944 a 440 2944 a SDict begin 12 H.A end 440 2944 a 440 2944 a SDict begin [/View [/XYZ H.V]/Dest (Item.6) cvn /DEST pdfmark end 440 2944 a 541 3066 a Fz(2.)42 b(Next,)31 b(let)f Ft(V)1048 3078 y Fy(2)1114 3066 y Fs(2)f(S)1254 3022 y Fy(\(5\))1248 3088 y(5)1374 3066 y Fz(con)n(tain)g(b)r(oth)i(c)n(haracteristics)26 b(of)31 b(the)g(form)e Ft(m)f Fz(=)3052 2949 y Fl(\024)3096 3015 y Fz(0)86 b Ft(\017)3096 3115 y Fz(0)c Ft(\016)3260 2949 y Fl(\025)648 3273 y Fz(and)27 b(of)h(the)g(form)e Ft(m)d Fz(=)1426 3156 y Fl(\024)1470 3223 y Fz(1)86 b Ft(\017)1470 3322 y Fz(0)d Ft(\016)1635 3156 y Fl(\025)1679 3273 y Fz(,)27 b(but)i(none)e(with)g Ft(m)2342 3285 y Fx(\003)2403 3273 y Fz(=)c(1.)648 3411 y(If)29 b(there)f(is)f(at)i (least)e(one)h(elemen)n(t)f Ft(m)d Fs(2)h Ft(V)2013 3423 y Fy(2)2079 3411 y Fz(suc)n(h)k(that)f Ft(m)2521 3381 y Fx(\003)2584 3411 y Fz(=)c(1)k(it)g(is)g(easy)f(to)i(see)648 3511 y(that)f(for)f(exactly)f(half)h(of)h(the)g(elemen)n(ts)e Ft(n)d Fs(2)h Ft(V)2183 3523 y Fy(2)2248 3511 y Fz(w)n(e)k(will)d(ha)n (v)n(e)h Ft(n)2769 3480 y Fx(\003)2831 3511 y Fz(=)d(1)k(while)f(for) 648 3610 y(the)34 b(other)f(half)h(w)n(e)f(will)e(ha)n(v)n(e)i Ft(n)1730 3580 y Fx(\003)1802 3610 y Fz(=)g(0.)56 b(Therefore,)34 b(using)g(\()2685 3610 y SDict begin H.S end 2685 3610 a Fz(3.8)2792 3557 y SDict begin H.R end 2792 3557 a 2792 3610 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.8) cvn H.B /ANN pdfmark end 2792 3610 a Fz(\))g(and)g(\()3058 3610 y SDict begin H.S end 3058 3610 a Fz(3.9)3164 3557 y SDict begin H.R end 3164 3557 a 3164 3610 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.9) cvn H.B /ANN pdfmark end 3164 3610 a Fz(\))g(to)648 3710 y(expand)27 b(all)e(theta)j (constan)n(ts,)f(w)n(e)g(ha)n(v)n(e)881 3874 y Ft(P)12 b Fz(\()p Ft(V)1026 3886 y Fy(2)1064 3874 y Fz(\))23 b(=)g(2)1249 3840 y Fy(16)1319 3874 y Ft(q)1359 3840 y Fy(2)1446 3795 y Fl(Y)1411 3973 y Fp(m)p Fx(2)p Fp(V)1554 3981 y Ff(2)1410 4034 y Fp(m)1469 4009 y Fd(\003)1504 4034 y Fy(=0)1601 3874 y Ft(\022)1640 3889 y Fp(\031)r Fy(\()p Fp(m)p Fy(\))1796 3874 y Fz(\()p Ft(\034)5 b(;)14 b Fz(0\))2021 3795 y Fl(Y)1995 3973 y Fp(n)p Fx(2)p Fp(V)2120 3981 y Ff(2)1994 4034 y Fp(n)2035 4009 y Fd(\003)2070 4034 y Fy(=1)2168 3874 y Ft(\022)2207 3889 y Fp(\031)r Fy(\()p Fp(n)p Fy(\))2345 3874 y Fz(\()p Ft(\034)5 b(;)2465 3818 y(z)p 2465 3855 43 4 v 2465 3931 a Fz(2)2517 3874 y(\))19 b(+)f Ft(O)r Fz(\()p Ft(q)2788 3840 y Fy(3)2826 3874 y Fz(\))3092 3874 y SDict begin H.S end 3092 3874 a 3092 3874 a SDict begin 15 H.A end 3092 3874 a 3092 3874 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.15) cvn /DEST pdfmark end 3092 3874 a Fz(\(3.15\))648 4154 y(Similar)i(to)25 b(case)e(1\))i(ab)r(o)n(v)n(e,)f(the)h Ft(\031)s Fz(\()p Ft(m)p Fz(\))h(are)e(pairwise)d(equal)j(and)g(the)h(corresp)r(ond-)648 4254 y(ing)h(pairs)g(of)h Ft(m)h Fz(di\013er)e(only)h(in)f(the)i(comp)r (onen)n(t)f Ft(m)2319 4224 y Fx(\003)2357 4254 y Fz(.)37 b(Th)n(us,)27 b(w)n(e)g(end)h(up)g(with)1076 4445 y Ft(P)12 b Fz(\()p Ft(V)1221 4457 y Fy(2)1258 4445 y Fz(\))1300 4388 y Ff(1)p 1301 4397 29 3 v 1301 4430 a(2)1366 4445 y Fz(=)23 b(2)1496 4410 y Fy(8)1533 4445 y Ft(q)1573 4333 y Fl(s)p 1656 4333 889 4 v 1737 4366 a(Y)1656 4548 y Fp(m)p Fx(2)p Fp(\031)r Fy(\()p Fp(V)1866 4556 y Ff(2)1898 4548 y Fy(\))1937 4445 y Ft(\022)1976 4457 y Fp(m)2040 4445 y Fz(\()p Ft(\034)5 b(;)14 b Fz(0\))g Ft(\022)2277 4457 y Fp(m)2339 4445 y Fz(\()p Ft(\034)5 b(;)2460 4389 y(z)p 2460 4426 43 4 v 2460 4502 a Fz(2)2512 4445 y(\))19 b(+)f Ft(O)r Fz(\()p Ft(q)2783 4410 y Fy(2)2821 4445 y Fz(\))p Ft(:)3092 4445 y SDict begin H.S end 3092 4445 a 3092 4445 a SDict begin 15 H.A end 3092 4445 a 3092 4445 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.16) cvn /DEST pdfmark end 3092 4445 a Fz(\(3.16\))648 4697 y(Also,)32 b(recalling)c(that)33 b(all)d Ft(m)h Fs(2)h Ft(\031)s Fz(\()p Ft(V)1827 4709 y Fy(2)1865 4697 y Fz(\))h(are)f(ev)n(en)g(and)g (applying)e(the)j(theta)f(heat)648 4797 y(equation)26 b(w)n(e)h(\014nd)785 4958 y Ft(@)834 4928 y Fy(2)871 4958 y Ft(P)12 b Fz(\()p Ft(V)1016 4970 y Fy(2)1054 4958 y Fz(\))1096 4906 y Ff(1)p 1096 4915 29 3 v 1096 4948 a(2)p 785 4995 354 4 v 843 5071 a Ft(@)5 b(z)931 5083 y Fp(i)958 5071 y Ft(@)g(z)1046 5083 y Fp(j)1148 4869 y Fl(\014)1148 4919 y(\014)1148 4969 y(\014)1148 5018 y(\014)1148 5068 y(\014)1176 5122 y Fp(z)r Fy(=0)1321 5014 y Fz(=)23 b(2)1451 4980 y Fy(5)1488 5014 y Ft(q)1615 4935 y Fl(X)1541 5117 y Fp(m)p Fx(2)p Fp(\031)r Fy(\()p Fp(V)1751 5125 y Ff(2)1784 5117 y Fy(\))1858 4958 y Ft(@)1907 4928 y Fy(2)1944 4958 y Ft(\022)1983 4970 y Fp(m)p 1833 4995 238 4 v 1833 5071 a Ft(@)5 b(z)1921 5083 y Fp(i)1948 5071 y Ft(@)g(z)2036 5083 y Fp(j)2117 4935 y Fl(Y)2094 5114 y Fp(n)p Fx(6)p Fy(=)p Fp(m)2259 5014 y Ft(\022)2298 5026 y Fp(n)2362 5014 y Fz(+)18 b Ft(O)r Fz(\()p Ft(q)2582 4980 y Fy(2)2620 5014 y Fz(\))3092 5014 y SDict begin H.S end 3092 5014 a 3092 5014 a SDict begin 15 H.A end 3092 5014 a 3092 5014 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.17) cvn /DEST pdfmark end 3092 5014 a Fz(\(3.17\))1321 5287 y(=)23 b(64)p Ft(\031)s(i)p Fz(\(1)17 b(+)h Ft(\016)1783 5299 y Fp(ij)1841 5287 y Fz(\))p Ft(q)2002 5208 y Fl(X)1927 5390 y Fp(m)p Fx(2)p Fp(\031)r Fy(\()p Fp(V)2137 5398 y Ff(2)2170 5390 y Fy(\))2219 5231 y Ft(@)5 b(\022)2307 5243 y Fp(m)p 2219 5268 151 4 v 2223 5344 a Ft(@)g(\034)2308 5356 y Fp(ij)2416 5208 y Fl(Y)2394 5387 y Fp(n)p Fx(6)p Fy(=)p Fp(m)2559 5287 y Ft(\022)2598 5299 y Fp(n)2661 5287 y Fz(+)18 b Ft(O)r Fz(\()p Ft(q)2881 5253 y Fy(2)2919 5287 y Fz(\))p Ft(:)3092 5287 y SDict begin H.S end 3092 5287 a 3092 5287 a SDict begin 15 H.A end 3092 5287 a 3092 5287 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.18) cvn /DEST pdfmark end 3092 5287 a Fz(\(3.18\))1831 5662 y(10)p eop end end %%Page: 11 11 TeXDict begin HPSdict begin 11 10 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 432 133 a SDict begin H.S end 432 133 a 432 133 a SDict begin H.R end 432 133 a 432 133 a SDict begin [/View [/XYZ H.V]/Dest (page.11) cvn /DEST pdfmark end 432 133 a 648 531 a Fz(Note)23 b(that)g Ft(P)12 b Fz(\()p Ft(V)1164 543 y Fy(2)1202 531 y Fz(\))1244 479 y Ff(1)p 1244 488 29 3 v 1244 521 a(2)1310 531 y Fz(is)22 b(an)h(ev)n(en)g(function)f(of) h Ft(z)k Fz(and)c(th)n(us)g(the)h(o)r(dd)f(partial)d(deriv)-5 b(a-)648 631 y(tiv)n(es)26 b(v)-5 b(anish)26 b(\(up)i(to)g Ft(O)r Fz(\()p Ft(q)1489 601 y Fy(2)1527 631 y Fz(\)\).)440 659 y SDict begin H.S end 440 659 a 440 659 a SDict begin 12 H.A end 440 659 a 440 659 a SDict begin [/View [/XYZ H.V]/Dest (Item.7) cvn /DEST pdfmark end 440 659 a 541 734 a Fz(3.)42 b(Last,)24 b(w)n(e)h(consider)d(subspaces)i(con)n (taining,)e(in)i(addition)e(to)j(c)n(haracteristics)20 b(con-)648 833 y(tained)31 b(in)g(subspaces)g(from)g(case)g(2\))h(ab)r (o)n(v)n(e,)g(c)n(haracteristics)27 b(of)32 b(the)h(form)d Ft(m)h Fz(=)648 866 y Fl(\024)691 932 y Fz(0)86 b Ft(\017)691 1032 y Fz(1)d Ft(\016)856 866 y Fl(\025)900 983 y Fz(.)35 b(These)21 b(do)h(not)f(ha)n(v)n(e)g(the)h(simple)d(pairings)f(observ)n (ed)i(ab)r(o)n(v)n(e,)i(but)g(w)n(e)g(can)648 1127 y(still)k(expand)i (the)i(theta)f(constan)n(ts)f(and)g(obtain)g(the)h(similar)24 b(expression)j(b)r(elo)n(w,)648 1227 y(but)j(it)g(cannot)g(b)r(e)h (simpli\014ed)26 b(as)k(easily)-7 b(.)42 b(This,)30 b(ho)n(w)n(ev)n (er,)f(will)f(turn)i(out)g(not)g(to)648 1327 y(b)r(e)i(necessary)d(for) i(our)g(purp)r(oses.)48 b(The)32 b(16)f(factors)f(of)i Ft(e)2510 1297 y Fp(\031)r(im)2633 1305 y Fd(\003)2703 1327 y Fz(together)f(yield)f(1,)648 1426 y(and)d(w)n(e)g(end)h(up)g (with)842 1634 y Ft(P)12 b Fz(\()p Ft(V)987 1646 y Fy(3)1025 1634 y Fz(\))1067 1577 y Ff(1)p 1067 1586 29 3 v 1067 1620 a(2)1132 1634 y Fz(=)23 b(2)1262 1600 y Fy(8)1299 1634 y Ft(q)1339 1519 y Fl(v)1339 1565 y(u)1339 1615 y(u)1339 1665 y(t)p 1426 1519 1140 4 v 1462 1555 a(Y)1427 1733 y Fp(m)p Fx(2)p Fp(V)1570 1741 y Ff(3)1426 1794 y Fp(m)1485 1769 y Fd(\003)1520 1794 y Fy(=0)1618 1634 y Ft(\022)1657 1649 y Fp(\031)r Fy(\()p Fp(m)p Fy(\))1813 1634 y Fz(\()p Ft(\034)5 b(;)14 b Fz(0\))2037 1555 y Fl(Y)2012 1733 y Fp(n)p Fx(2)p Fp(V)2137 1741 y Ff(3)2011 1794 y Fp(n)2052 1769 y Fd(\003)2086 1794 y Fy(=1)2184 1634 y Ft(\022)2223 1649 y Fp(\031)r Fy(\()p Fp(n)p Fy(\))2361 1634 y Fz(\()p Ft(\034)5 b(;)2481 1578 y(z)p 2481 1615 43 4 v 2481 1691 a Fz(2)2534 1634 y(\))18 b(+)g Ft(O)r Fz(\()p Ft(q)2804 1600 y Fy(2)2843 1634 y Fz(\))p Ft(:)3092 1634 y SDict begin H.S end 3092 1634 a 3092 1634 a SDict begin 15 H.A end 3092 1634 a 3092 1634 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.19) cvn /DEST pdfmark end 3092 1634 a Fz(\(3.19\))648 1951 y(F)-7 b(or)32 b(an)n(y)h(gen)n(us)g Ft(g)j Fz(there)d(will)e(b)r(e)j(at)f(least)f(2)2125 1921 y Fp(g)r Fx(\000)p Fy(2)2282 1951 y Fz(o)r(dd)i(c)n (haracteristics)29 b(in)k Ft(\031)s Fz(\()p Ft(V)3234 1963 y Fy(3)3272 1951 y Fz(\))648 2051 y(when)27 b Ft(V)912 2063 y Fy(3)978 2051 y Fz(is)f(of)i(this)f(t)n(yp)r(e.)37 b(Therefore,)26 b(w)n(e)h(ha)n(v)n(e)916 2225 y Ft(@)5 b(P)12 b Fz(\()p Ft(V)1110 2237 y Fy(3)1147 2225 y Fz(\))1189 2173 y Ff(1)p 1189 2182 29 3 v 1189 2215 a(2)p 916 2262 317 4 v 1016 2338 a Ft(@)5 b(z)1104 2350 y Fp(i)1242 2136 y Fl(\014)1242 2186 y(\014)1242 2236 y(\014)1242 2286 y(\014)1242 2335 y(\014)1269 2389 y Fp(z)r Fy(=0)1415 2282 y Fz(=)1522 2225 y Ft(@)1571 2195 y Fy(2)1608 2225 y Ft(P)12 b Fz(\()p Ft(V)1753 2237 y Fy(3)1791 2225 y Fz(\))1833 2173 y Ff(1)p 1833 2182 29 3 v 1833 2215 a(2)p 1522 2262 354 4 v 1580 2338 a Ft(@)5 b(z)1668 2350 y Fp(i)1695 2338 y Ft(@)g(z)1783 2350 y Fp(j)1886 2136 y Fl(\014)1886 2186 y(\014)1886 2236 y(\014)1886 2286 y(\014)1886 2335 y(\014)1913 2389 y Fp(z)r Fy(=0)2059 2282 y Fz(=)2172 2225 y Ft(@)2221 2195 y Fy(3)2258 2225 y Ft(P)12 b Fz(\()p Ft(V)2403 2237 y Fy(3)2441 2225 y Fz(\))2483 2173 y Ff(1)p 2483 2182 29 3 v 2483 2215 a(2)p 2166 2262 366 4 v 2166 2338 a Ft(@)5 b(z)2254 2350 y Fp(i)2281 2338 y Ft(@)g(z)2369 2350 y Fp(j)2404 2338 y Ft(@)g(z)2492 2350 y Fp(k)2542 2136 y Fl(\014)2542 2186 y(\014)2542 2236 y(\014)2542 2286 y(\014)2542 2335 y(\014)2569 2389 y Fp(z)r Fy(=0)2715 2282 y Fz(=)22 b(0)3092 2282 y SDict begin H.S end 3092 2282 a 3092 2282 a SDict begin 15 H.A end 3092 2282 a 3092 2282 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.20) cvn /DEST pdfmark end 3092 2282 a Fz(\(3.20\))648 2526 y(at)27 b(least)f(up)i(to)g Ft(O)r Fz(\()p Ft(q)1302 2496 y Fy(2)1340 2526 y Fz(\).)440 2634 y SDict begin H.S end 440 2634 a 440 2634 a SDict begin 12 H.A end 440 2634 a 440 2634 a SDict begin [/View [/XYZ H.V]/Dest (subsubsection.3.1.2) cvn /DEST pdfmark end 440 2634 a 186 x Fr(3.1.2.)45 b(The)31 b(expr)l(ession)f(for)h Ft(G)1454 2777 y Fy(\(5\))1454 2842 y(5)440 3037 y Fz(Let)d Fs(V)640 3049 y Fp(k)708 3037 y Fz(b)r(e)g(the)g(subset)g(of)g Fs(S)1370 2994 y Fy(\(5\))1364 3059 y(5)1487 3037 y Fz(con)n(taining)d (all)g(subspaces)i(from)f(case)h Ft(k)s Fz(\))h(ab)r(o)n(v)n(e.)35 b(Note)440 3154 y(that)24 b Ft(\031)s Fz(\()p Ft(V)c Fz(\),)25 b(for)e Ft(V)42 b Fs(62)24 