├── README.md ├── algebra.tex ├── all.bib ├── all.pdf ├── all.tex ├── analysis.tex ├── butterfly.eps ├── ccr.eps ├── cevaconic.eps ├── coharmonic.eps ├── conicexer.eps ├── conictangents.eps ├── cross-notes.tex ├── cross.pdf ├── cross.tex ├── csp-append.tex ├── csp-notes.pdf ├── csp-notes.tex ├── csp.tex ├── desargues.eps ├── equal.eps ├── fourpairs.eps ├── ineq.tex ├── logstar.pdf ├── logstar.tex ├── north.eps ├── octa.eps ├── octa1.eps ├── octa2.eps ├── orthogonal.eps ├── pascal.eps ├── polar.eps ├── polartang.eps ├── preamble.tex ├── quad.eps ├── quadratic.tex ├── righthyperbola.eps ├── selfpolar.eps ├── sheaf-coh.bib ├── sheaf-coh.pdf ├── sheaf-coh.tex ├── sumproduct.pdf ├── sumproduct.tex ├── triangulargrid.eps ├── weil.pdf └── weil.tex /README.md: -------------------------------------------------------------------------------- 1 | # all 2 | I'm trying to collect all my notes in one place, in as open a format as possible. The most recent draft is located at https://raw.githubusercontent.com/notzeb/all/master/all.pdf. 3 | 4 | I view these notes as my own personalized form of wikipedia's math pages: wikipedia's math pages are a great resource, but they never present things the way I would present them to myself. Latex code doesn't take much hard drive space, so it seems reasonable for every mathematician to maintain their own personalized compilation of the foundations of math, with their favorite clever (or straightforward) proofs of whichever results they think are most fundamental or interesting. 5 | 6 | However, if every mathematician were to write out their own personal Bourbaki from scratch, then there would be a huge amount of duplicated effort. My suggestion is for anyone who embarks on such a project to make their latex code publicly available, and to feel free to copy the latex code for results that they have no special insight on from others. Since notes are frequently updated, it may not make sense to post these projects to the arxiv - so I'm trying github as a repository, instead. 7 | 8 | My personal style of writing notes probably won't fit most people's tastes. If you think that my notes would be improved by being more verbose, or being more organized, or by including more examples, or better pictures: please feel free to copy them and change them to your liking! 9 | 10 | A very similar project can be found at https://github.com/vEnhance/napkin - he's done a much better job than me, actually (mine are at best 10% of the way to where I want them, his seem almost finished - thankfully it isn't a competition). Another similar project is the stacks project: https://github.com/stacks/stacks-project - it certainly seems like people find github a natural repository for these things! (Yet another one, this time for logic: https://openlogicproject.org/) 11 | 12 | # CSP notes 13 | 14 | The notes I'm currently actively working on are the notes on constraint satisfaction problems, which you can download directly via the following link: https://raw.githubusercontent.com/notzeb/all/master/csp-notes.pdf. These notes contain some material which hasn't been published. 15 | -------------------------------------------------------------------------------- /algebra.tex: -------------------------------------------------------------------------------- 1 | \section{Noncommutative rings} 2 | 3 | \begin{defn} If $R$ is a ring, then the \emph{Jacobson radical} $J(R)$ (sometimes written $\rad(R)$) is the intersection of the annihilators of all simple left $R$-modules. 4 | \end{defn} 5 | 6 | \begin{defn} A submodule $N$ of $M$ is \emph{superfluous}, written $N \subseteq_s M$ or $N \ll M$, if for all $H$ we have $N+H = M\ \implies\ H = M$. 7 | \end{defn} 8 | 9 | \begin{thm} We can replace ``left'' by ``right'' in the definition of the Jacobson radical of a ring. Furthermore, we have the following equivalent definitions: 10 | \begin{itemize} 11 | \item $J(R)$ is the intersection of all maximal left ideals of $R$, 12 | \item $J(R)$ is the sum of all superfluous left ideals of $R$, 13 | \item $J(R)$ is the maximal left ideal of $R$ such that for all $x \in J(R)$, $1-x$ has a left inverse, 14 | \item $J(R) = \{x \in R \mid 1+RxR \subseteq R^\times\}$. 15 | \end{itemize} 16 | \end{thm} 17 | 18 | \begin{lem}[Nakayama's Lemma] If $M$ is a finitely generated left $R$-module with $M = J(R)M$, then $M=0$. 19 | \end{lem} 20 | \begin{proof} Consider a minimal generating set $x_1, ..., x_n$ of $M$, and use $\sum x_i \in J(R)M$ to write $x_n$ as a linear combination of $x_1, ..., x_{n-1}$. 21 | \end{proof} 22 | 23 | \begin{prop} $J(R/J(R)) = 0$. 24 | \end{prop} 25 | 26 | \subsection{Artinian Rings} 27 | 28 | \begin{prop} If $R$, considered as a left $R$-module over itself, has a composition series of length $k$, then $J(R)^k = 0$. 29 | \end{prop} 30 | 31 | \begin{thm}[Hopkins' Theorem] If $M$ is a left module over a left Artinian ring, then the following are equivalent: 32 | \begin{itemize} 33 | \item $M$ is finitely generated, 34 | \item $M$ has finite length, 35 | \item $M$ is Noetherian, 36 | \item $M$ is Artinian. 37 | \end{itemize} 38 | \end{thm} 39 | 40 | \begin{thm}[Hopkins-Levitzki] If $R$ is \emph{semiprimary} - that is, if $R/J(R)$ is semisimple and $J(R)$ is nilpotent - then for left $R$-modules, being Noetherian, being Artinian, and having a composition series are equivalent. 41 | \end{thm} 42 | 43 | \begin{prop} If $J(R) = 0$, then every minimal left ideal of $R$ is a direct summand of $R$. 44 | \end{prop} 45 | 46 | \begin{thm} $R$ is semisimple if and only if it is left Artinian and has $J(R) = 0$. 47 | \end{thm} 48 | 49 | % TODO: Schur-Weyl, maybe via https://mathoverflow.net/questions/255492/how-to-constructively-combinatorially-prove-schur-weyl-duality 50 | 51 | 52 | 53 | 54 | 55 | \section{Commutative Algebra} 56 | 57 | \begin{defn} If $R$ is a commutative ring, then $I \lhd R$ means that $I$ is an ideal of $R$. 58 | \end{defn} 59 | 60 | \begin{defn} If $I,J \lhd R$, set $(I:J) = \{r \in R \mid rJ \subseteq I\}$. If $a \in R$, we abbreviate $(I:(a))$ to $(I:a)$. 61 | \end{defn} 62 | 63 | \subsection{Primary Ideals} 64 | 65 | \begin{defn} $Q \lhd R$ is \emph{primary} if $\forall a,b\in R$ with $ab \in Q$, either $b \in Q$ or $\exists n$ such that $a^n \in Q$. 66 | \end{defn} 67 | 68 | \begin{defn} If $I \lhd R$, then $\rad(I) = \{r \in R \mid \exists n\ r^n \in I\}$. 69 | \end{defn} 70 | 71 | \begin{prop} $Q$ is primary if and only if $\rad(Q)$ is prime. If $Q_1, Q_2$ are primary and $\rad(Q_1) = \rad(Q_2)$, then $Q_1 \cap Q_2$ is primary. If $R$ is Noetherian and $Q \lhd R$, then $\exists n$ such that $\rad(Q)^n \subseteq Q$. 72 | \end{prop} 73 | 74 | \begin{thm}[Primary Decomposition] If $R$ is Noetherian and $I \lhd R$, then $\exists k$ and $Q_1, ..., Q_k \lhd R$ primary such that $I = Q_1 \cap \cdots \cap Q_k$. 75 | \end{thm} 76 | \begin{proof} By $R$ Noetherian, $\forall a\in R\ \exists n$ with $(I:a^n) = (I:a^{n+1})$, and for this $n$ we have $(I+(a^n))\cap (I:a) = I$, so either $I$ is already primary or we can write $I$ as an intersection of bigger ideals, and apply Noetherian induction. 77 | \end{proof} 78 | 79 | \begin{lem} If $R$ is Noetherian, then for any $I \lhd R$ and $r \in R \setminus I$, there exists $s \in R$ such that $(I:rs)$ is prime. 80 | \end{lem} 81 | 82 | \begin{thm}[Uniqueness of radicals] If $R$ is Noetherian, $I = Q_1 \cap \cdots \cap Q_k$ with $Q_i \lhd R$ primary and no $Q_i$ containing $\cap_{j \ne i} Q_j$, and if $\fp \lhd R$ is prime, then $\exists r \in R$ with $(I:r) = \fp$ if and only if there is an $i$ with $\rad(Q_i) = \fp$. In particular, the set $\{\rad(Q_i)\}_{i \le k}$ is uniquely determined by $I$. 83 | \end{thm} 84 | 85 | \begin{thm}[Uniqueness of primaries with minimal radical] If $R$ is Noetherian, $I = Q_1 \cap \cdots \cap Q_k$ with $Q_i \lhd R$ primary and $\rad(Q_i) \not\subseteq \rad(Q_1)$ for $i > 1$, then for $n$ sufficiently large we have $(I:\rad(Q_2)^n \cdots \rad(Q_k)^n) = Q_1$, so $Q_1$ is uniquely determined by $I$ and $\rad(Q_1)$. 86 | \end{thm} 87 | 88 | % TODO: Group theory! Nielsen reduction, Frobenius groups, Frattini subgroups, Frattini's argument, primitive groups, CFSG, ... 89 | 90 | -------------------------------------------------------------------------------- /all.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/notzeb/all/97969592a92b61f13beb5e90dc945e29e8cd5ebd/all.pdf -------------------------------------------------------------------------------- /all.tex: -------------------------------------------------------------------------------- 1 | \input{preamble.tex} 2 | 3 | \title{Notes} 4 | \date{} 5 | \author{} 6 | \maketitle 7 | 8 | \tableofcontents 9 | 10 | 11 | \part{Basics} 12 | 13 | \chapter{Cross Ratios} 14 | 15 | \input{cross-notes.tex} 16 | 17 | 18 | \chapter{Inequalities} 19 | 20 | \input{ineq.tex} 21 | 22 | 23 | 24 | \part{Foundational Material} 25 | 26 | \chapter{Analysis} 27 | 28 | \input{analysis.tex} 29 | 30 | 31 | \chapter{Algebra} 32 | 33 | \input{algebra.tex} 34 | 35 | 36 | \chapter{Sheaf Cohomology} 37 | 38 | \input{sheaf-coh.tex} 39 | 40 | 41 | 42 | \part{Number Theory} 43 | 44 | \chapter{Weil bounds} 45 | 46 | \input{weil.tex} 47 | 48 | 49 | \chapter{The Sum-Product Theorem} 50 | 51 | \input{sumproduct.tex} 52 | 53 | 54 | \part{Constraints and Polymorphisms} 55 | 56 | \input{csp.tex} 57 | 58 | \input{csp-append.tex} 59 | 60 | 61 | \bibliographystyle{plain} 62 | \bibliography{all} 63 | 64 | \end{document} 65 | 66 | -------------------------------------------------------------------------------- /butterfly.eps: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/notzeb/all/97969592a92b61f13beb5e90dc945e29e8cd5ebd/butterfly.eps -------------------------------------------------------------------------------- /ccr.eps: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/notzeb/all/97969592a92b61f13beb5e90dc945e29e8cd5ebd/ccr.eps -------------------------------------------------------------------------------- /cevaconic.eps: -------------------------------------------------------------------------------- 1 | %!PS-Adobe-2.0 EPSF-2.0 2 | %%BoundingBox: 124 127 471 716 3 | %%HiResBoundingBox: 124.500000 127.500000 470.500000 715.500000 4 | %%Creator: Qt 3.3.8b 5 | %%CreationDate: Sun Oct 4 23:50:23 2015 6 | %%DocumentFonts: 7 | 8 | %%EndComments 9 | % EPSF created by ps2eps 1.68 10 | %%BeginProlog 11 | save 12 | countdictstack 13 | mark 14 | newpath 15 | /showpage {} def 16 | /setpagedevice {pop} def 17 | %%EndProlog 18 | %%Page 1 1 19 | %%BeginProlog 20 | % Prolog copyright 1994-2006 Trolltech. 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247 | 2764 1060 2801 1052 DL 248 | [0.125 0 0 0.125 138 118]ST 249 | 2727 1071 2764 1060 DL 250 | [0.125 0 0 0.125 138 118]ST 251 | 2690 1084 2727 1071 DL 252 | [0.125 0 0 0.125 138 118]ST 253 | 2653 1099 2690 1084 DL 254 | [0.125 0 0 0.125 138 118]ST 255 | 2634 1107 2653 1099 DL 256 | [0.125 0 0 0.125 138 118]ST 257 | 2615 1116 2634 1107 DL 258 | [0.125 0 0 0.125 138 118]ST 259 | 2577 1135 2615 1116 DL 260 | [0.125 0 0 0.125 138 118]ST 261 | 2539 1157 2577 1135 DL 262 | [0.125 0 0 0.125 138 118]ST 263 | 2499 1183 2539 1157 DL 264 | [0.125 0 0 0.125 138 118]ST 265 | 2459 1212 2499 1183 DL 266 | [0.125 0 0 0.125 138 118]ST 267 | 2415 1247 2459 1212 DL 268 | [0.125 0 0 0.125 138 118]ST 269 | 2368 1289 2415 1247 DL 270 | [0.125 0 0 0.125 138 118]ST 271 | 2311 1349 2368 1289 DL 272 | [0.125 0 0 0.125 138 118]ST 273 | 2272 1394 2311 1349 DL 274 | [0.125 0 0 0.125 138 118]ST 275 | 2244 1430 2272 1394 DL 276 | [0.125 0 0 0.125 138 118]ST 277 | 2223 1459 2244 1430 DL 278 | [0.125 0 0 0.125 138 118]ST 279 | 2207 1483 2223 1459 DL 280 | [0.125 0 0 0.125 138 118]ST 281 | 2195 1503 2207 1483 DL 282 | [0.125 0 0 0.125 138 118]ST 283 | 2184 1519 2195 1503 DL 284 | [0.125 0 0 0.125 138 118]ST 285 | 2176 1534 2184 1519 DL 286 | [0.125 0 0 0.125 138 118]ST 287 | 2169 1546 2176 1534 DL 288 | [0.125 0 0 0.125 138 118]ST 289 | 2163 1557 2169 1546 DL 290 | [0.125 0 0 0.125 138 118]ST 291 | 2157 1567 2163 1557 DL 292 | [0.125 0 0 0.125 138 118]ST 293 | 2153 1575 2157 1567 DL 294 | [0.125 0 0 0.125 138 118]ST 295 | 2149 1583 2153 1575 DL 296 | [0.125 0 0 0.125 138 118]ST 297 | 2145 1589 2149 1583 DL 298 | [0.125 0 0 0.125 138 118]ST 299 | 2142 1595 2145 1589 DL 300 | [0.125 0 0 0.125 138 118]ST 301 | 2139 1601 2142 1595 DL 302 | [0.125 0 0 0.125 138 118]ST 303 | 2137 1606 2139 1601 DL 304 | [0.125 0 0 0.125 138 118]ST 305 | 2135 1611 2137 1606 DL 306 | [0.125 0 0 0.125 138 118]ST 307 | 2132 1615 2135 1611 DL 308 | [0.125 0 0 0.125 138 118]ST 309 | 2131 1619 2132 1615 DL 310 | [0.125 0 0 0.125 138 118]ST 311 | 2129 1623 2131 1619 DL 312 | [0.125 0 0 0.125 138 118]ST 313 | 2127 1626 2129 1623 DL 314 | [0.125 0 0 0.125 138 118]ST 315 | 2126 1629 2127 1626 DL 316 | [0.125 0 0 0.125 138 118]ST 317 | 2124 1632 2126 1629 DL 318 | [0.125 0 0 0.125 138 118]ST 319 | 2123 1635 2124 1632 DL 320 | [0.125 0 0 0.125 138 118]ST 321 | 2122 1638 2123 1635 DL 322 | [0.125 0 0 0.125 138 118]ST 323 | 2088 1721 2122 1638 DL 324 | [0.125 0 0 0.125 138 118]ST 325 | 2060 1804 2088 1721 DL 326 | [0.125 0 0 0.125 138 118]ST 327 | 2060 1806 1804 VL 328 | [0.125 0 0 0.125 138 118]ST 329 | 2059 1809 2060 1806 DL 330 | [0.125 0 0 0.125 138 118]ST 331 | 2058 1812 2059 1809 DL 332 | [0.125 0 0 0.125 138 118]ST 333 | 2057 1815 2058 1812 DL 334 | [0.125 0 0 0.125 138 118]ST 335 | 2056 1819 2057 1815 DL 336 | [0.125 0 0 0.125 138 118]ST 337 | 2055 1822 2056 1819 DL 338 | [0.125 0 0 0.125 138 118]ST 339 | 2054 1826 2055 1822 DL 340 | [0.125 0 0 0.125 138 118]ST 341 | 2053 1830 2054 1826 DL 342 | [0.125 0 0 0.125 138 118]ST 343 | 2052 1835 2053 1830 DL 344 | [0.125 0 0 0.125 138 118]ST 345 | 2051 1840 2052 1835 DL 346 | [0.125 0 0 0.125 138 118]ST 347 | 2049 1846 2051 1840 DL 348 | [0.125 0 0 0.125 138 118]ST 349 | 2048 1852 2049 1846 DL 350 | [0.125 0 0 0.125 138 118]ST 351 | 2046 1859 2048 1852 DL 352 | [0.125 0 0 0.125 138 118]ST 353 | 2044 1867 2046 1859 DL 354 | [0.125 0 0 0.125 138 118]ST 355 | 2043 1875 2044 1867 DL 356 | [0.125 0 0 0.125 138 118]ST 357 | 2041 1885 2043 1875 DL 358 | [0.125 0 0 0.125 138 118]ST 359 | 2038 1896 2041 1885 DL 360 | [0.125 0 0 0.125 138 118]ST 361 | 2036 1908 2038 1896 DL 362 | [0.125 0 0 0.125 138 118]ST 363 | 2033 1922 2036 1908 DL 364 | [0.125 0 0 0.125 138 118]ST 365 | 2031 1939 2033 1922 DL 366 | [0.125 0 0 0.125 138 118]ST 367 | 2028 1959 2031 1939 DL 368 | [0.125 0 0 0.125 138 118]ST 369 | 2025 1984 2028 1959 DL 370 | [0.125 0 0 0.125 138 118]ST 371 | 2022 2013 2025 1984 DL 372 | [0.125 0 0 0.125 138 118]ST 373 | 2019 2050 2022 2013 DL 374 | [0.125 0 0 0.125 138 118]ST 375 | 2018 2098 2019 2050 DL 376 | [0.125 0 0 0.125 138 118]ST 377 | 2020 2162 2018 2098 DL 378 | [0.125 0 0 0.125 138 118]ST 379 | 2024 2210 2020 2162 DL 380 | [0.125 0 0 0.125 138 118]ST 381 | 2030 2250 2024 2210 DL 382 | [0.125 0 0 0.125 138 118]ST 383 | 2036 2285 2030 2250 DL 384 | [0.125 0 0 0.125 138 118]ST 385 | 2043 2318 2036 2285 DL 386 | [0.125 0 0 0.125 138 118]ST 387 | 2059 2376 2043 2318 DL 388 | [0.125 0 0 0.125 138 118]ST 389 | 2078 2428 2059 2376 DL 390 | [0.125 0 0 0.125 138 118]ST 391 | 2098 2476 2078 2428 DL 392 | [0.125 0 0 0.125 138 118]ST 393 | 2121 2520 2098 2476 DL 394 | [0.125 0 0 0.125 138 118]ST 395 | 2146 2561 2121 2520 DL 396 | [0.125 0 0 0.125 138 118]ST 397 | 2173 2600 2146 2561 DL 398 | [0.125 0 0 0.125 138 118]ST 399 | 2202 2635 2173 2600 DL 400 | [0.125 0 0 0.125 138 118]ST 401 | 2233 2669 2202 2635 DL 402 | [0.125 0 0 0.125 138 118]ST 403 | 2266 2699 2233 2669 DL 404 | [0.125 0 0 0.125 138 118]ST 405 | 2300 2727 2266 2699 DL 406 | [0.125 0 0 0.125 138 118]ST 407 | 2337 2752 2300 2727 DL 408 | [0.125 0 0 0.125 138 118]ST 409 | 2375 2775 2337 2752 DL 410 | [0.125 0 0 0.125 138 118]ST 411 | 2415 2795 2375 2775 DL 412 | [0.125 0 0 0.125 138 118]ST 413 | 2456 2811 2415 2795 DL 414 | [0.125 0 0 0.125 138 118]ST 415 | 2499 2825 2456 2811 DL 416 | [0.125 0 0 0.125 138 118]ST 417 | 2543 2836 2499 2825 DL 418 | [0.125 0 0 0.125 138 118]ST 419 | 2588 2843 2543 2836 DL 420 | [0.125 0 0 0.125 138 118]ST 421 | 2635 2847 2588 2843 DL 422 | [0.125 0 0 0.125 138 118]ST 423 | 2681 2848 2635 2847 DL 424 | [0.125 0 0 0.125 138 118]ST 425 | 2729 2845 2681 2848 DL 426 | [0.125 0 0 0.125 138 118]ST 427 | 2777 2840 2729 2845 DL 428 | [0.125 0 0 0.125 138 118]ST 429 | 2824 2831 2777 2840 DL 430 | [0.125 0 0 0.125 138 118]ST 431 | 2872 2818 2824 2831 DL 432 | [0.125 0 0 0.125 138 118]ST 433 | 2919 2803 2872 2818 DL 434 | [0.125 0 0 0.125 138 118]ST 435 | 2966 2784 2919 2803 DL 436 | [0.125 0 0 0.125 138 118]ST 437 | 3012 2763 2966 2784 DL 438 | [0.125 0 0 0.125 138 118]ST 439 | 3057 2738 