Merge branch 'master' of git:research/first_order_singularities/paper

1 files changed, 374 insertions, 0 deletions

diff --git a/figs/F_theta_singularities.eps b/figs/F_theta_singularities.eps
new file mode 100644
index 0000000..cba0e16
--- /dev/null
+++ b/figs/F_theta_singularities.eps
@@ -0,0 +1,374 @@
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: cairo 1.15.4 (http://cairographics.org)
+%%CreationDate: Tue May 16 17:20:36 2023
+%%Pages: 1
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%Creator: Wolfram Mathematica 13.2.1.0 for Linux x86 (64-bit) (January 31, 2023)
+%%BoundingBox: 0 0 360 220
+%%EndComments
+%%BeginProlog
+save
+50 dict begin
+/q { gsave } bind def
+/Q { grestore } bind def
+/cm { 6 array astore concat } bind def
+/w { setlinewidth } bind def
+/J { setlinecap } bind def
+/j { setlinejoin } bind def
+/M { setmiterlimit } bind def
+/d { setdash } bind def
+/m { moveto } bind def
+/l { lineto } bind def
+/c { curveto } bind def
+/h { closepath } bind def
+/re { exch dup neg 3 1 roll 5 3 roll moveto 0 rlineto
+      0 exch rlineto 0 rlineto closepath } bind def
+/S { stroke } bind def
+/f { fill } bind def
+/f* { eofill } bind def
+/n { newpath } bind def
+/W { clip } bind def
+/W* { eoclip } bind def
+/BT { } bind def
+/ET { } bind def
+/pdfmark where { pop globaldict /?pdfmark /exec load put }
+    { globaldict begin /?pdfmark /pop load def /pdfmark
+    /cleartomark load def end } ifelse
+/BDC { mark 3 1 roll /BDC pdfmark } bind def
+/EMC { mark /EMC pdfmark } bind def
+/cairo_store_point { /cairo_point_y exch def /cairo_point_x exch def } def
+/Tj { show currentpoint cairo_store_point } bind def
+/TJ {
+  {
+    dup
+    type /stringtype eq
+    { show } { -0.001 mul 0 cairo_font_matrix dtransform rmoveto } ifelse
+  } forall
+  currentpoint cairo_store_point
+} bind def
+/cairo_selectfont { cairo_font_matrix aload pop pop pop 0 0 6 array astore
+    cairo_font exch selectfont cairo_point_x cairo_point_y moveto } bind def
+/Tf { pop /cairo_font exch def /cairo_font_matrix where
+      { pop cairo_selectfont } if } bind def
+/Td { matrix translate cairo_font_matrix matrix concatmatrix dup
+      /cairo_font_matrix exch def dup 4 get exch 5 get cairo_store_point
+      /cairo_font where { pop cairo_selectfont } if } bind def
+/Tm { 2 copy 8 2 roll 6 array astore /cairo_font_matrix exch def
+      cairo_store_point /cairo_font where { pop cairo_selectfont } if } bind def
+/g { setgray } bind def
+/rg { setrgbcolor } bind def
+/d1 { setcachedevice } bind def
+/cairo_flush_ascii85_file { cairo_ascii85_file status { cairo_ascii85_file flushfile } if } def
+/cairo_image { image cairo_flush_ascii85_file } def
+/cairo_imagemask { imagemask cairo_flush_ascii85_file } def
+%%EndProlog