b(V)1189 3166 y Fy(3)1226 3154 y Fz(,)h(is)e(a)g(totally)f(isotropic)f(elemen)n(t)h(of) i Fs(S)2465 3111 y Fy(\(4\))2459 3176 y(4)2554 3154 y Fz(,)h(and)f(in)f(fact)i Ft(\031)s Fz(\()p Fs(V)3147 3166 y Fy(1)3184 3154 y Fz(\))f(=)440 3271 y Ft(\031)s Fz(\()p Fs(V)573 3283 y Fy(2)611 3271 y Fz(\))k(is)e(the)i(set)g(of)f (all)f(totally)f(isotropic)f(elemen)n(ts)i(of)h Fs(S)2329 3228 y Fy(\(4\))2323 3293 y(4)2419 3271 y Fz(,)h(so)f Ft(G)2637 3228 y Fy(\(5\))2637 3293 y(5)2749 3271 y Fz(=)c Ft(G)2902 3228 y Fy(\(4\))2902 3293 y(4)3010 3271 y Fz(+)18 b Ft(O)r Fz(\()p Ft(q)s Fz(\).)440 3462 y(No)n(w,)27 b(com)n(bining)d(the)k(results)e(from)g(the)i(previous)e(subsection,) 547 3635 y Ft(G)612 3592 y Fy(\(5\))612 3657 y(5)725 3635 y Fz(=)863 3556 y Fl(X)812 3756 y Fp(V)15 b Fx(2S)956 3725 y Ff(\(5\))952 3774 y(5)1047 3635 y Ft(P)d Fz(\()p Ft(V)19 b Fz(\))1253 3578 y Ff(1)p 1253 3587 29 3 v 1253 3621 a(2)725 4035 y Fz(=)j Ft(G)877 3992 y Fy(\(4\))877 4057 y(4)985 4035 y Fz(+)c(2)1110 4001 y Fy(8)1161 4035 y Ft(q)1215 3819 y Fl(0)1215 3965 y(B)1215 4015 y(B)1215 4068 y(@)1352 3956 y(X)1301 4156 y Fp(V)d Fx(2S)1445 4126 y Ff(\(4\))1441 4174 y(4)1536 3916 y Fl(s)p 1619 3916 594 4 v 1644 3956 a(Y)1619 4135 y Fp(m)p Fx(2)p Fp(V)1790 4035 y Ft(\022)1829 4047 y Fp(m)1905 4035 y Ft(\022)1944 4047 y Fp(m)2008 4035 y Fz(\()p Ft(\034)5 b(;)2128 3979 y(z)p 2128 4016 43 4 v 2128 4092 a Fz(2)2180 4035 y(\))19 b Fs(\000)f Fz(2)2356 4001 y Fx(\000)p Fy(7)2477 3956 y Fl(X)2458 4135 y Fp(m)p Fx(2)p Fp(V)2639 3979 y Ft(\022)2680 3949 y Fy(2)2678 4000 y Fp(m)2741 3979 y Fz(\()p Ft(\034)5 b(;)14 b(z)t Fz(\))p 2639 4016 287 4 v 2732 4092 a Ft(\022)2773 4068 y Fy(2)2771 4113 y Fp(m)2966 3956 y Fl(Y)2950 4135 y Fp(n)p Fx(2)p Fp(V)3103 4035 y Ft(\022)3142 4047 y Fp(n)891 4471 y Fz(+)1005 4392 y Fl(X)970 4570 y Fp(V)1009 4578 y Ff(3)1041 4570 y Fx(2V)1128 4578 y Ff(3)1174 4355 y Fl(v)1174 4402 y(u)1174 4452 y(u)1174 4502 y(t)p 1262 4355 956 4 v 1298 4392 a(Y)1263 4570 y Fp(m)p Fx(2)p Fp(V)1406 4578 y Ff(3)1262 4631 y Fp(m)1321 4606 y Fd(\003)1356 4631 y Fy(=0)1453 4471 y Ft(\022)1492 4486 y Fp(\031)r Fy(\()p Fp(m)p Fy(\))1689 4392 y Fl(Y)1663 4570 y Fp(n)p Fx(2)p Fp(V)1788 4578 y Ff(3)1662 4631 y Fp(n)1703 4606 y Fd(\003)1738 4631 y Fy(=1)1836 4471 y Ft(\022)1875 4486 y Fp(\031)r Fy(\()p Fp(n)p Fy(\))2013 4471 y Fz(\()p Ft(\034)5 b(;)2133 4415 y(z)p 2133 4452 43 4 v 2133 4528 a Fz(2)2185 4471 y(\))2217 4254 y Fl(1)2217 4400 y(C)2217 4450 y(C)2217 4503 y(A)2309 4471 y Fz(+)18 b Ft(O)r Fz(\()p Ft(q)2529 4437 y Fy(2)2567 4471 y Fz(\))p Ft(:)3092 4471 y SDict begin H.S end 3092 4471 a 3092 4471 a SDict begin 15 H.A end 3092 4471 a 3092 4471 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.21) cvn /DEST pdfmark end 3092 4471 a Fz(\(3.21\))440 4778 y(Also,)26 b(this)h(giv)n(es)f(us)687 4953 y Ft(@)736 4923 y Fy(2)773 4953 y Ft(G)838 4910 y Fy(\(5\))838 4975 y(5)p 687 4990 241 4 v 689 5066 a Ft(@)5 b(z)777 5078 y Fp(i)804 5066 y Ft(@)g(z)892 5078 y Fp(j)938 4864 y Fl(\014)938 4913 y(\014)938 4963 y(\014)938 5013 y(\014)938 5063 y(\014)965 5117 y Fp(z)r Fy(=0)1111 5009 y Fz(=)22 b(56)p Ft(\031)s(i)p Fz(\(1)17 b(+)h Ft(\016)1572 5021 y Fp(ij)1631 5009 y Fz(\))p Ft(q)1767 4930 y Fl(X)1717 5130 y Fp(V)c Fx(2S)1860 5100 y Ff(\(4\))1856 5148 y(4)1970 4930 y Fl(X)1951 5108 y Fp(m)p Fx(2)p Fp(V)2132 4953 y Ft(@)5 b(\022)2220 4965 y Fp(m)p 2132 4990 151 4 v 2136 5066 a Ft(@)g(\034)2221 5078 y Fp(ij)2329 4930 y Fl(Y)2307 5109 y Fp(n)p Fx(6)p Fy(=)p Fp(m)2472 5009 y Ft(\022)2511 5021 y Fp(n)2574 5009 y Fz(+)18 b Ft(O)r Fz(\()p Ft(q)2794 4975 y Fy(2)2833 5009 y Fz(\))3092 5009 y SDict begin H.S end 3092 5009 a 3092 5009 a SDict begin 15 H.A end 3092 5009 a 3092 5009 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.22) cvn /DEST pdfmark end 3092 5009 a Fz(\(3.22\))1111 5332 y(=)k(56)p Ft(\031)s(i)p Fz(\(1)17 b(+)h Ft(\016)1572 5344 y Fp(ij)1631 5332 y Fz(\))p Ft(q)1713 5276 y(@)5 b(G)1827 5233 y Fy(\(4\))1827 5298 y(4)p 1713 5313 204 4 v 1743 5389 a Ft(@)g(\034)1828 5401 y Fp(ij)1945 5332 y Fz(+)18 b Ft(O)r Fz(\()p Ft(q)2165 5298 y Fy(2)2203 5332 y Fz(\))p Ft(:)3092 5332 y SDict begin H.S end 3092 5332 a 3092 5332 a SDict begin 15 H.A end 3092 5332 a 3092 5332 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.23) cvn /DEST pdfmark end 3092 5332 a Fz(\(3.23\))1831 5662 y(11)p eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 440 133 a SDict begin H.S end 440 133 a 440 133 a SDict begin H.R end 440 133 a 440 133 a SDict begin [/View [/XYZ H.V]/Dest (page.12) cvn /DEST pdfmark end 440 133 a 398 x Fz(And)32 b(\014nally)-7 b(,)30 b(as)h(the)g(con)n(tribution)e (from)h(all)f Ft(V)1994 543 y Fy(3)2061 531 y Fs(2)h(V)2197 543 y Fy(3)2265 531 y Fz(will)f(v)-5 b(anish)30 b(in)h Ft(z)h Fz(=)d(0)i(b)r(ecause)440 631 y Ft(\031)s Fz(\()p Ft(V)570 643 y Fy(3)608 631 y Fz(\))d(con)n(tains)e(o)r(dd)i(c)n (haracteristics,)23 b(w)n(e)k(can)g(see)h(that)1226 839 y Ft(G)1291 795 y Fy(\(5\))1291 861 y(5)1381 743 y Fl(\014)1381 793 y(\014)1381 843 y(\014)1409 897 y Fp(z)r Fy(=0)1554 839 y Fz(=)23 b(\(1)18 b(+)g(224)c Ft(q)s Fz(\))e Ft(G)2106 795 y Fy(\(4\))2106 861 y(4)2214 839 y Fz(+)18 b Ft(O)r Fz(\()p Ft(q)2434 804 y Fy(2)2472 839 y Fz(\))p Ft(:)3092 839 y SDict begin H.S end 3092 839 a 3092 839 a SDict begin 15 H.A end 3092 839 a 3092 839 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.24) cvn /DEST pdfmark end 3092 839 a Fz(\(3.24\))440 1075 y(Note)28 b(that,)h(b)r(ecause)f(the)h (\014rst)f(terms)f(from)g(the)i(expansion)d(of)j Ft(G)2589 1032 y Fy(\(1\))2589 1097 y(1)2678 1075 y Fz(\()p Ft(\025)p Fz(\))g(are)f(1)18 b(+)h(224)14 b Ft(q)s Fz(,)440 1192 y(this)27 b(is)f(consisten)n(t)g(with)i(the)g(factorization)c(prop)r (ert)n(y)i(for)h Ft(G)2417 1149 y Fy(\()p Fp(g)r Fy(\))2417 1202 y Fp(g)2508 1192 y Fz(.)440 1312 y SDict begin H.S end 440 1312 a 440 1312 a SDict begin 12 H.A end 440 1312 a 440 1312 a SDict begin [/View [/XYZ H.V]/Dest (subsection.3.2) cvn /DEST pdfmark end 440 1312 a 187 x Fr(3.2.)44 b(The)31 b(exp)l(ansion)f(of)h Ft(#)1317 1456 y Fy(\(5\))1317 1521 y(5)440 1732 y Fz(W)-7 b(e)39 b(will)d(no)n(w)i(do)g(the)h(same)f (for)g Ft(#)1633 1689 y Fy(\(5\))1633 1754 y(5)1761 1732 y Fz(as)g(done)g(ab)r(o)n(v)n(e)f(for)i Ft(G)2533 1689 y Fy(\(5\))2533 1754 y(5)2622 1732 y Fz(,)j(that)d(is,)h(tak)n(e)e(the) 440 1832 y(F)-7 b(ourier-Jacobi)23 b(expansion)j(and)h(calculate)e(the) j Ft(z)2062 1844 y Fp(i)2089 1832 y Ft(;)14 b(z)2165 1844 y Fp(j)2227 1832 y Fz(deriv)-5 b(ativ)n(es)25 b(of)i(the)h (\014rst)g(terms.)440 2023 y(Note)g(that)g(as)e Ft(#)971 2035 y Fy(5)1009 2023 y Fz(\()p Ft(\034)9 b Fz(\))1118 1993 y Fy(\()p Fp(g)r Fy(\))1233 2023 y Fz(:=)1343 1960 y Fl(P)1431 2048 y Fp(p)1465 2056 y Ff(1)1498 2048 y Fp(;:::)n(;p)1630 2056 y Fk(g)1665 2048 y Fx(2)p Fy(\003)1755 2056 y Ff(5)1806 2023 y Ft(e)1845 1993 y Fp(\031)r(i)p Fy(\()p Fp(p)1969 2002 y Fk(k)2005 1993 y Fx(\001)p Fp(p)2059 2002 y Fk(l)2083 1993 y Fy(\))p Fp(\034)2140 2002 y Fk(k)q(l)2200 2023 y Fz(,)28 b(w)n(e)f(can)g(write)653 2276 y Ft(#)702 2288 y Fy(5)753 2159 y Fl(\022)814 2225 y Ft(\025)84 b(z)989 2195 y Fp(t)818 2325 y Ft(z)102 b(\034)1017 2159 y Fl(\023)1102 2276 y Fz(=)1306 2197 y Fl(X)1189 2375 y Fp(p)1223 2383 y Ff(1)1256 2375 y Fp(;:::)o(;p)1389 2383 y Ff(5)1421 2375 y Fx(2)p Fy(\003)1511 2383 y Ff(5)1557 2276 y Ft(e)1596 2241 y Fp(\031)r(ip)1694 2249 y Ff(1)1727 2241 y Fx(\001)p Fp(p)1781 2249 y Ff(1)1814 2241 y Fp(\025)1857 2276 y Ft(e)1896 2241 y Fy(2)p Fp(\031)r(i)2005 2198 y Fe(P)2074 2260 y Fk(i)2112 2241 y Fp(p)2146 2249 y Ff(1)2179 2241 y Fp(p)2213 2249 y Fk(i)2239 2241 y Fp(z)2271 2249 y Fk(i)2301 2276 y Ft(e)2340 2240 y Fp(\031)r(i)2416 2196 y Fe(P)2485 2215 y Ff(5)2485 2259 y Fk(i;j)r(>)p Ff(1)2642 2240 y Fp(p)2676 2248 y Fk(i)2703 2240 y Fp(p)2737 2248 y Fk(j)2768 2240 y Fp(\034)2799 2248 y Fk(ij)2856 2276 y Ft(:)3092 2276 y SDict begin H.S end 3092 2276 a 3092 2276 a SDict begin 15 H.A end 3092 2276 a 3092 2276 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.25) cvn /DEST pdfmark end 3092 2276 a Fz(\(3.25\))440 2544 y(The)39 b(\014rst)h(term)e(in)g(the)i Ft(q)s Fz(-expansion)d(is)h(easy)h(to)g (obtain,)i(and)e(w)n(e)g(will)e(obtain)h(the)440 2643 y Ft(q)s Fz(-linear)25 b(term)h(as)h(in)g([)1159 2643 y SDict begin H.S end 1159 2643 a Fz(27)1243 2643 y SDict begin 12 H.L end 1243 2643 a 1243 2643 a SDict begin [/Subtype /Link/Dest (cite.matone2010getting) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1243 2643 a -1 w Fz(])h(b)n(y)f(writing)603 2842 y Ft(F)668 2808 y Fy(\()p Fp(g)r Fy(\))759 2842 y Fz(\()p Ft(\034)5 b(;)14 b(z)t Fz(\))23 b(:=)1319 2764 y Fl(X)1077 2953 y Fp(p)1111 2961 y Ff(1)1144 2953 y Fp(;:::)o(;p)1277 2961 y Fk(g)1311 2953 y Fx(2)p Fy(\()p Fp(D)1436 2961 y Ff(8)1469 2953 y Fx(\010)p Fp(D)1575 2961 y Ff(8)1607 2953 y Fy(\))1633 2930 y Ff(+)1694 2842 y Ft(e)1733 2806 y Fp(\031)r(i)1808 2763 y Fe(P)1878 2778 y Fk(g)1878 2825 y(i;j)r Ff(=1)2033 2806 y Fp(p)2067 2814 y Fk(i)2093 2806 y Fp(p)2127 2814 y Fk(j)2158 2806 y Fp(\034)2189 2814 y Fk(ij)2286 2764 y Fl(X)2266 2941 y Fy(~)-39 b Fp(p)p Fx(\001)6 b Fy(~)-39 b Fp(p)p Fy(=2)2446 2842 y Ft(e)2485 2808 y Fy(2)p Fp(\031)r(i)2594 2765 y Fe(P)2663 2780 y Fk(g)2663 2827 y(i)p Ff(=1)2778 2808 y Fy(~)g Fp(pp)2840 2816 y Fk(i)2867 2808 y Fp(z)2899 2816 y Fk(i)3092 2842 y SDict begin H.S end 3092 2842 a 3092 2842 a SDict begin 15 H.A end 3092 2842 a 3092 2842 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.26) cvn /DEST pdfmark end 3092 2842 a Fz(\(3.26\))440 3136 y(Clearly)-7 b(,)17 b(the)i(norm)e(2)h(v)n(ectors)f(are)g(\()p Ft(:)d(:)g(:)g(;)g Fs(\006)p Fz(1)p Ft(;)g(:)g(:)g(:)f(;)h Fs(\006)p Fz(1)p Ft(;)g(:)g(:)g(:)e(;)i Fz(0)2344 3106 y Fy(8)2381 3136 y Fz(\))19 b(and)f(\(0)2658 3106 y Fy(8)2695 3136 y Ft(;)c(:)g(:)g(:)f (;)h Fs(\006)p Fz(1)p Ft(;)g(:)g(:)g(:)f(;)h Fs(\006)p Fz(1)p Ft(;)g(:)g(:)g(:)o Fz(\),)440 3236 y(where)22 b Ft(:)14 b(:)g(:)23 b Fz(denotes)g(a)f(p)r(ossibly)f(empt)n(y)g (sequence)i(of)g(zero)r(es.)34 b(There)22 b(are)g(2)9 b Fs(\001)g Fz(4)g Fs(\001)2961 3169 y Fl(\000)2998 3199 y Fy(8)2998 3264 y(2)3031 3169 y Fl(\001)3092 3236 y Fz(=)23 b(224)440 3335 y(of)28 b(those.)440 3526 y(No)n(w)f(the)h (\014rst)f(terms)g(of)g(the)h(series)e(in)h Ft(q)j Fz(will)25 b(b)r(e:)1048 3754 y Ft(#)1097 3711 y Fy(\(5\))1097 3776 y(5)1200 3637 y Fl(\022)1261 3704 y Ft(\025)83 b(z)1435 3674 y Fp(t)1264 3803 y Ft(z)102 b(\034)1464 3637 y Fl(\023)1548 3754 y Fz(=)23 b Ft(#)1685 3711 y Fy(\(4\))1685 3776 y(5)1774 3754 y Fz(\()p Ft(\034)9 b Fz(\))20 b(+)e Ft(q)s(F)2091 3720 y Fy(\(4\))2180 3754 y Fz(\()p Ft(\034)5 b(;)14 b(z)t Fz(\))k(+)g Ft(O)r Fz(\()p Ft(q)2603 3720 y Fy(2)2641 3754 y Fz(\))p Ft(:)3092 3754 y SDict begin H.S end 3092 3754 a 3092 3754 a SDict begin 15 H.A end 3092 3754 a 3092 3754 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.27) cvn /DEST pdfmark end 3092 3754 a Fz(\(3.27\))440 4090 y(No)n(w)28 b(w)n(e)g(will)e(express)h(the)i Ft(z)1384 4102 y Fp(i)1412 4090 y Ft(z)1451 4102 y Fp(j)1485 4090 y Fz(-deriv)-5 