3012 2763 DL 440 | [0.125 0 0 0.125 138 118]ST 441 | 3101 2711 3057 2738 DL 442 | [0.125 0 0 0.125 138 118]ST 443 | 3144 2682 3101 2711 DL 444 | [0.125 0 0 0.125 138 118]ST 445 | 3185 2651 3144 2682 DL 446 | [0.125 0 0 0.125 138 118]ST 447 | 3224 2617 3185 2651 DL 448 | [0.125 0 0 0.125 138 118]ST 449 | 3261 2581 3224 2617 DL 450 | [0.125 0 0 0.125 138 118]ST 451 | 3297 2544 3261 2581 DL 452 | [0.125 0 0 0.125 138 118]ST 453 | -11438 38415 16230 -36648 DL 454 | [0.125 0 0 0.125 138 118]ST 455 | 40000 2848 -40000 HL 456 | [0.125 0 0 0.125 138 118]ST 457 | 1 16 B 0 0 PE 458 | -38097 -12226 37480 14004 DL 459 | [0.125 0 0 0.125 138 118]ST 460 | 217 -40078 4781 39792 DL 461 | [0.125 0 0 0.125 138 118]ST 462 | 1 24 B 0 0 PE 463 | 31468 24724 -29874 -26628 DL 464 | [0.125 0 0 0.125 138 118]ST 465 | 1 16 B 0 0 PE 466 | -26024 30544 30544 -26024 DL 467 | [0.125 0 0 0.125 138 118]ST 468 | 1 24 B 0 0 PE 469 | 3334 2502 3299 2542 DL 470 | [0.125 0 0 0.125 138 118]ST 471 | 3368 2459 3334 2502 DL 472 | [0.125 0 0 0.125 138 118]ST 473 | 3401 2412 3368 2459 DL 474 | [0.125 0 0 0.125 138 118]ST 475 | 3433 2361 3401 2412 DL 476 | [0.125 0 0 0.125 138 118]ST 477 | 3464 2307 3433 2361 DL 478 | [0.125 0 0 0.125 138 118]ST 479 | 3493 2248 3464 2307 DL 480 | [0.125 0 0 0.125 138 118]ST 481 | 3519 2186 3493 2248 DL 482 | [0.125 0 0 0.125 138 118]ST 483 | 3543 2121 3519 2186 DL 484 | [0.125 0 0 0.125 138 118]ST 485 | 3563 2052 3543 2121 DL 486 | [0.125 0 0 0.125 138 118]ST 487 | 3579 1980 3563 2052 DL 488 | [0.125 0 0 0.125 138 118]ST 489 | 3590 1906 3579 1980 DL 490 | [0.125 0 0 0.125 138 118]ST 491 | 3596 1830 3590 1906 DL 492 | [0.125 0 0 0.125 138 118]ST 493 | 3597 1753 3596 1830 DL 494 | [0.125 0 0 0.125 138 118]ST 495 | 3591 1676 3597 1753 DL 496 | [0.125 0 0 0.125 138 118]ST 497 | 3586 1637 3591 1676 DL 498 | [0.125 0 0 0.125 138 118]ST 499 | 3579 1599 3586 1637 DL 500 | [0.125 0 0 0.125 138 118]ST 501 | 3571 1561 3579 1599 DL 502 | [0.125 0 0 0.125 138 118]ST 503 | 3561 1524 3571 1561 DL 504 | [0.125 0 0 0.125 138 118]ST 505 | 3549 1487 3561 1524 DL 506 | [0.125 0 0 0.125 138 118]ST 507 | 3536 1451 3549 1487 DL 508 | [0.125 0 0 0.125 138 118]ST 509 | 3520 1416 3536 1451 DL 510 | [0.125 0 0 0.125 138 118]ST 511 | 3504 1382 3520 1416 DL 512 | [0.125 0 0 0.125 138 118]ST 513 | 3485 1349 3504 1382 DL 514 | [0.125 0 0 0.125 138 118]ST 515 | 3465 1317 3485 1349 DL 516 | [0.125 0 0 0.125 138 118]ST 517 | 3444 1287 3465 1317 DL 518 | [0.125 0 0 0.125 138 118]ST 519 | 3421 1258 3444 1287 DL 520 | [0.125 0 0 0.125 138 118]ST 521 | 3396 1230 3421 1258 DL 522 | [0.125 0 0 0.125 138 118]ST 523 | 3370 1204 3396 1230 DL 524 | [0.125 0 0 0.125 138 118]ST 525 | 3343 1180 3370 1204 DL 526 | [0.125 0 0 0.125 138 118]ST 527 | 3315 1157 3343 1180 DL 528 | [0.125 0 0 0.125 138 118]ST 529 | 3286 1137 3315 1157 DL 530 | [0.125 0 0 0.125 138 118]ST 531 | 3255 1118 3286 1137 DL 532 | [0.125 0 0 0.125 138 118]ST 533 | 3224 1101 3255 1118 DL 534 | [0.125 0 0 0.125 138 118]ST 535 | 3191 1086 3224 1101 DL 536 | [0.125 0 0 0.125 138 118]ST 537 | 3158 1073 3191 1086 DL 538 | [0.125 0 0 0.125 138 118]ST 539 | 3124 1062 3158 1073 DL 540 | [0.125 0 0 0.125 138 118]ST 541 | 3090 1053 3124 1062 DL 542 | [0.125 0 0 0.125 138 118]ST 543 | 3055 1046 3090 1053 DL 544 | [0.125 0 0 0.125 138 118]ST 545 | 3019 1041 3055 1046 DL 546 | [0.125 0 0 0.125 138 118]ST 547 | 2984 1038 3019 1041 DL 548 | [0.125 0 0 0.125 138 118]ST 549 | 2948 1036 2984 1038 DL 550 | [0.125 0 0 0.125 138 118]ST 551 | 2929 1037 2948 1036 DL 552 | [0.125 0 0 0.125 138 118]ST 553 | 2911 1037 2929 HL 554 | [0.125 0 0 0.125 138 118]ST 555 | 2875 1040 2911 1037 DL 556 | [0.125 0 0 0.125 138 118]ST 557 | 2838 1045 2875 1040 DL 558 | [0.125 0 0 0.125 138 118]ST 559 | 2801 1052 2838 1045 DL 560 | [0.125 0 0 0.125 138 118]ST 561 | 2764 1060 2801 1052 DL 562 | [0.125 0 0 0.125 138 118]ST 563 | 2727 1071 2764 1060 DL 564 | [0.125 0 0 0.125 138 118]ST 565 | 2690 1084 2727 1071 DL 566 | [0.125 0 0 0.125 138 118]ST 567 | 2653 1099 2690 1084 DL 568 | [0.125 0 0 0.125 138 118]ST 569 | 2634 1107 2653 1099 DL 570 | [0.125 0 0 0.125 138 118]ST 571 | 2615 1116 2634 1107 DL 572 | [0.125 0 0 0.125 138 118]ST 573 | 2577 1135 2615 1116 DL 574 | [0.125 0 0 0.125 138 118]ST 575 | 2539 1157 2577 1135 DL 576 | [0.125 0 0 0.125 138 118]ST 577 | 2499 1183 2539 1157 DL 578 | [0.125 0 0 0.125 138 118]ST 579 | 2459 1212 2499 1183 DL 580 | [0.125 0 0 0.125 138 118]ST 581 | 2415 1247 2459 1212 DL 582 | [0.125 0 0 0.125 138 118]ST 583 | 2368 1289 2415 1247 DL 584 | [0.125 0 0 0.125 138 118]ST 585 | 2311 1349 2368 1289 DL 586 | [0.125 0 0 0.125 138 118]ST 587 | 2272 1394 2311 1349 DL 588 | [0.125 0 0 0.125 138 118]ST 589 | 2244 1430 2272 1394 DL 590 | [0.125 0 0 0.125 138 118]ST 591 | 2223 1459 2244 1430 DL 592 | [0.125 0 0 0.125 138 118]ST 593 | 2207 1483 2223 1459 DL 594 | [0.125 0 0 0.125 138 118]ST 595 | 2195 1503 2207 1483 DL 596 | [0.125 0 0 0.125 138 118]ST 597 | 2184 1519 2195 1503 DL 598 | [0.125 0 0 0.125 138 118]ST 599 | 2176 1534 2184 1519 DL 600 | [0.125 0 0 0.125 138 118]ST 601 | 2169 1546 2176 1534 DL 602 | [0.125 0 0 0.125 138 118]ST 603 | 2163 1557 2169 1546 DL 604 | [0.125 0 0 0.125 138 118]ST 605 | 2157 1567 2163 1557 DL 606 | [0.125 0 0 0.125 138 118]ST 607 | 2153 1575 2157 1567 DL 608 | [0.125 0 0 0.125 138 118]ST 609 | 2149 1583 2153 1575 DL 610 | [0.125 0 0 0.125 138 118]ST 611 | 2145 1589 2149 1583 DL 612 | [0.125 0 0 0.125 138 118]ST 613 | 2142 1595 2145 1589 DL 614 | [0.125 0 0 0.125 138 118]ST 615 | 2139 1601 2142 1595 DL 616 | [0.125 0 0 0.125 138 118]ST 617 | 2137 1606 2139 1601 DL 618 | [0.125 0 0 0.125 138 118]ST 619 | 2135 1611 2137 1606 DL 620 | [0.125 0 0 0.125 138 118]ST 621 | 2132 1615 2135 1611 DL 622 | [0.125 0 0 0.125 138 118]ST 623 | 2131 1619 2132 1615 DL 624 | [0.125 0 0 0.125 138 118]ST 625 | 2129 1623 2131 1619 DL 626 | [0.125 0 0 0.125 138 118]ST 627 | 2127 1626 2129 1623 DL 628 | [0.125 0 0 0.125 138 118]ST 629 | 2126 1629 2127 1626 DL 630 | [0.125 0 0 0.125 138 118]ST 631 | 2124 1632 2126 1629 DL 632 | [0.125 0 0 0.125 138 118]ST 633 | 2123 1635 2124 1632 DL 634 | [0.125 0 0 0.125 138 118]ST 635 | 2122 1638 2123 1635 DL 636 | [0.125 0 0 0.125 138 118]ST 637 | 2088 1721 2122 1638 DL 638 | [0.125 0 0 0.125 138 118]ST 639 | 2060 1804 2088 1721 DL 640 | [0.125 0 0 0.125 138 118]ST 641 | 2060 1806 1804 VL 642 | [0.125 0 0 0.125 138 118]ST 643 | 2059 1809 2060 1806 DL 644 | [0.125 0 0 0.125 138 118]ST 645 | 2058 1812 2059 1809 DL 646 | [0.125 0 0 0.125 138 118]ST 647 | 2057 1815 2058 1812 DL 648 | [0.125 0 0 0.125 138 118]ST 649 | 2056 1819 2057 1815 DL 650 | [0.125 0 0 0.125 138 118]ST 651 | 2055 1822 2056 1819 DL 652 | [0.125 0 0 0.125 138 118]ST 653 | 2054 1826 2055 1822 DL 654 | [0.125 0 0 0.125 138 118]ST 655 | 2053 1830 2054 1826 DL 656 | [0.125 0 0 0.125 138 118]ST 657 | 2052 1835 2053 1830 DL 658 | [0.125 0 0 0.125 138 118]ST 659 | 2051 1840 2052 1835 DL 660 | [0.125 0 0 0.125 138 118]ST 661 | 2049 1846 2051 1840 DL 662 | [0.125 0 0 0.125 138 118]ST 663 | 2048 1852 2049 1846 DL 664 | [0.125 0 0 0.125 138 118]ST 665 | 2046 1859 2048 1852 DL 666 | [0.125 0 0 0.125 138 118]ST 667 | 2044 1867 2046 1859 DL 668 | [0.125 0 0 0.125 138 118]ST 669 | 2043 1875 2044 1867 DL 670 | [0.125 0 0 0.125 138 118]ST 671 | 2041 1885 2043 1875 DL 672 | [0.125 0 0 0.125 138 118]ST 673 | 2038 1896 2041 1885 DL 674 | [0.125 0 0 0.125 138 118]ST 675 | 2036 1908 2038 1896 DL 676 | [0.125 0 0 0.125 138 118]ST 677 | 2033 1922 2036 1908 DL 678 | [0.125 0 0 0.125 138 118]ST 679 | 2031 1939 2033 1922 DL 680 | [0.125 0 0 0.125 138 118]ST 681 | 2028 1959 2031 1939 DL 682 | [0.125 0 0 0.125 138 118]ST 683 | 2025 1984 2028 1959 DL 684 | [0.125 0 0 0.125 138 118]ST 685 | 2022 2013 2025 1984 DL 686 | [0.125 0 0 0.125 138 118]ST 687 | 2019 2050 2022 2013 DL 688 | [0.125 0 0 0.125 138 118]ST 689 | 2018 2098 2019 2050 DL 690 | [0.125 0 0 0.125 138 118]ST 691 | 2020 2162 2018 2098 DL 692 | [0.125 0 0 0.125 138 118]ST 693 | 2024 2210 2020 2162 DL 694 | [0.125 0 0 0.125 138 118]ST 695 | 2030 2250 2024 2210 DL 696 | [0.125 0 0 0.125 138 118]ST 697 | 2036 2285 2030 2250 DL 698 | [0.125 0 0 0.125 138 118]ST 699 | 2043 2318 2036 2285 DL 700 | [0.125 0 0 0.125 138 118]ST 701 | 2059 2376 2043 2318 DL 702 | [0.125 0 0 0.125 138 118]ST 703 | 2078 2428 2059 2376 DL 704 | [0.125 0 0 0.125 138 118]ST 705 | 2098 2476 2078 2428 DL 706 | [0.125 0 0 0.125 138 118]ST 707 | 2121 2520 2098 2476 DL 708 | [0.125 0 0 0.125 138 118]ST 709 | 2146 2561 2121 2520 DL 710 | [0.125 0 0 0.125 138 118]ST 711 | 2173 2600 2146 2561 DL 712 | [0.125 0 0 0.125 138 118]ST 713 | 2202 2635 2173 2600 DL 714 | [0.125 0 0 0.125 138 118]ST 715 | 2233 2669 2202 2635 DL 716 | [0.125 0 0 0.125 138 118]ST 717 | 2266 2699 2233 2669 DL 718 | [0.125 0 0 0.125 138 118]ST 719 | 2300 2727 2266 2699 DL 720 | [0.125 0 0 0.125 138 118]ST 721 | 2337 2752 2300 2727 DL 722 | [0.125 0 0 0.125 138 118]ST 723 | 2375 2775 2337 2752 DL 724 | [0.125 0 0 0.125 138 118]ST 725 | 2415 2795 2375 2775 DL 726 | [0.125 0 0 0.125 138 118]ST 727 | 2456 2811 2415 2795 DL 728 | [0.125 0 0 0.125 138 118]ST 729 | 2499 2825 2456 2811 DL 730 | [0.125 0 0 0.125 138 118]ST 731 | 2543 2836 2499 2825 DL 732 | [0.125 0 0 0.125 138 118]ST 733 | 2588 2843 2543 2836 DL 734 | [0.125 0 0 0.125 138 118]ST 735 | 2635 2847 2588 2843 DL 736 | [0.125 0 0 0.125 138 118]ST 737 | 2681 2848 2635 2847 DL 738 | [0.125 0 0 0.125 138 118]ST 739 | 2729 2845 2681 2848 DL 740 | [0.125 0 0 0.125 138 118]ST 741 | 2777 2840 2729 2845 DL 742 | [0.125 0 0 0.125 138 118]ST 743 | 2824 2831 2777 2840 DL 744 | [0.125 0 0 0.125 138 118]ST 745 | 2872 2818 2824 2831 DL 746 | [0.125 0 0 0.125 138 118]ST 747 | 2919 2803 2872 2818 DL 748 | [0.125 0 0 0.125 138 118]ST 749 | 2966 2784 2919 2803 DL 750 | [0.125 0 0 0.125 138 118]ST 751 | 3012 2763 2966 2784 DL 752 | [0.125 0 0 0.125 138 118]ST 753 | 3057 2738 3012 2763 DL 754 | [0.125 0 0 0.125 138 118]ST 755 | 3101 2711 3057 2738 DL 756 | [0.125 0 0 0.125 138 118]ST 757 | 3144 2682 3101 2711 DL 758 | [0.125 0 0 0.125 138 118]ST 759 | 3185 2651 3144 2682 DL 760 | [0.125 0 0 0.125 138 118]ST 761 | 3224 2617 3185 2651 DL 762 | [0.125 0 0 0.125 138 118]ST 763 | 3261 2581 3224 2617 DL 764 | [0.125 0 0 0.125 138 118]ST 765 | 3297 2544 3261 2581 DL 766 | [0.125 0 0 0.125 138 118]ST 767 | 0 0 B 0 0 PE 768 | 2504 472 64 64 E 769 | [0.125 0 0 0.125 138 118]ST 770 | 1640 2816 64 64 E 771 | [0.125 0 0 0.125 138 118]ST 772 | 5304 2816 64 64 E 773 | [0.125 0 0 0.125 138 118]ST 774 | 2584 1872 64 64 E 775 | [0.125 0 0 0.125 138 118]ST 776 | 2638 2816 64 64 E 777 | [0.125 0 0 0.125 138 118]ST 778 | 3310 1146 64 64 E 779 | [0.125 0 0 0.125 138 118]ST 780 | 2056 1689 64 64 E 781 | QP 782 | %%Trailer 783 | %%Pages: 1 784 | %%DocumentFonts: 785 | %%Trailer 786 | cleartomark 787 | countdictstack 788 | exch sub { end } repeat 789 | restore 790 | %%EOF 791 | -------------------------------------------------------------------------------- /coharmonic.eps: -------------------------------------------------------------------------------- 1 | %!PS-Adobe-2.0 EPSF-2.0 2 | %%BoundingBox: 142 127 444 716 3 | %%HiResBoundingBox: 142.000000 127.500000 443.500000 715.500000 4 | %%Creator: Qt 3.3.8b 5 | %%CreationDate: Sun Oct 4 21:25:25 2015 6 | %%DocumentFonts: 7 | 8 | %%EndComments 9 | % EPSF created by ps2eps 1.68 10 | %%BeginProlog 11 | save 12 | countdictstack 13 | mark 14 | newpath 15 | /showpage {} def 16 | /setpagedevice {pop} def 17 | %%EndProlog 18 | %%Page 1 1 19 | %%BeginProlog 20 | % Prolog copyright 1994-2006 Trolltech. You may copy this prolog in any way 21 | % that is directly related to this document. For other use of this prolog, 22 | % see your licensing agreement for Qt. 23 | /d/def load def/D{bind d}bind d/d2{dup dup}D/B{0 d2}D/W{255 d2}D/ED{exch d}D 24 | /D0{0 ED}D/LT{lineto}D/MT{moveto}D/S{stroke}D/F{setfont}D/SW{setlinewidth}D 25 | /CP{closepath}D/RL{rlineto}D/NP{newpath}D/CM{currentmatrix}D/SM{setmatrix}D 26 | /TR{translate}D/SD{setdash}D/SC{aload pop setrgbcolor}D/CR{currentfile read 27 | pop}D/i{index}D/bs{bitshift}D/scs{setcolorspace}D/DB{dict dup begin}D/DE{end 28 | d}D/ie{ifelse}D/sp{astore pop}D/BSt 0 d/LWi 1 d/PSt 1 d/Cx 0 d/Cy 0 d/WFi 29 | false d/OMo false d/BCol[1 1 1]d/PCol[0 0 0]d/BkCol[1 1 1]d/BDArr[0.94 0.88 30 | 0.63 0.50 0.37 0.12 0.06]d/defM matrix d/nS 0 d/GPS{PSt 1 ge PSt 5 le and{{ 31 | LArr PSt 1 sub 2 mul get}{LArr PSt 2 mul 1 sub get}ie}{[]}ie}D/QS{PSt 0 ne{ 32 | gsave LWi SW true GPS 0 SD S OMo PSt 1 ne and{BkCol SC false GPS dup 0 get 33 | SD S}if grestore}if}D/r28{{CR dup 32 gt{exit}if pop}loop 3{CR}repeat 0 4{7 34 | bs exch dup 128 gt{84 sub}if 42 sub 127 and add}repeat}D/rA 0 d/rL 0 d/rB{rL 35 | 0 eq{/rA r28 d/rL 28 d}if dup rL gt{rA exch rL sub rL exch/rA 0 d/rL 0 d rB 36 | exch bs add}{dup rA 16#fffffff 3 -1 roll bs not and exch dup rL exch sub/rL 37 | ED neg rA exch bs/rA ED}ie}D/uc{/rL 0 d 0{dup 2 i length ge{exit}if 1 rB 1 38 | eq{3 rB dup 3 ge{1 add dup rB 1 i 5 ge{1 i 6 ge{1 i 7 ge{1 i 8 ge{128 add}if 39 | 64 add}if 32 add}if 16 add}if 3 add exch pop}if 3 add exch 10 rB 1 add{dup 3 40 | i lt{dup}{2 i}ie 4 i 3 i 3 i sub 2 i getinterval 5 i 4 i 3 -1 roll 41 | putinterval dup 4 -1 roll add 3 1 roll 4 -1 roll exch sub dup 0 eq{exit}if 3 42 | 1 roll}loop pop pop}{3 rB 1 add{2 copy 8 rB put 1 add}repeat}ie}loop pop}D 43 | /sl D0/QCIgray D0/QCIcolor D0/QCIindex D0/QCI{/colorimage where{pop false 3 44 | colorimage}{exec/QCIcolor ED/QCIgray QCIcolor length 3 idiv string d 0 1 45 | QCIcolor length 3 idiv 1 sub{/QCIindex ED/x QCIindex 3 mul d QCIgray 46 | QCIindex QCIcolor x get 0.30 mul QCIcolor x 1 add get 0.59 mul QCIcolor x 2 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and{WFi{clip}{eoclip}ie defM SM pathbbox 3 i 3 i TR 4 60 | 2 roll 3 2 roll exch sub/h ED sub/w ED OMo{NP 0 0 MT 0 h RL w 0 RL 0 h neg 61 | RL CP BkCol SC fill}if BCol SC 0.3 SW NP BSt 9 eq BSt 11 eq or{0 4 h{dup 0 62 | exch MT w exch LT}for}if BSt 10 eq BSt 11 eq or{0 4 w{dup 0 MT h LT}for}if 63 | BSt 12 eq BSt 14 eq or{w h gt{0 6 w h add{dup 0 MT h sub h LT}for}{0 6 w h 64 | add{dup 0 exch MT w sub w exch LT}for}ie}if BSt 13 eq BSt 14 eq or{w h gt{0 65 | 6 w h add{dup h MT h sub 0 LT}for}{0 6 w h add{dup w exch MT w sub 0 exch LT 66 | }for}ie}if S}if BSt 24 eq{}if grestore}D/mat matrix d/ang1 D0/ang2 D0/w D0/h 67 | D0/x D0/y D0/ARC{/ang2 ED/ang1 ED/h ED/w ED/y ED/x ED mat CM pop x w 2 div 68 | add y h 2 div add TR 1 h w div neg scale ang2 0 ge{0 0 w 2 div ang1 ang1 69 | ang2 add arc}{0 0 w 2 div ang1 ang1 ang2 add arcn}ie mat SM}D/C D0/P{NP MT 70 | 0.5 0.5 rmoveto 0 -1 RL -1 0 RL 0 1 RL CP fill}D/M{/Cy ED/Cx ED}D/L{NP Cx Cy 71 | MT/Cy ED/Cx ED Cx Cy LT QS}D/DL{NP MT LT QS}D/HL{1 i 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ashow}ie}D/AT{ty MT 97 | 1 i dup length 2 div exch stringwidth pop 3 -1 roll exch sub exch div exch 0 98 | exch ashow}D/QI{/C save d pageinit/Cx 0 d/Cy 0 d/OMo false d}D/QP{C restore 99 | showpage}D/SPD{/setpagedevice where{1 DB 3 1 roll d end setpagedevice}{pop 100 | pop}ie}D/SV{BSt LWi PSt Cx Cy WFi OMo BCol PCol BkCol/nS nS 1 add d gsave}D 101 | /RS{nS 0 gt{grestore/BkCol ED/PCol ED/BCol ED/OMo ED/WFi ED/Cy ED/Cx ED/PSt 102 | ED/LWi ED/BSt ED/nS nS 1 sub d}if}D/CLSTART{/clipTmp matrix CM d defM SM NP} 103 | D/CLEND{clip NP clipTmp SM}D/CLO{grestore gsave defM SM}D 104 | /LArr[ [] [] [ 13.333 4.000 ] [ 4.000 13.333 ] [ 4.000 4.000 ] [ 4.000 4.000 ] [ 6.667 4.000 4.000 4.000 ] [ 4.000 6.667 4.000 4.000 ] [ 6.667 4.000 4.000 4.000 4.000 ] [ 4.000 6.667 4.000 4.000 4.000 4.000 ] ] d 105 | /pageinit { 106 | 36 24 translate 107 | % 184*280 mm (landscape) 108 | 90 rotate 0.75 -0.75 scale/defM matrix CM d } d 109 | %%EndProlog 110 | %%BeginSetup 111 | %%EndSetup 112 | %%Page: 1 1 113 | 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118]ST 502 | 3161 1333 3185 1300 DL 503 | [0.125 0 0 0.125 138 118]ST 504 | 3139 1365 3161 1333 DL 505 | [0.125 0 0 0.125 138 118]ST 506 | 3119 1399 3139 1365 DL 507 | [0.125 0 0 0.125 138 118]ST 508 | 3101 1432 3119 1399 DL 509 | [0.125 0 0 0.125 138 118]ST 510 | 3084 1467 3101 1432 DL 511 | [0.125 0 0 0.125 138 118]ST 512 | 3068 1501 3084 1467 DL 513 | [0.125 0 0 0.125 138 118]ST 514 | 3055 1536 3068 1501 DL 515 | [0.125 0 0 0.125 138 118]ST 516 | 3043 1570 3055 1536 DL 517 | [0.125 0 0 0.125 138 118]ST 518 | 3032 1606 3043 1570 DL 519 | [0.125 0 0 0.125 138 118]ST 520 | 3023 1641 3032 1606 DL 521 | [0.125 0 0 0.125 138 118]ST 522 | 3015 1676 3023 1641 DL 523 | [0.125 0 0 0.125 138 118]ST 524 | 3009 1712 3015 1676 DL 525 | [0.125 0 0 0.125 138 118]ST 526 | 3004 1748 3009 1712 DL 527 | [0.125 0 0 0.125 138 118]ST 528 | 3001 1784 3004 1748 DL 529 | [0.125 0 0 0.125 138 118]ST 530 | 2999 1820 3001 1784 DL 531 | [0.125 0 0 0.125 138 118]ST 532 | 2999 1856 1820 VL 533 | [0.125 0 0 0.125 138 118]ST 534 | 3000 1893 2999 1856 DL 535 | [0.125 0 0 0.125 138 118]ST 536 | 3003 1930 3000 1893 DL 537 | [0.125 0 0 0.125 138 118]ST 538 | 3008 1968 3003 1930 DL 539 | [0.125 0 0 0.125 138 118]ST 540 | 3014 2007 3008 1968 DL 541 | [0.125 0 0 0.125 138 118]ST 542 | 3022 2047 3014 2007 DL 543 | [0.125 0 0 0.125 138 118]ST 544 | 3033 2088 3022 2047 DL 545 | [0.125 0 0 0.125 138 118]ST 546 | 3046 2132 3033 2088 DL 547 | [0.125 0 0 0.125 138 118]ST 548 | 3064 2179 3046 2132 DL 549 | [0.125 0 0 0.125 138 118]ST 550 | 3086 2232 3064 2179 DL 551 | [0.125 0 0 0.125 138 118]ST 552 | 3121 2300 3086 2232 DL 553 | [0.125 0 0 0.125 138 118]ST 554 | 3149 2347 3121 2300 DL 555 | [0.125 0 0 0.125 138 118]ST 556 | 3172 2383 3149 2347 DL 557 | [0.125 0 0 0.125 138 118]ST 558 | 3192 2410 3172 2383 DL 559 | [0.125 0 0 0.125 138 118]ST 560 | 3208 2432 3192 2410 DL 561 | [0.125 0 0 0.125 138 118]ST 562 | 3222 2449 3208 2432 DL 563 | [0.125 0 0 0.125 138 118]ST 564 | 3234 2464 3222 2449 DL 565 | [0.125 0 0 0.125 138 118]ST 566 | 3245 2476 3234 2464 DL 567 | [0.125 0 0 0.125 138 118]ST 568 | 3254 2486 3245 2476 DL 569 | [0.125 0 0 0.125 138 118]ST 570 | 3262 2495 3254 2486 DL 571 | [0.125 0 0 0.125 138 118]ST 572 | 3269 2503 3262 2495 DL 573 | [0.125 0 0 0.125 138 118]ST 574 | 3275 2510 3269 2503 DL 575 | [0.125 0 0 0.125 138 118]ST 576 | 3281 2516 3275 2510 DL 577 | [0.125 0 0 0.125 138 118]ST 578 | 3286 2522 3281 2516 DL 579 | [0.125 0 0 0.125 138 118]ST 580 | 3291 2527 3286 2522 DL 581 | [0.125 0 0 0.125 138 118]ST 582 | 3295 2531 3291 2527 DL 583 | [0.125 0 0 0.125 138 118]ST 584 | 3299 2535 3295 2531 DL 585 | [0.125 0 0 0.125 138 118]ST 586 | 3303 2539 3299 2535 DL 587 | [0.125 0 0 0.125 138 118]ST 588 | 3306 2542 3303 2539 DL 589 | [0.125 0 0 0.125 138 118]ST 590 | 3309 2545 3306 2542 DL 591 | [0.125 0 0 0.125 138 118]ST 592 | 3312 2548 3309 2545 DL 593 | [0.125 0 0 0.125 138 118]ST 594 | 3315 2550 3312 2548 DL 595 | [0.125 0 0 0.125 138 118]ST 596 | 3317 2553 3315 2550 DL 597 | [0.125 0 0 0.125 138 118]ST 598 | 3320 2555 3317 2553 DL 599 | [0.125 0 0 0.125 138 118]ST 600 | 3322 2557 3320 2555 DL 601 | [0.125 0 0 0.125 138 118]ST 602 | 3393 2619 3322 2557 DL 603 | [0.125 0 0 0.125 138 118]ST 604 | 3471 2678 3393 2619 DL 605 | [0.125 0 0 0.125 138 118]ST 606 | 3474 2680 3471 2678 DL 607 | [0.125 0 0 0.125 138 118]ST 608 | 3477 2682 3474 2680 DL 609 | [0.125 0 0 0.125 138 118]ST 610 | 3480 2684 3477 2682 DL 611 | [0.125 0 0 0.125 138 118]ST 612 | 3484 2686 3480 2684 DL 613 | [0.125 0 0 0.125 138 118]ST 614 | 3487 2689 3484 2686 DL 615 | [0.125 0 0 0.125 138 118]ST 616 | 3491 2691 3487 2689 DL 617 | [0.125 0 0 0.125 138 118]ST 618 | 3496 2694 3491 2691 DL 619 | [0.125 0 0 0.125 138 118]ST 620 | 3501 2697 3496 2694 DL 621 | [0.125 0 0 0.125 138 118]ST 622 | 3506 2701 3501 2697 DL 623 | [0.125 0 0 0.125 138 118]ST 624 | 3512 2705 3506 2701 DL 625 | [0.125 0 0 0.125 138 118]ST 626 | 3518 2709 3512 2705 DL 627 | [0.125 0 0 0.125 138 118]ST 628 | 3526 2713 3518 2709 DL 629 | [0.125 0 0 0.125 138 118]ST 630 | 3534 2718 3526 2713 DL 631 | [0.125 0 0 0.125 138 118]ST 632 | 3543 2724 3534 2718 DL 633 | [0.125 0 0 0.125 138 118]ST 634 | 3554 2730 3543 2724 DL 635 | [0.125 0 0 0.125 138 118]ST 636 | 3566 2737 3554 2730 DL 637 | [0.125 0 0 0.125 138 118]ST 638 | 3581 2745 3566 2737 DL 639 | [0.125 0 0 0.125 138 118]ST 640 | 3598 2754 3581 2745 DL 641 | [0.125 0 0 0.125 138 118]ST 642 | 3619 2765 3598 2754 DL 643 | [0.125 0 0 0.125 138 118]ST 644 | 3644 2778 3619 2765 DL 645 | [0.125 0 0 0.125 138 118]ST 646 | 3676 2792 3644 2778 DL 647 | [0.125 0 0 0.125 138 118]ST 648 | 3718 2810 3676 2792 DL 649 | [0.125 0 0 0.125 138 118]ST 650 | 3773 2831 3718 2810 DL 651 | [0.125 0 0 0.125 138 118]ST 652 | 3850 2855 3773 2831 DL 653 | [0.125 0 0 0.125 138 118]ST 654 | 3964 2883 3850 2855 DL 655 | [0.125 0 0 0.125 138 118]ST 656 | 4059 2898 3964 2883 DL 657 | [0.125 0 0 0.125 138 118]ST 658 | 4146 2906 4059 2898 DL 659 | [0.125 0 0 0.125 138 118]ST 660 | 4228 2909 4146 2906 DL 661 | [0.125 0 0 0.125 138 118]ST 662 | 4308 2907 4228 2909 DL 663 | [0.125 0 0 0.125 138 118]ST 664 | 4386 2900 4308 2907 DL 665 | [0.125 0 0 0.125 138 118]ST 666 | 4463 2888 4386 2900 DL 667 | [0.125 0 0 0.125 138 118]ST 668 | 4501 2881 4463 2888 DL 669 | [0.125 0 0 0.125 138 118]ST 670 | 4539 2873 4501 2881 DL 671 | [0.125 0 0 0.125 138 118]ST 672 | 4614 2852 4539 2873 DL 673 | [0.125 0 0 0.125 138 118]ST 674 | 4687 2827 4614 2852 DL 675 | [0.125 0 0 0.125 138 118]ST 676 | 4760 2797 4687 2827 DL 677 | [0.125 0 0 0.125 138 118]ST 678 | 4830 2763 4760 2797 DL 679 | [0.125 0 0 0.125 138 118]ST 680 | 4898 2723 4830 2763 DL 681 | [0.125 0 0 0.125 138 118]ST 682 | 4964 2679 4898 2723 DL 683 | [0.125 0 0 0.125 138 118]ST 684 | 5026 2629 4964 2679 DL 685 | [0.125 0 0 0.125 138 118]ST 686 | 5085 2575 5026 2629 DL 687 | [0.125 0 0 0.125 138 118]ST 688 | 5140 2516 5085 2575 DL 689 | [0.125 0 0 0.125 138 118]ST 690 | 5165 2485 5140 2516 DL 691 | [0.125 0 0 0.125 138 118]ST 692 | 5189 2453 5165 2485 DL 693 | [0.125 0 0 0.125 138 118]ST 694 | 5212 2420 5189 2453 DL 695 | [0.125 0 0 0.125 138 118]ST 696 | 5234 2385 5212 2420 DL 697 | [0.125 0 0 0.125 138 118]ST 698 | 5254 2350 5234 2385 DL 699 | [0.125 0 0 0.125 138 118]ST 700 | 5272 2314 5254 2350 DL 701 | [0.125 0 0 0.125 138 118]ST 702 | 5289 2277 5272 2314 DL 703 | [0.125 0 0 0.125 138 118]ST 704 | 5304 2240 5289 2277 DL 705 | [0.125 0 0 0.125 138 118]ST 706 | 5317 2202 5304 2240 DL 707 | [0.125 0 0 0.125 138 118]ST 708 | 5329 2163 5317 2202 DL 709 | [0.125 0 0 0.125 138 118]ST 710 | 5339 2123 5329 2163 DL 711 | [0.125 0 0 0.125 138 118]ST 712 | 5347 2084 5339 2123 DL 713 | [0.125 0 0 0.125 138 118]ST 714 | 5353 2043 5347 2084 DL 715 | [0.125 0 0 0.125 138 118]ST 716 | 5357 2003 5353 2043 DL 717 | [0.125 0 0 0.125 138 118]ST 718 | 5360 1962 5357 2003 DL 719 | [0.125 0 0 0.125 138 118]ST 720 | 5361 1922 5360 1962 DL 721 | [0.125 0 0 0.125 138 118]ST 722 | 5359 1881 5361 1922 DL 723 | [0.125 0 0 0.125 138 118]ST 724 | 5356 1840 5359 1881 DL 725 | [0.125 0 0 0.125 138 118]ST 726 | 5351 1800 5356 1840 DL 727 | [0.125 0 0 0.125 138 118]ST 728 | 5344 1760 5351 1800 DL 729 | [0.125 0 0 0.125 138 118]ST 730 | 5336 1720 5344 1760 DL 731 | [0.125 0 0 0.125 138 118]ST 732 | 5326 1681 5336 1720 DL 733 | [0.125 0 0 0.125 138 118]ST 734 | 5314 1642 5326 1681 DL 735 | [0.125 0 0 0.125 138 118]ST 736 | 5300 1603 5314 1642 DL 737 | [0.125 0 0 0.125 138 118]ST 738 | 5268 1529 5300 1603 DL 739 | [0.125 0 0 0.125 138 118]ST 740 | 5230 1458 5268 1529 DL 741 | [0.125 0 0 0.125 138 118]ST 742 | 5187 1390 5230 1458 DL 743 | [0.125 0 0 0.125 138 118]ST 744 | 5140 1326 5187 1390 DL 745 | [0.125 0 0 0.125 138 118]ST 746 | 5088 1266 5140 1326 DL 747 | [0.125 0 0 0.125 138 118]ST 748 | 5033 1211 5088 1266 DL 749 | [0.125 0 0 0.125 138 118]ST 750 | 0 0 B 0 0 PE 751 | 1256 2018 64 64 E 752 | [0.125 0 0 0.125 138 118]ST 753 | 3113 1325 64 64 E 754 | [0.125 0 0 0.125 138 118]ST 755 | 2977 1944 64 64 E 756 | [0.125 0 0 0.125 138 118]ST 757 | 3361 2587 64 64 E 758 | [0.125 0 0 0.125 138 118]ST 759 | 4364 858 64 64 E 760 | [0.125 0 0 0.125 138 118]ST 761 | 5327 1842 64 64 E 762 | [0.125 0 0 0.125 138 118]ST 763 | 4383 2864 64 64 E 764 | QP 765 | %%Trailer 766 | %%Pages: 1 767 | %%DocumentFonts: 768 | %%Trailer 769 | cleartomark 770 | countdictstack 771 | exch sub { end } repeat 772 | restore 773 | %%EOF 774 | -------------------------------------------------------------------------------- /conicexer.eps: -------------------------------------------------------------------------------- 1 | %!PS-Adobe-2.0 EPSF-2.0 2 | %%BoundingBox: 124 127 471 716 3 | %%HiResBoundingBox: 124.500000 127.500000 470.500000 715.500000 4 | %%Creator: Qt 3.3.8b 5 | %%CreationDate: Wed Jun 10 14:19:05 2015 6 | %%DocumentFonts: 7 | 8 | %%EndComments 9 | % EPSF created by ps2eps 1.68 10 | %%BeginProlog 11 | save 12 | countdictstack 13 | mark 14 | newpath 15 | /showpage {} def 16 | /setpagedevice {pop} def 17 | %%EndProlog 18 | %%Page 1 1 19 | %%BeginProlog 20 | % Prolog copyright 1994-2006 Trolltech. You may copy this prolog in any way 21 | % that is directly related to this document. For other use of this prolog, 22 | % see your licensing agreement for Qt. 23 | /d/def load def/D{bind d}bind d/d2{dup dup}D/B{0 d2}D/W{255 d2}D/ED{exch d}D 24 | /D0{0 ED}D/LT{lineto}D/MT{moveto}D/S{stroke}D/F{setfont}D/SW{setlinewidth}D 25 | /CP{closepath}D/RL{rlineto}D/NP{newpath}D/CM{currentmatrix}D/SM{setmatrix}D 26 | /TR{translate}D/SD{setdash}D/SC{aload pop setrgbcolor}D/CR{currentfile read 27 | pop}D/i{index}D/bs{bitshift}D/scs{setcolorspace}D/DB{dict dup begin}D/DE{end 28 | d}D/ie{ifelse}D/sp{astore pop}D/BSt 0 d/LWi 1 d/PSt 1 d/Cx 0 d/Cy 0 d/WFi 29 | false d/OMo false d/BCol[1 1 1]d/PCol[0 0 0]d/BkCol[1 1 1]d/BDArr[0.94 0.88 30 | 0.63 0.50 0.37 0.12 0.06]d/defM matrix d/nS 0 d/GPS{PSt 1 ge PSt 5 le and{{ 31 | LArr PSt 1 sub 2 mul get}{LArr PSt 2 mul 1 sub get}ie}{[]}ie}D/QS{PSt 0 ne{ 32 | gsave LWi SW true GPS 0 SD S OMo PSt 1 ne and{BkCol SC false GPS dup 0 get 33 | SD S}if grestore}if}D/r28{{CR dup 32 gt{exit}if pop}loop 3{CR}repeat 0 4{7 34 | bs exch dup 128 gt{84 sub}if 42 sub 127 and add}repeat}D/rA 0 d/rL 0 d/rB{rL 35 | 0 eq{/rA r28 d/rL 28 d}if dup rL gt{rA exch rL sub rL exch/rA 0 d/rL 0 d rB 36 | exch bs add}{dup rA 16#fffffff 3 -1 roll bs not and exch dup rL exch sub/rL 37 | ED neg rA exch bs/rA ED}ie}D/uc{/rL 0 d 0{dup 2 i length ge{exit}if 1 rB 1 38 | eq{3 rB dup 3 ge{1 add dup rB 1 i 5 ge{1 i 6 ge{1 i 7 ge{1 i 8 ge{128 add}if 39 | 64 add}if 32 add}if 16 add}if 3 add exch pop}if 3 add exch 10 rB 1 add{dup 3 40 | i lt{dup}{2 i}ie 4 i 3 i 3 i sub 2 i getinterval 5 i 4 i 3 -1 roll 41 | putinterval dup 4 -1 roll add 3 1 roll 4 -1 roll exch sub dup 0 eq{exit}if 3 42 | 1 roll}loop pop pop}{3 rB 1 add{2 copy 8 rB put 1 add}repeat}ie}loop pop}D 43 | /sl D0/QCIgray D0/QCIcolor D0/QCIindex D0/QCI{/colorimage where{pop false 3 44 | colorimage}{exec/QCIcolor ED/QCIgray QCIcolor length 3 idiv string d 0 1 45 | QCIcolor length 3 idiv 1 sub{/QCIindex ED/x QCIindex 3 mul d QCIgray 46 | QCIindex QCIcolor x get 0.30 mul QCIcolor x 1 add get 0.59 mul QCIcolor x 2 47 | add get 0.11 mul add add cvi put}for QCIgray image}ie}D/di{gsave TR 1 i 1 eq 48 | {false eq{pop true 3 1 roll 4 i 4 i false 4 i 4 i imagemask BkCol SC 49 | imagemask}{pop false 3 1 roll imagemask}ie}{dup false ne{/languagelevel 50 | where{pop languagelevel 3 ge}{false}ie}{false}ie{/ma ED 8 eq{/dc[0 1]d 51 | /DeviceGray}{/dc[0 1 0 1 0 1]d/DeviceRGB}ie scs/im ED/mt ED/h ED/w ED/id 7 52 | DB/ImageType 1 d/Width w d/Height h d/ImageMatrix mt d/DataSource im d 53 | /BitsPerComponent 8 d/Decode dc d DE/md 7 DB/ImageType 1 d/Width w d/Height 54 | h d/ImageMatrix mt d/DataSource ma d/BitsPerComponent 1 d/Decode[0 1]d DE 4 55 | DB/ImageType 3 d/DataDict id d/MaskDict md d/InterleaveType 3 d end image}{ 56 | pop 8 4 1 roll 8 eq{image}{QCI}ie}ie}ie grestore}d/BF{gsave BSt 1 eq{BCol SC 57 | WFi{fill}{eofill}ie}if BSt 2 ge BSt 8 le and{BDArr BSt 2 sub get/sc ED BCol{ 58 | 1. exch sub sc mul 1. exch sub}forall 3 array astore SC WFi{fill}{eofill}ie} 59 | if BSt 9 ge BSt 14 le and{WFi{clip}{eoclip}ie defM SM pathbbox 3 i 3 i TR 4 60 | 2 roll 3 2 roll exch sub/h ED sub/w ED OMo{NP 0 0 MT 0 h RL w 0 RL 0 h neg 61 | RL CP BkCol SC fill}if BCol SC 0.3 SW NP BSt 9 eq BSt 11 eq or{0 4 h{dup 0 62 | exch MT w exch LT}for}if BSt 10 eq BSt 11 eq or{0 4 w{dup 0 MT h LT}for}if 63 | BSt 12 eq BSt 14 eq or{w h gt{0 6 w h add{dup 0 MT h sub h LT}for}{0 6 w h 64 | add{dup 0 exch MT w sub w exch LT}for}ie}if BSt 13 eq BSt 14 eq or{w h gt{0 65 | 6 w h add{dup h MT h sub 0 LT}for}{0 6 w h add{dup w exch MT w sub 0 exch LT 66 | }for}ie}if S}if BSt 24 eq{}if grestore}D/mat matrix d/ang1 D0/ang2 D0/w D0/h 67 | D0/x D0/y D0/ARC{/ang2 ED/ang1 ED/h ED/w ED/y ED/x ED mat CM pop x w 2 div 68 | add y h 2 div add TR 1 h w div neg scale ang2 0 ge{0 0 w 2 div ang1 ang1 69 | ang2 add arc}{0 0 w 2 div ang1 ang1 ang2 add arcn}ie mat SM}D/C D0/P{NP MT 70 | 0.5 0.5 rmoveto 0 -1 RL -1 0 RL 0 1 RL CP fill}D/M{/Cy ED/Cx ED}D/L{NP Cx Cy 71 | MT/Cy ED/Cx ED Cx Cy LT QS}D/DL{NP MT LT QS}D/HL{1 i DL}D/VL{2 i exch DL}D/R 72 | {/h ED/w ED/y ED/x ED NP x y MT 0 h RL w 0 RL 0 h neg RL CP BF QS}D/ACR{/h 73 | ED/w ED/y ED/x ED x y MT 0 h RL w 0 RL 0 h neg RL CP}D/xr D0/yr D0/rx D0/ry 74 | D0/rx2 D0/ry2 D0/RR{/yr ED/xr ED/h ED/w ED/y ED/x ED xr 0 le yr 0 le or{x y 75 | w h R}{xr 100 ge yr 100 ge or{x y w h E}{/rx xr w mul 200 div d/ry yr h mul 76 | 200 div d/rx2 rx 2 mul d/ry2 ry 2 mul d NP x rx add y MT x y rx2 ry2 180 -90 77 | x y h add ry2 sub rx2 ry2 270 -90 x w add rx2 sub y h add ry2 sub rx2 ry2 0 78 | -90 x w add rx2 sub y rx2 ry2 90 -90 ARC ARC ARC ARC CP BF QS}ie}ie}D/E{/h 79 | ED/w ED/y ED/x ED mat CM pop x w 2 div add y h 2 div add TR 1 h w div scale 80 | NP 0 0 w 2 div 0 360 arc mat SM BF QS}D/A{16 div exch 16 div exch NP ARC QS} 81 | D/PIE{/ang2 ED/ang1 ED/h ED/w ED/y ED/x ED NP x w 2 div add y h 2 div add MT 82 | x y w h ang1 16 div ang2 16 div ARC CP BF QS}D/CH{16 div exch 16 div exch NP 83 | ARC CP BF QS}D/BZ{curveto QS}D/CRGB{255 div 3 1 roll 255 div 3 1 roll 255 84 | div 3 1 roll}D/BC{CRGB BkCol sp}D/BR{CRGB BCol sp/BSt ED}D/WB{1 W BR}D/NB{0 85 | B BR}D/PE{setlinejoin setlinecap CRGB PCol sp/LWi ED/PSt ED LWi 0 eq{0.25 86 | /LWi ED}if PCol SC}D/P1{1 0 5 2 roll 0 0 PE}D/ST{defM SM concat}D/MF{true 87 | exch true exch{exch pop exch pop dup 0 get dup findfont dup/FontName get 3 88 | -1 roll eq{exit}if}forall exch dup 1 get/fxscale ED 2 get/fslant ED exch 89 | /fencoding ED[fxscale 0 fslant 1 0 0]makefont fencoding false eq{}{dup 90 | maxlength dict begin{1 i/FID ne{def}{pop pop}ifelse}forall/Encoding 91 | fencoding d currentdict end}ie definefont pop}D/MFEmb{findfont dup length 92 | dict begin{1 i/FID ne{d}{pop pop}ifelse}forall/Encoding ED currentdict end 93 | definefont pop}D/DF{findfont/fs 3 -1 roll d[fs 0 0 fs -1 mul 0 0]makefont d} 94 | D/ty 0 d/Y{/ty ED}D/Tl{gsave SW NP 1 i exch MT 1 i 0 RL S grestore}D/XYT{ty 95 | MT/xyshow where{pop pop xyshow}{exch pop 1 i dup length 2 div exch 96 | stringwidth pop 3 -1 roll exch sub exch div exch 0 exch ashow}ie}D/AT{ty MT 97 | 1 i dup length 2 div exch stringwidth pop 3 -1 roll exch sub exch div exch 0 98 | exch ashow}D/QI{/C save d pageinit/Cx 0 d/Cy 0 d/OMo false d}D/QP{C restore 99 | showpage}D/SPD{/setpagedevice where{1 DB 3 1 roll d end setpagedevice}{pop 100 | pop}ie}D/SV{BSt LWi PSt Cx Cy WFi OMo BCol PCol BkCol/nS nS 1 add d gsave}D 101 | /RS{nS 0 gt{grestore/BkCol ED/PCol ED/BCol ED/OMo ED/WFi ED/Cy ED/Cx ED/PSt 102 | ED/LWi ED/BSt ED/nS nS 1 sub d}if}D/CLSTART{/clipTmp matrix CM d defM SM NP} 103 | D/CLEND{clip NP clipTmp SM}D/CLO{grestore gsave defM SM}D 104 | /LArr[ [] [] [ 13.333 4.000 ] [ 4.000 13.333 ] [ 4.000 4.000 ] [ 4.000 4.000 ] [ 6.667 4.000 4.000 4.000 ] [ 4.000 6.667 4.000 4.000 ] [ 6.667 4.000 4.000 4.000 4.000 ] [ 4.000 6.667 4.000 4.000 4.000 4.000 ] ] d 105 | /pageinit { 106 | 36 24 translate 107 | % 184*280 mm (landscape) 108 | 90 rotate 0.75 -0.75 scale/defM matrix CM d } d 109 | %%EndProlog 110 | %%BeginSetup 111 | %%EndSetup 112 | %%Page: 1 1 113 | %%BeginPageSetup 114 | QI 115 | %%EndPageSetup 116 | [0.125 0 0 0.125 138 118]ST 117 | CLSTART 118 | 138 118 784 461 ACR 119 | CLEND 120 | 0 0 B 0 0 PE 121 | 1 B BR 122 | W BC 123 | 1584 1200 64 64 E 124 | [0.125 0 0 0.125 138 118]ST 125 | 2696 1128 64 64 E 126 | [0.125 0 0 0.125 138 118]ST 127 | 1 2 B 0 0 PE 128 | 40003 -1253 -39830 3916 DL 129 | [0.125 0 0 0.125 138 118]ST 130 | 0 0 B 0 0 PE 131 | 3388 1083 64 64 E 132 | [0.125 0 0 0.125 138 118]ST 133 | 4139 1035 64 64 E 134 | [0.125 0 0 0.125 138 118]ST 135 | 1816 2704 64 64 E 136 | [0.125 0 0 0.125 138 118]ST 137 | 2736 2760 64 64 E 138 | [0.125 0 0 0.125 138 118]ST 139 | 1 2 B 0 0 PE 140 | 39767 5044 -40085 184 DL 141 | [0.125 0 0 0.125 138 118]ST 142 | 0 0 B 0 0 PE 143 | 3469 2805 64 64 E 144 | [0.125 0 0 0.125 138 118]ST 145 | 1 2 B 0 0 PE 146 | 24219 31840 -23305 -32515 DL 147 | [0.125 0 0 0.125 138 118]ST 148 | 22074 -33487 -16928 36361 DL 149 | [0.125 0 0 0.125 138 118]ST 150 | 0 0 B 0 0 PE 151 | 2194 2026 64 64 E 152 | [0.125 0 0 0.125 138 118]ST 153 | 1 2 B 0 0 PE 154 | 18153 -35899 -10150 38927 DL 155 | [0.125 0 0 0.125 138 118]ST 156 | 0 0 B 0 0 PE 157 | 3839 1828 64 64 E 158 | [0.125 0 0 0.125 138 118]ST 159 | 1 2 B 0 0 PE 160 | -18689 -35443 22710 33012 DL 161 | [0.125 0 0 0.125 138 118]ST 162 | 0 0 B 0 0 PE 163 | 4466 2865 64 64 E 164 | [0.125 0 0 0.125 138 118]ST 165 | 3232 3432 64 64 E 166 | [0.125 0 0 0.125 138 118]ST 167 | 3041 510 64 64 E 168 | [0.125 0 0 0.125 138 118]ST 169 | 1 2 B 0 0 PE 170 | 1508 1575 1524 1488 DL 171 | [0.125 0 0 0.125 138 118]ST 172 | 1501 1620 1508 1575 DL 173 | [0.125 0 0 0.125 138 118]ST 174 | 1496 1667 1501 1620 DL 175 | [0.125 0 0 0.125 138 118]ST 176 | 1493 1714 1496 1667 DL 177 | [0.125 0 0 0.125 138 118]ST 178 | 1492 1763 1493 1714 DL 179 | [0.125 0 0 0.125 138 118]ST 180 | 1492 1814 1763 VL 181 | [0.125 0 0 0.125 138 118]ST 182 | 1495 1865 1492 1814 DL 183 | [0.125 0 0 0.125 138 118]ST 184 | 1500 1918 1495 1865 DL 185 | [0.125 0 0 0.125 138 118]ST 186 | 1506 1971 1500 1918 DL 187 | [0.125 0 0 0.125 138 118]ST 188 | 1515 2026 1506 1971 DL 189 | [0.125 0 0 0.125 138 118]ST 190 | 1527 2081 1515 2026 DL 191 | [0.125 0 0 0.125 138 118]ST 192 | 1541 2137 1527 2081 DL 193 | [0.125 0 0 0.125 138 118]ST 194 | 1557 2194 1541 2137 DL 195 | [0.125 0 0 0.125 138 118]ST 196 | 1577 2252 1557 2194 DL 197 | [0.125 0 0 0.125 138 118]ST 198 | 1598 2309 1577 2252 DL 199 | [0.125 0 0 0.125 138 118]ST 200 | 1623 2368 1598 2309 DL 201 | [0.125 0 0 0.125 138 118]ST 202 | 1650 2426 1623 2368 DL 203 | [0.125 0 0 0.125 138 118]ST 204 | 1680 2484 1650 2426 DL 205 | [0.125 0 0 0.125 138 118]ST 206 | 1714 2542 1680 2484 DL 207 | [0.125 0 0 0.125 138 118]ST 208 | 1749 2599 1714 2542 DL 209 | [0.125 0 0 0.125 138 118]ST 210 | 1788 2656 1749 2599 DL 211 | [0.125 0 0 0.125 138 118]ST 212 | 1829 2712 1788 2656 DL 213 | [0.125 0 0 0.125 138 118]ST 214 | 1874 2767 1829 2712 DL 215 | [0.125 0 0 0.125 138 118]ST 216 | 1920 2822 1874 2767 DL 217 | [0.125 0 0 0.125 138 118]ST 218 | 1970 2874 1920 2822 DL 219 | [0.125 0 0 0.125 138 118]ST 220 | 2021 2926 1970 2874 DL 221 | [0.125 0 0 0.125 138 118]ST 222 | 2076 2976 2021 2926 DL 223 | [0.125 0 0 0.125 138 118]ST 224 | 2132 3023 2076 2976 DL 225 | [0.125 0 0 0.125 138 118]ST 226 | 2190 3069 2132 3023 DL 227 | [0.125 0 0 0.125 138 118]ST 228 | 2250 3113 2190 3069 DL 229 | [0.125 0 0 0.125 138 118]ST 230 | 2312 3155 2250 3113 DL 231 | [0.125 0 0 0.125 138 118]ST 232 | 2376 3194 2312 3155 DL 233 | [0.125 0 0 0.125 138 118]ST 234 | 2440 3231 2376 3194 DL 235 | [0.125 0 0 0.125 138 118]ST 236 | 2506 3265 2440 3231 DL 237 | [0.125 0 0 0.125 138 118]ST 238 | 2573 3297 2506 3265 DL 239 | [0.125 0 0 0.125 138 118]ST 240 | 2640 3325 2573 3297 DL 241 | [0.125 0 0 0.125 138 118]ST 242 | 2708 3352 2640 3325 DL 243 | [0.125 0 0 0.125 138 118]ST 244 | 2776 3375 2708 3352 DL 245 | [0.125 0 0 0.125 138 118]ST 246 | 2844 3396 2776 3375 DL 247 | [0.125 0 0 0.125 138 118]ST 248 | 2912 3413 2844 3396 DL 249 | [0.125 0 0 0.125 138 118]ST 250 | 2980 3429 2912 3413 DL 251 | [0.125 0 0 0.125 138 118]ST 252 | 3047 3441 2980 3429 DL 253 | [0.125 0 0 0.125 138 118]ST 254 | 3114 3451 3047 3441 DL 255 | [0.125 0 0 0.125 138 118]ST 256 | 3180 3458 3114 3451 DL 257 | [0.125 0 0 0.125 138 118]ST 258 | 3244 3463 3180 3458 DL 259 | [0.125 0 0 0.125 138 118]ST 260 | 3308 3465 3244 3463 DL 261 | [0.125 0 0 0.125 138 118]ST 262 | 3370 3465 3308 HL 263 | [0.125 0 0 0.125 138 118]ST 264 | 3432 3463 3370 3465 DL 265 | [0.125 0 0 0.125 138 118]ST 266 | 3491 3458 3432 3463 DL 267 | [0.125 0 0 0.125 138 118]ST 268 | 3549 3452 3491 3458 DL 269 | [0.125 0 0 0.125 138 118]ST 270 | 3606 3444 3549 3452 DL 271 | [0.125 0 0 0.125 138 118]ST 272 | 3661 3434 3606 3444 DL 273 | [0.125 0 0 0.125 138 118]ST 274 | 3714 3422 3661 3434 DL 275 | [0.125 0 0 0.125 138 118]ST 276 | 3766 3409 3714 3422 DL 277 | [0.125 0 0 0.125 138 118]ST 278 | 3815 3394 3766 3409 DL 279 | [0.125 0 0 0.125 138 118]ST 280 | 3863 3378 3815 3394 DL 281 | [0.125 0 0 0.125 138 118]ST 282 | 3910 3361 3863 3378 DL 283 | [0.125 0 0 0.125 138 118]ST 284 | 3997 3323 3910 3361 DL 285 | [0.125 0 0 0.125 138 118]ST 286 | 4077 3281 3997 3323 DL 287 | [0.125 0 0 0.125 138 118]ST 288 | 4150 3237 4077 3281 DL 289 | [0.125 0 0 0.125 138 118]ST 290 | 4217 3190 4150 3237 DL 291 | [0.125 0 0 0.125 138 118]ST 292 | 4278 3142 4217 3190 DL 293 | [0.125 0 0 0.125 138 118]ST 294 | 4333 3093 4278 3142 DL 295 | [0.125 0 0 0.125 138 118]ST 296 | 4382 3043 4333 3093 DL 297 | [0.125 0 0 0.125 138 118]ST 298 | 4426 2993 4382 3043 DL 299 | [0.125 0 0 0.125 138 118]ST 300 | 4466 2943 4426 2993 DL 301 | [0.125 0 0 0.125 138 118]ST 302 | 4501 2893 4466 2943 DL 303 | [0.125 0 0 0.125 138 118]ST 304 | 4532 2843 4501 2893 DL 305 | [0.125 0 0 0.125 138 118]ST 306 | 4560 2795 4532 2843 DL 307 | [0.125 0 0 0.125 138 118]ST 308 | 4584 2747 4560 2795 DL 309 | [0.125 0 0 0.125 138 118]ST 310 | 4605 2701 4584 2747 DL 311 | [0.125 0 0 0.125 138 118]ST 312 | 4623 2655 4605 2701 DL 313 | [0.125 0 0 0.125 138 118]ST 314 | 4639 2611 4623 2655 DL 315 | [0.125 0 0 0.125 138 118]ST 316 | 4652 2568 4639 2611 DL 317 | [0.125 0 0 0.125 138 118]ST 318 | 4673 2485 4652 2568 DL 319 | [0.125 0 0 0.125 138 118]ST 320 | 4687 2407 4673 2485 DL 321 | [0.125 0 0 0.125 138 118]ST 322 | 4696 2334 4687 2407 DL 323 | [0.125 0 0 0.125 138 118]ST 324 | 4700 2266 4696 2334 DL 325 | [0.125 0 0 0.125 138 118]ST 326 | 4701 2234 4700 2266 DL 327 | [0.125 0 0 0.125 138 118]ST 328 | 4701 2203 2234 VL 329 | [0.125 0 0 0.125 138 118]ST 330 | 4699 2143 4701 2203 DL 331 | [0.125 0 0 0.125 138 118]ST 332 | 4695 2087 4699 2143 DL 333 | [0.125 0 0 0.125 138 118]ST 334 | 4688 2036 4695 2087 DL 335 | [0.125 0 0 0.125 138 118]ST 336 | 4685 2011 4688 2036 DL 337 | [0.125 0 0 0.125 138 118]ST 338 | 4681 1987 4685 2011 DL 339 | [0.125 0 0 0.125 138 118]ST 340 | 4663 1899 4681 1987 DL 341 | [0.125 0 0 0.125 138 118]ST 342 | 4642 1821 4663 1899 DL 343 | [0.125 0 0 0.125 138 118]ST 344 | 4620 1753 4642 1821 DL 345 | [0.125 0 0 0.125 138 118]ST 346 | 4598 1692 4620 1753 DL 347 | [0.125 0 0 0.125 138 118]ST 348 | 4553 1588 4598 1692 DL 349 | [0.125 0 0 0.125 138 118]ST 350 | 4509 1503 4553 1588 DL 351 | [0.125 0 0 0.125 138 118]ST 352 | 4468 1432 4509 1503 DL 353 | [0.125 0 0 0.125 138 118]ST 354 | 4429 1371 4468 1432 DL 355 | [0.125 0 0 0.125 138 118]ST 356 | 4391 1317 4429 1371 DL 357 | [0.125 0 0 0.125 138 118]ST 358 | 4353 1267 4391 1317 DL 359 | [0.125 0 0 0.125 138 118]ST 360 | 4311 1216 4353 1267 DL 361 | [0.125 0 0 0.125 138 118]ST 362 | 4285 1186 4311 1216 DL 363 | [0.125 0 0 0.125 138 118]ST 364 | 4267 1166 4285 1186 DL 365 | [0.125 0 0 0.125 138 118]ST 366 | 4254 1152 4267 1166 DL 367 | [0.125 0 0 0.125 138 118]ST 368 | 4171 1067 4254 1152 DL 369 | [0.125 0 0 0.125 138 118]ST 370 | 4085 990 4171 1067 DL 371 | [0.125 0 0 0.125 138 118]ST 372 | 4071 978 4085 990 DL 373 | [0.125 0 0 0.125 138 118]ST 374 | 4052 963 4071 978 DL 375 | [0.125 0 0 0.125 138 118]ST 376 | 4024 940 4052 963 DL 377 | [0.125 0 0 0.125 138 118]ST 378 | 4001 922 4024 940 DL 379 | [0.125 0 0 0.125 138 118]ST 380 | 3979 906 4001 922 DL 381 | [0.125 0 0 0.125 138 118]ST 382 | 3891 845 3979 906 DL 383 | [0.125 0 0 0.125 138 118]ST 384 | 3790 783 3891 845 DL 385 | [0.125 0 0 0.125 138 118]ST 386 | 3729 750 3790 783 DL 387 | [0.125 0 0 0.125 138 118]ST 388 | 3660 715 3729 750 DL 389 | [0.125 0 0 0.125 138 118]ST 390 | 3578 678 3660 715 DL 391 | [0.125 0 0 0.125 138 118]ST 392 | 3481 640 3578 678 DL 393 | [0.125 0 0 0.125 138 118]ST 394 | 3425 621 3481 640 DL 395 | [0.125 0 0 0.125 138 118]ST 396 | 3363 602 3425 621 DL 397 | [0.125 0 0 0.125 138 118]ST 398 | 3295 583 3363 602 DL 399 | [0.125 0 0 0.125 138 118]ST 400 | 3219 566 3295 583 DL 401 | [0.125 0 0 0.125 138 118]ST 402 | 3134 550 3219 566 DL 403 | [0.125 0 0 0.125 138 118]ST 404 | 3039 538 3134 550 DL 405 | [0.125 0 0 0.125 138 118]ST 406 | 2933 530 3039 538 DL 407 | [0.125 0 0 0.125 138 118]ST 408 | 2813 528 2933 530 DL 409 | [0.125 0 0 0.125 138 118]ST 410 | 2748 531 2813 528 DL 411 | [0.125 0 0 0.125 138 118]ST 412 | 2679 537 2748 531 DL 413 | [0.125 0 0 0.125 138 118]ST 414 | 2607 546 2679 537 DL 415 | [0.125 0 0 0.125 138 118]ST 416 | 2530 560 2607 546 DL 417 | [0.125 0 0 0.125 138 118]ST 418 | 2450 578 2530 560 DL 419 | [0.125 0 0 0.125 138 118]ST 420 | 2366 603 2450 578 DL 421 | [0.125 0 0 0.125 138 118]ST 422 | 2322 618 2366 603 DL 423 | [0.125 0 0 0.125 138 118]ST 424 | 2278 635 2322 618 DL 425 | [0.125 0 0 0.125 138 118]ST 426 | 2233 654 2278 635 DL 427 | [0.125 0 0 0.125 138 118]ST 428 | 2187 675 2233 654 DL 429 | [0.125 0 0 0.125 138 118]ST 430 | 2141 698 2187 675 DL 431 | [0.125 0 0 0.125 138 118]ST 432 | 2094 724 2141 698 DL 433 | [0.125 0 0 0.125 138 118]ST 434 | 2047 753 2094 724 DL 435 | [0.125 0 0 0.125 138 118]ST 436 | 2000 785 2047 753 DL 437 | [0.125 0 0 0.125 138 118]ST 438 | 1953 820 2000 785 DL 439 | [0.125 0 0 0.125 138 118]ST 440 | 1906 858 1953 820 DL 441 | [0.125 0 0 0.125 138 118]ST 442 | 1860 900 1906 858 DL 443 | [0.125 0 0 0.125 138 118]ST 444 | 1815 946 1860 900 DL 445 | [0.125 0 0 0.125 138 118]ST 446 | 1770 996 1815 946 DL 447 | [0.125 0 0 0.125 138 118]ST 448 | 1728 1049 1770 996 DL 449 | [0.125 0 0 0.125 138 118]ST 450 | 1687 1108 1728 1049 DL 451 | [0.125 0 0 0.125 138 118]ST 452 | 1649 1170 1687 1108 DL 453 | [0.125 0 0 0.125 138 118]ST 454 | 1631 1203 1649 1170 DL 455 | [0.125 0 0 0.125 138 118]ST 456 | 1614 1237 1631 1203 DL 457 | [0.125 0 0 0.125 138 118]ST 458 | 1582 1308 1614 1237 DL 459 | [0.125 0 0 0.125 138 118]ST 460 | 1567 1346 1582 1308 DL 461 | [0.125 0 0 0.125 138 118]ST 462 | 1554 1384 1567 1346 DL 463 | [0.125 0 0 0.125 138 118]ST 464 | 1541 1424 1554 1384 DL 465 | [0.125 0 0 0.125 138 118]ST 466 | 1530 1464 1541 1424 DL 467 | [0.125 0 0 0.125 138 118]ST 468 | 3501 2837 2728 1160 DL 469 | [0.125 0 0 0.125 138 118]ST 470 | 3420 1115 2768 2792 DL 471 | [0.125 0 0 0.125 138 118]ST 472 | 3871 1860 2226 2058 DL 473 | [0.125 0 0 0.125 138 118]ST 474 | 40003 -1253 -39830 3916 DL 475 | [0.125 0 0 0.125 138 118]ST 476 | 39767 5044 -40085 184 DL 477 | [0.125 0 0 0.125 138 118]ST 478 | 24219 31840 -23305 -32515 DL 479 | [0.125 0 0 0.125 138 118]ST 480 | 22074 -33487 -16928 36361 DL 481 | [0.125 0 0 0.125 138 118]ST 482 | 18153 -35899 -10150 38927 DL 483 | [0.125 0 0 0.125 138 118]ST 484 | -18689 -35443 22710 33012 DL 485 | [0.125 0 0 0.125 138 118]ST 486 | 1508 1575 1524 1488 DL 487 | [0.125 0 0 0.125 138 118]ST 488 | 1501 1620 1508 1575 DL 489 | [0.125 0 0 0.125 138 118]ST 490 | 1496 1667 1501 1620 DL 491 | [0.125 0 0 0.125 138 118]ST 492 | 1493 1714 1496 1667 DL 493 | [0.125 0 0 0.125 138 118]ST 494 | 1492 1763 1493 1714 DL 495 | [0.125 0 0 0.125 138 118]ST 496 | 1492 1814 1763 VL 497 | [0.125 0 0 0.125 138 118]ST 498 | 1495 1865 1492 1814 DL 499 | [0.125 0 0 0.125 138 118]ST 500 | 1500 1918 1495 1865 DL 501 | [0.125 0 0 0.125 138 118]ST 502 | 1506 1971 1500 1918 DL 503 | [0.125 0 0 0.125 138 118]ST 504 | 1515 2026 1506 1971 DL 505 | [0.125 0 0 0.125 138 118]ST 506 | 1527 2081 1515 2026 DL 507 | [0.125 0 0 0.125 138 118]ST 508 | 1541 2137 1527 2081 DL 509 | [0.125 0 0 0.125 138 118]ST 510 | 1557 2194 1541 2137 DL 511 | [0.125 0 0 0.125 138 118]ST 512 | 1577 2252 1557 2194 DL 513 | [0.125 0 0 0.125 138 118]ST 514 | 1598 2309 1577 2252 DL 515 | [0.125 0 0 0.125 138 118]ST 516 | 1623 2368 1598 2309 DL 517 | [0.125 0 0 0.125 138 118]ST 518 | 1650 2426 1623 2368 DL 519 | [0.125 0 0 0.125 138 118]ST 520 | 1680 2484 1650 2426 DL 521 | [0.125 0 0 0.125 138 118]ST 522 | 1714 2542 1680 2484 DL 523 | [0.125 0 0 0.125 138 118]ST 524 | 1749 2599 1714 2542 DL 525 | [0.125 0 0 0.125 138 118]ST 526 | 1788 2656 1749 2599 DL 527 | [0.125 0 0 0.125 138 118]ST 528 | 1829 2712 1788 2656 DL 529 | [0.125 0 0 0.125 138 118]ST 530 | 1874 2767 1829 2712 DL 531 | [0.125 0 0 0.125 138 118]ST 532 | 1920 2822 1874 2767 DL 533 | [0.125 0 0 0.125 138 118]ST 534 | 1970 2874 1920 2822 DL 535 | [0.125 0 0 0.125 138 118]ST 536 | 2021 2926 1970 2874 DL 537 | [0.125 0 0 0.125 138 118]ST 538 | 2076 2976 2021 2926 DL 539 | [0.125 0 0 0.125 138 118]ST 540 | 2132 3023 2076 2976 DL 541 | [0.125 0 0 0.125 138 118]ST 542 | 2190 3069 2132 3023 DL 543 | [0.125 0 0 0.125 138 118]ST 544 | 2250 3113 2190 3069 DL 545 | [0.125 0 0 0.125 138 118]ST 546 | 2312 3155 2250 3113 DL 547 | [0.125 0 0 0.125 138 118]ST 548 | 2376 3194 2312 3155 DL 549 | [0.125 0 0 0.125 138 118]ST 550 | 2440 3231 2376 3194 DL 551 | [0.125 0 0 0.125 138 118]ST 552 | 2506 3265 2440 3231 DL 553 | [0.125 0 0 0.125 138 118]ST 554 | 2573 3297 2506 3265 DL 555 | [0.125 0 0 0.125 138 118]ST 556 | 2640 3325 2573 3297 DL 557 | [0.125 0 0 0.125 138 118]ST 558 | 2708 3352 2640 3325 DL 559 | [0.125 0 0 0.125 138 118]ST 560 | 2776 3375 2708 3352 DL 561 | [0.125 0 0 0.125 138 118]ST 562 | 2844 3396 2776 3375 DL 563 | [0.125 0 0 0.125 138 118]ST 564 | 2912 3413 2844 3396 DL 565 | [0.125 0 0 0.125 138 118]ST 566 | 2980 3429 2912 3413 DL 567 | [0.125 0 0 0.125 138 118]ST 568 | 3047 3441 2980 3429 DL 569 | [0.125 0 0 0.125 138 118]ST 570 | 3114 3451 3047 3441 DL 571 | [0.125 0 0 0.125 138 118]ST 572 | 3180 3458 3114 3451 DL 573 | [0.125 0 0 0.125 138 118]ST 574 | 3244 3463 3180 3458 DL 575 | [0.125 0 0 0.125 138 118]ST 576 | 3308 3465 3244 3463 DL 577 | [0.125 0 0 0.125 138 118]ST 578 | 3370 3465 3308 HL 579 | [0.125 0 0 0.125 138 118]ST 580 | 3432 3463 3370 3465 DL 581 | [0.125 0 0 0.125 138 118]ST 582 | 3491 3458 3432 3463 DL 583 | [0.125 0 0 0.125 138 118]ST 584 | 3549 3452 3491 3458 DL 585 | [0.125 0 0 0.125 138 118]ST 586 | 3606 3444 3549 3452 DL 587 | [0.125 0 0 0.125 138 118]ST 588 | 3661 3434 3606 3444 DL 589 | [0.125 0 0 0.125 138 118]ST 590 | 3714 3422 3661 3434 DL 591 | [0.125 0 0 0.125 138 118]ST 592 | 3766 3409 3714 3422 DL 593 | [0.125 0 0 0.125 138 118]ST 594 | 3815 3394 3766 3409 DL 595 | [0.125 0 0 0.125 138 118]ST 596 | 3863 3378 3815 3394 DL 597 | [0.125 0 0 0.125 138 118]ST 598 | 3910 3361 3863 3378 DL 599 | [0.125 0 0 0.125 138 118]ST 600 | 3997 3323 3910 3361 DL 601 | [0.125 0 0 0.125 138 118]ST 602 | 4077 3281 3997 3323 DL 603 | [0.125 0 0 0.125 138 118]ST 604 | 4150 3237 4077 3281 DL 605 | [0.125 0 0 0.125 138 118]ST 606 | 4217 3190 4150 3237 DL 607 | [0.125 0 0 0.125 138 118]ST 608 | 4278 3142 4217 3190 DL 609 | [0.125 0 0 0.125 138 118]ST 610 | 4333 3093 4278 3142 DL 611 | [0.125 0 0 0.125 138 118]ST 612 | 4382 3043 4333 3093 DL 613 | [0.125 0 0 0.125 138 118]ST 614 | 4426 2993 4382 3043 DL 615 | [0.125 0 0 0.125 138 118]ST 616 | 4466 2943 4426 2993 DL 617 | [0.125 0 0 0.125 138 118]ST 618 | 4501 2893 4466 2943 DL 619 | [0.125 0 0 0.125 138 118]ST 620 | 4532 2843 4501 2893 DL 621 | [0.125 0 0 0.125 138 118]ST 622 | 4560 2795 4532 2843 DL 623 | [0.125 0 0 0.125 138 118]ST 624 | 4584 2747 4560 2795 DL 625 | [0.125 0 0 0.125 138 118]ST 626 | 4605 2701 4584 2747 DL 627 | [0.125 0 0 0.125 138 118]ST 628 | 4623 2655 4605 2701 DL 629 | [0.125 0 0 0.125 138 118]ST 630 | 4639 2611 4623 2655 DL 631 | [0.125 0 0 0.125 138 118]ST 632 | 4652 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https://raw.githubusercontent.com/notzeb/all/97969592a92b61f13beb5e90dc945e29e8cd5ebd/cross.pdf -------------------------------------------------------------------------------- /cross.tex: -------------------------------------------------------------------------------- 1 | \documentclass[letterpaper,11pt]{article} 2 | \usepackage{amsfonts,amssymb,amsmath,amsthm,latexsym} 3 | \usepackage[all]{xy} 4 | \usepackage{fullpage} 5 | \usepackage{graphicx} 6 | \usepackage{epstopdf} 7 | \usepackage{epsfig} 8 | \usepackage{hyperref} 9 | \usepackage{verbatim} 10 | 11 | \hypersetup{ 12 | colorlinks=true, 13 | linkcolor=blue, 14 | filecolor=magenta, 15 | urlcolor=cyan, 16 | citecolor=blue, 17 | } 18 | 19 | \newtheorem{thm}{Theorem} 20 | \newtheorem{lem}{Lemma} 21 | \newtheorem{cor}{Corollary} 22 | \newtheorem{prop}{Proposition} 23 | 24 | \theoremstyle{definition} 25 | \newtheorem{defn}{Definition} 26 | 27 | \theoremstyle{remark} 28 | \newtheorem{ex}{Example} 29 | \newtheorem{rem}{Remark} 30 | \newtheorem{exer}{Exercise} 31 | 32 | \begin{document} 33 | 34 | \newcommand{\CC}{\mathbb{C}} 35 | \newcommand{\RR}{\mathbb{R}} 36 | \newcommand{\ZZ}{\mathbb{Z}} 37 | \newcommand{\PP}{\mathbb{P}} 38 | \newcommand{\fish}{\reflectbox{$\alpha$}} 39 | \newcommand{\Bl}{\text{\rm Bl}} 40 | \newcommand{\PGL}{\text{\rm PGL}} 41 | 42 | \title{Cross Ratios} 43 | \date{} 44 | \maketitle 45 | 46 | \tableofcontents 47 | 48 | \input{cross-notes.tex} 49 | 50 | \bibliographystyle{plain} 51 | \bibliography{all} 52 | 53 | \end{document} 54 | 55 | -------------------------------------------------------------------------------- /csp-notes.