+%%BeginSetup
+%%BeginResource: font f-0-0
+%!FontType1-1.1 f-0-0 1.0
+11 dict begin
+/FontName /f-0-0 def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/FontBBox {-12 -14 908 692                                   } readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 48 /zero put
+dup 73 /I put
+dup 76 /L put
+dup 82 /R put
+dup 89 /Y put
+dup 101 /e put
+dup 109 /m put
+readonly def
+currentdict end
+currentfile eexec
+f983ef0097ece636fb4a96c74d26ab84185f6dfa4a16a7a1c27bbe3f1156aea698df336d20b467
+b10e7f33846656653c5ac6962759d3056cbdb3190bac614b984bf5a132dc418192443014ba63de
+800d392b6fea026574bb2535fd7bb5338f35bf15a88ea328fdaa49670c7852e3d060f3c5d6b07f
+2ef6d0f22646c5d18e19a2ae3ee120390f6dd96f76dcf1e127de5e9299077a00c17c0d71e36e5b
+9d5ec58fceda57739a6a4214d4b79d6c48d2784b60c320323c7acddddf34db833cac0cf109f799
+69d114a330d372e5c978a66acc84e3fe5557f6240856a013ffaa0199444e5c5036f775eba4a5c5
+8cde66cf604b9aca2178431127b8a1ff7ed633a65c04600af5f573483112251ca40352c6e2284e
+344947523a70f7b109147467f7576cb1df156dca2f25ccb577c1669626620a113ccdc0d69bb530
+dcedcb09d669e42789d3d1f1eb2fba73343f65132cb03065d412c4d987b79778555f98cbc72ab7
+18a72c35e371bf684a718b599d9f12c1a856afb05fa3bd876d6a55dcc6279feec01485b499c3da
+d50b200814e31ce340178663202f0a0324b15a23d545a665ada375c999b9af688f6f64738c2b03
+3fad0ff51c135ab5fc976cac8f6ee6fd117af76aede659c3d7c3ef848e783a3d7fc2ffd6dfb65d
+6fec08c4a48386ae22093696e4a4149206687406201494f7c92d83695ede02da0ec2efc745388b
+d4e96b9bf02196c1313f7b64670b3012e1b26b62b4c4398f62988c452011a3012866807ac3e958
+f6a33e22de6dad25663512af2f744cb4b2bccf96491c7dabfa4a6f3809714ce5e12c0ec56411de
+8d2d1ce34f9869137188615c0f4fbdfbea4b6aa825293167df8a44ecd5367cc0d134374c602f68
+cb24646d0a40cd42b9eaaa594b27274ae56eca9966b295b52c551e9431db92a8490105b9fca3cf
+abf2e646e6178f99068ffc46e80d0e6b91f6f7753ef194c42e47f2e32cfdbeff85b184ae87b96c
+ba03c00d2bb80c124412d6eac22d919a886efaafd1e1f0539f0b2277d7a3a570c352d18e8b9098
+262a6012482668a2836e6c631e98f1ffdfbd7e6a8327c434da366e42b10b3a04e6020dbad97dbf
+44a4eeb858298517e1b653d5719207f977d26cb9faf9285db28ec7bb55ecdf88173ed6a7542538
+75b780cffa546283ca777009c8bde4c70cd590dd3eca81b89e295c43771d95b90337e38f8f9d62
+501c3f0250ee10607517107eb9b8686aa2b2e73618e6ec508658efdfd4a9fbeff0a5f794ecb851
+81b07fb69e57d0ba45b34b1ddf3985403bf1ddbe9e24e73214cb9e3417f1569784c7585e54ce17
+993a0b1834c6e8641e48f9d2ed82931b358ce1638f893f198549802d03b5f77c30cdb701565094
+9cbc86327e59767775ffe0232e8fb09d9a865d66031371a7c4f279e83b7bbd2a9d3d5c04b876de
+303044b11569247c0a341892b60d49bf5a3d90cff8ecb498d5221203b27e9f7a0e2a0f5c83b803
+141289b187e0e40330f366ffe1889280fe6e24c8fe248cdd7b505c38a12f9aea4e0604d3a39b85
+dfa73f752e4deea83932b5b06717fe4ff53bc029968c48bf2eb97e7a4e0e462e6524fa833c2544