b(ativ)n(es)25 b(of)k Ft(F)2088 4059 y Fy(\(4\))2177 4090 y Fz(,)g(the)g Ft(q)s Fz(-linear)c(term)j(from)f(\() 3101 4090 y SDict begin H.S end 3101 4090 a Fz(3.27)3249 4036 y SDict begin H.R end 3249 4036 a 3249 4090 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.27) cvn H.B /ANN pdfmark end 3249 4090 a Fz(\),)440 4208 y(as)f(done)g(ab)r(o)n(v)n(e) f(for)g Ft(G)1162 4165 y Fy(\(5\))1162 4230 y(5)1252 4208 y Fz(.)36 b(Because)26 b(the)g(norm)f(2)h(v)n(ectors)e(are)i(the)g (same)f(as)h(those)g(from)440 4308 y Ft(D)509 4320 y Fy(8)546 4308 y Fz(,)i(w)n(e)f(can)g(use)h(the)g(fact)f(that)1354 4424 y Fl(X)1123 4613 y Fy(~)-39 b Fp(p)q Fx(2)p Fy(\()o Fp(D)1276 4621 y Ff(8)1309 4613 y Fx(\010)p Fp(D)1415 4621 y Ff(8)1448 4613 y Fy(\))1474 4590 y Ff(+)1520 4613 y Fy(:)6 b(~)-39 b Fp(p)p Fx(\001)6 b Fy(~)-39 b Fp(p)p Fy(=2)1711 4503 y Fz(\()p Ft(p)1785 4515 y Fp(i)1832 4503 y Fs(\001)25 b Fz(~)-49 b Ft(p)p Fz(\)\()p Ft(p)2021 4515 y Fp(j)2075 4503 y Fs(\001)25 b Fz(~)-49 b Ft(p)p Fz(\))23 b(=)g(28)14 b Ft(p)2441 4515 y Fp(i)2486 4503 y Fs(\001)k Ft(p)2569 4515 y Fp(j)2604 4503 y Ft(;)3092 4503 y SDict begin H.S end 3092 4503 a 3092 4503 a SDict begin 15 H.A end 3092 4503 a 3092 4503 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.28) cvn /DEST pdfmark end 3092 4503 a Fz(\(3.28\))440 4789 y(whic)n(h)27 b(is)f(men)n(tioned)g (and)h(used)h(in)e([)1635 4789 y SDict begin H.S end 1635 4789 a Fz(27)1719 4789 y SDict begin 12 H.L end 1719 4789 a 1719 4789 a SDict begin [/Subtype /Link/Dest (cite.matone2010getting) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1719 4789 a Fz(].)37 b(W)-7 b(e)28 b(then)g(obtain)677 4975 y Ft(@)726 4945 y Fy(2)763 4975 y Ft(F)828 4945 y Fy(\(4\))p 677 5012 240 4 v 678 5088 a Ft(@)5 b(z)766 5100 y Fp(i)793 5088 y Ft(@)g(z)881 5100 y Fp(j)927 4911 y Fl(\014)927 4961 y(\014)927 5010 y(\014)927 5060 y(\014)954 5116 y Fp(z)r Fy(=0)1100 5031 y Fz(=)1305 4952 y Fl(X)1187 5130 y Fp(p)1221 5138 y Ff(1)1254 5130 y Fp(;:::)o(;p)1387 5138 y Ff(4)1419 5130 y Fx(2)p Fy(\003)1509 5138 y Ff(5)1556 5031 y Ft(e)1595 4995 y Fp(\031)r(i)1670 4952 y Fe(P)1740 4967 y Fk(g)1740 5014 y(i;j)r Ff(=1)1894 4995 y Fp(p)1928 5003 y Fk(i)1955 4995 y Fp(p)1989 5003 y Fk(j)2020 4995 y Fp(\034)2051 5003 y Fk(ij)2148 4952 y Fl(X)2128 5130 y Fy(~)-39 b Fp(p)p Fx(\001)6 b Fy(~)-39 b Fp(p)p Fy(=2)2294 5031 y Fz(\(2)p Ft(\031)s(i)p Fz(\))2479 4997 y Fy(2)2516 5031 y Fz(\()7 b(~)-49 b Ft(pp)2632 5043 y Fp(i)2660 5031 y Fz(\)\()7 b(~)-49 b Ft(pp)2808 5043 y Fp(j)2843 5031 y Fz(\))3092 5031 y SDict begin H.S end 3092 5031 a 3092 5031 a SDict begin 15 H.A end 3092 5031 a 3092 5031 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.29) cvn /DEST pdfmark end 3092 5031 a Fz(\(3.29\))1100 5326 y(=)22 b(56)p Ft(\031)s(i)p Fz(\(1)c(+)g Ft(\016)1562 5338 y Fp(ij)1620 5326 y Fz(\))1686 5270 y Ft(@)5 b(F)1800 5240 y Fy(\(4\))p 1686 5307 203 4 v 1716 5383 a Ft(@)g(\034)1801 5395 y Fp(ij)1899 5206 y Fl(\014)1899 5256 y(\014)1899 5305 y(\014)1899 5355 y(\014)1926 5411 y Fp(z)r Fy(=0)2072 5326 y Fz(=)22 b(56)p Ft(\031)s(i)p Fz(\(1)c(+)g Ft(\016)2534 5338 y Fp(ij)2592 5326 y Fz(\))2634 5270 y Ft(@)5 b(#)2732 5227 y Fy(\(4\))2732 5292 y(5)p 2634 5307 187 4 v 2656 5383 a Ft(@)g(\034)2741 5395 y Fp(ij)2831 5326 y Ft(:)3092 5326 y SDict begin H.S end 3092 5326 a 3092 5326 a SDict begin 15 H.A end 3092 5326 a 3092 5326 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.30) cvn /DEST pdfmark end 3092 5326 a Fz(\(3.30\))1831 5662 y(12)p eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 432 133 a SDict begin H.S end 432 133 a 432 133 a SDict begin H.R end 432 133 a 432 133 a SDict begin [/View [/XYZ H.V]/Dest (page.13) cvn /DEST pdfmark end 432 133 a 440 448 a SDict begin H.S end 440 448 a 440 448 a SDict begin 12 H.A end 440 448 a 440 448 a SDict begin [/View [/XYZ H.V]/Dest (subsection.3.3) cvn /DEST pdfmark end 440 448 a 83 x Fr(3.3.)44 b(The)31 b(\014nal)f(expr)l(ession)440 751 y Fz(Let)e(no)n(w,)f(for)g(brevit)n(y)-7 b(,)26 b Ft(f)1262 721 y Fy(\()p Fp(g)r Fy(\))1352 751 y Ft(;)14 b(f)1439 708 y Fy(\()p Fp(g)r Fy(\))1430 773 y(0)1557 751 y Fz(and)27 b Ft(f)1768 708 y Fy(\()p Fp(g)r Fy(\))1759 773 y(1)1886 751 y Fz(b)r(e)h(de\014ned)g(b)n(y)1704 938 y Ft(f)1754 904 y Fy(\()p Fp(g)r Fy(\))1867 938 y Fz(:=)23 b Ft(#)2027 895 y Fy(\()p Fp(g)r Fy(\))2027 960 y(5)2136 938 y Fs(\000)18 b Ft(G)2284 904 y Fy(\()p Fp(g)r Fy(\))2284 959 y Fp(g)3092 938 y SDict begin H.S end 3092 938 a 3092 938 a SDict begin 15 H.A end 3092 938 a 3092 938 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.31) cvn /DEST pdfmark end 3092 938 a Fz(\(3.31\))1370 1090 y Ft(f)1420 1055 y Fy(\()p Fp(g)r Fy(\))1533 1090 y Fz(=)23 b Ft(f)1671 1047 y Fy(\()p Fp(g)r Fy(\))1662 1112 y(0)1779 1090 y Fz(+)18 b Ft(q)s(f)1952 1047 y Fy(\()p Fp(g)r Fy(\))1943 1112 y(1)2061 1090 y Fz(+)g Ft(O)r Fz(\()p Ft(q)2281 1055 y Fy(2)2319 1090 y Fz(\))p Ft(:)3092 1090 y SDict begin H.S end 3092 1090 a 3092 1090 a SDict begin 15 H.A end 3092 1090 a 3092 1090 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.32) cvn /DEST pdfmark end 3092 1090 a Fz(\(3.32\))440 1272 y(W)-7 b(e)36 b(no)n(w)f(dev)n(elop) f Ft(f)1130 1241 y Fy(\(5\))1254 1272 y Fz(as)h(a)g(function)g(of)h Ft(s)p Fz(.)61 b(Applying)34 b(form)n(ula)e(\()2712 1272 y SDict begin H.S end 2712 1272 a Fz(3.4)2819 1218 y SDict begin H.R end 2819 1218 a 2819 1272 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.4) cvn H.B /ANN pdfmark end 2819 1272 a Fz(\))k(to)f Ft(f)3046 1241 y Fy(\(5\))3171 1272 y Fz(and)440 1387 y(noting)26 b(that)i Ft(f)928 1344 y Fy(\(5\))919 1410 y(0)1040 1387 y Fz(=)23 b Ft(f)1178 1357 y Fy(\(4\))1266 1387 y Fz(,)28 b(w)n(e)f(ha)n(v)n(e)563 1658 y Ft(f)613 1624 y Fy(\(5\))702 1658 y Fz(\()p Ft(\034)770 1670 y Fp(s)806 1658 y Fz(\))c(=)g Ft(f)999 1624 y Fy(\(4\))1088 1658 y Fz(\()p Ft(\034)1156 1670 y Fy(0)1193 1658 y Fz(\))c(+)f Ft(s)1380 1491 y Fl(0)1380 1641 y(@)1453 1658 y Ft(f)1503 1615 y Fy(\(5\))1494 1680 y(1)1591 1658 y Fz(\()p Ft(\034)1659 1670 y Fy(0)1697 1658 y Ft(;)c(z)t Fz(\))k(+)1910 1579 y Fl(X)1917 1757 y Fp(i)p Fx(\024)p Fp(j)2054 1602 y Ft(@)5 b(f)2153 1572 y Fy(\(4\))p 2054 1639 188 4 v 2076 1715 a Ft(@)g(\034)2161 1727 y Fp(ij)2251 1658 y Ft(\033)2298 1670 y Fp(ij)2357 1658 y Fz(\()p Ft(p;)14 b(p)2510 1624 y Fx(0)2533 1658 y Fz(\))2565 1491 y Fl(1)2565 1641 y(A)2656 1658 y Fz(+)19 b Ft(O)r Fz(\()p Ft(s)2876 1624 y Fy(2)2914 1658 y Fz(\))p Ft(:)3092 1658 y SDict begin H.S end 3092 1658 a 3092 1658 a SDict begin 15 H.A end 3092 1658 a 3092 1658 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.33) cvn /DEST pdfmark end 3092 1658 a Fz(\(3.33\))440 2019 y(No)n(w,)31 b(w)n(e)g(expand)g(this)f(using)h(\()1493 2019 y SDict begin H.S end 1493 2019 a Fz(3.7)1599 1965 y SDict begin H.R end 1599 1965 a 1599 2019 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.7) cvn H.B /ANN pdfmark end 1599 2019 a Fz(\),)h(letting)e Ft(u)e Fz(:=)g Ft(x)p Fz(\()p Ft(p)p Fz(\))22 b Fs(\000)e Ft(x)p Fz(\()p Ft(p)2526 1989 y Fx(0)2550 2019 y Fz(\))31 b(for)g(a)f(lo)r(cal)f(c)n (hart)h Ft(x)p Fz(.)440 2118 y(F)-7 b(or)27 b(brevit)n(y)f(w)n(e)h (write)1049 2336 y Ft(S)1100 2348 y Fp(ij)1182 2336 y Fz(:=)1302 2280 y Ft(u)1350 2250 y Fy(2)p 1302 2317 85 4 v 1324 2393 a Fz(4)1407 2280 y Ft(@)5 b(!)1508 2292 y Fp(i)1535 2280 y Fz(\()p Ft(p)p Fz(\))p 1407 2317 235 4 v 1476 2393 a Ft(@)g(x)1661 2280 y(@)g(!)1762 2292 y Fp(j)1797 2280 y Fz(\()p Ft(p)p Fz(\))p 1661 2317 242 4 v 1734 2393 a Ft(@)g(x)1931 2336 y Fz(+)2024 2280 y Ft(u)2072 2250 y Fy(3)p 2024 2317 85 4 v 2046 2393 a Fz(2)2129 2280 y Ft(@)2178 2250 y Fy(2)2215 2280 y Ft(!)2267 2292 y Fp(i)2294 2280 y Fz(\()p Ft(p)p Fz(\))p 2129 2317 272 4 v 2198 2393 a Ft(@)g(x)2294 2369 y Fy(2)2420 2280 y Ft(@)g(!)2521 2292 y Fp(j)2556 2280 y Fz(\()p Ft(p)p Fz(\))p 2420 2317 242 4 v 2493 2393 a Ft(@)g(x)2672 2336 y(:)3092 2336 y SDict begin H.S end 3092 2336 a 3092 2336 a SDict begin 15 H.A end 3092 2336 a 3092 2336 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.34) cvn /DEST pdfmark end 3092 2336 a Fz(\(3.34\))440 2539 y(Remem)n(b)r(er)26 b(that)h Ft(\033)1083 2551 y Fp(ij)1142 2539 y Fz(\()p Ft(p;)14 b(q)s Fz(\))24 b(=)e Ft(S)1487 2551 y Fp(ij)1564 2539 y Fz(+)c Ft(O)r Fz(\()p Ft(u)1792 2509 y Fy(4)1830 2539 y Fz(\).)37 b(Then,)440 2787 y Ft(f)490 2752 y Fy(\(5\))579 2787 y Fz(\()p Ft(\034)647 2799 y Fp(s)683 2787 y Fz(\))23 b(=)g Ft(f)876 2752 y Fy(\(4\))964 2787 y Fz(\()p Ft(\034)1032 2799 y Fy(0)1070 2787 y Fz(\))c(+)f Ft(s)1257 2708 y Fl(X)1264 2886 y Fp(i)p Fx(\024)p Fp(j)1391 2645 y Fl( )1456 2787 y Ft(u)1504 2752 y Fy(2)1557 2730 y Ft(@)1606 2700 y Fy(2)1643 2730 y Ft(f)1693 2687 y Fy(\(5\))1684 2752 y(1)p 1551 2767 238 4 v 1551 2843 a Ft(@)5 b(z)1639 2855 y Fp(i)1666 2843 y Ft(@)g(z)1754 2855 y Fp(j)1798 2787 y Ft(!)1850 2799 y Fp(i)1878 2787 y Fz(\()p Ft(p)p Fz(\))p Ft(!)2036 2799 y Fp(j)2071 2787 y Fz(\()p Ft(p)p Fz(\))18 b(+)2288 2730 y Ft(@)5 b(f)2387 2700 y Fy(\(4\))p 2288 2767 188 4 v 2310 2843 a Ft(@)g(\034)2395 2855 y Fp(ij)2486 2787 y Ft(S)2537 2799 y Fp(ij)2614 2787 y Fz(+)18 b Ft(O)r Fz(\()p Ft(u)2842 2752 y Fy(4)2879 2787 y Fz(\))2911 2645 y Fl(!)2996 2787 y Fz(+)g Ft(O)r Fz(\()p Ft(s)3215 2752 y Fy(2)3253 2787 y Fz(\))p Ft(:)3092 2983 y SDict begin H.S end 3092 2983 a 3092 2983 a SDict begin 15 H.A end 3092 2983 a 3092 2983 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.35) cvn /DEST pdfmark end 3092 2983 a Fz(\(3.35\))440 3197 y(By)31 b(\()606 3197 y SDict begin H.S end 606 3197 a Fz(3.23)754 3143 y SDict begin H.R end 754 3143 a 754 3197 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.23) cvn H.B /ANN pdfmark end 754 3197 a Fz(\))f(and)f(\()1011 3197 y SDict begin H.S end 1011 3197 a Fz(3.30)1159 3143 y SDict begin H.R end 1159 3143 a 1159 3197 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.30) cvn H.B /ANN pdfmark end 1159 3197 a Fz(\))h(w)n(e)g(kno)n(w)f (that)1762 3153 y Fp(@)1801 3128 y Ff(2)1834 3153 y Fp(f)1873 3123 y Ff(\(5\))1866 3171 y(1)p 1757 3178 199 4 v 1757 3226 a Fp(@)t(z)1828 3234 y Fk(i)1854 3226 y Fp(@)t(z)1925 3234 y Fk(j)1992 3197 y Fz(=)e(56)p Ft(\031)s(i)p Fz(\(1)18 b(+)i Ft(\016)2461 3209 y Fp(ij)2519 3197 y Fz(\))2561 3160 y Fp(@)t(f)2639 3135 y Ff(\(4\))p 2562 3178 156 4 v 2578 3226 a Fp(@)t(\034)2648 3234 y Fk(ij)2727 3197 y Fz(.)43 b(This)29 b(lea)n(v)n(es)e(us)440 3308 y(with)440 3516 y Ft(f)490 3482 y Fy(\(5\))579 3516 y Fz(\()p Ft(\034)647 3528 y Fp(s)683 3516 y Fz(\))c(=)g Ft(f)876 3482 y Fy(\(4\))964 3516 y Fz(\()p Ft(\034)1032 3528 y Fy(0)1070 3516 y Fz(\))c(+)f Ft(s)1257 3437 y Fl(X)1264 3615 y Fp(i)p Fx(\024)p Fp(j)1401 3460 y Ft(@)5 b(f)1500 3430 y Fy(\(4\))p 1401 3497 188 4 v 1423 3573 a Ft(@)g(\034)1508 3585 y Fp(ij)1612 3449 y Fl(\000)1650 3516 y Fz(56)p Ft(\031)s(i)p Fz(\(1)17 b(+)h Ft(\016)2024 3528 y Fp(ij)2082 3516 y Fz(\))p Ft(u)2162 3482 y Fy(2)2199 3516 y Ft(!)2251 3528 y Fp(i)2279 3516 y Fz(\()p Ft(p)p Fz(\))p Ft(!)2437 3528 y Fp(j)2472 3516 y Fz(\()p Ft(p)p Fz(\))h(+)f Ft(S)2731 3528 y Fp(ij)2807 3516 y