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/notzeb/all/97969592a92b61f13beb5e90dc945e29e8cd5ebd/csp-notes.pdf -------------------------------------------------------------------------------- /csp-notes.tex: -------------------------------------------------------------------------------- 1 | \input{preamble.tex} 2 | 3 | \title{Notes on CSPs and Polymorphisms} 4 | \date{} 5 | \author{} 6 | \maketitle 7 | 8 | \tableofcontents 9 | 10 | \input{csp.tex} 11 | 12 | \bibliographystyle{plain} 13 | \bibliography{all} 14 | 15 | \input{csp-append.tex} 16 | 17 | \end{document} 18 | 19 | -------------------------------------------------------------------------------- /desargues.eps: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/notzeb/all/97969592a92b61f13beb5e90dc945e29e8cd5ebd/desargues.eps -------------------------------------------------------------------------------- /equal.eps: -------------------------------------------------------------------------------- 1 | %!PS-Adobe-2.0 EPSF-2.0 2 | %%BoundingBox: 124 127 471 716 3 | %%HiResBoundingBox: 124.500000 127.500000 470.500000 715.500000 4 | %%Creator: Qt 3.3.8b 5 | %%CreationDate: Wed Jun 10 13:39:34 2015 6 | %%DocumentFonts: LiberationSans 7 | 8 | %%EndComments 9 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248 51 295 51 347 _c 249 | 51 401 58 449 73 493 _c 250 | 87 536 109 573 138 603 _c 251 | 166 633 201 657 243 673 _c 252 | 285 689 332 698 386 698 _c 253 | 459 698 520 683 569 655 _c 254 | 618 626 654 584 678 528 _c 255 | 589 499 _l 256 | 583 515 574 530 563 545 _c 257 | 551 559 537 572 521 584 _c 258 | 505 596 485 605 463 612 _c 259 | 441 618 415 622 387 622 _c 260 | _cl}_e}_d 261 | /D{{722 0 82 0 674 688 _sc 262 | 674 351 _m 263 | 674 293 665 243 648 199 _c 264 | 631 155 608 119 578 89 _c 265 | 548 59 512 37 471 22 _c 266 | 430 7 386 0 339 0 _c 267 | 82 0 _l 268 | 82 688 _l 269 | 310 688 _l 270 | 362 688 411 681 456 668 _c 271 | 500 654 539 634 571 607 _c 272 | 603 579 629 544 647 502 _c 273 | 665 460 674 409 674 351 _c 274 | 581 351 _m 275 | 581 397 574 437 561 470 _c 276 | 547 503 528 530 504 551 _c 277 | 480 572 451 588 418 598 _c 278 | 384 608 348 613 308 613 _c 279 | 175 613 _l 280 | 175 75 _l 281 | 329 75 _l 282 | 365 75 398 80 429 92 _c 283 | 459 104 486 121 508 144 _c 284 | 530 167 548 196 561 231 _c 285 | 574 265 581 305 581 351 _c 286 | _cl}_e}_d 287 | /E{667 0 82 0 624 688 _sc 288 | 82 0 _m 289 | 82 688 _l 290 | 604 688 _l 291 | 604 612 _l 292 | 175 612 _l 293 | 175 391 _l 294 | 575 391 _l 295 | 575 316 _l 296 | 175 316 _l 297 | 175 76 _l 298 | 624 76 _l 299 | 624 0 _l 300 | 82 0 _l 301 | _cl}_d 302 | /F{611 0 82 0 571 688 _sc 303 | 175 612 _m 304 | 175 356 _l 305 | 559 356 _l 306 | 559 279 _l 307 | 175 279 _l 308 | 175 0 _l 309 | 82 0 _l 310 | 82 688 _l 311 | 571 688 _l 312 | 571 612 _l 313 | 175 612 _l 314 | _cl}_d 315 | /G{{778 0 50 -9 703 698 _sc 316 | 50 347 _m 317 | 50 401 57 449 72 493 _c 318 | 86 536 108 573 137 603 _c 319 | 165 633 201 657 244 673 _c 320 | 286 689 336 698 393 698 _c 321 | 435 698 472 694 504 687 _c 322 | 536 679 563 669 587 655 _c 323 | 610 641 630 624 646 604 _c 324 | 662 584 676 561 688 536 _c 325 | 599 510 _l 326 | 590 526 580 542 568 556 _c 327 | 556 570 542 581 525 591 _c 328 | 508 601 488 608 466 614 _c 329 | 444 619 418 622 390 622 _c 330 | 348 622 312 615 281 602 _c 331 | 250 589 225 571 205 547 _c 332 | 185 523 170 494 160 460 _c 333 | 150 426 145 388 145 347 _c 334 | 145 305 150 267 161 233 _c 335 | 171 199 187 169 209 144 _c 336 | 230 119 256 100 288 86 _c 337 | 319 72 355 66 397 66 _c 338 | 424 66 449 68 472 73 _c 339 | 494 77 515 83 533 91 _c 340 | 551 98 568 106 582 115 _c 341 | 596 123 607 132 617 142 _c 342 | 617 266 _l 343 | 412 266 _l 344 | 412 344 _l 345 | 703 344 _l 346 | 703 107 _l 347 | 687 91 669 76 649 62 _c 348 | 628 48 605 35 579 25 _c 349 | 553 14 525 6 495 0 _c 350 | 464 -6 431 -9 397 -9 _c 351 | 339 -9 288 0 245 17 _c 352 | 201 34 165 59 137 91 _c 353 | 108 123 86 160 72 204 _c 354 | 57 248 50 295 50 347 _c 355 | _cl}_e}_d 356 | /H{722 0 82 0 641 688 _sc 357 | 547 0 _m 358 | 547 319 _l 359 | 175 319 _l 360 | 175 0 _l 361 | 82 0 _l 362 | 82 688 _l 363 | 175 688 _l 364 | 175 397 _l 365 | 547 397 _l 366 | 547 688 _l 367 | 641 688 _l 368 | 641 0 _l 369 | 547 0 _l 370 | _cl}_d 371 | /P{667 0 82 0 614 688 _sc 372 | 614 481 _m 373 | 614 451 609 423 599 397 _c 374 | 589 371 574 348 554 329 _c 375 | 534 310 510 295 481 284 _c 376 | 451 273 417 268 377 268 _c 377 | 175 268 _l 378 | 175 0 _l 379 | 82 0 _l 380 | 82 688 _l 381 | 372 688 _l 382 | 412 688 448 683 478 673 _c 383 | 508 663 533 649 553 631 _c 384 | 573 613 589 591 599 565 _c 385 | 609 539 614 511 614 481 _c 386 | 521 480 _m 387 | 521 523 507 556 480 579 _c 388 | 453 601 413 613 360 613 _c 389 | 175 613 _l 390 | 175 342 _l 391 | 364 342 _l 392 | 418 342 457 353 483 377 _c 393 | 508 401 521 435 521 480 _c 394 | _cl}_d 395 | /Q{{778 0 47 -188 730 698 _sc 396 | 730 347 _m 397 | 730 298 723 254 711 214 _c 398 | 698 174 680 138 657 108 _c 399 | 633 78 604 53 570 34 _c 400 | 536 15 497 3 453 -2 _c 401 | 459 -22 467 -40 476 -55 _c 402 | 484 -69 494 -81 505 -91 _c 403 | 516 -100 528 -107 542 -111 _c 404 | 556 -115 571 -118 588 -118 _c 405 | 597 -118 607 -117 617 -116 _c 406 | 627 -114 636 -113 644 -112 _c 407 | 644 -177 _l 408 | 631 -180 617 -183 603 -185 _c 409 | 588 -187 573 -188 557 -188 _c 410 | 529 -188 504 -183 484 -175 _c 411 | 463 -167 445 -155 429 -140 _c 412 | 413 -124 399 -105 388 -83 _c 413 | 376 -60 366 -35 358 -7 _c 414 | 306 -4 261 6 222 24 _c 415 | 183 42 151 67 125 99 _c 416 | 99 130 79 167 66 209 _c 417 | 53 251 47 297 47 347 _c 418 | 47 401 54 449 69 493 _c 419 | 84 536 106 573 135 603 _c 420 | 164 633 200 657 243 673 _c 421 | 285 689 334 698 389 698 _c 422 | 443 698 492 689 535 673 _c 423 | 577 656 613 632 642 602 _c 424 | 670 572 692 535 707 492 _c 425 | 722 448 730 400 730 347 _c 426 | 635 347 _m 427 | 635 388 629 426 619 460 _c 428 | 608 494 592 523 572 547 _c 429 | 551 571 525 589 495 602 _c 430 | 464 615 429 622 389 622 _c 431 | 348 622 312 615 281 602 _c 432 | 250 589 224 571 204 547 _c 433 | 183 523 167 494 157 460 _c 434 | 147 426 142 388 142 347 _c 435 | 142 305 147 267 158 233 _c 436 | 168 199 184 169 205 144 _c 437 | 225 119 251 100 282 86 _c 438 | 312 72 348 66 388 66 _c 439 | }_e{431 66 468 73 499 87 _c 440 | 530 101 556 120 576 145 _c 441 | 596 170 610 200 620 234 _c 442 | 630 268 635 306 635 347 _c 443 | _cl}_e}_d 444 | /X{667 0 22 0 646 688 _sc 445 | 543 0 _m 446 | 336 301 _l 447 | 125 0 _l 448 | 22 0 _l 449 | 284 357 _l 450 | 42 688 _l 451 | 146 688 _l 452 | 337 418 _l 453 | 523 688 _l 454 | 626 688 _l 455 | 391 361 _l 456 | 646 0 _l 457 | 543 0 _l 458 | _cl}_d 459 | /Y{667 0 22 0 645 688 _sc 460 | 379 285 _m 461 | 379 0 _l 462 | 287 0 _l 463 | 287 285 _l 464 | 22 688 _l 465 | 125 688 _l 466 | 334 360 _l 467 | 542 688 _l 468 | 645 688 _l 469 | 379 285 _l 470 | _cl}_d 471 | /Z{611 0 32 0 580 688 _sc 472 | 580 0 _m 473 | 32 0 _l 474 | 32 70 _l 475 | 451 612 _l 476 | 67 612 _l 477 | 67 688 _l 478 | 557 688 _l 479 | 557 620 _l 480 | 138 76 _l 481 | 580 76 _l 482 | 580 0 _l 483 | _cl}_d 484 | end readonly def 485 | /BuildGlyph 486 | {exch begin 487 | CharStrings exch 488 | 2 copy known not{pop /.notdef}if 489 | true 3 1 roll get exec 490 | end}_d 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/setpagedevice {pop} def 17 | %%EndProlog 18 | %%Page 1 1 19 | %%BeginProlog 20 | % Prolog copyright 1994-2006 Trolltech. You may copy this prolog in any way 21 | % that is directly related to this document. For other use of this prolog, 22 | % see your licensing agreement for Qt. 23 | /d/def load def/D{bind d}bind d/d2{dup dup}D/B{0 d2}D/W{255 d2}D/ED{exch d}D 24 | /D0{0 ED}D/LT{lineto}D/MT{moveto}D/S{stroke}D/F{setfont}D/SW{setlinewidth}D 25 | /CP{closepath}D/RL{rlineto}D/NP{newpath}D/CM{currentmatrix}D/SM{setmatrix}D 26 | /TR{translate}D/SD{setdash}D/SC{aload pop setrgbcolor}D/CR{currentfile read 27 | pop}D/i{index}D/bs{bitshift}D/scs{setcolorspace}D/DB{dict dup begin}D/DE{end 28 | d}D/ie{ifelse}D/sp{astore pop}D/BSt 0 d/LWi 1 d/PSt 1 d/Cx 0 d/Cy 0 d/WFi 29 | false d/OMo false d/BCol[1 1 1]d/PCol[0 0 0]d/BkCol[1 1 1]d/BDArr[0.94 0.88 30 | 0.63 0.50 0.37 0.12 0.06]d/defM matrix d/nS 0 d/GPS{PSt 1 ge PSt 5 le and{{ 31 | LArr PSt 1 sub 2 mul get}{LArr PSt 2 mul 1 sub get}ie}{[]}ie}D/QS{PSt 0 ne{ 32 | gsave LWi SW true GPS 0 SD S OMo PSt 1 ne and{BkCol SC false GPS dup 0 get 33 | SD S}if grestore}if}D/r28{{CR dup 32 gt{exit}if pop}loop 3{CR}repeat 0 4{7 34 | bs exch dup 128 gt{84 sub}if 42 sub 127 and add}repeat}D/rA 0 d/rL 0 d/rB{rL 35 | 0 eq{/rA r28 d/rL 28 d}if dup rL gt{rA exch rL sub rL exch/rA 0 d/rL 0 d rB 36 | exch bs add}{dup rA 16#fffffff 3 -1 roll bs not and exch dup rL exch sub/rL 37 | ED neg rA exch bs/rA ED}ie}D/uc{/rL 0 d 0{dup 2 i length ge{exit}if 1 rB 1 38 | eq{3 rB dup 3 ge{1 add dup rB 1 i 5 ge{1 i 6 ge{1 i 7 ge{1 i 8 ge{128 add}if 39 | 64 add}if 32 add}if 16 add}if 3 add exch pop}if 3 add exch 10 rB 1 add{dup 3 40 | i lt{dup}{2 i}ie 4 i 3 i 3 i sub 2 i getinterval 5 i 4 i 3 -1 roll 41 | putinterval dup 4 -1 roll add 3 1 roll 4 -1 roll exch sub dup 0 eq{exit}if 3 42 | 1 roll}loop pop pop}{3 rB 1 add{2 copy 8 rB put 1 add}repeat}ie}loop pop}D 43 | /sl D0/QCIgray D0/QCIcolor D0/QCIindex D0/QCI{/colorimage where{pop false 3 44 | colorimage}{exec/QCIcolor ED/QCIgray QCIcolor length 3 idiv string d 0 1 45 | QCIcolor length 3 idiv 1 sub{/QCIindex ED/x QCIindex 3 mul d QCIgray 46 | QCIindex QCIcolor x get 0.30 mul QCIcolor x 1 add get 0.59 mul QCIcolor x 2 47 | add get 0.11 mul add add cvi put}for QCIgray image}ie}D/di{gsave TR 1 i 1 eq 48 | {false eq{pop true 3 1 roll 4 i 4 i false 4 i 4 i imagemask BkCol SC 49 | imagemask}{pop false 3 1 roll imagemask}ie}{dup false ne{/languagelevel 50 | where{pop languagelevel 3 ge}{false}ie}{false}ie{/ma ED 8 eq{/dc[0 1]d 51 | /DeviceGray}{/dc[0 1 0 1 0 1]d/DeviceRGB}ie scs/im ED/mt ED/h ED/w ED/id 7 52 | DB/ImageType 1 d/Width w d/Height h d/ImageMatrix mt d/DataSource im d 53 | /BitsPerComponent 8 d/Decode dc d DE/md 7 DB/ImageType 1 d/Width w d/Height 54 | h d/ImageMatrix mt d/DataSource ma d/BitsPerComponent 1 d/Decode[0 1]d DE 4 55 | DB/ImageType 3 d/DataDict id d/MaskDict md d/InterleaveType 3 d end image}{ 56 | pop 8 4 1 roll 8 eq{image}{QCI}ie}ie}ie grestore}d/BF{gsave BSt 1 eq{BCol SC 57 | WFi{fill}{eofill}ie}if BSt 2 ge BSt 8 le and{BDArr BSt 2 sub get/sc ED BCol{ 58 | 1. exch sub sc mul 1. exch sub}forall 3 array astore SC WFi{fill}{eofill}ie} 59 | if BSt 9 ge BSt 14 le and{WFi{clip}{eoclip}ie defM SM pathbbox 3 i 3 i TR 4 60 | 2 roll 3 2 roll exch sub/h ED sub/w ED OMo{NP 0 0 MT 0 h RL w 0 RL 0 h neg 61 | RL CP BkCol SC fill}if BCol SC 0.3 SW NP BSt 9 eq BSt 11 eq or{0 4 h{dup 0 62 | exch MT w exch LT}for}if BSt 10 eq BSt 11 eq or{0 4 w{dup 0 MT h LT}for}if 63 | BSt 12 eq BSt 14 eq or{w h gt{0 6 w h add{dup 0 MT h sub h LT}for}{0 6 w h 64 | add{dup 0 exch MT w sub w exch LT}for}ie}if BSt 13 eq BSt 14 eq or{w h gt{0 65 | 6 w h add{dup h MT h sub 0 LT}for}{0 6 w h add{dup w exch MT w sub 0 exch LT 66 | }for}ie}if S}if BSt 24 eq{}if grestore}D/mat matrix d/ang1 D0/ang2 D0/w D0/h 67 | D0/x D0/y D0/ARC{/ang2 ED/ang1 ED/h ED/w ED/y ED/x ED mat CM pop x w 2 div 68 | add y h 2 div add TR 1 h w div neg scale ang2 0 ge{0 0 w 2 div ang1 ang1 69 | ang2 add arc}{0 0 w 2 div ang1 ang1 ang2 add arcn}ie mat SM}D/C D0/P{NP MT 70 | 0.5 0.5 rmoveto 0 -1 RL -1 0 RL 0 1 RL CP fill}D/M{/Cy ED/Cx ED}D/L{NP Cx Cy 71 | MT/Cy ED/Cx ED Cx Cy LT QS}D/DL{NP MT LT QS}D/HL{1 i DL}D/VL{2 i exch DL}D/R 72 | {/h ED/w ED/y ED/x ED NP x y MT 0 h RL w 0 RL 0 h neg RL CP BF QS}D/ACR{/h 73 | ED/w ED/y ED/x ED x y MT 0 h RL w 0 RL 0 h neg RL CP}D/xr D0/yr D0/rx D0/ry 74 | D0/rx2 D0/ry2 D0/RR{/yr ED/xr ED/h ED/w ED/y ED/x ED xr 0 le yr 0 le or{x y 75 | w h R}{xr 100 ge yr 100 ge or{x y w h E}{/rx xr w mul 200 div d/ry yr h mul 76 | 200 div d/rx2 rx 2 mul d/ry2 ry 2 mul d NP x rx add y MT x y rx2 ry2 180 -90 77 | x y h add ry2 sub rx2 ry2 270 -90 x w add rx2 sub y h add ry2 sub rx2 ry2 0 78 | -90 x w add rx2 sub y rx2 ry2 90 -90 ARC ARC ARC ARC CP BF QS}ie}ie}D/E{/h 79 | ED/w ED/y ED/x ED mat CM pop x w 2 div add y h 2 div add TR 1 h w div scale 80 | NP 0 0 w 2 div 0 360 arc mat SM BF QS}D/A{16 div exch 16 div exch NP ARC QS} 81 | D/PIE{/ang2 ED/ang1 ED/h ED/w ED/y ED/x ED NP x w 2 div add y h 2 div add MT 82 | x y w h ang1 16 div ang2 16 div ARC CP BF QS}D/CH{16 div exch 16 div exch NP 83 | ARC CP BF QS}D/BZ{curveto QS}D/CRGB{255 div 3 1 roll 255 div 3 1 roll 255 84 | div 3 1 roll}D/BC{CRGB BkCol sp}D/BR{CRGB BCol sp/BSt ED}D/WB{1 W BR}D/NB{0 85 | B BR}D/PE{setlinejoin setlinecap CRGB PCol sp/LWi ED/PSt ED LWi 0 eq{0.25 86 | /LWi ED}if PCol SC}D/P1{1 0 5 2 roll 0 0 PE}D/ST{defM SM concat}D/MF{true 87 | exch true exch{exch pop exch pop dup 0 get dup findfont dup/FontName get 3 88 | -1 roll eq{exit}if}forall exch dup 1 get/fxscale ED 2 get/fslant ED exch 89 | /fencoding ED[fxscale 0 fslant 1 0 0]makefont fencoding false eq{}{dup 90 | maxlength dict begin{1 i/FID ne{def}{pop pop}ifelse}forall/Encoding 91 | fencoding d currentdict end}ie definefont pop}D/MFEmb{findfont dup length 92 | dict begin{1 i/FID ne{d}{pop pop}ifelse}forall/Encoding ED currentdict end 93 | definefont pop}D/DF{findfont/fs 3 -1 roll d[fs 0 0 fs -1 mul 0 0]makefont d} 94 | D/ty 0 d/Y{/ty ED}D/Tl{gsave SW NP 1 i exch MT 1 i 0 RL S grestore}D/XYT{ty 95 | MT/xyshow where{pop pop xyshow}{exch pop 1 i dup length 2 div exch 96 | stringwidth pop 3 -1 roll exch sub exch div exch 0 exch ashow}ie}D/AT{ty MT 97 | 1 i dup length 2 div exch stringwidth pop 3 -1 roll exch sub exch div exch 0 98 | exch ashow}D/QI{/C save d pageinit/Cx 0 d/Cy 0 d/OMo false d}D/QP{C restore 99 | showpage}D/SPD{/setpagedevice where{1 DB 3 1 roll d end setpagedevice}{pop 100 | pop}ie}D/SV{BSt LWi PSt Cx Cy WFi OMo BCol PCol BkCol/nS nS 1 add d gsave}D 101 | /RS{nS 0 gt{grestore/BkCol ED/PCol ED/BCol ED/OMo ED/WFi ED/Cy ED/Cx ED/PSt 102 | ED/LWi ED/BSt ED/nS nS 1 sub d}if}D/CLSTART{/clipTmp matrix CM d defM SM NP} 103 | D/CLEND{clip NP clipTmp SM}D/CLO{grestore gsave defM SM}D 104 | /LArr[ [] [] [ 13.333 4.000 ] [ 4.000 13.333 ] [ 4.000 4.000 ] [ 4.000 4.000 ] [ 6.667 4.000 4.000 4.000 ] [ 4.000 6.667 4.000 4.000 ] [ 6.667 4.000 4.000 4.000 4.000 ] [ 4.000 6.667 4.000 4.000 4.000 4.000 ] ] d 105 | /pageinit { 106 | 36 24 translate 107 | % 184*280 mm (landscape) 108 | 90 rotate 0.75 -0.75 scale/defM matrix CM d } d 109 | %%EndProlog 110 | %%BeginSetup 111 | %%EndSetup 112 | %%Page: 1 1 113 | %%BeginPageSetup 114 | QI 115 | %%EndPageSetup 116 | [0.125 0 0 0.125 138 118]ST 117 | CLSTART 118 | 138 118 784 461 ACR 119 | CLEND 120 | 0 0 B 0 0 PE 121 | 1 B BR 122 | W BC 123 | 1912 800 64 64 E 124 | [0.125 0 0 0.125 138 118]ST 125 | 3296 336 64 64 E 126 | [0.125 0 0 0.125 138 118]ST 127 | 1720 2960 64 64 E 128 | [0.125 0 0 0.125 138 118]ST 129 | 4224 2816 64 64 E 130 | [0.125 0 0 0.125 138 118]ST 131 | 4616 504 64 64 E 132 | [0.125 0 0 0.125 138 118]ST 133 | 1 2 B 0 0 PE 134 | 2156 3130 2048 3102 DL 135 | [0.125 0 0 0.125 138 118]ST 136 | 2267 3153 2156 3130 DL 137 | [0.125 0 0 0.125 138 118]ST 138 | 2381 3171 2267 3153 DL 139 | [0.125 0 0 0.125 138 118]ST 140 | 2498 3184 2381 3171 DL 141 | [0.125 0 0 0.125 138 118]ST 142 | 2615 3193 2498 3184 DL 143 | [0.125 0 0 0.125 138 118]ST 144 | 2734 3197 2615 3193 DL 145 | [0.125 0 0 0.125 138 118]ST 146 | 2853 3196 2734 3197 DL 147 | [0.125 0 0 0.125 138 118]ST 148 | 2971 3190 2853 3196 DL 149 | [0.125 0 0 0.125 138 118]ST 150 | 3089 3181 2971 3190 DL 151 | [0.125 0 0 0.125 138 118]ST 152 | 3205 3167 3089 3181 DL 153 | [0.125 0 0 0.125 138 118]ST 154 | 3319 3150 3205 3167 DL 155 | [0.125 0 0 0.125 138 118]ST 156 | 3431 3129 3319 3150 DL 157 | [0.125 0 0 0.125 138 118]ST 158 | 3541 3105 3431 3129 DL 159 | [0.125 0 0 0.125 138 118]ST 160 | 3647 3078 3541 3105 DL 161 | [0.125 0 0 0.125 138 118]ST 162 | 3750 3048 3647 3078 DL 163 | [0.125 0 0 0.125 138 118]ST 164 | 3850 3016 3750 3048 DL 165 | [0.125 0 0 0.125 138 118]ST 166 | 4037 2946 3850 3016 DL 167 | [0.125 0 0 0.125 138 118]ST 168 | 4210 2871 4037 2946 DL 169 | [0.125 0 0 0.125 138 118]ST 170 | 4366 2791 4210 2871 DL 171 | [0.125 0 0 0.125 138 118]ST 172 | 4507 2708 4366 2791 DL 173 | [0.125 0 0 0.125 138 118]ST 174 | 4632 2625 4507 2708 DL 175 | [0.125 0 0 0.125 138 118]ST 176 | 4743 2541 4632 2625 DL 177 | [0.125 0 0 0.125 138 118]ST 178 | 4841 2459 4743 2541 DL 179 | [0.125 0 0 0.125 138 118]ST 180 | 4927 2379 4841 2459 DL 181 | [0.125 0 0 0.125 138 118]ST 182 | 5001 2301 4927 2379 DL 183 | [0.125 0 0 0.125 138 118]ST 184 | 5066 2226 5001 2301 DL 185 | [0.125 0 0 0.125 138 118]ST 186 | 5121 2154 5066 2226 DL 187 | [0.125 0 0 0.125 138 118]ST 188 | 5168 2084 5121 2154 DL 189 | [0.125 0 0 0.125 138 118]ST 190 | 5208 2018 5168 2084 DL 191 | [0.125 0 0 0.125 138 118]ST 192 | 5242 1956 5208 2018 DL 193 | [0.125 0 0 0.125 138 118]ST 194 | 5271 1896 5242 1956 DL 195 | [0.125 0 0 0.125 138 118]ST 196 | 5294 1839 5271 1896 DL 197 | [0.125 0 0 0.125 138 118]ST 198 | 5313 1785 5294 1839 DL 199 | [0.125 0 0 0.125 138 118]ST 200 | 5329 1735 5313 1785 DL 201 | [0.125 0 0 0.125 138 118]ST 202 | 5341 1686 5329 1735 DL 203 | [0.125 0 0 0.125 138 118]ST 204 | 