+99918cfc9b5c58f63e8bafb50d95ca777708fb85e9fd4babeb223ba13b0d5325efaf7813ead314
+ba40c2317c875af7430deaab519f75944b416da03bc21b407842090e3499d799ed64c606ae45cc
+de54f08502d814e5cd8cb51c31d0c7c591ea9e77342183be87d82d0bdbccf16c8edef1c1499d14
+4bff2d5ba6173f20939c2f5662bdb688061117662e55fa0d8aa7080bdc88b7a24b462ccffc833e
+1e75fd5537be5f7e7079349eacdec1709dd1889595a5092327dbbce7a3a8fca00d74fce608e932
+9721f45d89f7ccc9ee55a8f4082b95ef9fe1787ce123cdddbb83af12376b602e43b5a8cfcf4550
+0e418a92789eea9e01719ad69ffd7ed668660165e0d959ec6695fac6f428d6d777d8610000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+cleartomark
+%%EndResource
+%%BeginResource: font f-1-0
+%!FontType1-1.1 f-1-0 1.0
+11 dict begin
+/FontName /f-1-0 def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/FontBBox {29 -169 437 799                                   } readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 40 /parenleft put
+dup 41 /parenright put
+readonly def
+currentdict end
+currentfile eexec
+f983ef0097ece636fb4a96c74d26ab84185f6dfa4a16a7a1c27bbe3f1156aea698df336d20b467
+b10e7f33846656653c5ac6962759d3056cbdb3190bac614b984bf5a132dc418192443014ba63de
+800d392b6fea026574bb2535fd7bb5338f35bf15a88ea328fdaa49670c7852e3d060f3c5d6b07f
+2ef6d0f22646c5d18e19a2ae3ee120390f6dd96f76dcf1e127de5e9299077a00c17c0d71e36e5b
+9d5ec58fceda57739a6a4214d4b79d6c48d2784b60c320323c7acddddf34db833cac0cf109f799
+69d114a330d372e5c978a66acc84e3fe5557f6240856a013ffaa0199444e5c5036f775eba4a5c5
+8cde66cf604b9aca2178431127b8a1ff7ed633a65c04600af5f573483112251ca90bc9cf8825c7
+41412dda705317b7cb229dd377ff3d18e94daa9ba3bbd4ccd2ecde18a1c220179cca13293a42f7
+1d167cd6ac624b8f8573fc281c165eeb093c7705f0c9d1ce383c8701d14fcde12ac700699f81e8
+412c2e7e4c95c1440662af1652abdb4ce6be4b5f976724c26d20f166e17532da87ef8b24e90d2b
+ca09313d345112c202c1d9528e8c8e15bdfd68385cef36d711b99ee54d324b8778dae18d9af490
+41fbe748f7910913a3460e4af7353d7952f1fa9fde2bb171a7717e297425e0c08ca9800b01b2f5
+d645e0dbbc3a76bdfc773c19f4917354ed597d131dcc12985dd83ff833cc665a3ed3f11ec3814b
+3a8be6d7aef97d0b12b552d8da18f6cea65a8e284355941aae2da895a0fee8c8c2d4c22664f827
+5fb1ca0fbda2da6ff7fee9a610c58ec5d0c82dbb9b890d86f03ce85b69a56821a477e957df28b4
+b7afc889cbe637525896dc572b943c7f55648deb0894a36d8a1f7cfe05e48b6abb21a9949c7070
+43ecfa2c9ed9ed591f98e5fb1581bfb12e97b35bab44504370459fd28ca0c151ded5634089598c
+3d20720000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+cleartomark
+%%EndResource
+%%BeginResource: font f-1-1
+%!FontType1-1.1 f-1-1 1.0
+11 dict begin
+/FontName /f-1-1 def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/FontBBox {28 -13 453 799                                    } readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 1 /uni03B8 put
+readonly def
+currentdict end