Fz(+)g Ft(O)r Fz(\()p Ft(u)3035 3482 y Fy(4)3073 3516 y Fz(\))3105 3449 y Fl(\001)3162 3516 y Fz(+)g Ft(O)r Fz(\()p Ft(s)3381 3482 y Fy(2)3419 3516 y Fz(\))p Ft(:)3092 3713 y SDict begin H.S end 3092 3713 a 3092 3713 a SDict begin 15 H.A end 3092 3713 a 3092 3713 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.36) cvn /DEST pdfmark end 3092 3713 a Fz(\(3.36\))440 3908 y(No)n(w,)33 b(let)f Ft(J)837 3878 y Fy(\()p Fp(g)r Fy(\))958 3908 y Fz(:=)f Ft(#)1126 3865 y Fy(\()p Fp(g)r Fy(\))1126 3930 y(6)1238 3908 y Fs(\000)22 b Ft(#)1374 3865 y Fy(\()p Fp(g)r Fy(\))1374 3930 y(7)1464 3908 y Fz(.)52 b(Because)31 b Ft(f)1911 3878 y Fy(\(4\))2031 3908 y Fz(=)2136 3875 y Fy(3)p 2136 3889 34 4 v 2136 3936 a(7)2179 3908 y Ft(J)2233 3878 y Fy(\(4\))2322 3908 y Fz(,)j(from)d([)2603 3908 y SDict begin H.S end 2603 3908 a Fz(36)2687 3908 y SDict begin 12 H.L end 2687 3908 a 2687 3908 a SDict begin [/Subtype /Link/Dest (cite.latticevsriemann) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 2687 3908 a -1 w Fz(],)j(w)n(e)e(can)g(rewrite)440 4007 y(the)c(ab)r(o)n(v)n (e)e(as)h(follo)n(ws:)440 4216 y Ft(f)490 4181 y Fy(\(5\))579 4216 y Fz(\()p Ft(\034)647 4228 y Fp(s)683 4216 y Fz(\))c(=)836 4159 y(3)p 836 4197 42 4 v 836 4273 a(7)887 4216 y Ft(J)941 4181 y Fy(\(4\))1030 4216 y Fz(\()p Ft(\034)1098 4228 y Fy(0)1136 4216 y Fz(\))c(+)1280 4159 y(3)p Ft(s)p 1280 4197 81 4 v 1299 4273 a Fz(7)1384 4137 y Fl(X)1391 4315 y Fp(i)p Fx(\024)p Fp(j)1528 4159 y Ft(@)5 b(J)1631 4129 y Fy(\(4\))p 1528 4197 192 4 v 1552 4273 a Ft(@)g(\034)1637 4285 y Fp(ij)1744 4148 y Fl(\000)1782 4216 y Fz(56)p Ft(\031)s(i)p Fz(\(1)17 b(+)h Ft(\016)2156 4228 y Fp(ij)2214 4216 y Fz(\))p Ft(u)2294 4181 y Fy(2)2331 4216 y Ft(!)2383 4228 y Fp(i)2411 4216 y Fz(\()p Ft(p)p Fz(\))p Ft(!)2569 4228 y Fp(j)2604 4216 y Fz(\()p Ft(p)p Fz(\))g(+)g Ft(S)2862 4228 y Fp(ij)2939 4216 y Fz(+)g Ft(O)r Fz(\()p Ft(u)3167 4181 y Fy(4)3205 4216 y Fz(\))3237 4148 y Fl(\001)3294 4216 y Fz(+)g Ft(O)r Fz(\()p Ft(s)3513 4181 y Fy(2)3551 4216 y Fz(\))p Ft(:)3092 4413 y SDict begin H.S end 3092 4413 a 3092 4413 a SDict begin 15 H.A end 3092 4413 a 3092 4413 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.37) cvn /DEST pdfmark end 3092 4413 a Fz(\(3.37\))440 4674 y(In)40 b([)579 4674 y SDict begin H.S end 579 4674 a Fz(34)663 4674 y SDict begin 12 H.L end 663 4674 a 663 4674 a SDict begin [/Subtype /Link/Dest (cite.grushevsky2011superstring) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 663 4674 a -1 w Fz(,)k(p.)c(16-17])e (Grushevsky)h(and)i(Salv)-5 b(ati)38 b(Manni)h(obtain)g(a)g(similar)d (expression)440 4774 y(for)31 b(the)g(degeneration)e(of)j Ft(J)1363 4744 y Fy(\(5\))1452 4774 y Fz(,)g(di\013ering)d(only)h(in)h (the)g(n)n(umerical)d(co)r(e\016cien)n(ts.)46 b(They)440 4887 y(sho)n(w)30 b(that)i(the)f Ft(!)1031 4899 y Fp(i)1058 4887 y Fz(\()p Ft(p)p Fz(\))p Ft(!)1216 4899 y Fp(j)1251 4887 y Fz(\()p Ft(p)p Fz(\))h(term)e(v)-5 b(anishes)30 b(and)h(that)2269 4825 y Fl(P)2357 4912 y Fp(i)p Fx(\024)p Fp(j)2491 4855 y(@)t(J)2572 4830 y Ff(\(4\))p 2491 4869 159 4 v 2509 4916 a Fp(@)t(\034)2579 4924 y Fk(ij)2659 4887 y Ft(S)2710 4899 y Fp(ij)2800 4887 y Fz(cannot)g(v)-5 b(anish)440 5014 y(ev)n(erywhere)27 b(due)i(to)g(the)g(fact)g(that)g Ft(J)1678 4984 y Fy(\(4\))1796 5014 y Fz(is)f(the)h(Sc)n(hottky)g (form.)39 b(W)-7 b(e)29 b(refer)f(to)h([)3070 5014 y SDict begin H.S end 3070 5014 a Fz(34)3154 5014 y SDict begin 12 H.L end 3154 5014 a 3154 5014 a SDict begin [/Subtype /Link/Dest (cite.grushevsky2011superstring) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 3154 5014 a -1 w Fz(])g(for)440 5114 y(details.)38 b(This)28 b(sho)n(ws)g(that)h Ft(f)1401 5084 y Fy(\(5\))1490 5114 y Fz(\()p Ft(\034)1558 5126 y Fp(s)1594 5114 y Fz(\))g(do)r(es)f(not)h (v)-5 b(anish)28 b(ev)n(erywhere.)39 b(Th)n(us,)29 b(the)g(ab)r(o)n(v)n (e)440 5214 y(leads)d(to)i(the)f(conclusion)1670 5383 y Ft(#)1719 5340 y Fy(\(5\))1719 5405 y(5)1832 5383 y Fs(6\021)22 b Ft(G)1984 5340 y Fy(\(5\))1984 5405 y(5)3092 5383 y SDict begin H.S end 3092 5383 a 3092 5383 a SDict begin 15 H.A end 3092 5383 a 3092 5383 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.38) cvn /DEST pdfmark end 3092 5383 a Fz(\(3.38\))1831 5662 y(13)p eop end end %%Page: 14 14 TeXDict begin HPSdict begin 14 13 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 440 133 a SDict begin H.S end 440 133 a 440 133 a SDict begin H.R end 440 133 a 440 133 a SDict begin [/View [/XYZ H.V]/Dest (page.14) cvn /DEST pdfmark end 440 133 a 398 x Fz(when)28 b(restricted)e(to)h Fs(J)1184 543 y Fy(5)1222 531 y Fz(,)g(as)g(promised.)p 3244 531 4 57 v 3248 479 50 4 v 3248 531 V 3297 531 4 57 v 440 734 a(The)20 b(factorization)c(prop)r(erties)i(of)h Ft(G)1611 691 y Fy(\()p Fp(g)r Fy(\))1611 744 y Fp(g)1721 734 y Fz(and)h Ft(#)1924 691 y Fy(\()p Fp(g)r Fy(\))1924 756 y(5)2034 734 y Fz(no)n(w)f(imply)e(that)j(this)f(holds)f(for)h(higher) 440 851 y(genera)27 b(as)g(w)n(ell)f(-)i(assuming)d Ft(G)1457 808 y Fy(\()p Fp(g)r Fy(\))1457 861 y Fp(g)1576 851 y Fz(is)i(w)n(ell-de\014ned)f(for)i Ft(g)e Fs(\025)e Fz(6,)j(the)i(ab)r (o)n(v)n(e)e(implies)d(that)440 951 y Ft(f)490 920 y Fy(\()p Fp(g)r Fy(\))608 951 y Fz(do)r(es)j(not)h(v)-5 b(anish)26 b(iden)n(tically)d(on)28 b Fs(J)1772 963 y Fp(g)1811 951 y Fz(,)f(for)g(all)f Ft(g)f Fs(\025)e Fz(5.)440 1108 y SDict begin H.S end 440 1108 a 440 1108 a SDict begin 12 H.A end 440 1108 a 440 1108 a SDict begin [/View [/XYZ H.V]/Dest (section.4) cvn /DEST pdfmark end 440 1108 a 168 x Fu(4.)47 b(The)32 b(trace)h(of)f Fc(f)1163 1245 y Fb(\(5\))440 1526 y Fz(Here)27 b(w)n(e)f(will)f(lo)r(ok)g(at)i (the)g(trace)g(of)g Ft(f)1688 1496 y Fy(\(5\))1776 1526 y Fz(,)h(de\014ned)f(as)2214 1464 y Fl(P)2301 1551 y Fp(m)2378 1526 y Ft(f)2428 1496 y Fy(\(5\))2517 1526 y Fz([)p Ft(m)p Fz(],)g(b)r(ecause)g(it)f(o)r(ccurs)440 1626 y(in)h(the)h(cosmological)21 b(constan)n(t)27 b(and)g(is)g(th)n (us)g(of)h(in)n(terest)e(for)h(the)h(gen)n(us)f(5)g(measure.)440 1817 y(The)33 b(de\014nition)e(of)h Ft(f)1139 1786 y Fy(\(5\))1228 1817 y Fz([)p Ft(m)p Fz(])h(is)f(as)g(follo)n(ws:)44 b(for)32 b(an)n(y)g(mo)r(dular)e(form)h Ft(f)41 b Fz(and)33 b(for)f Ft(\015)3145 1829 y Fp(m)3240 1817 y Fz(=)440 1849 y Fl(\022)503 1915 y Ft(A)86 b(B)501 2015 y(C)j(D)720 1849 y Fl(\023)812 1966 y Fz(suc)n(h)30 b(that)1185 1849 y Fl(\024)1234 1915 y Fz(diag\()p Ft(A)1482 1885 y Fp(T)1534 1915 y Ft(C)6 b Fz(\))1229 2015 y(diag)o(\()p Ft(B)1481 1985 y Fp(T)1534 2015 y Ft(D)r Fz(\))1637 1849 y Fl(\025)1709 1966 y Fz(=)28 b Ft(m)p Fz(,)j(w)n(e)f(ha)n(v)n(e)f Ft(f)9 b Fz([)p Ft(m)p Fz(])28 b(:=)f(\()p Ft(f)9 b Fs(j)2665 1978 y Fy(8)2716 1966 y Ft(\015)2759 1978 y Fp(m)2822 1966 y Fz(\).)46 b(When)31 b Ft(f)39 b Fz(is)440 2116 y(a)25 b(mo)r(dular)e(form)h(with)h(resp)r(ect)h(to)f(\000\(1)p Ft(;)14 b Fz(2\),)25 b Ft(f)9 b Fz([)p Ft(m)p Fz(])26 b(do)r(es)f(not)g(dep)r(end)i(on)e(the)h(particular)440 2215 y(c)n(hoice)g(of)h Ft(\015)824 2227 y Fp(m)887 2215 y Fz(.)440 2418 y(In)35 b([)574 2418 y SDict begin H.S end 574 2418 a Fz(34)658 2418 y SDict begin 12 H.L end 658 2418 a 658 2418 a SDict begin [/Subtype /Link/Dest (cite.grushevsky2011superstring) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 658 2418 a -1 w Fz(])f(Grushevsky)f(and)h (Salv)-5 b(ati)33 b(Manni)g(calculate)f(the)i(traces)g(of)g(the)h (forms)d Ft(G)3190 2375 y Fy(\()p Fp(g)r Fy(\))3190 2427 y Fp(p)3281 2418 y Fz(.)440 2517 y(They)22 b(use)h(a)f(di\013eren)n(t)f (notation:)33 b(their)21 b Ft(S)1780 2529 y Fp(i)1830 2517 y Fz(equals)g(2)2119 2487 y Fx(\000)p Fp(i)2212 2455 y Fl(P)2299 2542 y Fp(m)2376 2517 y Ft(G)2441 2529 y Fp(i)2469 2517 y Fz([)p Ft(m)p Fz(].)35 b(Their)21 b(result)g(sho)n(ws)440 2637 y(that)28 b(all)736 2575 y Fl(P)823 2662 y Fp(m)900 2637 y Ft(G)965 2594 y Fy(\()p Fp(g)r Fy(\))965 2647 y Fp(p)1084 2637 y Fz(can)g(b)r(e)g(recursiv)n (ely)c(expressed)j(through)2451 2575 y Fl(P)2538 2662 y Fp(m)2615 2637 y Ft(G)2680 2594 y Fy(\()p Fp(g)r Fy(\))2680 2659 y(0)2799 2637 y Fz(and)2961 2575 y Fl(P)3049 2662 y Fp(m)3125 2637 y Ft(G)3190 2594 y Fy(\()p Fp(g)r Fy(\))3190 2659 y(1)3281 2637 y Fz(.)440 2757 y(Note)g(that)g(this)f(form)n(ula)e (is)i(only)g(v)-5 b(alid)24 b(for)j(the)g Ft(G)2082 2714 y Fy(\()p Fp(g)r Fy(\))2082 2767 y Fp(p)2200 2757 y Fz(with)f Ft(p)d Fs(\024)g Ft(g)s Fz(,)j(b)r(ecause)h(the)g(others)440 2857 y(v)-5 b(anish)26 b(iden)n(tically)-7 b(,)24 b(and)j(for)g(1)c Fs(\024)g Ft(n)f Fs(\024)h Fz(4.)553 3069 y(2)609 2990 y Fl(X)638 3164 y Fp(m)742 3069 y Ft(G)807 3025 y Fy(\()p Fp(g)r Fy(\))807 3091 y Fp(n)p Fy(+1)936 3069 y Fz([)p Ft(m)p Fz(])g(=)1276 3012 y(2)1318 2982 y Fy(2\()p Fp(g)r Fx(\000)p Fp(n)p Fy(+1\))1637 3012 y Fs(\000)18 b Fz(1)p 1176 3049 687 4 v 1176 3125 a(\(2)1250 3101 y Fp(n)1313 3125 y Fs(\000)g Fz(1\))c(\(2)1558 3101 y Fp(n)p Fy(+1)1705 3125 y Fs(\000)k Fz(1\))1886 2990 y Fl(X)1917 3164 y Fp(m)2020 3069 y Ft(G)2085 3025 y Fy(\()p Fp(g)r Fy(\))2085 3091 y Fp(n)p Fx(\000)p Fy(1)2215 3069 y Fz([)p Ft(m)p Fz(])h Fs(\000)2582 3012 y Fz(3)p 2446 3049 314 4 v 2446 3125 a(2)2488 3101 y Fp(n)p Fy(+1)2635 3125 y Fs(\000)f Fz(1)2783 2990 y Fl(X)2814 3164 y Fp(m)2917 3069 y Ft(G)2982 3034 y Fy(\()p Fp(g)r Fy(\))2982 3089 y Fp(n)3073 3069 y Fz([)p Ft(m)p Fz(])3134 3250 y SDict begin H.S end 3134 3250 a 3134 3250 a SDict begin 15 H.A end 3134 3250 a 3134 3250 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.1) cvn /DEST pdfmark end 3134 3250 a Fz(\(4.1\))440 3433 y(Because)f Ft(G)813 3390 y Fy(\(5\))813 3455 y(0)903 3433 y Fz([)p Ft(m)p Fz(])23 b(=)f Ft(\022)1173 3403 y Fy(16)1171 3453 y Fp(m)1263 3433 y Fz(and)c Ft(G)1480 3390 y Fy(\(5\))1480 3455 y(1)1569 3433 y Fz([)p Ft(m)p Fz(])23 b(=)g Ft(\022)1840 3403 y Fy(8)1838 3453 y Fp(m)1915 3370 y Fl(P)2002 3458 y Fp(n)p Fx(6)p Fy(=0)2145 3433 y Ft(\022)2186 3403 y Fy(8)2184 3453 y Fp(m)p Fy(+)p Fp(n)2340 3433 y Fz(,)d(w)n(e)e(see)g(that)2792 3370 y Fl(P)2879 3458 y Fp(m)2956 3433 y Ft(G)3021 3390 y Fy(\(5\))3021 3455 y(0)3110 3433 y Fz([)p Ft(m)p Fz(])24 b(=)440 3502 y Fl(P)528 3590 y Fp(m)605 3565 y Ft(\022)646 3535 y Fy(16)644 3585 y Fp(m)744 3565 y Fz(=)k Ft(#)886 3577 y Fy(7)924 3565 y Fz(,)j(and)1143 3502 y Fl(P)1231 3590 y Fp(m)1307 3565 y Ft(G)1372 3522 y Fy(\(5\))1372 3587 y(1)1462 3565 y Fz([)p Ft(m)p Fz(])d(=)1702 3497 y Fl(\000)1740 3502 y(P)1828 3590 y Fp(m)1905 3565 y Ft(\022)1946 3535 y Fy(8)1944 3585 y Fp(m)2007 3497 y Fl(\001)2045 3515 y Fy(2)2102 3565 y Fs(\000)2188 3502 y Fl(P)2275 3590 y Fp(m)2352 3565 y Ft(\022)2393 3535 y Fy(16)2391 3585 y Fp(m)2492 3565 y Fz(=)g Ft(#)2634 3577 y Fy(6)2692 3565 y Fs(\000)20 b Ft(#)2826 3577 y Fy(7)2863 3565 y Fz(.)47 b(Therefore,)440 3664 y(w)n(e)27 b(can)g(easily)e(obtain)1157 3777 y Fl(X)1188 3951 y Fp(m)1291 3856 y Ft(G)1356 3813 y Fy(\(5\))1356 3878 y(5)1445 3856 y Fz([)p Ft(m)p Fz(])e(=)1706 3800 y(32)p 1685 3837 125 4 v 1685 3913 a(217)1833 3764 y Fl(\020)1883 3856 y Fz(950)14 b Ft(#)2072 3813 y Fy(\(5\))2072 3878 y(6)2178 3856 y Fs(\000)k Fz(733)c Ft(#)2450 3813 y Fy(\(5\))2450 3878 y(7)2538 