5350 1641 5341 1686 DL 205 | [0.125 0 0 0.125 138 118]ST 206 | 5357 1597 5350 1641 DL 207 | [0.125 0 0 0.125 138 118]ST 208 | 5361 1556 5357 1597 DL 209 | [0.125 0 0 0.125 138 118]ST 210 | 5364 1518 5361 1556 DL 211 | [0.125 0 0 0.125 138 118]ST 212 | 5365 1481 5364 1518 DL 213 | [0.125 0 0 0.125 138 118]ST 214 | 5364 1446 5365 1481 DL 215 | [0.125 0 0 0.125 138 118]ST 216 | 5363 1413 5364 1446 DL 217 | [0.125 0 0 0.125 138 118]ST 218 | 5360 1382 5363 1413 DL 219 | [0.125 0 0 0.125 138 118]ST 220 | 5356 1352 5360 1382 DL 221 | [0.125 0 0 0.125 138 118]ST 222 | 5352 1324 5356 1352 DL 223 | [0.125 0 0 0.125 138 118]ST 224 | 5347 1297 5352 1324 DL 225 | [0.125 0 0 0.125 138 118]ST 226 | 5341 1271 5347 1297 DL 227 | [0.125 0 0 0.125 138 118]ST 228 | 5335 1247 5341 1271 DL 229 | [0.125 0 0 0.125 138 118]ST 230 | 5321 1202 5335 1247 DL 231 | [0.125 0 0 0.125 138 118]ST 232 | 5307 1161 5321 1202 DL 233 | [0.125 0 0 0.125 138 118]ST 234 | 5291 1123 5307 1161 DL 235 | [0.125 0 0 0.125 138 118]ST 236 | 5275 1089 5291 1123 DL 237 | [0.125 0 0 0.125 138 118]ST 238 | 5259 1057 5275 1089 DL 239 | [0.125 0 0 0.125 138 118]ST 240 | 5242 1028 5259 1057 DL 241 | [0.125 0 0 0.125 138 118]ST 242 | 5226 1002 5242 1028 DL 243 | [0.125 0 0 0.125 138 118]ST 244 | 5210 977 5226 1002 DL 245 | [0.125 0 0 0.125 138 118]ST 246 | 5178 933 5210 977 DL 247 | [0.125 0 0 0.125 138 118]ST 248 | 5163 914 5178 933 DL 249 | [0.125 0 0 0.125 138 118]ST 250 | 5148 895 5163 914 DL 251 | [0.125 0 0 0.125 138 118]ST 252 | 5119 862 5148 895 DL 253 | [0.125 0 0 0.125 138 118]ST 254 | 5091 833 5119 862 DL 255 | [0.125 0 0 0.125 138 118]ST 256 | 5065 807 5091 833 DL 257 | [0.125 0 0 0.125 138 118]ST 258 | 5040 784 5065 807 DL 259 | [0.125 0 0 0.125 138 118]ST 260 | 4994 744 5040 784 DL 261 | [0.125 0 0 0.125 138 118]ST 262 | 4953 712 4994 744 DL 263 | [0.125 0 0 0.125 138 118]ST 264 | 4879 660 4953 712 DL 265 | [0.125 0 0 0.125 138 118]ST 266 | 4811 619 4879 660 DL 267 | [0.125 0 0 0.125 138 118]ST 268 | 4774 598 4811 619 DL 269 | [0.125 0 0 0.125 138 118]ST 270 | 4751 586 4774 598 DL 271 | [0.125 0 0 0.125 138 118]ST 272 | 4735 577 4751 586 DL 273 | [0.125 0 0 0.125 138 118]ST 274 | 4648 536 4735 577 DL 275 | [0.125 0 0 0.125 138 118]ST 276 | 4556 498 4648 536 DL 277 | [0.125 0 0 0.125 138 118]ST 278 | 4538 492 4556 498 DL 279 | [0.125 0 0 0.125 138 118]ST 280 | 4511 482 4538 492 DL 281 | [0.125 0 0 0.125 138 118]ST 282 | 4467 467 4511 482 DL 283 | [0.125 0 0 0.125 138 118]ST 284 | 4379 441 4467 467 DL 285 | [0.125 0 0 0.125 138 118]ST 286 | 4274 414 4379 441 DL 287 | [0.125 0 0 0.125 138 118]ST 288 | 4130 386 4274 414 DL 289 | [0.125 0 0 0.125 138 118]ST 290 | 4034 372 4130 386 DL 291 | [0.125 0 0 0.125 138 118]ST 292 | 3914 360 4034 372 DL 293 | [0.125 0 0 0.125 138 118]ST 294 | 3760 351 3914 360 DL 295 | [0.125 0 0 0.125 138 118]ST 296 | 3554 352 3760 351 DL 297 | [0.125 0 0 0.125 138 118]ST 298 | 3424 359 3554 352 DL 299 | [0.125 0 0 0.125 138 118]ST 300 | 3270 375 3424 359 DL 301 | [0.125 0 0 0.125 138 118]ST 302 | 3087 403 3270 375 DL 303 | [0.125 0 0 0.125 138 118]ST 304 | 2867 449 3087 403 DL 305 | [0.125 0 0 0.125 138 118]ST 306 | 2741 483 2867 449 DL 307 | [0.125 0 0 0.125 138 118]ST 308 | 2603 526 2741 483 DL 309 | [0.125 0 0 0.125 138 118]ST 310 | 2453 580 2603 526 DL 311 | [0.125 0 0 0.125 138 118]ST 312 | 2290 648 2453 580 DL 313 | [0.125 0 0 0.125 138 118]ST 314 | 2204 689 2290 648 DL 315 | [0.125 0 0 0.125 138 118]ST 316 | 2115 734 2204 689 DL 317 | [0.125 0 0 0.125 138 118]ST 318 | 2023 785 2115 734 DL 319 | [0.125 0 0 0.125 138 118]ST 320 | 1929 841 2023 785 DL 321 | [0.125 0 0 0.125 138 118]ST 322 | 1834 904 1929 841 DL 323 | [0.125 0 0 0.125 138 118]ST 324 | 1737 974 1834 904 DL 325 | [0.125 0 0 0.125 138 118]ST 326 | 1641 1051 1737 974 DL 327 | [0.125 0 0 0.125 138 118]ST 328 | 1545 1137 1641 1051 DL 329 | [0.125 0 0 0.125 138 118]ST 330 | 1499 1182 1545 1137 DL 331 | [0.125 0 0 0.125 138 118]ST 332 | 1453 1230 1499 1182 DL 333 | [0.125 0 0 0.125 138 118]ST 334 | 1408 1280 1453 1230 DL 335 | [0.125 0 0 0.125 138 118]ST 336 | 1365 1333 1408 1280 DL 337 | [0.125 0 0 0.125 138 118]ST 338 | 1344 1360 1365 1333 DL 339 | [0.125 0 0 0.125 138 118]ST 340 | 1324 1387 1344 1360 DL 341 | [0.125 0 0 0.125 138 118]ST 342 | 1284 1444 1324 1387 DL 343 | [0.125 0 0 0.125 138 118]ST 344 | 1247 1503 1284 1444 DL 345 | [0.125 0 0 0.125 138 118]ST 346 | 1212 1564 1247 1503 DL 347 | [0.125 0 0 0.125 138 118]ST 348 | 1181 1627 1212 1564 DL 349 | [0.125 0 0 0.125 138 118]ST 350 | 1152 1692 1181 1627 DL 351 | [0.125 0 0 0.125 138 118]ST 352 | 1140 1725 1152 1692 DL 353 | [0.125 0 0 0.125 138 118]ST 354 | 1128 1758 1140 1725 DL 355 | [0.125 0 0 0.125 138 118]ST 356 | 1117 1792 1128 1758 DL 357 | [0.125 0 0 0.125 138 118]ST 358 | 1108 1827 1117 1792 DL 359 | [0.125 0 0 0.125 138 118]ST 360 | 1099 1862 1108 1827 DL 361 | [0.125 0 0 0.125 138 118]ST 362 | 1092 1897 1099 1862 DL 363 | [0.125 0 0 0.125 138 118]ST 364 | 1086 1932 1092 1897 DL 365 | [0.125 0 0 0.125 138 118]ST 366 | 1081 1968 1086 1932 DL 367 | [0.125 0 0 0.125 138 118]ST 368 | 1078 2004 1081 1968 DL 369 | [0.125 0 0 0.125 138 118]ST 370 | 1076 2040 1078 2004 DL 371 | [0.125 0 0 0.125 138 118]ST 372 | 1076 2076 2040 VL 373 | [0.125 0 0 0.125 138 118]ST 374 | 1077 2113 1076 2076 DL 375 | [0.125 0 0 0.125 138 118]ST 376 | 1079 2150 1077 2113 DL 377 | [0.125 0 0 0.125 138 118]ST 378 | 1083 2186 1079 2150 DL 379 | [0.125 0 0 0.125 138 118]ST 380 | 1089 2223 1083 2186 DL 381 | [0.125 0 0 0.125 138 118]ST 382 | 1096 2260 1089 2223 DL 383 | [0.125 0 0 0.125 138 118]ST 384 | 1105 2297 1096 2260 DL 385 | [0.125 0 0 0.125 138 118]ST 386 | 1116 2333 1105 2297 DL 387 | [0.125 0 0 0.125 138 118]ST 388 | 1128 2370 1116 2333 DL 389 | [0.125 0 0 0.125 138 118]ST 390 | 1142 2406 1128 2370 DL 391 | [0.125 0 0 0.125 138 118]ST 392 | 1158 2442 1142 2406 DL 393 | [0.125 0 0 0.125 138 118]ST 394 | 1176 2478 1158 2442 DL 395 | [0.125 0 0 0.125 138 118]ST 396 | 1195 2513 1176 2478 DL 397 | [0.125 0 0 0.125 138 118]ST 398 | 1216 2548 1195 2513 DL 399 | [0.125 0 0 0.125 138 118]ST 400 | 1239 2582 1216 2548 DL 401 | [0.125 0 0 0.125 138 118]ST 402 | 1264 2616 1239 2582 DL 403 | [0.125 0 0 0.125 138 118]ST 404 | 1291 2649 1264 2616 DL 405 | [0.125 0 0 0.125 138 118]ST 406 | 1319 2682 1291 2649 DL 407 | [0.125 0 0 0.125 138 118]ST 408 | 1349 2714 1319 2682 DL 409 | [0.125 0 0 0.125 138 118]ST 410 | 1381 2745 1349 2714 DL 411 | [0.125 0 0 0.125 138 118]ST 412 | 1415 2776 1381 2745 DL 413 | [0.125 0 0 0.125 138 118]ST 414 | 1450 2805 1415 2776 DL 415 | [0.125 0 0 0.125 138 118]ST 416 | 1487 2834 1450 2805 DL 417 | [0.125 0 0 0.125 138 118]ST 418 | 1526 2862 1487 2834 DL 419 | [0.125 0 0 0.125 138 118]ST 420 | 1566 2889 1526 2862 DL 421 | [0.125 0 0 0.125 138 118]ST 422 | 1608 2915 1566 2889 DL 423 | [0.125 0 0 0.125 138 118]ST 424 | 1695 2964 1608 2915 DL 425 | [0.125 0 0 0.125 138 118]ST 426 | 1788 3008 1695 2964 DL 427 | [0.125 0 0 0.125 138 118]ST 428 | 1886 3048 1788 3008 DL 429 | [0.125 0 0 0.125 138 118]ST 430 | 1988 3084 1886 3048 DL 431 | [0.125 0 0 0.125 138 118]ST 432 | 30576 25798 -29721 -26779 DL 433 | [0.125 0 0 0.125 138 118]ST 434 | -11220 -38510 16817 36416 DL 435 | [0.125 0 0 0.125 138 118]ST 436 | -17987 35857 23203 -32724 DL 437 | [0.125 0 0 0.125 138 118]ST 438 | 32716 -23267 -28298 28476 DL 439 | [0.125 0 0 0.125 138 118]ST 440 | 0 0 B 0 0 PE 441 | 3789 1655 64 64 E 442 | [0.125 0 0 0.125 138 118]ST 443 | 3073 1812 64 64 E 444 | [0.125 0 0 0.125 138 118]ST 445 | 2 2 B 0 0 PE 446 | 39595 -6194 -38532 11017 DL 447 | [0.125 0 0 0.125 138 118]ST 448 | 0 0 B 0 0 PE 449 | 2308 1981 64 64 E 450 | [0.125 0 0 0.125 138 118]ST 451 | 1 2 B 0 0 PE 452 | -11219 -38427 14213 37423 DL 453 | [0.125 0 0 0.125 138 118]ST 454 | 26656 -30102 -20017 34872 DL 455 | [0.125 0 0 0.125 138 118]ST 456 | 0 0 B 0 0 PE 457 | 2705 3165 64 64 E 458 | [0.125 0 0 0.125 138 118]ST 459 | 1 2 B 0 0 PE 460 | 2156 3130 2048 3102 DL 461 | [0.125 0 0 0.125 138 118]ST 462 | 2267 3153 2156 3130 DL 463 | [0.125 0 0 0.125 138 118]ST 464 | 2381 3171 2267 3153 DL 465 | [0.125 0 0 0.125 138 118]ST 466 | 2498 3184 2381 3171 DL 467 | [0.125 0 0 0.125 138 118]ST 468 | 2615 3193 2498 3184 DL 469 | [0.125 0 0 0.125 138 118]ST 470 | 2734 3197 2615 3193 DL 471 | [0.125 0 0 0.125 138 118]ST 472 | 2853 3196 2734 3197 DL 473 | [0.125 0 0 0.125 138 118]ST 474 | 2971 3190 2853 3196 DL 475 | [0.125 0 0 0.125 138 118]ST 476 | 3089 3181 2971 3190 DL 477 | [0.125 0 0 0.125 138 118]ST 478 | 3205 3167 3089 3181 DL 479 | [0.125 0 0 0.125 138 118]ST 480 | 3319 3150 3205 3167 DL 481 | [0.125 0 0 0.125 138 118]ST 482 | 3431 3129 3319 3150 DL 483 | [0.125 0 0 0.125 138 118]ST 484 | 3541 3105 3431 3129 DL 485 | [0.125 0 0 0.125 138 118]ST 486 | 3647 3078 3541 3105 DL 487 | [0.125 0 0 0.125 138 118]ST 488 | 3750 3048 3647 3078 DL 489 | [0.125 0 0 0.125 138 118]ST 490 | 3850 3016 3750 3048 DL 491 | [0.125 0 0 0.125 138 118]ST 492 | 4037 2946 3850 3016 DL 493 | [0.125 0 0 0.125 138 118]ST 494 | 4210 2871 4037 2946 DL 495 | [0.125 0 0 0.125 138 118]ST 496 | 4366 2791 4210 2871 DL 497 | [0.125 0 0 0.125 138 118]ST 498 | 4507 2708 4366 2791 DL 499 | [0.125 0 0 0.125 138 118]ST 500 | 4632 2625 4507 2708 DL 501 | [0.125 0 0 0.125 138 118]ST 502 | 4743 2541 4632 2625 DL 503 | [0.125 0 0 0.125 138 118]ST 504 | 4841 2459 4743 2541 DL 505 | [0.125 0 0 0.125 138 118]ST 506 | 4927 2379 4841 2459 DL 507 | [0.125 0 0 0.125 138 118]ST 508 | 5001 2301 4927 2379 DL 509 | [0.125 0 0 0.125 138 118]ST 510 | 5066 2226 5001 2301 DL 511 | [0.125 0 0 0.125 138 118]ST 512 | 5121 2154 5066 2226 DL 513 | [0.125 0 0 0.125 138 118]ST 514 | 5168 2084 5121 2154 DL 515 | [0.125 0 0 0.125 138 118]ST 516 | 5208 2018 5168 2084 DL 517 | [0.125 0 0 0.125 138 118]ST 518 | 5242 1956 5208 2018 DL 519 | [0.125 0 0 0.125 138 118]ST 520 | 5271 1896 5242 1956 DL 521 | [0.125 0 0 0.125 138 118]ST 522 | 5294 1839 5271 1896 DL 523 | [0.125 0 0 0.125 138 118]ST 524 | 5313 1785 5294 1839 DL 525 | [0.125 0 0 0.125 138 118]ST 526 | 5329 1735 5313 1785 DL 527 | [0.125 0 0 0.125 138 118]ST 528 | 5341 1686 5329 1735 DL 529 | [0.125 0 0 0.125 138 118]ST 530 | 5350 1641 5341 1686 DL 531 | [0.125 0 0 0.125 138 118]ST 532 | 5357 1597 5350 1641 DL 533 | [0.125 0 0 0.125 138 118]ST 534 | 5361 1556 5357 1597 DL 535 | [0.125 0 0 0.125 138 118]ST 536 | 5364 1518 5361 1556 DL 537 | [0.125 0 0 0.125 138 118]ST 538 | 5365 1481 5364 1518 DL 539 | [0.125 0 0 0.125 138 118]ST 540 | 5364 1446 5365 1481 DL 541 | [0.125 0 0 0.125 138 118]ST 542 | 5363 1413 5364 1446 DL 543 | [0.125 0 0 0.125 138 118]ST 544 | 5360 1382 5363 1413 DL 545 | [0.125 0 0 0.125 138 118]ST 546 | 5356 1352 5360 1382 DL 547 | [0.125 0 0 0.125 138 118]ST 548 | 5352 1324 5356 1352 DL 549 | [0.125 0 0 0.125 138 118]ST 550 | 5347 1297 5352 1324 DL 551 | [0.125 0 0 0.125 138 118]ST 552 | 5341 1271 5347 1297 DL 553 | [0.125 0 0 0.125 138 118]ST 554 | 5335 1247 5341 1271 DL 555 | [0.125 0 0 0.125 138 118]ST 556 | 5321 1202 5335 1247 DL 557 | [0.125 0 0 0.125 138 118]ST 558 | 5307 1161 5321 1202 DL 559 | [0.125 0 0 0.125 138 118]ST 560 | 5291 1123 5307 1161 DL 561 | [0.125 0 0 0.125 138 118]ST 562 | 5275 1089 5291 1123 DL 563 | [0.125 0 0 0.125 138 118]ST 564 | 5259 1057 5275 1089 DL 565 | [0.125 0 0 0.125 138 118]ST 566 | 5242 1028 5259 1057 DL 567 | 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| %%DocumentFonts: 796 | %%Trailer 797 | cleartomark 798 | countdictstack 799 | exch sub { end } repeat 800 | restore 801 | %%EOF 802 | -------------------------------------------------------------------------------- /polar.eps: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/notzeb/all/97969592a92b61f13beb5e90dc945e29e8cd5ebd/polar.eps -------------------------------------------------------------------------------- /preamble.tex: -------------------------------------------------------------------------------- 1 | \documentclass[letterpaper,11pt]{report} 2 | \usepackage{amsfonts,amssymb,amsmath,amsthm,amsopn,latexsym} 3 | \usepackage{stmaryrd} 4 | \usepackage[all]{xy} 5 | \usepackage{fullpage} 6 | \usepackage{graphicx} 7 | \usepackage{epstopdf} 8 | \usepackage{epsfig} 9 | \usepackage{hyperref} 10 | \usepackage{enumitem,todonotes} 11 | \usepackage{verbatim} 12 | \usepackage{mathdots} 13 | 14 | \usepackage{algorithm} 15 | \usepackage[noend]{algpseudocode} 16 | \usepackage{graphicx} 17 | \usepackage[nottoc,notlot,notlof]{tocbibind} 18 | %\usepackage{fullpage} 19 | 20 | % from stack exchange 21 | \usepackage{sistyle} 22 | \usepackage{mathtools} 23 | \usepackage{stmaryrd} 24 | \SIthousandsep{,} 25 | 26 | \usepackage{tikz} 27 | \usetikzlibrary{topaths,calc,decorations.pathmorphing,quotes,positioning,fit,shapes} 28 | 29 | \hypersetup{ 30 | colorlinks=true, 31 | linkcolor=blue, 32 | filecolor=magenta, 33 | urlcolor=cyan, 34 | citecolor=blue, 35 | } 36 | 37 | \usepackage{appendix} 38 | 39 | \DeclareFontFamily{U}{matha}{\hyphenchar\font45} 40 | \DeclareFontShape{U}{matha}{m}{n}{ 41 | <5> <6> <7> <8> <9> <10> gen * matha 42 | <10.95> matha10 <12> <14.4> <17.28> <20.74> <24.88> matha12 43 | }{} 44 | \DeclareSymbolFont{matha}{U}{matha}{m}{n} 45 | \DeclareFontSubstitution{U}{matha}{m}{n} 46 | \DeclareMathSymbol{\thinsubset}{3}{matha}{"80} 47 | \DeclareMathSymbol{\thinsupset}{3}{matha}{"81} 48 | 49 | \makeatletter 50 | \@addtoreset{chapter}{part} 51 | \@addtoreset{@ppsaveapp}{part} 52 | \makeatother 53 | 54 | \newcommand{\fish}{\reflectbox{$\alpha$}} 55 | 56 | \DeclareMathOperator*{\argmax}{argmax} 57 | \DeclareMathOperator*{\argmin}{argmin} 58 | 59 | \DeclareMathOperator{\tw}{tw} 60 | \DeclareMathOperator{\Spec}{Spec} 61 | \DeclareMathOperator{\Mor}{Mor} 62 | \DeclareMathOperator{\Frob}{Frob} 63 | \DeclareMathOperator{\ord}{ord} 64 | \DeclareMathOperator{\Nm}{Nm} 65 | \DeclareMathOperator{\Gal}{Gal} 66 | \DeclareMathOperator{\Fix}{Fix} 67 | \DeclareMathOperator{\Div}{Div} 68 | \DeclareMathOperator{\divf}{div} 69 | \DeclareMathOperator{\Pic}{Pic} 70 | 71 | \DeclareMathOperator\Supp{Supp} 72 | \DeclareMathOperator\Tr{Tr} 73 | 74 | \DeclareMathOperator{\Clo}{Clo} 75 | \DeclareMathOperator{\Pol}{Pol} 76 | \DeclareMathOperator{\Inv}{Inv} 77 | \DeclareMathOperator{\lcm}{lcm} 78 | \DeclareMathOperator{\CSP}{CSP} 79 | \DeclareMathOperator{\GenSAT}{GenSAT} 80 | \DeclareMathOperator{\Sg}{Sg} 81 | \DeclareMathOperator{\Cg}{Cg} 82 | \DeclareMathOperator{\Sig}{Sig} 83 | \DeclareMathOperator{\Forks}{Forks} 84 | \DeclareMathOperator{\Aut}{Aut} 85 | \DeclareMathOperator{\End}{End} 86 | \DeclareMathOperator{\Con}{Con} 87 | \DeclareMathOperator{\Hom}{Hom} 88 | \DeclareMathOperator{\Sn}{Sn} 89 | \DeclareMathOperator{\typ}{typ} 90 | \DeclareMathOperator{\arity}{ar} 91 | 92 | \DeclareMathOperator{\rad}{rad} 93 | \DeclareMathOperator{\diam}{diam} 94 | \DeclareMathOperator{\inter}{int} 95 | \DeclareMathOperator{\supp}{supp} 96 | \DeclareMathOperator{\maj}{maj} 97 | \DeclareMathOperator{\aff}{aff} 98 | 99 | \newcommand{\F}{\mathbf{F}} 100 | % \newcommand{\RR}{\mathbf{R}} 101 | % \newcommand{\CC}{\mathbf{C}} 102 | %\newcommand{\G}{\mathbf{G}} 103 | \newcommand{\tr}[0]{\operatorname{tr}} 104 | \newcommand{\wt}[1]{\widetilde{#1}} 105 | \newcommand{\frob}[0]{\operatorname{Frob}} 106 | \newcommand{\card}[0]{\#} 107 | \newcommand{\pmat}[4]{\begin{pmatrix}#1 & #2 \\ #3 & #4\end{pmatrix}} 108 | \newcommand{\pderiv}[2]{\frac{\partial #1}{\partial #2}} 109 | \newcommand{\Q}{\mathbf{Q}} 110 | \newcommand{\OO}{\mathcal{O}} 111 | \newcommand{\Z}{\mathbf{Z}} 112 | \newcommand{\leg}[2]{\left(\frac{#1}{#2}\right)} 113 | \newcommand{\mf}[1]{\mathfrak{#1}} 114 | \newcommand{\nm}{\operatorname{N}_{K/\Q}} 115 | \newcommand{\Cl}{\operatorname{Cl}} 116 | \newcommand{\sgn}{\operatorname{sgn}} 117 | % \newcommand{\Gal}{\operatorname{Gal}} 118 | \newcommand{\Mat}{\operatorname{Mat}} 119 | \newcommand{\cl}{\overline} 120 | %\newcommand{\cal}[1]{\mathcal{#1}} 121 | %\newcommand{\R}{\mathbf{R}} 122 | %\newcommand{\HH}{\mathbf{H}} 123 | \newcommand{\ul}[1]{\underline{#1}} 124 | \newcommand{\ol}[1]{\overline{#1}} 125 | \newcommand{\wh}[1]{\widehat{#1}} 126 | %\newcommand{\scr}[1]{\mathscr{#1}} 127 | \newcommand{\PS}{\mathbf{P}} 128 | \newcommand{\mbb}[1]{\mathbf{#1}} 129 | \newcommand{\newpar}[1]{\noindent\textbf{#1}} 130 | \newcommand{\Cal}[1]{\mathcal{#1}} 131 | \newcommand{\A}{\mathbf{A}} 132 | \DeclareMathOperator{\et}{\acute{e}t} 133 | \newcommand{\mbf}[1]{\mathbf{#1}} 134 | \newcommand{\rmono}[2]{\underset{#2}{\stackrel{#1}{\hookrightarrow}}} 135 | \newcommand{\lmono}[2]{\underset{#2}{\stackrel{#1}{\hookleftarrow}}} 136 | \newcommand{\rrat}[2]{\underset{#2}{\stackrel{#1}{\dashrightarrow}}} 137 | \newcommand{\lrat}[2]{\underset{#2}{\stackrel{#1}{\dashleftarrow}}} 138 | \newcommand{\bu}{\bullet} 139 | %\newcommand{\E}{\mathbb{E}} 140 | \newcommand{\mrm}[1]{\mathrm{#1}} 141 | 142 | \DeclareMathOperator{\GL}{GL} 143 | \DeclareMathOperator{\SL}{SL} 144 | \DeclareMathOperator{\PSL}{PSL} 145 | % \DeclareMathOperator{\Frob}{Frob} 146 | \DeclareMathOperator{\ab}{ab} 147 | \DeclareMathOperator{\coker}{coker} 148 | \DeclareMathOperator{\cyc}{cyc} 149 | \DeclareMathOperator{\N}{\mathbf{N}} 150 | % \DeclareMathOperator{\Tr}{Tr} 151 | \DeclareMathOperator{\Enc}{Enc} 152 | \DeclareMathOperator{\Dec}{Dec} 153 | \DeclareMathOperator{\PGL}{PGL} 154 | \DeclareMathOperator{\Sp}{Sp} 155 | \DeclareMathOperator{\SO}{SO} 156 | \DeclareMathOperator{\Ort}{O} 157 | %\DeclareMathOperator{\Hom}{Hom} 158 | \DeclareMathOperator{\Ind}{Ind} 159 | \DeclareMathOperator{\Ima}{Im\,} 160 | \DeclareMathOperator{\rank}{rank} 161 | \DeclareMathOperator{\Jac}{Jac} 162 | % \DeclareMathOperator{\ord}{ord} 163 | % \DeclareMathOperator{\Aut}{Aut} 164 | \DeclareMathOperator{\Rep}{Rep} 165 | % \DeclareMathOperator{\Nm}{Nm} 166 | % \DeclareMathOperator{\Spec}{Spec\,} 167 | \DeclareMathOperator{\Art}{Art} 168 | \DeclareMathOperator{\Lie}{Lie} 169 | %\DeclareMathOperator{\End}{End} 170 | \DeclareMathOperator{\Isom}{Isom} 171 | \DeclareMathOperator{\Inn}{Inn} 172 | \DeclareMathOperator{\Out}{Out} 173 | \DeclareMathOperator{\ad}{ad} 174 | \DeclareMathOperator{\Ver}{Ver} 175 | \DeclareMathOperator{\Br}{Br} 176 | \DeclareMathOperator{\unr}{unr} 177 | \DeclareMathOperator{\Res}{Res} 178 | \DeclareMathOperator{\Frac}{Frac} 179 | % \DeclareMathOperator{\Div}{Div} 180 | \DeclareMathOperator{\cusp}{cusp} 181 | \DeclareMathOperator{\Stab}{Stab} 182 | \DeclareMathOperator{\Bun}{Bun} 183 | \DeclareMathOperator{\Ext}{Ext} 184 | % \DeclareMathOperator{\Pic}{Pic} 185 | \DeclareMathOperator{\ch}{ch\,} 186 | \DeclareMathOperator{\Id}{Id} 187 | \DeclareMathOperator{\Ad}{Ad} 188 | \DeclareMathOperator{\Gr}{Gr} 189 | \DeclareMathOperator{\Shtuka}{Shtuka} 190 | \DeclareMathOperator{\Quot}{Quot} 191 | \DeclareMathOperator{\Jet}{Jet} 192 | \DeclareMathOperator{\Sym}{Sym} 193 | \DeclareMathOperator{\codim}{codim} 194 | \DeclareMathOperator{\bfet}{\textbf{f\'{e}t}} 195 | \DeclareMathOperator{\disc}{disc} 196 | \DeclareMathOperator{\vol}{vol} 197 | \DeclareMathOperator{\Span}{Span} 198 | \DeclareMathOperator{\cond}{cond} 199 | % \DeclareMathOperator{\Fix}{Fix} 200 | \DeclareMathOperator{\val}{val} 201 | \DeclareMathOperator{\reg}{reg} 202 | \DeclareMathOperator{\Rat}{Rat} 203 | \DeclareMathOperator{\Hilb}{Hilb} 204 | \DeclareMathOperator{\Def}{Def} 205 | \DeclareMathOperator{\Chow}{Chow} 206 | \DeclareMathOperator{\Proj}{Proj} 207 | \DeclareMathOperator{\vir}{vir} 208 | \DeclareMathOperator{\ob}{ob} 209 | \DeclareMathOperator{\Ob}{Ob} 210 | \DeclareMathOperator{\intr}{intr} 211 | \DeclareMathOperator{\virdim}{vir. dim} 212 | \DeclareMathOperator{\red}{red} 213 | %\DeclareMathOperator{\supp}{supp} 214 | \DeclareMathOperator{\Bl}{Bl} 215 | \DeclareMathOperator{\cone}{cone} 216 | \DeclareMathOperator{\Ball}{Ball} 217 | \DeclareMathOperator{\genus}{genus} 218 | \DeclareMathOperator{\measure}{measure} 219 | \DeclareMathOperator{\dR}{dR} 220 | 221 | \begin{document} 222 | 223 | \makeatletter 224 | \newtheorem*{rep@theorem}{\rep@title} 225 | \newcommand{\newreptheorem}[2]{% 226 | \newenvironment{rep#1}[1]{% 227 | \def\rep@title{#2 \ref{##1}}% 228 | \begin{rep@theorem}}% 229 | {\end{rep@theorem}}} 230 | \makeatother 231 | 232 | \newtheorem{thm}{Theorem}[section] 233 | \newreptheorem{thm}{Theorem} 234 | \newtheorem{prop}[thm]{Proposition} 235 | \newtheorem{cor}[thm]{Corollary} 236 | \newtheorem{lem}[thm]{Lemma} 237 | \newreptheorem{lem}{Lemma} 238 | 239 | \theoremstyle{definition} 240 | \newtheorem{defn}[thm]{Definition} 241 | \newtheorem{claim}{Claim} 242 | \newtheorem{conj}{Conjecture}[section] 243 | \newtheorem{prob}{Problem}[section] 244 | 245 | \theoremstyle{remark} 246 | \newtheorem{rem}{Remark}[section] 247 | \newtheorem{ex}{Example}[section] 248 | \newtheorem{exer}{Exercise}[section] 249 | 250 | \newcommand{\llhd}{{\mathrlap{<}\;\lhd}} 251 | 252 | \newcommand{\Rho}{\mathrm{P}} 253 | \newcommand{\cS}{\mathcal{S}} 254 | \newcommand{\cM}{\mathcal{M}} 255 | \newcommand{\cN}{\mathcal{N}} 256 | \newcommand{\gk}{\kappa} 257 | \newcommand{\gS}{\Sigma} 258 | \newcommand{\gl}{\lambda} 259 | \newcommand{\gt}{\theta} 260 | 261 | %hmm... 262 | %\newcommand{\bb}[1]{\mathbb #1} 263 | %\newcommand{\cal}[1]{\mathcal #1} 264 | %\newcommand{\bf}[1]{\mathbf #1} 265 | 266 | \newcommand{\cF}{\mathcal{F}} 267 | \newcommand{\cE}{\mathcal{E}} 268 | \newcommand{\cG}{\mathcal{G}} 269 | \newcommand{\cP}{\mathcal{P}} 270 | \newcommand{\cV}{\mathcal{V}} 271 | \newcommand{\cW}{\mathcal{W}} 272 | \newcommand{\cB}{\mathcal{B}} 273 | \newcommand{\cA}{\mathcal{A}} 274 | \newcommand{\cH}{\mathcal{H}} 275 | \newcommand{\ZZ}{\mathbb{Z}} 276 | \newcommand{\NN}{\mathbb{N}} 277 | \newcommand{\QQ}{\mathbb{Q}} 278 | \newcommand{\bA}{\mathbb{A}} 279 | \newcommand{\bB}{\mathbb{B}} 280 | \newcommand{\bC}{\mathbb{C}} 281 | \newcommand{\bD}{\mathbb{D}} 282 | \newcommand{\bE}{\mathbb{E}} 283 | \newcommand{\bF}{\mathbb{F}} 284 | \newcommand{\bG}{\mathbb{G}} 285 | \newcommand{\bI}{\mathbb{I}} 286 | \newcommand{\bL}{\mathbb{L}} 287 | \newcommand{\bM}{\mathbb{M}} 288 | \newcommand{\bN}{\mathbb{N}} 289 | \newcommand{\bP}{\mathbb{P}} 290 | \newcommand{\bQ}{\mathbb{Q}} 291 | \newcommand{\bS}{\mathbb{S}} 292 | \newcommand{\bX}{\mathbb{X}} 293 | \newcommand{\fA}{\mathbf{A}} 294 | \newcommand{\fB}{\mathbf{B}} 295 | \newcommand{\fC}{\mathbf{C}} 296 | \newcommand{\fD}{\mathbf{D}} 297 | \newcommand{\fG}{\mathbf{G}} 298 | \newcommand{\fP}{\mathbf{P}} 299 | \newcommand{\fR}{\mathbf{R}} 300 | \newcommand{\fS}{\mathbf{S}} 301 | \newcommand{\fT}{\mathbf{T}} 302 | \newcommand{\fX}{\mathbf{X}} 303 | \newcommand{\fY}{\mathbf{Y}} 304 | \newcommand{\fZ}{\mathbf{Z}} 305 | 306 | \newcommand{\RR}{\mathbb{R}} 307 | \newcommand{\CC}{\mathbb{C}} 308 | \newcommand{\FF}{\mathbb{F}} 309 | \newcommand{\HH}{\mathbb{H}} 310 | \newcommand{\PP}{\mathbb{P}} 311 | \newcommand{\EE}{\mathbb{E}} 312 | 313 | \newcommand{\cC}{\mathcal{C}} 314 | \newcommand{\cD}{\mathcal{D}} 315 | \newcommand{\cK}{\mathcal{K}} 316 | \newcommand{\cL}{\mathcal{L}} 317 | \newcommand{\cO}{\mathcal{O}} 318 | \newcommand{\cT}{\mathcal{T}} 319 | \newcommand{\cU}{\mathcal{U}} 320 | 321 | \newcommand{\fp}{\mathfrak{p}} 322 | 323 | \newcommand{\dotcup}{\ensuremath{\mathaccent\cdot\cup}} 324 | 325 | \newcommand{\rvline}{\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}} 326 | 327 | \def\D{\mathrm{d}} 328 | 329 | -------------------------------------------------------------------------------- /quad.eps: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/notzeb/all/97969592a92b61f13beb5e90dc945e29e8cd5ebd/quad.eps -------------------------------------------------------------------------------- /quadratic.tex: -------------------------------------------------------------------------------- 1 | \documentclass[letterpaper,11pt]{article} 2 | \usepackage{amsfonts,amssymb,amsmath,amsthm,latexsym,fullpage} 3 | 4 | \newcommand{\logm}{\text{LM}} 5 | 6 | \begin{document} 7 | 8 | \newtheorem*{thm}{Theorem} 9 | 10 | \theoremstyle{remark} 11 | \newtheorem*{exer}{Exercise} 12 | 13 | \title{Keep completing the square!} 14 | \date{} 15 | \maketitle 16 | 17 | Suppose someone hands you a quadratic polynomial in several variables, such as 18 | \[ 19 | x^2 + 2xy - 2xz + 2y^2 + 2yz + 6z^2 - z + 1, 20 | \] 21 | and asks you to check whether it is always $\ge 0$. How do you do it? 22 | 23 | The trick to this is a slight generalization of the high school procedure known as ``completing the square'', which I like to call ``keep completing the square'' (I stumbled on this method after meditating on what the Cholesky decomposition really \emph{meant} in terms of quadratic polynomials). We start by trying to write down a square that agrees with our polynomial at least as far as $x$ is concerned, that is, we try to solve the equation 24 | \[ 25 | (x + Ay + Bz + C)^2 = x^2 + 2xy - 2xz + ..., 26 | \] 27 | for $A,B,C$ (and ignoring the $...$, since it doesn't involve $x$). In this case, we can take $A = 1, B = -1, C = 0$, and we get 28 | \[ 29 | (x + y - z)^2 = x + 2xy - 2xz + y^2 - 2yz + z^2. 30 | \] 31 | Since that doesn't completely match our polynomial, we look at the difference: 32 | \[ 33 | (x^2 + 2xy - 2xz + 2y^2 + 2yz + 6z^2 - z + 1) - (x + y - z)^2 = y^2 + 4yz + 5z^2 - z + 1. 34 | \] 35 | Now we complete the square again, this time with $y$, and so on. Writing the whole process in one string of equalities, we get 36 | \begin{align*} 37 | x^2 + 2xy - 2xz + 2y^2 + 2yz + 6z^2 - 2z + 1 &= (x + y - z)^2 + y^2 + 4yz + 5z^2 - z + 1\\ 38 | &= (x + y - z)^2 + (y + 2z)^2 + z^2 - z + 1\\ 39 | &= (x + y - z)^2 + (y + 2z)^2 + (z - \tfrac{1}{2})^2 + \tfrac{3}{4}, 40 | \end{align*} 41 | and this is clearly positive, since it is a sum of squares. 42 | 43 | Let's do a more complicated example (the previous example was clearly chosen to let you avoid taking any square roots). What if we are faced with something like 44 | \[ 45 | 6x^2 - 4xy + 2xz + 3y^2 - 4yz + 2z^2? 46 | \] 47 | At the very first step, it seems like we'll have to take the square root of $6$. What a mess! Here's how to avoid the mess: instead of starting with a square like 48 | \[ 49 | (\sqrt{6}x + Ay + Bz)^2, 50 | \] 51 | instead we start by looking for something like 52 | \[ 53 | 6(x + Ay + Bz)^2. 54 | \] 55 | Now we can find $A, B$ by simple division, and we get $A = -\frac{1}{3}, B = \frac{1}{6}$. Continuing, we get 56 | \begin{align*} 57 | 6x^2 - 4xy + 2xz + 3y^2 - 4yz + 2z^2 &= 6(x - \tfrac{1}{3}y + \tfrac{1}{6}z)^2 + \tfrac{7}{3}y^2 - \tfrac{10}{3}yz + \tfrac{11}{6}z^2\\ 58 | &= 6(x - \tfrac{1}{3}y + \tfrac{1}{6}z)^2 + \tfrac{7}{3}(y - \tfrac{5}{7}z)^2 + \tfrac{9}{14}z^2, 59 | \end{align*} 60 | which is again obviously positive since it has been written as a sum of squares with positive coefficients. (By the way, I came up this polynomial by expanding out $(x-y)^2 + (x+y-z)^2 + (2x-y+z)^2$ - so we see that there can be multiple ways to write the same polynomial as a sum of squares. If we had processed the variables in a different order, we could come up with yet another way to write it as a sum of squares!) 61 | 62 | What happens if we try to do this to a quadratic polynomial which \emph{isn't} always $\ge 0$? Obviously, something has to go wrong. Let's try the polynomial 63 | \[ 64 | x^2 - 4xy + 2xz + y^2 - 2yz + 2z^2. 65 | \] 66 | The first step goes just fine: we get 67 | \[ 68 | x^2 - 4xy + 2xz + y^2 - 2yz + 2z^2 = (x - 2y + z)^2 - 3y^2 + 2yz + z^2. 69 | \] 70 | But now we have a problem: the coefficient of $y^2$ is negative. Could our polynomial still be $\ge 0$? Maybe the $z^2$ and the $(x - 2y + z)^2$ somehow always conspire to be larger than $3y^2$? Nope! To see why, just set $z$ to $0$, and choose $x$ to make $x - 2y + z$ equal to $0$, for instance, take $z = 0, y = 1, x = 2$. 71 | 72 | In the previous example, we had a problem because the coefficient of $y^2$ was negative. What if the coefficient of $y^2$ comes out to exactly $0$? For an example, let's consider the polynomial 73 | \[ 74 | x^2 - 2xy - 2xz + y^2 - 2yz + 10z^2. 75 | \] 76 | After the first step, we get 77 | \[ 78 | x^2 - 2xy - 2xz + y^2 - 2yz + 2z^2 = (x - y - z)^2 - 4yz + 9z^2. 79 | \] 80 | To show that this sometimes goes negative, we will take $z$ to be whatever nonzero value we like - say, take $z = 1$ - and then pick $y$ to make $-4yz + 9z^2$ come out negative (we can do this since, for any fixed nonzero $z$, $-4yz + 9z^2$ is a linear function of $y$ with a nonzero $y$-coefficient), and finally pick $x$ to make $x-y-z$ equal to $0$. For instance, we can take $z = 1, y = 3, x = 4$. 81 | 82 | At the end of the day, we have a procedure that starts with a quadratic polynomial in any number of variables, and either writes it as a sum of squares with positive coefficients, or spits out a point where it is negative! We summarize in the following theorem. 83 | 84 | \begin{thm} Suppose that $Q(x_1, ..., x_n) = \sum_{i,j} a_{ij} x_ix_j + \sum_i a_ix_i + a$, where $a_{ij}, a_i, a$ are some coefficients. Then either we can write $Q$ in the form 85 | \[ 86 | Q(x_1, ..., x_n) = \sum_{i=1}^n c_i(x_i + b_{i(i+1)}x_{i+1} + \cdots + b_{in}x_n + b_i)^2 + c 87 | \] 88 | with $c_i \ge 0$ for all $i$ and $c \ge 0$, or else we can find a point $(x_1, ..., x_n)$ such that $Q(x_1, ..., x_n) < 0$. 89 | \end{thm} 90 | 91 | In the case of homogeneous quadratic polynomials, people often like to represent their coefficients in a symmetric matrix. In the three variable case, the matrix 92 | \[ 93 | \begin{bmatrix} a & b & d\\ 94 | b & c & e\\ 95 | d & e & f 96 | \end{bmatrix} 97 | \] 98 | corresponds to the polynomial 99 | \[ 100 | ax^2 + 2bxy + cy^2 + 2dxz + 2eyz + fz^2. 101 | \] 102 | Why the random factors of $2$? This is because we have the nice formula 103 | \[ 104 | \begin{bmatrix} x & y & z \end{bmatrix} \begin{bmatrix} a & b & d\\ b & c & e\\ d & e & f \end{bmatrix} \begin{bmatrix} x\\ y\\ z\end{bmatrix} = ax^2 + 2bxy + cy^2 + 2dxz + 2eyz + fz^2. 105 | \] 106 | 107 | When we follow the ``keep completing the square'' procedure for this general three variable homogeneous quadratic, we get 108 | \begin{align*} 109 | ax^2 + 2bxy + cy^2 + 2dxz + 2eyz + fz^2 &= a(x + \tfrac{b}{a}y + \tfrac{d}{a}z)^2 + \tfrac{ac-b^2}{a}y^2 + 2\tfrac{ae-bd}{a}yz + \tfrac{af-d^2}{a}z^2\\ 110 | &= a(x + \tfrac{b}{a}y + \tfrac{d}{a}z)^2 + \tfrac{ac-b^2}{a}(y + \tfrac{ae-bd}{ac-b^2}z)^2 + \tfrac{(af-d^2)(ac-b^2)-(ae-bd)^2}{a(ac-b^2)}z^2\\ 111 | &= a(x + \tfrac{b}{a}y + \tfrac{d}{a}z)^2 + \tfrac{ac-b^2}{a}(y + \tfrac{ae-bd}{ac-b^2}z)^2 + \tfrac{acf + 2bde - ae^2 - b^2f - cd^2}{ac-b^2}z^2. 112 | \end{align*} 113 | Curiously, the coefficients in that last formula happen to be ratios of determinants: 114 | \begin{align*} 115 | \det \begin{bmatrix} a\end{bmatrix} &= a,\\ 116 | \det \begin{bmatrix} a & b\\ b & c\end{bmatrix} &= ac - b^2,\\ 117 | \det \begin{bmatrix} a & b & d\\ b & c & e\\ d & e & f \end{bmatrix} &= acf + 2bde - ae^2 - b^2f - cd^2. 118 | \end{align*} 119 | So we've proved that a three variable homogeneous quadratic is $\ge 0$ if those three determinants are all positive! 120 | 121 | \bigskip 122 | 123 | \begin{exer} Generalize this determinant formula to any number of variables. 124 | \end{exer} 125 | 126 | \end{document} 127 | 128 | -------------------------------------------------------------------------------- /righthyperbola.eps: -------------------------------------------------------------------------------- 1 | %!PS-Adobe-2.0 EPSF-2.0 2 | %%BoundingBox: 124 127 471 716 3 | %%HiResBoundingBox: 124.500000 127.500000 470.500000 715.500000 4 | %%Creator: Qt 3.3.8b 5 | %%CreationDate: Sun Oct 4 22:02:38 2015 6 | %%DocumentFonts: 7 | 8 | %%EndComments 9 | % EPSF created by ps2eps 1.68 10 | %%BeginProlog 11 | save 12 | countdictstack 13 | mark 14 | newpath 15 | /showpage {} def 16 | /setpagedevice {pop} def 17 | %%EndProlog 18 | %%Page 1 1 19 | %%BeginProlog 20 | % Prolog copyright 1994-2006 Trolltech. 