+currentfile eexec
+f983ef0097ece636fb4a96c74d26ab84185f6dfa4a16a7a1c27bbe3f1156aea698df336d20b467
+b10e7f33846656653c5ac6962759d3056cbdb3190bac614b984bf5a132dc418192443014ba63de
+800d392b6fea026574bb2535fd7bb5338f35bf15a88ea328fdaa49670c7852e3d060f3c5d6b07f
+2ef6d0f22646c5d18e19a2ae3ee120390f6dd96f76dcf1e127de5e9299077a00c17c0d71e36e5b
+9d5ec58fceda57739a6a4214d4b79d6c48d2784b60c320323c7acddddf34db833cac0cf109f799
+69d114a330d372e5c978a66acc84e3fe5557f6240856a013ffaa0199444e5c5036f775eba4a5c5
+8cde66cf604b9aca2178431127b8a1ff7ed633a65c04600af5f573483112251cae134058d6a192
+3c6b565d778d9618b31789d664bb543a0e576f202df308890d53425311c0df16d70e7088bc233c
+2aeafd23aed96e18251b90593258f09f68b68c844c8e5bf170e06e331e14147fa538f8473eeb1e
+c84dbfec93c3410e50135635b99de6891224a561e7c815e270995932d4ca76c8cc14b02767bb66
+7e6acad1ab52cbc965dc9fc5d35b7f55bd68178f09c7128011331eb172346c01b77767a131365b
+3cb1e76bf2ef8772818a52fde15db663b8b91d9d22148c044c1f17c084466eee12d35401b0d206
+447bd9cb8b6e927f74a828d01e7ce0e8504701d21d52e83a0f908b3eaf18af033d7fdc4519160d
+a1f3998dc14198ecb66970ab1422655946606979ebc3920ec6406caae936b284318ce59d7fd021
+6da6916dccf7cddaa08c95fe84fe0a36eff860ec858faee6eae9b234fd7f6b7922b6407fe66548
+4ee8a12cf4b24800049489170c1c089065cfd05e7611f514854ffb249e79b19db4e86d95818819
+599eccc923f9ec10ebcf73f55e7828ba032aff285278971bd99c0a3ddf0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+cleartomark
+%%EndResource
+%%BeginResource: font f-2-0
+%!FontType1-1.1 f-2-0 1.0
+11 dict begin
+/FontName /f-2-0 def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/FontBBox {0 -12 363 715                                     } readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 105 /i put
+readonly def
+currentdict end
+currentfile eexec
+f983ef0097ece636fb4a96c74d26ab84185f6dfa4a16a7a1c27bbe3f1156aea698df336d20b467
+b10e7f33846656653c5ac6962759d3056cbdb3190bac614b984bf5a132dc418192443014ba63de
+800d392b6fea026574bb2535fd7bb5338f35bf15a88ea328fdaa49670c7852e3d060f3c5d6b07f
+2ef6d0f22646c5d18e19a2ae3ee120390f6dd96f76dcf1e127de5e9299077a00c17c0d71e36e5b
+9d5ec58fceda57739a6a4214d4b79d6c48d2784b60c320323c7acddddf34db833cac0cf109f799
+69d114a330d372e5c978a66acc84e3fe5557f6240856a013ffaa0199444e5c5036f775eba4a5c5
+8cde66cf604b9aca2178431127b8a1ff7ed633a65c04600af5f573483112251cae134058d6a192
+3c6b565d778d9618b31789d664bb543a0e576f252e8520c2d0698c337b75c663f61d3deead2935
+71dc9621dfa39dc5f30363291678a7d9dfdeaa2de43d090fdf9945858d9b5b3a5df2a5865b3028
+dc4b7ac7fe8270eda10a09d5669f164eb5fa6e4ca848129c7f59349ba575bbdf02dfbd8f5ed330
+d9554dd1eb8b856cdef62fe801b421cc107aa678dbf1124aab4cbde909fd8611cafc587bb34560
+f3428a01d0fd050e1c1ae785787bd5c142500e345326b240d6d37c6801da6061e4be60b75cc4a0