3764 y Fl(\021)3134 3856 y SDict begin H.S end 3134 3856 a 3134 3856 a SDict begin 15 H.A end 3134 3856 a 3134 3856 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.2) cvn /DEST pdfmark end 3134 3856 a Fz(\(4.2\))1180 4024 y Fl(X)1211 4198 y Fp(m)1314 4102 y Ft(G)1379 4059 y Fy(\(4\))1379 4125 y(4)1468 4102 y Fz([)p Ft(m)p Fz(])23 b(=)g Fs(\000)1773 4046 y Fz(16)p 1773 4083 84 4 v 1793 4159 a(7)1879 4010 y Fl(\020)1929 4102 y Fz(22)14 b Ft(#)2076 4059 y Fy(\(4\))2076 4125 y(6)2183 4102 y Fs(\000)k Fz(29)c Ft(#)2413 4059 y Fy(\(4\))2413 4125 y(7)2501 4010 y Fl(\021)2564 4102 y Ft(:)3134 4102 y SDict begin H.S end 3134 4102 a 3134 4102 a SDict begin 15 H.A end 3134 4102 a 3134 4102 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.3) cvn /DEST pdfmark end 3134 4102 a Fz(\(4.3\))440 4334 y(F)-7 b(rom)26 b([)681 4334 y SDict begin H.S end 681 4334 a Fz(27)765 4334 y SDict begin 12 H.L end 765 4334 a 765 4334 a SDict begin [/Subtype /Link/Dest (cite.matone2010getting) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 765 4334 a -1 w Fz(,)i(p.)g(28])f(w)n(e)g(learn)f (that)1274 4437 y Fl(X)1305 4612 y Fp(m)1408 4516 y Ft(#)1457 4473 y Fy(\()p Fp(g)r Fy(\))1457 4538 y(5)1547 4516 y Fz([)p Ft(m)p Fz(])e(=)e(2)1819 4482 y Fp(g)r Fx(\000)p Fy(1)1956 4424 y Fl(\020)2006 4516 y Ft(#)2055 4473 y Fy(\(5\))2055 4538 y(6)2162 4516 y Fz(+)c Ft(#)2294 4473 y Fy(\(5\))2294 4538 y(7)2384 4424 y Fl(\021)2447 4516 y Ft(:)3134 4516 y SDict begin H.S end 3134 4516 a 3134 4516 a SDict begin 15 H.A end 3134 4516 a 3134 4516 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.4) cvn /DEST pdfmark end 3134 4516 a Fz(\(4.4\))440 4743 y(Com)n(bining)24 b(the)k(ab)r(o)n(v)n(e)e(facts,)h(w)n(e)g(obtain)f(the)i(follo)n(wing) 23 b(expressions)i(for)h(the)i(gen)n(us)f(4)440 4843 y(and)g(gen)n(us)g(5)g(trace)g(of)h Ft(f)1252 4813 y Fy(\()p Fp(g)r Fy(\))1342 4843 y Fz(:)1415 4970 y Fl(X)1445 5144 y Fp(m)1549 5049 y Ft(f)1599 5014 y Fy(\(4\))1687 5049 y Fz([)p Ft(m)p Fz(])c(=)e Fs(\000)1992 4992 y Fz(2)2034 4962 y Fy(3)2089 4992 y Fs(\001)c Fz(3)g Fs(\001)h Fz(17)p 1992 5029 324 4 v 2133 5105 a(7)2325 5049 y Ft(J)2379 5014 y Fy(\(4\))3134 5049 y SDict begin H.S end 3134 5049 a 3134 5049 a SDict begin 15 H.A end 3134 5049 a 3134 5049 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.5) cvn 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a(7)1103 764 y Ft(s)1156 685 y Fl(X)1163 863 y Fp(i)p Fx(\024)p Fp(j)1300 708 y Ft(@)5 b(J)1403 678 y Fy(\(4\))p 1300 745 192 4 v 1324 821 a Ft(@)g(\034)1409 833 y Fp(ij)1515 647 y Fl(\022)1576 764 y Fz(56)p Ft(u)1708 730 y Fy(2)1744 764 y Fz(\(1)18 b(+)g Ft(\016)1956 776 y Fp(ij)2015 764 y Fz(\))p Ft(!)2099 776 y Fp(i)2126 764 y Fz(\()p Ft(q)s Fz(\))p Ft(!)2282 776 y Fp(j)2318 764 y Fz(\()p Ft(q)s Fz(\))h(+)f Ft(u)2572 730 y Fy(2)2618 708 y Fz(1)p 2618 745 42 4 v 2618 821 a(4)2680 708 y Ft(@)5 b(!)2781 720 y Fp(i)p 2680 745 129 4 v 2696 821 a Ft(@)g(x)2818 764 y Fz(\()p Ft(q)s Fz(\))2932 708 y Ft(@)g(!)3033 720 y Fp(j)p 2932 745 136 4 v 2952 821 a Ft(@)g(x)3078 764 y Fz(\()p Ft(q)s Fz(\))1723 1040 y(+)1798 984 y(1)p 1798 1021 42 4 v 1798 1097 a(2)1849 1040 y Ft(u)1897 1006 y Fy(3)1944 984 y Ft(@)1993 954 y Fy(2)2030 984 y Ft(!)2082 996 y Fp(i)2109 984 y Fz(\()p Ft(q)s Fz(\))p 1944 1021 270 4 v 2012 1097 a Ft(@)g(x)2108 1073 y Fy(2)2234 984 y Ft(@)g(!)2335 996 y Fp(j)2369 984 y Fz(\()p Ft(q)s Fz(\))p 2234 1021 240 4 v 2306 1097 a Ft(@)g(x)2502 1040 y Fz(+)18 b Ft(O)r Fz(\()p Ft(u)2730 1006 y Fy(4)2768 1040 y Fz(\))2800 923 y Fl(\023)2880 1040 y Fz(+)g Ft(O)r Fz(\()p Ft(s)3099 1006 y Fy(2)3137 1040 y Fz(\))p Ft(;)3134 1197 y SDict begin H.S end 3134 1197 a 3134 1197 a SDict begin 15 H.A end 3134 1197 a 3134 1197 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.3) cvn /DEST pdfmark end 3134 1197 a Fz(\(5.3\))440 1376 y(w)n(e)33 b(can)h(compare)d(it)j (with)f(the)h(v)n(ery)f(similar)c(expression)i(found)j(in)f([)2738 1376 y SDict begin H.S end 2738 1376 a Fz(34)2822 1376 y SDict begin 12 H.L end 2822 1376 a 2822 1376 a SDict begin [/Subtype /Link/Dest (cite.grushevsky2011superstring) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 2822 1376 a -1 w Fz(])h(for)f(the)i(\014rst)440 1475 y(terms)26 b(in)h Ft(u)g Fz(in)g(the)h Ft(s)p Fz(-linear)d(term)h (when)i(taking)e(the)i(same)e(degeneration)f(for)i Ft(J)3073 1445 y Fy(\(5\))3162 1475 y Fz(,)493 1708 y Ft(J)547 1674 y Fy(\(5\))659 1708 y Fz(=)c Ft(J)801 1674 y Fy(\(4\))908 1708 y Fz(+)18 b Ft(s)1044 1629 y Fl(X)1051 1807 y Fp(i)p Fx(\024)p Fp(j)1188 1652 y Ft(@)5 b(J)1291 1622 y Fy(\(4\))p 1188 1689 192 4 v 1212 1765 a Ft(@)g(\034)1297 1777 y Fp(ij)1404 1591 y Fl(\022)1465 1708 y Fz(30)p Ft(u)1597 1674 y Fy(2)1632 1708 y Fz(\(1)19 b(+)f Ft(\016)1845 1720 y Fp(ij)1903 1708 y Fz(\))p Ft(!)1987 1720 y Fp(i)2015 1708 y Fz(\()p Ft(q)s Fz(\))p Ft(!)2171 1720 y Fp(j)2206 1708 y Fz(\()p Ft(q)s Fz(\))h(+)f Ft(u)2460 1674 y Fy(2)2507 1652 y Fz(1)p 2507 1689 42 4 v 2507 1765 a(4)2568 1652 y Ft(@)5 b(!)2669 1664 y Fp(i)p 2568 1689 129 4 v 2584 1765 a Ft(@)g(x)2706 1708 y Fz(\()p Ft(q)s Fz(\))2820 1652 y Ft(@)g(!)2921 1664 y Fp(j)p 2821 1689 136 4 v 2840 1765 a Ft(@)g(x)2966 1708 y Fz(\()p Ft(q)s Fz(\))1588 1984 y(+)1686 1928 y(1)p 1686 1965 42 4 v 1686 2041 a(2)1738 1984 y Ft(u)1786 1950 y Fy(3)1832 1928 y Ft(@)1881 1898 y Fy(2)1918 1928 y Ft(!)1970 1940 y Fp(i)1998 1928 y Fz(\()p Ft(q)s Fz(\))p 1832 1965 270 4 v 1901 2041 a Ft(@)g(x)1997 2017 y Fy(2)2122 1928 y Ft(@)g(!)2223 1940 y Fp(j)2258 1928 y Fz(\()p Ft(q)s Fz(\))p 2122 1965 240 4 v 2194 2041 a Ft(@)g(x)2391 1984 y Fz(+)18 b Ft(O)r Fz(\()p Ft(u)2619 1950 y Fy(4)2656 1984 y Fz(\))2688 1867 y Fl(\023)2768 1984 y Fz(+)g Ft(O)r Fz(\()p Ft(s)2987 1950 y Fy(2)3025 1984 y Fz(\))p Ft(:)3134 1984 y SDict begin H.S end 3134 1984 a 3134 1984 a SDict begin 15 H.A end 3134 1984 a 3134 1984 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.4) cvn /DEST pdfmark end 3134 1984 a Fz(\(5.4\))440 2212 y(Because,)27 b(from)f([)1000 2212 y SDict begin H.S end 1000 2212 a Fz(34)1084 2212 y SDict begin 12 H.L end 1084 2212 a 1084 2212 a SDict begin [/Subtype /Link/Dest (cite.grushevsky2011superstring) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1084 2212 a -1 w Fz(],)1242 2370 y Fl(X)1249 2548 y Fp(i)p Fx(\024)p Fp(j)1386 2393 y Ft(@)5 b(J)1489 2363 y Fy(\(4\))p 1386 2430 192 4 v 1410 2506 a Ft(@)g(\034)1495 2518 y Fp(ij)1587 2449 y Ft(u)1635 2415 y Fy(2)1672 2449 y Fz(\(1)18 b(+)g Ft(\016)1884 2461 y Fp(ij)1943 2449 y Fz(\))p Ft(!)2027 2461 y Fp(i)2054 2449 y Fz(\()p Ft(q)s Fz(\))p Ft(!)2210 2461 y Fp(j)2246 2449 y Fz(\()p Ft(q)s Fz(\))23 b(=)g(0)3134 2449 y SDict begin H.S end 3134 2449 a 3134 2449 a SDict begin 15 H.A end 3134 2449 a 3134 2449 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.5) cvn /DEST pdfmark end 3134 2449 a Fz(\(5.5\))2025 2711 y Ft(f)2075 2677 y Fy(\(4\))2187 2711 y Fz(=)2285 2655 y(3)p 2285 2692 42 4 v 2285 2768 a(7)2336 2711 y Ft(J)2390 2677 y Fy(\(4\))2479 2711 y Ft(;)3134 2711 y SDict begin H.S end 3134 2711 a 3134 2711 a SDict begin 15 H.A end 3134 2711 a 3134 2711 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.6) cvn /DEST pdfmark end 3134 2711 a Fz(\(5.6\))440 2928 y(w)n(e)29 b(conclude)e(that)j(the)f(only)f(linear)e(com)n (bination)f(of)30 b Ft(f)2269 2898 y Fy(\(5\))2386 2928 y Fz(and)f Ft(J)2603 2898 y Fy(\(5\))2722 2928 y Fz(that)g(v)-5 b(anishes)27 b(at)440 3028 y(\014rst)i(order)f(along)f(the)j(b)r (oundary)f(is)f Ft(f)1711 2998 y Fy(\(5\))1819 3028 y Fs(\000)1913 2995 y Fy(3)p 1913 3009 34 4 v 1913 3056 a(7)1957 3028 y Ft(J)2011 2998 y Fy(\(5\))2100 3028 y Fz(,)i(a)f(mo)r(dular)d(form)i(with)h(resp)r(ect)g(to)440 3138 y(\000\(1)p Ft(;)14 b Fz(2\))677 3150 y Fy(5)714 3138 y Fz(.)48 b(But)32 b(w)n(e)f(kno)n(w)f(from)g(equation)g(\()1875 3138 y SDict begin H.S end 1875 3138 a Fz(5.2)1982 3085 y SDict begin H.R end 1982 3085 a 1982 3138 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.5.2) cvn H.B /ANN pdfmark end 1982 3138 a Fz(\))h(that)h Ft(f)2279 3108 y Fy(\(5\))2397 3138 y Fs(6)p Fz(=)2501 3106 y Fy(3)p 2501 3120 34 4 v 2501 3167 a(7)2544 3138 y Ft(J)2598 3108 y Fy(\(5\))2719 3138 y Fz(on)f(the)h(Jacobian)440 3249 y(lo)r(cus.)k(Therefore,)26 b Ft(f)1132 3219 y Fy(\(5\))1248 3249 y Fz(cannot)i(b)r(e)g(a)f(m)n (ultiple)d(of)j Ft(J)2176 3219 y Fy(\(5\))2293 3249 y Fz(ev)n(erywhere)f(on)h Fs(J)2896 3261 y Fy(5)2934 3249 y Fz(.)440 3412 y SDict begin H.S end 440 3412 a 440 3412 a SDict begin 12 H.A end 440 3412 a 440 3412 a SDict begin [/View [/XYZ H.V]/Dest (section.6) cvn /DEST pdfmark end 440 3412 a 175 x Fu(6.)47 b(The)32 b(t)m(w)m(o-p)s(oin)m(t)g (function)g(in)f(gen)m(us)g(4)440 3849 y Fz(Matone)38 b(and)h(V)-7 b(olpato)37 b(sho)n(w)h(in)g([)1594 3849 y SDict begin H.S end 1594 3849 a Fz(27)1678 3849 y SDict begin 12 H.L end 1678 3849 a 1678 3849 a SDict begin [/Subtype /Link/Dest (cite.matone2010getting) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1678 3849 a -1 w Fz(])h(that)g(it)g(is)e(not)i(p)r(ossible)e(to)h (mak)n(e)f(a)i(gen)n(us)f(5)440 3949 y(measure)24 b(from)i(the)g(OPSMY) h(forms)e(that)h(satis\014es)f(all)f(requiremen)n(ts,)g(assuming)f (their)440 4049 y(metho)r(d)d(of)h(obtaining)e(the)i(gen)n(us)f(4)h(t)n (w)n(o-p)r(oin)n(t)e(function)h(is)g(correct.)33 b(T)-7 b(o)21 b(b)r(e)g(precise,)g(the)440 4148 y(degeneration)26 b(to)i(gen)n(us)g(4)g(yields)e(a)j(not)f(iden)n(tically)c(v)-5 b(anishing)26 b(t)n(w)n(o-p)r(oin)n(t)h(function)h(if)440 4248 y(the)21 b(gen)n(us)e(5)h(cosmological)13 b(constan)n(t)20 b(is)f(made)g(to)h(v)-5 b(anish,)20 b(i.e.)34 b(requiremen)n(t)17 b(c\))j(from)f(the)440 4348 y(in)n(tro)r(duction)e(is)g(not)i (satis\014ed.)33 b(Therefore,)19 b(one)g(ma)n(y)e(ask)h(whether)h(b)n (y)f(com)n(bining)d(these)440 4459 y(forms)24 b(with)h Ft(G)919 4416 y Fy(\(5\))919 4481 y(5)1034 4459 y Fz(one)h(can)f (construct)g(a)g(measure)f(that)i(do)r(es)f(satisfy)g(these)g(prop)r (erties.)440 4558 y(The)j(answ)n(er)e(is)g(y)n(es.)440 4749 y(In)f(order)f(to)h(obtain)f(the)h(gen)n(us)g(4)f(t)n(w)n(o-p)r (oin)n(t)f(function)i(from)f(the)h(gen)n(us)f(5)h(measure,)f(w)n(e)440 4849 y(follo)n(w)j(the)k(pro)r(cedure)e(set)h(b)n(y)g(Matone)g(and)g(V) -7 b(olpato.)43 b(Therefore,)29 b(all)f(results)h(in)g(this)440 4949 y(section)d(dep)r(end)j(on)e(their)g(pro)r(cedure)f(b)r(eing)h (correct.)36 b(A)28 b(discussion)c(of)k(its)f(v)-5 b(alidit)n(y)24 b(is,)440 5048 y(ho)n(w)n(ev)n(er,)34 b(b)r(ey)n(ond)h(the)g(scop)r(e)f (of)g(the)h(presen)n(t)f(pap)r(er.)58 b(F)-7 b(or)34 b(details,)g(w)n(e)h(refer)e(to)i(the)440 5148 y(original)23 b(pap)r(er)k([)996 5148 y SDict begin H.S end 996 5148 a Fz(27)1080 5148 y SDict begin 12 H.L end 1080 5148 a 1080 5148 a SDict begin [/Subtype /Link/Dest (cite.matone2010getting) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1080 5148 a -1 w Fz(].)37 b(That)28 b(is,)e(consider)961 5376 y SDict begin H.S end 961 5376 a 961 5376 a SDict begin 12 H.A end 961 5376 a 961 5376 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.1) cvn /DEST pdfmark