From 9448d2889b2ae8a6e73511e9293458ca21e17c96 Mon Sep 17 00:00:00 2001
From: minco <mail@drchi.co.kr>
Date: Tue, 22 Jul 2025 04:58:42 +0900
Subject: [PATCH] asd

---
 main.aux         |   2 +
 main.fdb_latexmk | 179 +++++++++++++++++
 main.fls         | 286 ++++++++++++++++++++++++++
 main.log         | 510 +++++++++++++++++++++++++++++++++++++++++++++++
 main.pdf         | Bin 0 -> 139650 bytes
 main.synctex.gz  | Bin 0 -> 10706 bytes
 main.tex         | 141 +++++++++++++
 7 files changed, 1118 insertions(+)
 create mode 100644 main.aux
 create mode 100644 main.fdb_latexmk
 create mode 100644 main.fls
 create mode 100644 main.log
 create mode 100644 main.pdf
 create mode 100644 main.synctex.gz
 create mode 100644 main.tex

diff --git a/main.aux b/main.aux
new file mode 100644
index 0000000..d490c35
--- /dev/null
+++ b/main.aux
@@ -0,0 +1,2 @@
+\relax 
+\gdef \@abspage@last{2}
diff --git a/main.fdb_latexmk b/main.fdb_latexmk
new file mode 100644
index 0000000..3455f99
--- /dev/null
+++ b/main.fdb_latexmk
@@ -0,0 +1,179 @@
+# Fdb version 4
+["pdflatex"] 1753127848.43771 "main.tex" "main.pdf" "main" 1753127848.89585 0
+  "/nix/store/dmm8f43zw2rwzgr1xs03ac1r98m8zp5i-texlive-combined-full-2024-final/share/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1 5467273 8adf9322fbd4564c966dec2fb8102906 ""
+  "/nix/store/dmm8f43zw2rwzgr1xs03ac1r98m8zp5i-texlive-combined-full-2024-final/share/texmf-var/web2c/pdftex/pdflatex.fmt" 1 3353366 6b55f6472b88ee4260a36562e619a9e4 ""
+  "/nix/store/dmm8f43zw2rwzgr1xs03ac1r98m8zp5i-texlive-combined-full-2024-final/share/texmf-var/web2c/texmf.cnf" 1 44167 843349d21bfcf9445e4ce4f7d9a9f0d0 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/map/fontname/texfonts.map" 1 3524 cb3e574dea2d1052e39280babc910dc8 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1 1004 54797486969f23fa377b128694d548df ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1 916 f87d7c45f9c908e672703b83b72241a3 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1 928 2dc8d444221b7a635bb58038579b861a ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm" 1 908 2921f8a10601f252058503cc6570e581 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1 940 228d6584342e91276bf566bcf9716b83 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/cm/cmbx12.tfm" 1 1324 c910af8c371558dc20f2d7822f66fe64 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/cm/cmex10.tfm" 1 992 662f679a0b3d2d53c1b94050fdaa3f50 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/cm/cmmi12.tfm" 1 1524 4414a8315f39513458b80dfc63bff03a ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/cm/cmmi6.tfm" 1 1512 f21f83efb36853c0b70002322c1ab3ad ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/cm/cmmi8.tfm" 1 1520 eccf95517727cb11801f4f1aee3a21b4 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/cm/cmr12.tfm" 1 1288 655e228510b4c2a1abe905c368440826 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/cm/cmr17.tfm" 1 1292 296a67155bdbfc32aa9c636f21e91433 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/cm/cmr6.tfm" 1 1300 b62933e007d01cfd073f79b963c01526 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/cm/cmr8.tfm" 1 1292 21c1c5bfeaebccffdb478fd231a0997d ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/cm/cmsy10.tfm" 1 1124 6c73e740cf17375f03eec0ee63599741 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/cm/cmsy6.tfm" 1 1116 933a60c408fc0a863a92debe84b2d294 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/cm/cmsy8.tfm" 1 1120 8b7d695260f3cff42e636090a8002094 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/tfm/public/cm/cmti12.tfm" 1 1484 ed72f8f5cf654cda15ecc8e32bfcbee5 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmbx12.pfb" 1 32080 340ef9bf63678554ee606688e7b5339d ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1 30251 6afa5cb1d0204815a708a080681d4674 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmmi12.pfb" 1 36741 fa121aac0049305630cf160b86157ee4 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmmi6.pfb" 1 37166 8ab3487cbe3ab49ebce74c29ea2418db ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmmi8.pfb" 1 35469 70d41d2b9ea31d5d813066df7c99281c ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmr12.pfb" 1 32722 d7379af29a190c3f453aba36302ff5a9 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmr17.pfb" 1 32362 179c33bbf43f19adbb3825bb4e36e57a ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmr6.pfb" 1 32734 69e00a6b65cedb993666e42eedb3d48f ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1 32726 0a1aea6fcd6468ee2cf64d891f5c43c8 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1 32569 5e5ddc8df908dea60932f3c484a54c0d ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmsy8.pfb" 1 32626 4f5c1b83753b1dd3a97d1b399a005b4b ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmti12.pfb" 1 36118 fad905eba93cff5bce1e185fe980a177 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/symbols/msam10.pfb" 1 31764 459c573c03a4949a528c2cc7f557e217 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/context/base/mkii/supp-pdf.mkii" 1 71627 94eb9990bed73c364d7f53f960cc8c5b ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/iftex/iftex.sty" 1 7984 7dbb9280f03c0a315425f1b4f35d43ee ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/iftex/ifvtex.sty" 1 1057 525c2192b5febbd8c1f662c9468335bb ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/basiclayer/pgfcore.code.tex" 1 1016 1c2b89187d12a2768764b83b4945667c ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex" 1 43820 1fef971b75380574ab35a0d37fd92608 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex" 1 19324 f4e4c6403dd0f1605fd20ed22fa79dea ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex" 1 6038 ccb406740cc3f03bbfb58ad504fe8c27 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex" 1 6911 f6d4cf5a3fef5cc879d668b810e82868 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex" 1 4883 42daaf41e27c3735286e23e48d2d7af9 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex" 1 2544 8c06d2a7f0f469616ac9e13db6d2f842 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex" 1 44195 5e390c414de027626ca5e2df888fa68d ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex" 1 17311 2ef6b2e29e2fc6a2fc8d6d652176e257 ""