+06ffe4ca26e456a9313b7f02bd60218dde34c309130b9ee274c413f042863100acb2ad54244397
+b1ccf64df275ba901ee414216a2d7c659e80f9eb06bcb6f8d4b9e005690000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+0000000000000000000000000000000000000000000000000000000000000000
+cleartomark
+%%EndResource
+%%EndSetup
+%%Page: 1 1
+%%BeginPageSetup
+%%PageBoundingBox: 0 0 360 220
+%%EndPageSetup
+q 0 0 360 220 rectclip
+1 0 0 -1 0 221 cm q
+0 g
+0.36 w
+0 J
+0 j
+[ 2.88 2.88] 0 d
+3.25 M 0.18 118.234 m 330.754 118.234 l S
+BT
+10.08 0 0 -10.08 336.51375 121.472907 Tm
+/f-0-0 1 Tf
+(Re)Tj
+10.8 0 0 -10.8 348.18375 121.472907 Tm
+/f-1-0 1 Tf
+(\()Tj
+/f-1-1 1 Tf
+<01>Tj
+/f-1-0 1 Tf
+[()-14(\))]TJ
+ET
+Q q
+165 16 1 204.801 re W n
+0 g
+0.36 w
+0 J
+0 j
+[ 2.88 2.88] 0 d
+3.25 M 165.469 220.387 m 165.469 16.078 l S
+Q q
+0 g
+BT
+10.08 0 0 -10.08 153.148125 9.12 Tm
+/f-0-0 1 Tf
+(Im)Tj
+10.8 0 0 -10.8 165.969375 9.12 Tm
+/f-1-0 1 Tf
+(\()Tj
+/f-1-1 1 Tf
+<01>Tj
+/f-1-0 1 Tf
+[()-14(\))]TJ
+ET
+Q q
+164 16 3 52 re W n
+0 g
+1.44 w
+2 J
+0 j
+[] 0.0 d
+3.25 M 165.469 67.156 m 165.469 -47.055 l S
+Q q
+164 168 3 52.801 re W n
+0 g
+1.44 w
+2 J
+0 j
+[] 0.0 d
+3.25 M 165.469 169.309 m 165.469 283.52 l S
+Q q
+0 g
+162.547 64.234 5.844 5.844 re f
+162.547 166.387 5.844 5.844 re f
+1.44 w
+2 J
+0 j
+[] 0.0 d
+3.25 M 0.18 118.234 m 41.5 118.234 l S
+45.633 118.234 m 45.633 117.137 45.199 116.086 44.422 115.312 c 43.648 
+114.535 42.598 114.102 41.5 114.102 c 40.406 114.102 39.355 114.535 38.578
+ 115.312 c 37.805 116.086 37.371 117.137 37.371 118.234 c 37.371 119.328
+ 37.805 120.379 38.578 121.156 c 39.355 121.93 40.406 122.363 41.5 122.363
+ c 42.598 122.363 43.648 121.93 44.422 121.156 c 45.199 120.379 45.633 119.328
+ 45.633 118.234 c h
+45.633 118.234 m f*
+Q q
+288 117 43 2 re W n
+0 g
+1.44 w
+2 J
+0 j
+[] 0.0 d
+3.25 M 330.754 118.234 m 289.434 118.234 l S
+Q q
+0 g
+293.562 118.234 m 293.562 117.137 293.129 116.086 292.355 115.312 c 291.578
+ 114.535 290.527 114.102 289.434 114.102 c 288.336 114.102 287.285 114.535
+ 286.512 115.312 c 285.734 116.086 285.301 117.137 285.301 118.234 c 285.301
+ 119.328 285.734 120.379 286.512 121.156 c 287.285 121.93 288.336 122.363
+ 289.434 122.363 c 290.527 122.363 291.578 121.93 292.355 121.156 c 293.129
+ 120.379 293.562 119.328 293.562 118.234 c h
+293.562 118.234 m f*
+0.36 w
+2 J
+0 j
+[] 0.0 d
+3.25 M 289.434 118.234 m 289.434 109.969 l S
+165.469 67.156 m 173.73 67.156 l S
+BT
+10.08 0 0 -10.08 177.725859 70.396453 Tm
+/f-2-0 1 Tf
+(i)Tj
+10.8 0 0 -10.8 181.385859 70.396453 Tm
+/f-1-1 1 Tf
+<01>Tj
+7.2 0 0 -7.2 186.125859 72.102703 Tm
+/f-0-0 1 Tf
+(YL)Tj
+10.8 0 0 -10.8 285.038906 104.944219 Tm
+/f-1-1 1 Tf
+<01>Tj
+7.2 0 0 -7.2 289.778906 106.650469 Tm
+/f-0-0 1 Tf
+(0)Tj
+ET
+Q Q
+showpage
+%%Trailer
+end restore
+%%EOF