end 961 5376 a Ft(X)1030 5388 y Fp(N)6 b(S)1137 5376 y Fz([\()p Ft(\017;)14 b(\016)s Fz(\)])23 b(:=)1502 5320 y(1)p 1502 5357 42 4 v 1502 5433 a(2)1567 5259 y Fl(\022)1635 5355 y Fz(~)1628 5376 y(\004)1683 5341 y Fy(\()p Fp(g)r Fy(+1\))1872 5259 y Fl(\024)1916 5325 y Fz(0)86 b Ft(\017)1916 5425 y Fz(0)d Ft(\016)2081 5259 y Fl(\025)2143 5376 y Fz(+)2233 5355 y(~)2226 5376 y(\004)2281 5341 y Fy(\()p Fp(g)r Fy(+1\))2470 5259 y Fl(\024)2513 5325 y Fz(0)j Ft(\017)2513 5425 y Fz(1)d Ft(\016)2678 5259 y Fl(\025\023)3134 5376 y Fz(\(6.1\))1831 5662 y(16)p eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 432 133 a SDict begin H.S end 432 133 a 432 133 a SDict begin H.R end 432 133 a 432 133 a SDict begin [/View [/XYZ H.V]/Dest (page.17) cvn /DEST pdfmark end 432 133 a 440 531 a Fz(and)34 b(con)n(tract)g(one)g(handle)g(from)f(a)h(family) e(of)i(curv)n(es,)h(where)f(then)i(the)f(term)e(linear)440 631 y(in)26 b(the)g(p)r(erturbation)f(parameter)f(will)g(b)r(e)i(the)h (t)n(w)n(o-p)r(oin)n(t)d(function.)36 b(As)27 b(the)f(argumen)n(t)440 731 y(from)32 b([)665 731 y SDict begin H.S end 665 731 a Fz(27)749 731 y SDict begin 12 H.L end 749 731 a 749 731 a SDict begin [/Subtype /Link/Dest (cite.matone2010getting) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 749 731 a Fz(])h(is)g(quite)g(detailed,)g(w)n(e)h (will)d(just)j(lo)r(ok)e(at)i(what)f(happ)r(ens)h(with)g(the)g(terms) 440 830 y Ft(c)476 842 y Fp(J)522 830 y Ft(J)576 800 y Fy(\(5\))683 830 y Fz(+)18 b Ft(c)802 842 y Fp(f)845 830 y Ft(f)895 800 y Fy(\(5\))1011 830 y Fz(whic)n(h)26 b(w)n(e)h(w)n(ould)f(lik)n(e)f(to)i(add)g(to)g(the)h(measure,)d (instead)h(of)h Fs(\000)p Ft(B)3124 842 y Fy(5)3161 830 y Ft(J)3215 800 y Fy(\(5\))440 930 y Fz(as)33 b(originally)c(prop)r (osed,)34 b(where)g Ft(B)1616 942 y Fy(5)1687 930 y Fz(is)f(the)i(co)r (e\016cien)n(t)e(of)h Ft(J)2476 900 y Fy(\(5\))2599 930 y Fz(in)f(the)i(cosmological)440 1029 y(constan)n(t)c(from)f(the)i ('plain')e(OPSMY)h(ansatz.)48 b(F)-7 b(rom)30 b(the)i(degeneration)d (in)i(the)h(limit)440 1129 y Ft(s)23 b Fs(!)g Fz(0,)e(w)n(e)e(obtain)g (a)g(surface)g(with)g(t)n(w)n(o)g(no)r(des)g Ft(a)h Fz(and)f Ft(b)p Fz(.)34 b(No)n(w,)21 b(let)e Ft(\027)2621 1099 y Fy(2)2616 1150 y Fx(\003)2658 1129 y Fz(\()p Ft(c)p Fz(\))24 b(=)f Ft(@)2914 1141 y Fp(i)2941 1129 y Ft(\022)2980 1141 y Fx(\003)3018 1129 y Fz(\(0\))p Ft(!)3176 1141 y Fp(i)3204 1129 y Fz(\()p Ft(c)p Fz(\))440 1229 y(for)k(an)g(o)r(dd)h (theta)g(c)n(haracteristic)23 b Fs(\003)28 b Fz(and)f(de\014ne)1488 1438 y Ft(E)5 b Fz(\()p Ft(a;)14 b(b)p Fz(\))23 b(:=)1926 1382 y Ft(\022)1965 1394 y Fx(\003)2003 1382 y Fz(\()p Ft(A)2097 1394 y Fp(ab)2167 1382 y Fz(\))p 1879 1419 368 4 v 1879 1495 a Ft(\027)1920 1507 y Fx(\003)1958 1495 y Fz(\()p Ft(a)p Fz(\))p Ft(\027)2107 1507 y Fx(\003)2146 1495 y Fz(\()p Ft(b)p Fz(\))3134 1438 y SDict begin H.S end 3134 1438 a 3134 1438 a SDict 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Fz(\))k(+)g Ft(O)r Fz(\()p Ft(s)2508 1918 y Fy(2)2546 1952 y Fz(\))p Ft(;)3134 1952 y SDict begin H.S end 3134 1952 a 3134 1952 a SDict begin 15 H.A end 3134 1952 a 3134 1952 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.3) cvn /DEST pdfmark end 3134 1952 a Fz(\(6.3\))440 2135 y(from)23 b([)656 2135 y SDict begin H.S end 656 2135 a Fz(27)740 2135 y SDict begin 12 H.L end 740 2135 a 740 2135 a SDict begin [/Subtype /Link/Dest (cite.matone2010getting) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 740 2135 a -1 w Fz(].)37 b(F)-7 b(or)24 b(the)h(OPSMY)f(part)g(of)h(the)g(ansatz)f(w) n(e)g(will)e(stic)n(k)i(to)g(the)h(notation)f(from)440 2234 y(Matone)k(and)g(V)-7 b(olpato,)27 b(that)h(is,)g(w)n(e)g(will)d (write)i(\002)2084 2246 y Fp(k)2153 2234 y Fz(for)h(the)h(lattice)d (theta)i(series,)f(with)440 2334 y(a)k(di\013eren)n(t)g(n)n(um)n(b)r (ering)e(of)i(lattices)e(for)i Ft(k)i Fs(\024)c Fz(5,)j(so)f(that)h(it) f(is)f(easier)f(to)j(compare)d(the)440 2434 y(form)n(ulae.)34 b(Here)27 b(w)n(e)g(presen)n(t)g(a)g(translation)e(diagram:)p 948 2619 1848 4 v 947 2719 4 100 v 998 2689 a([)1021 2689 y SDict begin H.S end 1021 2689 a Fz(27)1105 2689 y SDict begin 0 H.L end 1105 2689 a 1105 2689 a SDict begin [/Subtype /Link/Dest (cite.matone2010getting) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1105 2689 a -1 w Fz(])j(notation)p 1507 2719 4 100 v 277 w(Lattice)p 2219 2719 V 277 w(Our)f(notation)p 2794 2719 V 948 2722 1848 4 v 947 2830 4 108 v 1178 2800 a(\002)1243 2812 y Fy(0)p 1507 2830 V 1649 2800 a Fz(\()p Ft(D)1750 2812 y Fy(8)1805 2800 y Fs(\010)18 b Ft(D)1957 2812 y Fy(8)1994 2800 y Fz(\))2027 2758 y Fy(+)p 2219 2830 V 2465 2800 a Ft(#)2514 2812 y Fy(5)p 2794 2830 V 948 2833 1848 4 v 947 2934 4 102 v 1178 2904 a Fz(\002)1243 2916 y Fy(1)p 1507 2934 V 1720 2904 a Fi(Z)12 b Fs(\010)18 b Ft(A)1939 2869 y Fy(+)1939 2927 y(15)p 2219 2934 V 2465 2904 a Ft(#)2514 2916 y Fy(4)p 2794 2934 V 948 2938 1848 4 v 947 3045 4 108 v 1178 3015 a Fz(\002)1243 3027 y Fy(2)p 1507 3045 V 1559 3015 a Fi(Z)1620 2985 y Fy(2)1670 3015 y Fs(\010)g Fz(\()p Ft(E)1846 3027 y Fy(7)1902 3015 y Fs(\010)g Ft(E)2046 3027 y Fy(7)2084 3015 y Fz(\))2116 2974 y Fy(+)p 2219 3045 V 2465 3015 a Ft(#)2514 3027 y Fy(3)p 2794 3045 V 948 3049 1848 4 v 947 3150 4 102 v 1178 3120 a Fz(\002)1243 3132 y Fy(3)p 1507 3150 V 1699 3120 a Fi(Z)1760 3090 y Fy(4)1810 3120 y Fs(\010)g Ft(D)1964 3085 y Fy(+)1962 3142 y(12)p 2219 3150 V 2465 3120 a Ft(#)2514 3132 y Fy(2)p 2794 3150 V 948 3153 1848 4 v 947 3253 4 100 v 1178 3223 a Fz(\002)1243 3235 y Fy(4)p 1507 3253 V 1719 3223 a Fi(Z)1780 3193 y Fy(8)1830 3223 y Fs(\010)g Ft(E)1974 3235 y Fy(8)p 2219 3253 V 2465 3223 a Ft(#)2514 3235 y Fy(1)p 2794 3253 V 948 3256 1848 4 v 947 3356 4 100 v 1178 3326 a Fz(\002)1243 3338 y Fy(5)p 1507 3356 V 1802 3326 a Fi(Z)1864 3296 y Fy(16)p 2219 3356 V 2465 3326 a Ft(#)2514 3338 y Fy(0)p 2794 3356 V 948 3359 1848 4 v 947 3459 4 100 v 1178 3429 a Fz(\002)1243 3441 y Fy(6)p 1507 3459 V 1716 3429 a Ft(E)1777 3441 y Fy(8)1833 3429 y Fs(\010)g Ft(E)1977 3441 y Fy(8)p 2219 3459 V 2465 3429 a Ft(#)2514 3441 y Fy(6)p 2794 3459 V 948 3462 1848 4 v 947 3564 4 102 v 1178 3534 a Fz(\002)1243 3546 y Fy(7)p 1507 3564 V 1796 3534 a Ft(D)1867 3498 y Fy(+)1865 3556 y(16)p 2219 3564 V 2465 3534 a Ft(#)2514 3546 y Fy(7)p 2794 3564 V 948 3567 1848 4 v 440 3791 a Fz(Let)29 b Ft(N)657 3803 y Fp(k)726 3791 y Fz(b)r(e)g(the)g(n)n(um)n(b)r(er)e(of)i(norm)e(t)n(w)n(o)h(v)n (ectors)f(in)h(the)h(lattice)e(corresp)r(onding)e(to)k(\002)3241 3803 y Fp(k)3281 3791 y Fz(.)440 3890 y(Let)35 b Ft(c)632 3850 y Fp(g)632 3915 y(k)707 3890 y Fz(b)r(e)g(the)f(co)r(e\016cien)n (t)g(of)g(\002)1537 3902 y Fp(k)1612 3890 y Fz(in)g(the)g(OPSMY)g (ansatz)g(for)g(gen)n(us)f Ft(g)s Fz(,)j(where)e(the)440 3990 y(same)26 b(normalization)c(as)27 b(in)f([)1395 3990 y SDict begin H.S end 1395 3990 a Fz(32)1479 3990 y SDict begin 12 H.L end 1479 3990 a 1479 3990 a SDict begin [/Subtype /Link/Dest (cite.grushevsky) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1479 3990 a -1 w Fz(])i(is)e(used)i(\()p Ft(c)1870 3950 y Fp(g)1870 4015 y(k)1938 3990 y Fz(is)e(2)2063 3960 y Fy(4)p Fp(g)2162 3990 y Fz(times)g(the)i(co)r(e\016cien)n(ts)e (from)g([)3166 3990 y SDict begin H.S end 3166 3990 a Fz(35)3250 3990 y SDict begin 12 H.L end 3250 3990 a 3250 3990 a SDict begin [/Subtype /Link/Dest (cite.oura2010modular) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 3250 3990 a -1 w Fz(]\))440 4089 y(for)h(easier)e (comparison.)440 4280 y(W)-7 b(e)28 b(ha)n(v)n(e,)e(for)i(the)g(OPSMY)f (ansatz,)g(from)f([)1902 4280 y SDict begin H.S end 1902 4280 a Fz(27)1986 4280 y SDict begin 12 H.L end 1986 4280 a 1986 4280 a SDict begin [/Subtype /Link/Dest (cite.matone2010getting) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1986 4280 a -1 w Fz(],)787 4542 y Ft(X)856 4554 y Fp(N)6 b(S)963 4542 y Fz([)p Ft(m)p Fz(]\()p Ft(s;)14 b Fz(\012)p Ft(;)g(z)26 b Fz(=)d(0\))g(=)1668 4438 y Fy(7)1625 4463 y Fl(X)1624 4642 y Fp(k)q Fy(=0)1759 4542 y Ft(c)1795 4508 y Fy(5)1795 4562 y Fp(k)1850 4475 y Fl(\000)1888 4542 y Fz(1)18 b(+)g Ft(N)2098 4554 y Fp(k)2138 4542 y Ft(s)g Fz(+)h Ft(O)r Fz(\()p Ft(s)2415 4508 y Fy(2)2453 4542 y Fz(\))2485 4475 y Fl(\001)2537 4542 y Fz(\002)2602 4499 y Fy(\(4\))2602 4567 y Fp(k)2690 4542 y Fz([)p Ft(m)p Fz(]\(\012\))p Ft(:)3134 4542 y SDict begin H.S end 3134 4542 a 3134 4542 a SDict begin 15 H.A end 3134 4542 a 3134 4542 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.4) cvn /DEST pdfmark end 3134 4542 a Fz(\(6.4\))440 4799 y(W)-7 b(e)28 b(will)d(write)907 4982 y Ft(X)976 4994 y Fp(N)6 b(S)1083 4982 y Fz([)p Ft(m)p Fz(]\()p Ft(s;)14 b(\034)5 b(;)14 b(z)t Fz(\))22 b(=)h Ft(T)1622 4994 y Fy(0)1659 4982 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b(z)t Fz(\))k(+)g Ft(s)c(T)2166 4994 y Fy(1)2203 4982 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b(z)t Fz(\))k(+)g Ft(O)r Fz(\()p Ft(s)2744 4947 y Fy(2)2782 4982 y Fz(\))p Ft(:)3134 4982 y SDict begin H.S end 3134 4982 a 3134 4982 a SDict begin 15 H.A end 3134 4982 a 3134 4982 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.5) cvn /DEST pdfmark end 3134 4982 a Fz(\(6.5\))440 5181 y(Note)j(that)h Ft(E)869 5193 y Fy(8)911 5181 y Fs(\010)5 b Ft(E)1042 5193 y Fy(8)1101 5181 y Fz(and)21 b Ft(D)1327 5145 y Fy(+)1325 5203 y(16)1416 5181 y Fz(con)n(tain)f(480) f(norm)h(2)h(v)n(ectors)e(and)i(\()p Ft(D)2653 5193 y Fy(8)2709 5181 y Fs(\010)d Ft(D)2861 5193 y Fy(8)2898 5181 y Fz(\))2930 5139 y Fy(+)3006 5181 y Fz(con)n(tains)440 5298 y(224)27 b(of)h(them.)39 b(Also,)27 b(the)i Ft(s)p Fz(-linear)c(term)i(from)g Ft(G)2049 5255 y Fy(\(5\))2049 5320 y(5)2138 5298 y Fz(,)i(form)n(ula)c(\()2527 5298 y SDict begin H.S end 2527 5298 a Fz(3.24)2675 5244 y SDict begin H.R end 2675 5244 a 2675 5298 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.24) cvn H.B /ANN pdfmark end 2675 5298 a Fz(\),)k(equals)d(244)14 b Ft(G)3216 5255 y Fy(\(4\))3216 5320 y(4)1831 5662 y Fz(17)p eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 440 133 a SDict begin H.S end 440 133 a 440 133 a SDict begin H.R end 440 133 a 440 133 a SDict begin [/View [/XYZ H.V]/Dest (page.18) cvn /DEST pdfmark end 440 133 a 398 x Fz(in)27 b Ft(z)f Fz(=)d(0.)36 b(Therefore,)27 b(w)n(e)g(ha)n(v)n(e)522 788 y Ft(T)571 800 y Fy(0)608 788 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b Fz(0\))23 b(=)1065 685 y Fy(5)1022 709 y Fl(X)1021 888 y Fp(k)q Fy(=0)1156 788 y Ft(c)1192 754 y Fy(5)1192 809 y Fp(k)1233 788 y Fz(\002)1298 745 y Fy(\(4\))1298 813 y Fp(k)1387 788 y Fz([)p Ft(m)p Fz(])18 b(+)g Ft(cJ)1697 754 y Fy(\(4\))1805 788 y Fz(+)g Ft(c)1924 800 y Fp(f)1967 788 y Ft(f)2017 754 y Fy(\(4\))2128 788 y Fz(=)2216 671 y Fl(\022)2277 788 y Ft(c)2313 800 y Fp(J)2378 788 y Fs(\000)2471 732 y Fz(2)2513 702 y Fy(5)2568 732 y Fs(\001)h Fz(3)p 2471 769 181 4 v 2540 845 a(7)2661 671 y Fl(\023)2736 788 y Ft(J)2790 754 y Fy(\(4\))2898 788 y Fz(+)f Ft(c)3017 800 y Fp(f)3060 788 y Ft(f)3110 754 y Fy(\(4\))3134 975 y SDict begin H.S end 3134 975 a 3134 975 a SDict begin 15 H.A