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+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex" 1 33335 dd1fa4814d4e51f18be97d88bf0da60c ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex" 1 2965 4c2b1f4e0826925746439038172e5d6f ""
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+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex" 1 8893 e851de2175338fdf7c17f3e091d94618 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecorations.code.tex" 1 5628 dc0ee4ba7f3e40acae5600067ce833de ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecorations.pathmorphing.code.tex" 1 321 cdd11262840e01e25374a2d458f15e99 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecorations.pathreplacing.code.tex" 1 1319 0b2de5126c6cbc295f0eb77f7344b34d ""
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+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex" 1 186782 af500404a9edec4d362912fe762ded92 ""
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+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/libraries/pgflibraryfpu.code.tex" 1 85938 8e4ba97c5906e1c0d158aea81fe29af7 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex" 1 32995 ac577023e12c0e4bd8aa420b2e852d1a ""
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+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/math/pgfmathcalc.code.tex" 1 13270 2e54f2ce7622437bf37e013d399743e3 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/math/pgfmathfloat.code.tex" 1 104717 9b2393fbf004a0ce7fa688dbce423848 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex" 1 10165 cec5fa73d49da442e56efc2d605ef154 ""
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+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/math/pgfmathfunctions.code.tex" 1 9649 85779d3d8d573bfd2cd4137ba8202e60 ""
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+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex" 1 3177 27d85c44fbfe09ff3b2cf2879e3ea434 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex" 1 11024 0179538121bc2dba172013a3ef89519f ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex" 1 7890 0a86dbf4edfd88d022e0d889ec78cc03 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex" 1 3379 781797a101f647bab82741a99944a229 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex" 1 92405 f515f31275db273f97b9d8f52e1b0736 ""
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+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/math/pgfmathutil.code.tex" 1 8471 c2883569d03f69e8e1cabfef4999cfd7 ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/modules/pgfmoduledecorations.code.tex" 1 71742 3da44a8be6626eef1c400c68776c7a0f ""
+  "/nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/generic/pgf/modules/pgfmodulematrix.code.tex" 1 21211 1e73ec76bd73964d84197cc3d2685b01 ""
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+  "main.aux" 1753127848.86231 32 044b7f8fc9779af7531264e0c5c84b6d "pdflatex"
+  "main.tex" 1753127846.84123 3197 f45a3479336ca31493b2b995d7bdf3ae ""
+  (generated)
+  "main.aux"
+  "main.log"
+  "main.pdf"
+  (rewritten before read)
diff --git a/main.fls b/main.fls
new file mode 100644
index 0000000..83cba0b
--- /dev/null
+++ b/main.fls
@@ -0,0 +1,286 @@
+PWD /home/allen/papers/pi3
+INPUT /nix/store/dmm8f43zw2rwzgr1xs03ac1r98m8zp5i-texlive-combined-full-2024-final/share/texmf-var/web2c/texmf.cnf
+INPUT /nix/store/dmm8f43zw2rwzgr1xs03ac1r98m8zp5i-texlive-combined-full-2024-final/share/texmf-var/web2c/pdftex/pdflatex.fmt
+INPUT main.tex
+OUTPUT main.log
+INPUT /nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/latex/base/article.cls
+INPUT /nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/latex/base/article.cls
+INPUT /nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/tex/latex/base/size12.clo
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+</nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmex10.pfb></nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmmi12.pfb></nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmmi6.pfb></nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmr12.pfb></nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmr17.pfb></nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmr6.pfb></nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmr8.pfb></nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmsy8.pfb></nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/cm/cmti12.pfb></nix/store/z5ws6797p4763ycdkmdpayprws409k1g-texlive-combined-full-2024-final-texmfdist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
+Output written on main.pdf (2 pages, 139650 bytes).