end 3134 975 a 3134 975 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.6) cvn /DEST pdfmark end 3134 975 a Fz(\(6.6\))522 1161 y Ft(T)571 1173 y Fy(1)608 1161 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b Fz(0\))23 b(=)f(128\004)1202 1118 y Fy(\(4\))1202 1186 y Fp(O)r(P)9 b(S)s(M)d(Y)1476 1161 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)j Fz(\))20 b(+)1807 1044 y Fl(\022)1868 1161 y Fz(480)p Ft(c)2030 1173 y Fp(J)2093 1161 y Fs(\000)2186 1105 y Fz(720)d Fs(\001)h Fz(2)2412 1075 y Fy(5)2468 1105 y Fs(\001)g Fz(3)p 2186 1142 365 4 v 2348 1218 a(7)2561 1044 y Fl(\023)2636 1161 y Ft(J)2690 1127 y Fy(\(4\))2797 1161 y Fz(+)g(224)p Ft(c)3042 1173 y Fp(f)3084 1161 y Ft(f)3134 1127 y Fy(\(4\))3134 1318 y SDict begin H.S end 3134 1318 a 3134 1318 a SDict begin 15 H.A end 3134 1318 a 3134 1318 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.7) cvn /DEST pdfmark end 3134 1318 a Fz(\(6.7\))440 1496 y(As)28 b Ft(s)23 b Fs(!)g Fz(0,)k(w)n(e)g(get)440 1753 y Ft(X)509 1765 y Fp(N)6 b(S)616 1753 y Fz([)p Ft(m)p Fz(])23 b(=)f Ft(s)942 1649 y Fy(4)898 1674 y Fl(X)921 1851 y Fp(i;j)1032 1753 y Fz(2)p Ft(\031)s(iE)5 b Fz(\()p Ft(a;)14 b(b)p Fz(\))1400 1719 y Fy(2)1437 1753 y Ft(!)1489 1765 y Fp(i)1516 1753 y Fz(\()p Ft(a)p Fz(\))p Ft(!)1676 1765 y Fp(j)1711 1753 y Fz(\()p Ft(b)p Fz(\)\(1)19 b(+)f Ft(\016)2024 1765 y Fp(ij)2082 1753 y Fz(\))2128 1636 y Fl(\022\022)2250 1753 y Ft(c)2286 1765 y Fp(J)2351 1753 y Fs(\000)2444 1697 y Fz(2)2486 1667 y Fy(5)2541 1697 y Fs(\001)h Fz(3)p 2444 1734 181 4 v 2513 1810 a(7)2634 1636 y Fl(\023)2719 1697 y Ft(@)5 b(J)2822 1667 y Fy(\(4\))p 2719 1734 192 4 v 2743 1810 a Ft(@)g(\034)2828 1822 y Fp(ij)2939 1753 y Fz(+)18 b Ft(c)3058 1765 y Fp(f)3111 1697 y Ft(@)5 b(f)3210 1667 y Fy(\(4\))p 3111 1734 188 4 v 3133 1810 a Ft(@)g(\034)3218 1822 y Fp(ij)3309 1636 y Fl(\023)919 1991 y Fz(+)18 b Ft(sT)1102 1948 y Fy(\(4\))1090 2013 y(1)1190 1991 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b(A)1481 2003 y Fp(ab)1551 1991 y Fz(\))19 b(+)f Ft(O)r Fz(\()p Ft(s)1821 1956 y Fy(2)1859 1991 y Fz(\))p Ft(:)3134 1991 y SDict begin H.S end 3134 1991 a 3134 1991 a SDict begin 15 H.A end 3134 1991 a 3134 1991 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.8) cvn /DEST pdfmark end 3134 1991 a Fz(\(6.8\))440 2173 y(Calculating)f Ft(T)925 2185 y Fy(1)962 2173 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b(A)1253 2185 y Fp(ab)1323 2173 y Fz(\))22 b(from)e Ft(T)1616 2185 y Fy(1)1652 2173 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b Fz(0\))22 b(can)f(b)r(e)g(done)g(using)f (the)i(fact)f(that)g Ft(T)3148 2185 y Fy(1)3185 2173 y Fz([)p Ft(m)p Fz(])440 2273 y(is)30 b(a)g(section)f(of)i Fs(j)p Fz(2\002)p Fs(j)p Fz(,)g(b)r(ecause)f(of)h(the)g(mo)r(dular)d (prop)r(erties)h(of)i Ft(X)2632 2285 y Fp(N)6 b(S)2738 2273 y Fz(.)47 b(Here)30 b(\002)g(is)g(the)440 2373 y(divisor)25 b(of)i Ft(\022)844 2385 y Fy(0)881 2373 y Fz(\()p Ft(z)t Fz(\).)37 b(Matone)27 b(and)h(V)-7 b(olpato)26 b(pro)n(v)n(e)g(that)h (from)g(that)h(fact)f(it)g(follo)n(ws)e(that)440 2825 y Ft(T)489 2837 y Fy(1)526 2825 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b(A)817 2837 y Fp(ab)887 2825 y Fz(\))23 b(=)g Ft(E)5 b Fz(\()p Ft(a;)14 b(b)p Fz(\))1277 2791 y Fy(2)1328 2658 y Fl(0)1328 2808 y(@)1400 2825 y Ft(T)1449 2837 y Fy(1)1486 2825 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b Fz(0\))p Ft(!)s Fz(\()p Ft(a;)g(b)p Fz(\))k(+)2136 2769 y(1)p 2136 2806 42 4 v 2136 2882 a(2)2245 2721 y Fy(4)2202 2746 y Fl(X)2225 2923 y Fp(i;j)2335 2825 y Ft(@)2379 2837 y Fp(i)2407 2825 y Ft(@)2451 2837 y Fp(j)2486 2825 y Ft(T)2535 2837 y Fy(1)2572 2825 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b Fz(0\))p Ft(!)2927 2837 y Fp(i)2954 2825 y Fz(\()p Ft(a)p Fz(\))p Ft(!)3114 2837 y Fp(j)3149 2825 y Fz(\()p Ft(b)p Fz(\))3249 2658 y Fl(1)3249 2808 y(A)3336 2825 y Ft(:)3134 3032 y SDict begin H.S end 3134 3032 a 3134 3032 a SDict begin 15 H.A end 3134 3032 a 3134 3032 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.9) cvn /DEST pdfmark end 3134 3032 a Fz(\(6.9\))440 3227 y(F)-7 b(rom)22 b([)677 3227 y SDict begin H.S end 677 3227 a Fz(36)761 3227 y SDict begin 12 H.L end 761 3227 a 761 3227 a SDict begin [/Subtype /Link/Dest (cite.latticevsriemann) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 761 3227 a -1 w Fz(],)j(w)n(e)f(ha)n(v)n (e)e Ft(f)1187 3197 y Fy(\(4\))1299 3227 y Fz(=)1397 3194 y Fy(3)p 1397 3208 34 4 v 1397 3255 a(7)1440 3227 y Ft(J)1494 3197 y Fy(\(4\))1607 3227 y Fz(whic)n(h)g(is)h(the)h(Sc)n (hottky)g(form)e(and)i(v)-5 b(anishes)22 b(on)i Fs(J)3244 3239 y Fy(4)3281 3227 y Fz(.)440 3350 y(Th)n(us)j(w)n(e)h(ha)n(v)n(e)e Ft(T)1013 3362 y Fy(1)1050 3350 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b Fz(0\))23 b(=)g(128)14 b(\004)1659 3307 y Fy(\(4\))1659 3375 y Fp(O)r(P)9 b(S)s(M)d(Y)1959 3350 y Fz(on)28 b(the)g(Jacobian)d(lo)r(cus.)35 b(Then,)28 b(w)n(e)f(get)554 3616 y Ft(A)616 3628 y Fy(2)654 3616 y Fz([)p Ft(m)p Fz(]\()p Ft(a;)14 b(b)p Fz(\))23 b(=)f(128)14 b(\004)1259 3581 y Fy(\(4\))1347 3616 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)9 b Fz(\))p Ft(!)s Fz(\()p Ft(a;)14 b(b)p Fz(\))20 b(+)1957 3512 y Fy(4)1914 3537 y Fl(X)1937 3714 y Fp(i;j)2047 3616 y Fz(2)p Ft(\031)s(i)p Fz(\(1)e(+)g Ft(\016)2380 3628 y Fp(ij)2438 3616 y Fz(\))p Ft(!)2522 3628 y Fp(i)2550 3616 y Fz(\()p Ft(a)p Fz(\))p Ft(!)2710 3628 y Fp(j)2745 3616 y Fz(\()p Ft(b)p Fz(\))c Fs(\001)1138 3896 y(\001)1180 3779 y Fl(\022\022)1302 3896 y Ft(c)1338 3908 y Fp(J)1403 3896 y Fs(\000)1496 3840 y Fz(2)1538 3810 y Fy(5)1593 3840 y Fs(\001)k Fz(3)p 1496 3877 181 4 v 1565 3953 a(7)1686 3779 y Fl(\023)1771 3840 y Ft(@)5 b(J)1874 3810 y Fy(\(4\))p 1771 3877 192 4 v 1795 3953 a Ft(@)g(\034)1880 3965 y Fp(ij)1991 3896 y Fz(+)18 b Ft(c)2110 3908 y Fp(f)2163 3840 y Ft(@)5 b(f)2262 3810 y Fy(\(4\))p 2163 3877 188 4 v 2185 3953 a Ft(@)g(\034)2270 3965 y Fp(ij)2379 3896 y Fz(+)2472 3840 y(1)p 2472 3877 42 4 v 2472 3953 a(2)2523 3896 y Ft(@)2567 3908 y Fp(i)2595 3896 y Ft(@)2639 3908 y Fp(j)2674 3896 y Ft(T)2735 3853 y Fy(\(4\))2723 3918 y(1)2823 3896 y Fz([)p Ft(e)p Fz(]\()p Ft(\034)g(;)14 b Fz(0\))3092 3779 y Fl(\023)3167 3896 y Ft(:)3092 4055 y SDict begin H.S end 3092 4055 a 3092 4055 a SDict begin 15 H.A end 3092 4055 a 3092 4055 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.10) cvn /DEST pdfmark end 3092 4055 a Fz(\(6.10\))440 4263 y(Denoting)28 b(b)n(y)g Ft(f)966 4220 y Fy(\(5\))957 4285 y(1)1084 4263 y Fz(the)h Ft(s)p Fz(-linear)d(term)h(from)h(the)h Ft(s)p Fz(-expansion)d(of)j Ft(f)2667 4233 y Fy(\(5\))2756 4263 y Fz(,)g(and)g(using)e(the)440 4363 y(functions)724 4545 y Ft(F)789 4502 y Fy(\()p Fp(g)r Fy(\))777 4570 y Fp(k)880 4545 y Fz(\()p Ft(\034)5 b(;)14 b(z)t Fz(\))23 b(:=)1319 4466 y Fl(X)1198 4644 y Fp(p)1232 4652 y Ff(1)1265 4644 y Fp(;:::)o(;p)1398 4652 y Fk(g)1433 4644 y Fx(2)p Fy(\003)1523 4653 y Fk(k)1573 4545 y Ft(e)1612 4509 y Fp(\031)r(i)1687 4466 y Fe(P)1757 4481 y Fk(g)1757 4528 y(i;j)r Ff(=1)1911 4509 y Fp(p)1945 4517 y Fk(i)1972 4509 y Fp(p)2006 4517 y Fk(j)2037 4509 y Fp(\034)2068 4517 y Fk(ij)2165 4466 y Fl(X)2145 4644 y Fy(~)-39 b Fp(p)p Fx(\001)6 b Fy(~)-39 b Fp(p)p Fy(=2)2325 4545 y Ft(e)2364 4511 y Fy(2)p Fp(\031)r(i)2473 4467 y Fe(P)2542 4483 y Fk(g)2542 4530 y(i)p Ff(=1)2657 4511 y Fy(~)g Fp(pp)2719 4519 y Fk(i)2746 4511 y Fp(z)2778 4519 y Fk(i)3092 4545 y SDict begin H.S end 3092 4545 a 3092 4545 a SDict begin 15 H.A end 3092 4545 a 3092 4545 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.11) cvn /DEST pdfmark end 3092 4545 a Fz(\(6.11\))440 4818 y(w)n(e)27 b(end)h(up)g(with)f(the)h (mo)r(di\014ed)e(form)n(ula)502 5075 y Ft(@)546 5087 y Fp(i)574 5075 y Ft(@)618 5087 y Fp(j)653 5075 y Ft(T)714 5032 y Fy(\(4\))702 5097 y(1)802 5075 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b Fz(0\))23 b(=)g Ft(@)1260 5087 y Fp(i)1287 5075 y Ft(@)1331 5087 y Fp(j)1380 4933 y Fl( )1490 4971 y Fy(5)1446 4996 y Fl(X)1446 5175 y Fp(k)q Fy(=0)1580 5075 y Ft(c)1616 5041 y Fy(5)1616 5096 y Fp(k)1657 5075 y Ft(F)1722 5032 y Fy(\(4\))1710 5100 y Fp(k)1811 5075 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b Fz(0\))19 b(+)f Ft(c)2252 5087 y Fp(J)2312 4983 y Fl(\020)2362 5075 y Ft(F)2427 5032 y Fy(\(4\))2415 5097 y(6)2534 5075 y Fs(\000)g Ft(F)2682 5032 y Fy(\(4\))2670 5097 y(7)2771 4983 y Fl(\021)2839 5075 y Fz(+)g Ft(c)2958 5087 y Fp(f)3001 5075 y Ft(f)3051 5032 y Fy(\(5\))3042 5097 y(1)3140 4933 y Fl(!)3219 5075 y Ft(:)3092 5261 y SDict begin H.S end 3092 5261 a 3092 5261 a SDict begin 15 H.A end 3092 5261 a 3092 5261 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.12) cvn /DEST pdfmark end 3092 5261 a Fz(\(6.12\))1831 5662 y(18)p eop end end %%Page: 19 19 TeXDict begin HPSdict begin 19 18 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 432 133 a SDict begin H.S end 432 133 a 432 133 a SDict begin H.R end 432 133 a 432 133 a SDict begin [/View [/XYZ H.V]/Dest (page.19) cvn /DEST pdfmark end 432 133 a 440 531 a Fz(Here,)31 b(Matone)e(and)h(V)-7 b(olpato)29 b(in)n(tro)r(duce)g(the)i(co)r(e\016cien)n(ts)d Ft(s)2422 491 y Fp(g)2422 556 y(k)2493 531 y Fz(and)j Ft(t)2688 491 y Fp(g)2688 556 y(k)2728 531 y Fz(,)g(de\014ned)g(b)n(y)f(the)440 631 y(follo)n(wing)24 b(form)n(ula:)479 825 y Ft(@)523 837 y Fp(i)550 825 y Ft(@)594 837 y Fp(j)629 825 y Ft(c)665 785 y Fp(g)r Fy(+1)665 850 y Fp(k)788 825 y Ft(F)853 782 y Fy(\()p Fp(g)r 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[/Subtype /Link/Dest (cite.matone2010getting) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1630 1133 a -1 w Fz(],)j(and)e(noting)e(that)j Ft(f)2380 1090 y Fy(\(5\))2371 1155 y(1)2502 1133 y Fz(has)f(the)g (prop)r(ert)n(y)f(that)456 1225 y Fp(@)495 1200 y Ff(2)527 1225 y Fp(f)566 1194 y Ff(\(5\))559 1243 y(1)p 450 1249 199 4 v 450 1297 a Fp(@)t(z)521 1305 y Fk(i)547 1297 y Fp(@)t(z)618 1305 y Fk(j)682 1268 y Fz(=)22 b(28\(2)p Ft(\031)s(i)p Fz(\)\(1)c(+)g Ft(\016)1250 1280 y Fp(ij)1308 1268 y Fz(\))1350 1231 y Fp(@)t(f)1428 1206 y Ff(\(4\))p 1350 1249 156 4 v 1367 1297 a Fp(@)t(\034)1437 1305 y Fk(ij)1543 1268 y Fz(\(see)28 b(form)n(ulae)c(\()2083 1268 y SDict begin H.S end 2083 1268 a Fz(3.23)2231 1215 y SDict begin H.R end 2231 1215 a 2231 1268 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.23) cvn H.B /ANN pdfmark end 2231 1268 a Fz(\))k(and)g(\()2485 1268 y SDict begin H.S end 2485 1268 a Fz(3.30)2632 1215 y SDict begin H.R end 2632 1215 a 2632 1268 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.30) cvn H.B /ANN pdfmark end 2632 1268 a Fz(\))q(\),)g(w)n(e)f(then)h(get)462 1554 y Ft(@)506 1566 y Fp(i)533 1554 y Ft(@)577 1566 y Fp(j)612 1554 y Ft(T)673 1511 y Fy(\(4\))661 1576 y(1)762 1554 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b Fz(0\))23 b(=)f(2)p Ft(\031)s(i)p Fz(\(1)c(+)g Ft(\016)1508 1566 y Fp(ij)1566 1554 y Fz(\))1656 1498 y Ft(@)p 1608 1535 144 4 v 1608 1611 a(@)5 b(\034)1693 1623 y Fp(ij)1776 1412 y Fl( )1886 1451 y Fy(5)1842 1475 y Fl(X)1842 1654 y Fp(k)q Fy(=0)1976 1554 y Ft(s)2015 1520 y Fy(4)2015 1575 y Fp(k)2056 1554 y Fz(\002)2121 1511 y Fy(\(4\))2121 1579 y Fp(k)2210 1554 y Fz([)p Ft(e)p Fz(]\()p Ft(\034)k Fz(\))19 b(+)f(60)p Ft(c)2626 1566 y Fp(J)2672 1554 y Ft(J)2726 1520 y Fy(\(4\))2833 1554 y Fz(+)g(28)p Ft(c)3036 1566 y Fp(f)3078 1554 y Ft(f)3128 1520 y Fy(\(4\))3217 1412 y Fl(!)1817 1845 y Fs(\000)1896 1703 y Fl( )2005 1741 y Fy(5)1962 1766 y Fl(X)1962 1945 y Fp(k)q Fy(=0)2096 1845 y Ft(t)2126 1811 y Fy(4)2126 1866 y Fp(k)2167 1845 y Fz(\002)2232 1802 y Fy(\(4\))2232 1870 y Fp(k)2321 1845 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)9 b Fz(\))2549 1703 y Fl(!)2630 1845 y Ft(@)2674 1857 y Fp(i)2701 1845 y Ft(@)2745 1857 y Fp(j)2794 1845 y Fz(log)k Ft(\022)r Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b Fz(0\))p Ft(:)3092 2031 y SDict begin H.S end 3092 2031 a 3092 2031 a SDict begin 15 H.A end 3092 2031 a 3092 2031 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.14) cvn /DEST pdfmark end 3092 2031 a Fz(\(6.14\))440 2215 y(And)33 b(further)f(follo)n(wing)d(the)j(calculations)d(from)i([) 2096 2215 y SDict begin H.S end 2096 2215 a Fz(27)2180 2215 y SDict begin 12 H.L end 2180 2215 a 2180 2215 a SDict begin [/Subtype /Link/Dest (cite.matone2010getting) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 2180 2215 a -1 w Fz(])h(the)h(\014rst)f(term)f(in)h(big)f (brac)n(k)n(ets)440 2314 y(can)c(b)r(e)h(written)f(as)491 2452 y Fy(5)448 2476 y Fl(X)447 2655 y Fp(k)q Fy(=0)582 2555 y Ft(s)621 2521 y Fy(4)621 2576 y Fp(k)662 2555 y Fz(\002)727 2512 y Fy(\(4\))727 2580 y Fp(k)815 2555 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)9 b Fz(\))20 b(+)e(60)p Ft(c)1266 2567 y Fp(J)1312 2555 y Ft(J)1366 2521 y Fy(\(4\))1473 2555 y Fz(+)g(28)p Ft(c)1676 2567 y Fp(f)1718 2555 y Ft(f)1768 2521 y Fy(\(4\))1880 2555 y Fz(=)23 b(32\004)2107 2521 y Fy(\(4\))2195 2555 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)9 b Fz(\))1875 2820 y(+)1958 2703 y Fl(\022)2019 2820 y Fz(60)p Ft(c)2139 2832 y Fp(J)2203 2820 y Fz(+)2296 2764 y(3)18 b Fs(\001)h Fz(28)p 2296 2801 185 4 v 2368 2877 a(7)2491 2820 y Ft(c)2527 2832 y Fp(f)2588 2820 y Fs(\000)2681 2764 y Fz(152)e Fs(\001)i Fz(2)2908 2734 y Fy(5)2963 2764 y Fs(\001)f Fz(3)p 2681 2801 365 4 v 2843 2877 a(7)3056 2703 y Fl(\023)3131 2820 y Ft(J)3185 2786 y Fy(\(4\))3274 2820 y Ft(:)3092 2977 y SDict begin H.S end 3092 2977 a 3092 2977 a SDict begin 15 H.A end 3092 2977 a 3092 2977 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.15) cvn /DEST pdfmark end 3092 2977 a Fz(\(6.15\))440 3175 y(So,)29 b(ha)n(ving)e(carried)f(the)k(mo)r(di\014ed)1619 3154 y(~)1612 3175 y(\004)f(through)f(the)i(degeneration,)c(w)n(e)j(end)g (up)g(with)g(a)440 3275 y(sligh)n(tly)24 b(di\013eren)n(t)j(t)n(w)n (o-p)r(oin)n(t)e(function,)531 3460 y Ft(A)593 3472 y Fy(2)631 3460 y Fz([)p Ft(m)p Fz(]\()p Ft(a;)14 b(b)p Fz(\))23 b(=)f(128\004)1222 3426 y Fy(\(4\))1310 3460 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)9 b Fz(\))p Ft(!)s Fz(\()p Ft(a;)14 b(b)p Fz(\))949 3673 y Fs(\000)1075 3569 y Fy(4)1032 3594 y Fl(X)1055 3771 y Fp(i;j)1166 3673 y Ft(!)1218 3685 y Fp(i)1245 3673 y Fz(\()p Ft(a)p Fz(\))p Ft(!)1405 3685 y Fp(j)1440 3673 y Fz(\()p Ft(b)p Fz(\))1540 3531 y Fl(")1589 3673 y Fz(128\004)1770 3638 y Fy(\(4\))1858 3673 y Fz([)p Ft(m)p Fz(]\()p Ft(\034)9 b Fz(\))p Ft(@)2130 3685 y Fp(i)2159 3673 y Ft(@)2203 3685 y Fp(j)2252 3673 y Fz(log)k Ft(\022)r Fz([)p Ft(m)p Fz(]\()p Ft(\034)5 b(;)14 b Fz(0\))1115 3969 y Fs(\000)k Fz(2)p Ft(\031)s(i)p Fz(\(1)g(+)g Ft(\016)1531 3981 y Fp(ij)1589 3969 y Fz(\))1679 3912 y Ft(@)p 1631 3950 144 4 v 1631 4026 a(@)5 b(\034)1716 4038 y Fp(ij)1785 3827 y Fl(\()1852 3969 y Fz(16\004)1991 3934 y Fy(\(4\))2079 3969 y Fz([)p Ft(e)p Fz(]\()p Ft(\034)k Fz(\))1281 4251 y(+)1364 4134 y Fl(\022)1425 4251 y Fz(\(30)18 b(+)g(1\))p Ft(c)1752 4263 y Fp(J)1817 4251 y Fz(+)g(\(6)g(+)g(1\))p Ft(c)2185 4263 y Fp(f)2246 4251 y Fs(\000)2339 4195 y Fz(\(76)g(+)g(1\))g Fs(\001)h Fz(2)2732 4165 y Fy(5)2787 4195 y Fs(\001)f Fz(3)p 2339 4232 531 4 v 2584 4308 a(7)2880 4134 y Fl(\023)2955 4251 y Ft(J)3009 4217 y Fy(\(4\))3098 4109 y Fl(\)#)3092 4433 y SDict begin H.S end 3092 4433 a 3092 4433 a SDict begin 15 H.A end 3092 4433 a 3092 4433 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.16) cvn /DEST pdfmark end 3092 4433 a Fz(\(6.16\))440 4616 y(The)k(last)e (step)i(of)f(the)h(pro)r(cedure)e(from)h([)1744 4616 y SDict begin H.S end 1744 4616 a Fz(27)1828 4616 y SDict begin 12 H.L end 1828 4616 a 1828 4616 a SDict begin [/Subtype /Link/Dest (cite.matone2010getting) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 0] H.B /ANN pdfmark end 1828 4616 a -1 w Fz(])g(is)g(to)g(sum)g(o)n(v)n(er)f(ev)n(en)h(c)n (haracteristics.)30 b(This)440 4716 y(pro)r(cedure)d(yields,)e (\014nally)-7 b(,)26 b(on)h Fs(J)1521 4728 y Fy(4)1558 4716 y Fz(,)628 4894 y Fl(X)658 5068 y Fp(m)761 4973 y Ft(A)823 4985 y Fy(2)861 4973 y Fz([)p Ft(m)p Fz(]\()p Ft(a;)14 b(b)p Fz(\))23 b(=)g(2)1314 4939 y Fy(3)1350 4973 y Fz(\(2)1424 4939 y Fy(4)1480 4973 y Fz(+)18 b(1\))c Ft(C)1710 4985 y Fy(4)1804 4869 y(4)1761 4894 y Fl(X)1784 5071 y Fp(i;j)1895 4973 y Ft(!)1947 4985 y Fp(i)1974 4973 y Fz(\()p Ft(a)p Fz(\))p Ft(!)2134 4985 y Fp(j)2169 4973 y Fz(\()p Ft(b)p Fz(\)2)p Ft(\031)s(i)p Fz(\(1)k(+)g Ft(\016)2602 4985 y Fp(ij)2660 4973 y Fz(\))2702 4917 y Ft(@)5 b(J)2805 4887 y Fy(\(4\))p 2703 4954 192 4 v 2727 5030 a Ft(@)g(\034)2812 5042 y Fp(ij)3092 4973 y SDict begin H.S end 3092 4973 a 3092 4973 a SDict begin 15 H.A end 3092 4973 a 3092 4973 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.17) cvn /DEST pdfmark end 3092 4973 a Fz(\(6.17\))1018 5247 y Ft(C)1077 5259 y Fy(4)1138 5247 y Fz(:)23 b(=)1272 5130 y Fl(\022)1333 5247 y Fz(16)p Ft(B)1480 5259 y Fy(4)1534 5247 y Fs(\000)18 b Fz(8)p Ft(D)1728 5259 y Fy(4)1783 5247 y Fs(\000)g Fz(77)1960 5191 y(2)2002 5161 y Fy(5)2057 5191 y Fs(\001)g Fz(3)p 1960 5228 181 4 v 2029 5304 a(7)2168 5247 y(+)g(31)p Ft(c)2371 5259 y Fp(J)2435 5247 y Fz(+)g(7)p Ft(c)2596 5259 y Fp(f)2638 5130 y Fl(\023)2713 5247 y Ft(:)3092 5247 y SDict begin H.S end 3092 5247 a 3092 5247 a SDict begin 15 H.A end 3092 5247 a 3092 5247 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.18) cvn /DEST pdfmark end 3092 5247 a Fz(\(6.18\))1831 5662 y(19)p eop end end %%Page: 20 20 TeXDict begin HPSdict begin 20 19 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 440 133 a SDict begin H.S end 440 133 a 440 133 a SDict begin H.R end 440 133 a 440 133 a SDict begin [/View [/XYZ H.V]/Dest (page.20) cvn /DEST pdfmark end 440 133 a 398 x Fz(So,)27 b(to)h(mak)n(e)897 469 y Fl(P)984 556 y Fp(m)1061 531 y Ft(A)1123 543 y Fy(2)1161 531 y Fz([)p Ft(m)p Fz(]\()p Ft(a;)14 b(b)p Fz(\))27 b(v)-5 b(anish,)27 b(w)n(e)g(w)n(ould)f(need)1005 771 y(31)p Ft(c)1125 783 y Fp(J)1188 771 y Fz(+)18 b(7)p Ft(c)1349 783 y Fp(f)1415 771 y Fz(=)23 b(77)1597 715 y(2)1639 685 y Fy(5)1693 715 y Fs(\001)18 b Fz(3)p 1597 752 181 4 v 1665 828 a(7)1804 771 y(+)g(8)1941 715 y(2)1983 685 y Fy(7)2038 715 y Fs(\001)h Fz(3)p 1939 752 185 4 v 1939 828 a(7)f Fs(\001)g Fz(17)2152 771 y Fs(\000)2245 715 y Fz(2)2287 685 y Fy(6)2342 715 y 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Ft(c)1418 1396 y Fy(5)1418 1450 y Fp(k)1458 1430 y Fz(\002)1523 1442 y Fp(k)1564 1430 y Fz([)p Ft(m)p Fz(])23 b(=)g Fs(\000)p Fz(2)1901 1396 y Fy(4)1937 1430 y Fz(\(2)2011 1396 y Fy(5)2066 1430 y Fz(+)18 b(1\))2233 1374 y(2)2275 1344 y Fy(5)2330 1374 y Fs(\001)h Fz(17)p 2233 1411 222 4 v 2252 1487 a(7)f Fs(\001)g Fz(11)2465 1430 y Ft(J)2519 1396 y Fy(\(5\))2608 1430 y Ft(:)3092 1430 y SDict begin H.S end 3092 1430 a 3092 1430 a SDict begin 15 H.A end 3092 1430 a 3092 1430 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.20) cvn /DEST pdfmark end 3092 1430 a Fz(\(6.20\))440 1703 y(F)-7 b(rom)26 b(Section)949 1703 y SDict begin H.S end 949 1703 a Fz(4)990 1650 y SDict begin H.R end 990 1650 a 990 1703 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.4) cvn H.B /ANN pdfmark end 990 1703 a Fz(,)i(w)n(e)f(ha)n(v)n(e)g(for)g(the)h(trace)e(of)i Ft(f)1977 1673 y Fy(\(5\))2066 1703 y Fz(:)1276 1855 y Fl(X)1307 2030 y Fp(m)1410 1934 y 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begin 15 H.A end 3092 2967 a 3092 2967 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.23) cvn /DEST pdfmark end 3092 2967 a Fz(\(6.23\))440 3178 y(Com)n(bining)24 b(the)k(ab)r(o)n(v)n(e)e(linear)f(equations)h(\()1875 3178 y SDict begin H.S end 1875 3178 a Fz(6.19)2023 3124 y SDict begin H.R end 2023 3124 a 2023 3178 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.19) cvn H.B /ANN pdfmark end 2023 3178 a Fz(\))i(and)f(\()2276 3178 y SDict begin H.S end 2276 3178 a Fz(6.23)2424 3124 y SDict begin H.R end 2424 3124 a 2424 3178 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.23) cvn H.B /ANN pdfmark end 2424 3178 a Fz(\),)h(w)n(e)f(\014nd)h(the)g(solution)1246 3399 y Ft(c)1282 3411 y Fp(J)1351 3399 y Fz(=)23 b Fs(\000)1514 3343 y Fz(222647008)p 1514 3380 374 4 v 1638 3456 a(217)1897 3399 y Ft(;)14 b(c)1970 3411 y Fp(f)2036 3399 y Fz(=)2133 3343 y(77245568)p 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c(of)28 b(impro)n(ving)23 b(it)k(using)f(our)h(\014ndings.)1831 5662 y(20)p eop end end %%Page: 21 21 TeXDict begin HPSdict begin 21 20 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 432 133 a SDict begin H.S end 432 133 a 432 133 a SDict begin H.R end 432 133 a 432 133 a SDict begin [/View [/XYZ H.V]/Dest (page.21) cvn /DEST pdfmark end 432 133 a 440 535 a Fz(Let)28 b(\004)644 492 y Fy(\(6\))644 545 y Fp(m)761 535 y Fz(b)r(e)g(the)g(Grushevsky)e(ansatz)h(for)g(gen)n(us) g(6)2125 535 y SDict begin H.S end 2125 535 a -30 x Fy(2)2162 535 y SDict begin 12 H.L end 2162 535 a 2162 535 a SDict begin [/Subtype /Link/Dest (Hfootnote.2) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [1 0 0] H.B /ANN pdfmark end 2162 535 a 27 w Fz(\(see)h([)2379 535 y SDict begin 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Fz(E.)h(D'Hok)n(er)g(and)g(D.)h(Phong,)f(\\The)g(geometry)e(of)j (string)e(p)r(erturbation)g(theory,")611 4749 y Fr(R)l(ev.)j(Mo)l(d.)h (Phys)p Fz(,)e(no.)e(60,)g(p.)g(917,)g(1987.)440 4915 y SDict begin H.S end 440 4915 a Fz([25])570 4915 y SDict begin 12 H.A end 570 4915 a 570 4915 a SDict begin [/View [/XYZ H.V]/Dest (cite.atiyah1971riemann) cvn /DEST pdfmark end 570 4915 a 41 w Fz(M.)34 b(F.)g(A)n(tiy)n(ah,)g(\\Riemann)d(surfaces)i (and)h(spin)f(structures,")h Fr(A)n(nn.)h(Scient.)g(Ec.)611 5015 y(Norm.)30 b(Sup)p Fz(,)d(v)n(ol.)f(4,)h(pp.)h(47{62,)e(1971.)440 5181 y SDict begin H.S end 440 5181 a Fz([26])570 5181 y SDict begin 12 H.A end 570 5181 a 570 5181 a SDict begin [/View [/XYZ H.V]/Dest (cite.dhokerphong12) cvn /DEST pdfmark end 570 5181 a 41 w Fz(E.)h(D'Hok)n(er)g(and)g(D.)h(Phong,)f(\\Lectures)f (on)i(Tw)n(o-Lo)r(op)d(Sup)r(erstrings,")h(2002.)1831 5662 y(23)p eop end end %%Page: 24 24 TeXDict begin HPSdict begin 24 23 bop 0 0 a SDict begin /product where{pop 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