+PDF statistics:
+ 79 PDF objects out of 1000 (max. 8388607)
+ 48 compressed objects within 1 object stream
+ 0 named destinations out of 1000 (max. 500000)
+ 13 words of extra memory for PDF output out of 10000 (max. 10000000)
+
diff --git a/main.pdf b/main.pdf
new file mode 100644
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diff --git a/main.tex b/main.tex
new file mode 100644
index 0000000..a47eff3
--- /dev/null
+++ b/main.tex
@@ -0,0 +1,141 @@
+\documentclass[12pt]{article}
+
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath, amssymb, amsthm}
+\usepackage{pgfplots}
+\usepackage{geometry}
+\geometry{margin=1in}
+
+\newtheorem{theorem}{Theorem}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{example}[theorem]{Example}
+
+\theoremstyle{remark}
+\newtheorem*{remark}{Remark}
+
+% Document
+\begin{document}
+
+\title{Simple Proof That \( \pi ^ {3} < 3^{\pi} \)}
+\author{Minco}
+\date{\today}
+\maketitle
+
+\section*{Introduction}
+This document presents a very simple mathematical proof that $\pi^3 < 3^{\pi}$.
+The proof illustrates the beauty of pure mathematics, which allows a simple question to be shown in verbose manner.
+
+\begin{theorem}
+\[
+  \pi^3 < 3^{\pi}
+\]
+\end{theorem}
+
+\begin{proposition}
+Since the both bases are over 1, we can exponentiate both sides of the inequality by \(\frac{1}{3 \pi}\).
+So we have to prove that:
+\[
+  \pi^{\frac{1}{\pi}} < 3^{\frac{1}{3}}
+\]
+\end{proposition}
+
+\section*{Limit Analysis}
+Consider the function
+\[
+  f(x) = x^{\frac{1}{x}}, \quad x > 0.
+\]
+
+First, take the logarithm on both sides.
+
+\[
+  \log {f(x)} = \frac{1}{x} \log{x} \quad (\text{log is a natural logarithm})
+\]
+\[
+  \therefore f(x) = e^{\frac{1}{x} \log{x}}
+\]
+
+Then analyze the limits at the boundaries of the domain:
+
+\[
+  \lim_{x \to 0^+} f(x) = \lim_{X \to - \infty} e^{X} = 0
+\]
+
+And
+
+\[
+  \lim_{x \to \infty} f(x) = \lim_{X \to 0^+} e^{X} = 1 .
+\]
+
+\section*{Derivative Analysis}
+
+To find the derivative of \(f(x)\), use logarithmic differentiation as in Limit Analysis:
+
+\[
+y = x^{\frac{1}{x}} \implies \log y = \frac{\ln x}{x}.
+\]
+
+Differentiating both sides:
+\[
+\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}\left(\frac{\log x}{x}\right) = \frac{1 - \log x}{x^2}.
+\]
+
+Since \(x > 0\),
+
+\[
+  \frac{1- \log x}{x^2} = 0 \quad \rm{iff.} \quad 1 - \log x = 0
+\]
+
+\[
+  \therefore x = e \quad \text{where} \quad \frac{d}{dx} f(x) = 0
+\]
+
+\section*{Graph}
+\begin{center}
+\begin{tikzpicture}
+  \begin{axis}[
+    domain=0.1:5, 
+    samples=200,
+    axis lines=middle,
+    xlabel=$x$, ylabel={$f(x)$},
+    ymin=0, ymax=2,
+    ]
+    \addplot [black, thick] {x^(1/x)};
+    \addplot[
+      only marks,
+      mark=*,
+      mark size=2pt,
+      black
+    ] coordinates {(3, {3^(1/3)})};
+    \draw[dotted] (axis cs:3,0) -- (axis cs:3,{3^(1/3)});
+    \draw[dotted] (axis cs:0,{3^(1/3)}) -- (axis cs:3,{3^(1/3)});
+    \node [anchor=south east] at (axis cs:3,{3^(1/3)}) {$\bigl(3,\,3^{\frac{1}{3}}\bigr)$};
+    
+    \addplot[
+      only marks,
+      mark=*,
+      mark size=2pt,
+      black
+    ] coordinates {(3.14, {3.14^(1/3.14)})};
+    \draw[dotted] (axis cs:3.14,0) -- (axis cs:3.14,{3.14^(1/3.14)});
+    \draw[dotted] (axis cs:0,{3.14^(1/3.14)}) -- (axis cs:3.14,{3.14^(1/3.14)});
+    \node [anchor=south west] at (axis cs:3.14,{3.14^(1/3.14)}) {$\bigl(\pi,\,\pi^{\frac{1}{\pi}}\bigr)$};
+  \end{axis}
+\end{tikzpicture}
+\end{center}
+
+\begin{proof}
+  As shown in the graph,
+
+  \[
+    \pi^{\frac{1}{\pi}} < 3^{\frac{1}{3}}
+  \]
+  \[
+    \therefore \boxed{\pi^3 < 3^{\pi}}
+  \]
+\end{proof}
+\end{document}