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5295 524 DL 280 | [0.125 0 0 0.125 138 118]ST 281 | 81 3604 79 3605 DL 282 | [0.125 0 0 0.125 138 118]ST 283 | 82 3604 81 HL 284 | [0.125 0 0 0.125 138 118]ST 285 | 83 3603 82 3604 DL 286 | [0.125 0 0 0.125 138 118]ST 287 | 86 3601 83 3603 DL 288 | [0.125 0 0 0.125 138 118]ST 289 | 89 3599 86 3601 DL 290 | [0.125 0 0 0.125 138 118]ST 291 | 95 3596 89 3599 DL 292 | [0.125 0 0 0.125 138 118]ST 293 | 101 3592 95 3596 DL 294 | [0.125 0 0 0.125 138 118]ST 295 | 112 3585 101 3592 DL 296 | [0.125 0 0 0.125 138 118]ST 297 | 123 3578 112 3585 DL 298 | [0.125 0 0 0.125 138 118]ST 299 | 146 3564 123 3578 DL 300 | [0.125 0 0 0.125 138 118]ST 301 | 168 3551 146 3564 DL 302 | [0.125 0 0 0.125 138 118]ST 303 | 211 3524 168 3551 DL 304 | [0.125 0 0 0.125 138 118]ST 305 | 252 3499 211 3524 DL 306 | [0.125 0 0 0.125 138 118]ST 307 | 332 3449 252 3499 DL 308 | [0.125 0 0 0.125 138 118]ST 309 | 408 3402 332 3449 DL 310 | [0.125 0 0 0.125 138 118]ST 311 | 548 3316 408 3402 DL 312 | [0.125 0 0 0.125 138 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138 118]ST 409 | 3910 2610 4012 2806 DL 410 | [0.125 0 0 0.125 138 118]ST 411 | 3834 2453 3910 2610 DL 412 | [0.125 0 0 0.125 138 118]ST 413 | 3777 2323 3834 2453 DL 414 | [0.125 0 0 0.125 138 118]ST 415 | 3734 2212 3777 2323 DL 416 | [0.125 0 0 0.125 138 118]ST 417 | 3717 2162 3734 2212 DL 418 | [0.125 0 0 0.125 138 118]ST 419 | 3703 2115 3717 2162 DL 420 | [0.125 0 0 0.125 138 118]ST 421 | 3692 2071 3703 2115 DL 422 | [0.125 0 0 0.125 138 118]ST 423 | 3684 2028 3692 2071 DL 424 | [0.125 0 0 0.125 138 118]ST 425 | 3681 2007 3684 2028 DL 426 | [0.125 0 0 0.125 138 118]ST 427 | 3678 1987 3681 2007 DL 428 | [0.125 0 0 0.125 138 118]ST 429 | 3675 1948 3678 1987 DL 430 | [0.125 0 0 0.125 138 118]ST 431 | 3674 1928 3675 1948 DL 432 | [0.125 0 0 0.125 138 118]ST 433 | 3675 1909 3674 1928 DL 434 | [0.125 0 0 0.125 138 118]ST 435 | 3677 1871 3675 1909 DL 436 | [0.125 0 0 0.125 138 118]ST 437 | 3679 1852 3677 1871 DL 438 | [0.125 0 0 0.125 138 118]ST 439 | 3683 1833 3679 1852 DL 440 | [0.125 0 0 0.125 138 118]ST 441 | 3687 1814 3683 1833 DL 442 | [0.125 0 0 0.125 138 118]ST 443 | 3692 1795 3687 1814 DL 444 | [0.125 0 0 0.125 138 118]ST 445 | 3698 1776 3692 1795 DL 446 | [0.125 0 0 0.125 138 118]ST 447 | 3705 1757 3698 1776 DL 448 | [0.125 0 0 0.125 138 118]ST 449 | 3713 1737 3705 1757 DL 450 | [0.125 0 0 0.125 138 118]ST 451 | 3722 1717 3713 1737 DL 452 | [0.125 0 0 0.125 138 118]ST 453 | 3745 1676 3722 1717 DL 454 | [0.125 0 0 0.125 138 118]ST 455 | 3774 1632 3745 1676 DL 456 | [0.125 0 0 0.125 138 118]ST 457 | 3811 1585 3774 1632 DL 458 | [0.125 0 0 0.125 138 118]ST 459 | 3833 1560 3811 1585 DL 460 | [0.125 0 0 0.125 138 118]ST 461 | 3858 1533 3833 1560 DL 462 | [0.125 0 0 0.125 138 118]ST 463 | 3919 1473 3858 1533 DL 464 | [0.125 0 0 0.125 138 118]ST 465 | 3956 1440 3919 1473 DL 466 | [0.125 0 0 0.125 138 118]ST 467 | 3998 1404 3956 1440 DL 468 | [0.125 0 0 0.125 138 118]ST 469 | 4103 1320 3998 1404 DL 470 | [0.125 0 0 0.125 138 118]ST 471 | 4246 1214 4103 1320 DL 472 | [0.125 0 0 0.125 138 118]ST 473 | 4764 863 4246 1214 DL 474 | [0.125 0 0 0.125 138 118]ST 475 | 5295 524 4764 863 DL 476 | [0.125 0 0 0.125 138 118]ST 477 | 6379 -150 5295 524 DL 478 | [0.125 0 0 0.125 138 118]ST 479 | 81 3604 79 3605 DL 480 | [0.125 0 0 0.125 138 118]ST 481 | 82 3604 81 HL 482 | [0.125 0 0 0.125 138 118]ST 483 | 83 3603 82 3604 DL 484 | [0.125 0 0 0.125 138 118]ST 485 | 86 3601 83 3603 DL 486 | [0.125 0 0 0.125 138 118]ST 487 | 89 3599 86 3601 DL 488 | [0.125 0 0 0.125 138 118]ST 489 | 95 3596 89 3599 DL 490 | [0.125 0 0 0.125 138 118]ST 491 | 101 3592 95 3596 DL 492 | [0.125 0 0 0.125 138 118]ST 493 | 112 3585 101 3592 DL 494 | [0.125 0 0 0.125 138 118]ST 495 | 123 3578 112 3585 DL 496 | [0.125 0 0 0.125 138 118]ST 497 | 146 3564 123 3578 DL 498 | [0.125 0 0 0.125 138 118]ST 499 | 168 3551 146 3564 DL 500 | [0.125 0 0 0.125 138 118]ST 501 | 211 3524 168 3551 DL 502 | [0.125 0 0 0.125 138 118]ST 503 | 252 3499 211 3524 DL 504 | [0.125 0 0 0.125 138 118]ST 505 | 332 3449 252 3499 DL 506 | [0.125 0 0 0.125 138 118]ST 507 | 408 3402 332 3449 DL 508 | [0.125 0 0 0.125 138 118]ST 509 | 548 3316 408 3402 DL 510 | [0.125 0 0 0.125 138 118]ST 511 | 673 3238 548 3316 DL 512 | [0.125 0 0 0.125 138 118]ST 513 | 891 3103 673 3238 DL 514 | [0.125 0 0 0.125 138 118]ST 515 | 1073 2989 891 3103 DL 516 | [0.125 0 0 0.125 138 118]ST 517 | 1359 2809 1073 2989 DL 518 | [0.125 0 0 0.125 138 118]ST 519 | 1574 2672 1359 2809 DL 520 | [0.125 0 0 0.125 138 118]ST 521 | 1875 2477 1574 2672 DL 522 | [0.125 0 0 0.125 138 118]ST 523 | 2075 2343 1875 2477 DL 524 | [0.125 0 0 0.125 138 118]ST 525 | 2322 2170 2075 2343 DL 526 | [0.125 0 0 0.125 138 118]ST 527 | 2467 2059 2322 2170 DL 528 | [0.125 0 0 0.125 138 118]ST 529 | 2561 1981 2467 2059 DL 530 | [0.125 0 0 0.125 138 118]ST 531 | 2627 1921 2561 1981 DL 532 | [0.125 0 0 0.125 138 118]ST 533 | 0 0 B 0 0 PE 534 | 1 B BR 535 | 2480 576 64 64 E 536 | [0.125 0 0 0.125 138 118]ST 537 | 1392 2736 64 64 E 538 | [0.125 0 0 0.125 138 118]ST 539 | 3960 2736 64 64 E 540 | [0.125 0 0 0.125 138 118]ST 541 | 2480 1991 64 64 E 542 | [0.125 0 0 0.125 138 118]ST 543 | 3880 1448 64 64 E 544 | QP 545 | %%Trailer 546 | %%Pages: 1 547 | %%DocumentFonts: 548 | %%Trailer 549 | cleartomark 550 | countdictstack 551 | exch sub { end } repeat 552 | restore 553 | %%EOF 554 | -------------------------------------------------------------------------------- /selfpolar.eps: -------------------------------------------------------------------------------- 1 | %!PS-Adobe-2.0 EPSF-2.0 2 | %%BoundingBox: 124 127 471 716 3 | %%HiResBoundingBox: 124.500000 127.500000 470.500000 715.500000 4 | %%Creator: Qt 3.3.8b 5 | %%CreationDate: Mon Oct 5 01:31:33 2015 6 | %%DocumentFonts: 7 | 8 | %%EndComments 9 | % EPSF created by ps2eps 1.68 10 | %%BeginProlog 11 | save 12 | countdictstack 13 | mark 14 | newpath 15 | /showpage {} def 16 | /setpagedevice {pop} def 17 | %%EndProlog 18 | %%Page 1 1 19 | %%BeginProlog 20 | % Prolog copyright 1994-2006 Trolltech. You may copy this prolog in any way 21 | % that is directly related to this document. For other use of this prolog, 22 | % see your licensing agreement for Qt. 23 | /d/def load def/D{bind d}bind d/d2{dup dup}D/B{0 d2}D/W{255 d2}D/ED{exch d}D 24 | /D0{0 ED}D/LT{lineto}D/MT{moveto}D/S{stroke}D/F{setfont}D/SW{setlinewidth}D 25 | /CP{closepath}D/RL{rlineto}D/NP{newpath}D/CM{currentmatrix}D/SM{setmatrix}D 26 | /TR{translate}D/SD{setdash}D/SC{aload pop setrgbcolor}D/CR{currentfile read 27 | pop}D/i{index}D/bs{bitshift}D/scs{setcolorspace}D/DB{dict dup begin}D/DE{end 28 | d}D/ie{ifelse}D/sp{astore pop}D/BSt 0 d/LWi 1 d/PSt 1 d/Cx 0 d/Cy 0 d/WFi 29 | false d/OMo false d/BCol[1 1 1]d/PCol[0 0 0]d/BkCol[1 1 1]d/BDArr[0.94 0.88 30 | 0.63 0.50 0.37 0.12 0.06]d/defM matrix d/nS 0 d/GPS{PSt 1 ge PSt 5 le and{{ 31 | LArr PSt 1 sub 2 mul get}{LArr PSt 2 mul 1 sub get}ie}{[]}ie}D/QS{PSt 0 ne{ 32 | gsave LWi SW true GPS 0 SD S OMo PSt 1 ne and{BkCol SC false GPS dup 0 get 33 | SD S}if grestore}if}D/r28{{CR dup 32 gt{exit}if pop}loop 3{CR}repeat 0 4{7 34 | bs exch dup 128 gt{84 sub}if 42 sub 127 and add}repeat}D/rA 0 d/rL 0 d/rB{rL 35 | 0 eq{/rA r28 d/rL 28 d}if dup rL gt{rA exch rL sub rL exch/rA 0 d/rL 0 d rB 36 | exch bs add}{dup rA 16#fffffff 3 -1 roll bs not and exch dup rL exch sub/rL 37 | ED neg rA exch bs/rA ED}ie}D/uc{/rL 0 d 0{dup 2 i length ge{exit}if 1 rB 1 38 | eq{3 rB dup 3 ge{1 add dup rB 1 i 5 ge{1 i 6 ge{1 i 7 ge{1 i 8 ge{128 add}if 39 | 64 add}if 32 add}if 16 add}if 3 add exch pop}if 3 add exch 10 rB 1 add{dup 3 40 | i lt{dup}{2 i}ie 4 i 3 i 3 i sub 2 i getinterval 5 i 4 i 3 -1 roll 41 | putinterval dup 4 -1 roll add 3 1 roll 4 -1 roll exch sub dup 0 eq{exit}if 3 42 | 1 roll}loop pop pop}{3 rB 1 add{2 copy 8 rB put 1 add}repeat}ie}loop pop}D 43 | /sl D0/QCIgray D0/QCIcolor D0/QCIindex D0/QCI{/colorimage where{pop false 3 44 | colorimage}{exec/QCIcolor ED/QCIgray QCIcolor length 3 idiv string d 0 1 45 | QCIcolor length 3 idiv 1 sub{/QCIindex ED/x QCIindex 3 mul d QCIgray 46 | QCIindex QCIcolor x get 0.30 mul QCIcolor x 1 add get 0.59 mul QCIcolor x 2 47 | add get 0.11 mul add add cvi put}for QCIgray image}ie}D/di{gsave TR 1 i 1 eq 48 | {false eq{pop true 3 1 roll 4 i 4 i false 4 i 4 i imagemask BkCol SC 49 | imagemask}{pop false 3 1 roll imagemask}ie}{dup false ne{/languagelevel 50 | where{pop languagelevel 3 ge}{false}ie}{false}ie{/ma ED 8 eq{/dc[0 1]d 51 | /DeviceGray}{/dc[0 1 0 1 0 1]d/DeviceRGB}ie scs/im ED/mt ED/h ED/w ED/id 7 52 | DB/ImageType 1 d/Width w d/Height h d/ImageMatrix mt d/DataSource im d 53 | /BitsPerComponent 8 d/Decode dc d DE/md 7 DB/ImageType 1 d/Width w d/Height 54 | h d/ImageMatrix mt d/DataSource ma d/BitsPerComponent 1 d/Decode[0 1]d DE 4 55 | DB/ImageType 3 d/DataDict id d/MaskDict md d/InterleaveType 3 d end image}{ 56 | pop 8 4 1 roll 8 eq{image}{QCI}ie}ie}ie grestore}d/BF{gsave BSt 1 eq{BCol SC 57 | WFi{fill}{eofill}ie}if BSt 2 ge BSt 8 le and{BDArr BSt 2 sub get/sc ED BCol{ 58 | 1. exch sub sc mul 1. exch sub}forall 3 array astore SC WFi{fill}{eofill}ie} 59 | if BSt 9 ge BSt 14 le and{WFi{clip}{eoclip}ie defM SM pathbbox 3 i 3 i TR 4 60 | 2 roll 3 2 roll exch sub/h ED sub/w ED OMo{NP 0 0 MT 0 h RL w 0 RL 0 h neg 61 | RL CP BkCol SC fill}if BCol SC 0.3 SW NP BSt 9 eq BSt 11 eq or{0 4 h{dup 0 62 | exch MT w exch LT}for}if BSt 10 eq BSt 11 eq or{0 4 w{dup 0 MT h LT}for}if 63 | BSt 12 eq BSt 14 eq or{w h gt{0 6 w h add{dup 0 MT h sub h LT}for}{0 6 w h 64 | add{dup 0 exch MT w sub w exch LT}for}ie}if BSt 13 eq BSt 14 eq or{w h gt{0 65 | 6 w h add{dup h MT h sub 0 LT}for}{0 6 w h add{dup w exch MT w sub 0 exch LT 66 | }for}ie}if S}if BSt 24 eq{}if grestore}D/mat matrix d/ang1 D0/ang2 D0/w D0/h 67 | D0/x D0/y D0/ARC{/ang2 ED/ang1 ED/h ED/w ED/y ED/x ED mat CM pop x w 2 div 68 | add y h 2 div add TR 1 h w div neg scale ang2 0 ge{0 0 w 2 div ang1 ang1 69 | ang2 add arc}{0 0 w 2 div ang1 ang1 ang2 add arcn}ie mat SM}D/C D0/P{NP MT 70 | 0.5 0.5 rmoveto 0 -1 RL -1 0 RL 0 1 RL CP fill}D/M{/Cy ED/Cx ED}D/L{NP Cx Cy 71 | MT/Cy ED/Cx ED Cx Cy LT QS}D/DL{NP MT LT QS}D/HL{1 i DL}D/VL{2 i exch DL}D/R 72 | {/h ED/w ED/y ED/x ED NP x y MT 0 h RL w 0 RL 0 h neg RL CP BF QS}D/ACR{/h 73 | ED/w ED/y ED/x ED x y MT 0 h RL w 0 RL 0 h neg RL CP}D/xr D0/yr D0/rx D0/ry 74 | D0/rx2 D0/ry2 D0/RR{/yr ED/xr ED/h ED/w ED/y ED/x ED xr 0 le yr 0 le or{x y 75 | w h R}{xr 100 ge yr 100 ge or{x y w h E}{/rx xr w mul 200 div d/ry yr h mul 76 | 200 div d/rx2 rx 2 mul d/ry2 ry 2 mul d NP x rx add y MT x y rx2 ry2 180 -90 77 | x y h add ry2 sub rx2 ry2 270 -90 x w add rx2 sub y h add ry2 sub rx2 ry2 0 78 | -90 x w add rx2 sub y rx2 ry2 90 -90 ARC ARC ARC ARC CP BF QS}ie}ie}D/E{/h 79 | ED/w ED/y ED/x ED mat CM pop x w 2 div add y h 2 div add TR 1 h w div scale 80 | NP 0 0 w 2 div 0 360 arc mat SM BF QS}D/A{16 div exch 16 div exch NP ARC QS} 81 | D/PIE{/ang2 ED/ang1 ED/h ED/w ED/y ED/x ED NP x w 2 div add y h 2 div add MT 82 | x y w h ang1 16 div ang2 16 div ARC CP BF QS}D/CH{16 div exch 16 div exch NP 83 | ARC CP BF QS}D/BZ{curveto QS}D/CRGB{255 div 3 1 roll 255 div 3 1 roll 255 84 | div 3 1 roll}D/BC{CRGB BkCol sp}D/BR{CRGB BCol sp/BSt ED}D/WB{1 W BR}D/NB{0 85 | B BR}D/PE{setlinejoin setlinecap CRGB PCol sp/LWi ED/PSt ED LWi 0 eq{0.25 86 | /LWi ED}if PCol SC}D/P1{1 0 5 2 roll 0 0 PE}D/ST{defM SM concat}D/MF{true 87 | exch true exch{exch pop exch pop dup 0 get dup findfont dup/FontName get 3 88 | -1 roll eq{exit}if}forall exch dup 1 get/fxscale ED 2 get/fslant ED exch 89 | /fencoding ED[fxscale 0 fslant 1 0 0]makefont fencoding false eq{}{dup 90 | maxlength dict begin{1 i/FID ne{def}{pop pop}ifelse}forall/Encoding 91 | fencoding d currentdict end}ie definefont pop}D/MFEmb{findfont dup length 92 | dict begin{1 i/FID ne{d}{pop pop}ifelse}forall/Encoding ED currentdict end 93 | definefont pop}D/DF{findfont/fs 3 -1 roll d[fs 0 0 fs -1 mul 0 0]makefont d} 94 | D/ty 0 d/Y{/ty ED}D/Tl{gsave SW NP 1 i exch MT 1 i 0 RL S grestore}D/XYT{ty 95 | MT/xyshow where{pop pop xyshow}{exch pop 1 i dup length 2 div exch 96 | stringwidth pop 3 -1 roll exch sub exch div exch 0 exch ashow}ie}D/AT{ty MT 97 | 1 i dup length 2 div exch stringwidth pop 3 -1 roll exch sub exch div exch 0 98 | exch ashow}D/QI{/C save d pageinit/Cx 0 d/Cy 0 d/OMo false d}D/QP{C restore 99 | showpage}D/SPD{/setpagedevice where{1 DB 3 1 roll d end setpagedevice}{pop 100 | pop}ie}D/SV{BSt LWi PSt Cx Cy WFi OMo BCol PCol BkCol/nS nS 1 add d gsave}D 101 | /RS{nS 0 gt{grestore/BkCol ED/PCol ED/BCol ED/OMo ED/WFi ED/Cy ED/Cx ED/PSt 102 | ED/LWi ED/BSt ED/nS nS 1 sub d}if}D/CLSTART{/clipTmp matrix CM d defM SM NP} 103 | D/CLEND{clip NP clipTmp SM}D/CLO{grestore gsave defM SM}D 104 | /LArr[ [] [] [ 13.333 4.000 ] [ 4.000 13.333 ] [ 4.000 4.000 ] [ 4.000 4.000 ] [ 6.667 4.000 4.000 4.000 ] [ 4.000 6.667 4.000 4.000 ] [ 6.667 4.000 4.000 4.000 4.000 ] [ 4.000 6.667 4.000 4.000 4.000 4.000 ] ] d 105 | /pageinit { 106 | 36 24 translate 107 | % 184*280 mm (landscape) 108 | 90 rotate 0.75 -0.75 scale/defM matrix CM d } d 109 | %%EndProlog 110 | %%BeginSetup 111 | %%EndSetup 112 | %%Page: 1 1 113 | %%BeginPageSetup 114 | QI 115 | %%EndPageSetup 116 | [0.125 0 0 0.125 138 118]ST 117 | CLSTART 118 | 138 118 784 461 ACR 119 | CLEND 120 | 0 0 B 0 0 PE 121 | 1 215 d2 BR 122 | W BC 123 | 2400 312 64 64 E 124 | [0.125 0 0 0.125 138 118]ST 125 | 1896 1560 64 64 E 126 | [0.125 0 0 0.125 138 118]ST 127 | 2784 1992 64 64 E 128 | [0.125 0 0 0.125 138 118]ST 129 | 3504 1456 64 64 E 130 | [0.125 0 0 0.125 138 118]ST 131 | 1 B BR 132 | 3000 432 64 64 E 133 | [0.125 0 0 0.125 138 118]ST 134 | 1 2 B 0 0 PE 135 | 1757 1172 1753 1134 DL 136 | [0.125 0 0 0.125 138 118]ST 137 | 1763 1211 1757 1172 DL 138 | [0.125 0 0 0.125 138 118]ST 139 | 1771 1252 1763 1211 DL 140 | [0.125 0 0 0.125 138 118]ST 141 | 1782 1295 1771 1252 DL 142 | [0.125 0 0 0.125 138 118]ST 143 | 1796 1339 1782 1295 DL 144 | [0.125 0 0 0.125 138 118]ST 145 | 1803 1361 1796 1339 DL 146 | [0.125 0 0 0.125 138 118]ST 147 | 1812 1384 1803 1361 DL 148 | [0.125 0 0 0.125 138 118]ST 149 | 1832 1430 1812 1384 DL 150 | [0.125 0 0 0.125 138 118]ST 151 | 1856 1477 1832 1430 DL 152 | [0.125 0 0 0.125 138 118]ST 153 | 1883 1525 1856 1477 DL 154 | [0.125 0 0 0.125 138 118]ST 155 | 1914 1573 1883 1525 DL 156 | [0.125 0 0 0.125 138 118]ST 157 | 1950 1621 1914 1573 DL 158 | [0.125 0 0 0.125 138 118]ST 159 | 1990 1669 1950 1621 DL 160 | [0.125 0 0 0.125 138 118]ST 161 | 2034 1715 1990 1669 DL 162 | [0.125 0 0 0.125 138 118]ST 163 | 2083 1761 2034 1715 DL 164 | [0.125 0 0 0.125 138 118]ST 165 | 2136 1804 2083 1761 DL 166 | [0.125 0 0 0.125 138 118]ST 167 | 2193 1845 2136 1804 DL 168 | [0.125 0 0 0.125 138 118]ST 169 | 2254 1883 2193 1845 DL 170 | [0.125 0 0 0.125 138 118]ST 171 | 2319 1918 2254 1883 DL 172 | [0.125 0 0 0.125 138 118]ST 173 | 2387 1948 2319 1918 DL 174 | [0.125 0 0 0.125 138 118]ST 175 | 2458 1974 2387 1948 DL 176 | [0.125 0 0 0.125 138 118]ST 177 | 2531 1995 2458 1974 DL 178 | [0.125 0 0 0.125 138 118]ST 179 | 2568 2004 2531 1995 DL 180 | [0.125 0 0 0.125 138 118]ST 181 | 2605 2011 2568 2004 DL 182 | [0.125 0 0 0.125 138 118]ST 183 | 2643 2016 2605 2011 DL 184 | [0.125 0 0 0.125 138 118]ST 185 | 2680 2021 2643 2016 DL 186 | [0.125 0 0 0.125 138 118]ST 187 | 2718 2023 2680 2021 DL 188 | [0.125 0 0 0.125 138 118]ST 189 | 2755 2025 2718 2023 DL 190 | [0.125 0 0 0.125 138 118]ST 191 | 2793 2025 2755 HL 192 | [0.125 0 0 0.125 138 118]ST 193 | 2829 2023 2793 2025 DL 194 | [0.125 0 0 0.125 138 118]ST 195 | 2866 2020 2829 2023 DL 196 | [0.125 0 0 0.125 138 118]ST 197 | 2902 2016 2866 2020 DL 198 | [0.125 0 0 0.125 138 118]ST 199 | 2937 2010 2902 2016 DL 200 | [0.125 0 0 0.125 138 118]ST 201 | 2972 2003 2937 2010 DL 202 | [0.125 0 0 0.125 138 118]ST 203 | 3006 1995 2972 2003 DL 204 | [0.125 0 0 0.125 138 118]ST 205 | 3039 1985 3006 1995 DL 206 | [0.125 0 0 0.125 138 118]ST 207 | 3072 1974 3039 1985 DL 208 | [0.125 0 0 0.125 138 118]ST 209 | 3103 1962 3072 1974 DL 210 | [0.125 0 0 0.125 138 118]ST 211 | 3134 1949 3103 1962 DL 212 | [0.125 0 0 0.125 138 118]ST 213 | 3164 1935 3134 1949 DL 214 | [0.125 0 0 0.125 138 118]ST 215 | 3192 1920 3164 1935 DL 216 | [0.125 0 0 0.125 138 118]ST 217 | 3220 1904 3192 1920 DL 218 | [0.125 0 0 0.125 138 118]ST 219 | 3246 1887 3220 1904 DL 220 | [0.125 0 0 0.125 138 118]ST 221 | 3271 1869 3246 1887 DL 222 | [0.125 0 0 0.125 138 118]ST 223 | 3295 1851 3271 1869 DL 224 | [0.125 0 0 0.125 138 118]ST 225 | 3318 1832 3295 1851 DL 226 | [0.125 0 0 0.125 138 118]ST 227 | 3340 1812 3318 1832 DL 228 | [0.125 0 0 0.125 138 118]ST 229 | 3361 1792 3340 1812 DL 230 | [0.125 0 0 0.125 138 118]ST 231 | 3380 1771 3361 1792 DL 232 | [0.125 0 0 0.125 138 118]ST 233 | 3398 1750 3380 1771 DL 234 | [0.125 0 0 0.125 138 118]ST 235 | 3416 1729 3398 1750 DL 236 | [0.125 0 0 0.125 138 118]ST 237 | 3432 1707 3416 1729 DL 238 | [0.125 0 0 0.125 138 118]ST 239 | 3447 1685 3432 1707 DL 240 | [0.125 0 0 0.125 138 118]ST 241 | 3461 1663 3447 1685 DL 242 | [0.125 0 0 0.125 138 118]ST 243 | 3485 1618 3461 1663 DL 244 | [0.125 0 0 0.125 138 118]ST 245 | 3496 1596 3485 1618 DL 246 | [0.125 0 0 0.125 138 118]ST 247 | 3506 1574 3496 1596 DL 248 | [0.125 0 0 0.125 138 118]ST 249 | 3523 1529 3506 1574 DL 250 | [0.125 0 0 0.125 138 118]ST 251 | 3537 1485 3523 1529 DL 252 | [0.125 0 0 0.125 138 118]ST 253 | 3547 1441 3537 1485 DL 254 | [0.125 0 0 0.125 138 118]ST 255 | 3555 1398 3547 1441 DL 256 | [0.125 0 0 0.125 138 118]ST 257 | 3558 1377 3555 1398 DL 258 | [0.125 0 0 0.125 138 118]ST 259 | 3560 1356 3558 1377 DL 260 | [0.125 0 0 0.125 138 118]ST 261 | 3562 1315 3560 1356 DL 262 | [0.125 0 0 0.125 138 118]ST 263 | 3563 1276 3562 1315 DL 264 | [0.125 0 0 0.125 138 118]ST 265 | 3561 1237 3563 1276 DL 266 | [0.125 0 0 0.125 138 118]ST 267 | 3558 1200 3561 1237 DL 268 | [0.125 0 0 0.125 138 118]ST 269 | 3553 1164 3558 1200 DL 270 | [0.125 0 0 0.125 138 118]ST 271 | 3547 1129 3553 1164 DL 272 | [0.125 0 0 0.125 138 118]ST 273 | 3539 1095 3547 1129 DL 274 | [0.125 0 0 0.125 138 118]ST 275 | 3531 1063 3539 1095 DL 276 | [0.125 0 0 0.125 138 118]ST 277 | 3510 1002 3531 1063 DL 278 | [0.125 0 0 0.125 138 118]ST 279 | 3487 945 3510 1002 DL 280 | [0.125 0 0 0.125 138 118]ST 281 | 3462 892 3487 945 DL 282 | [0.125 0 0 0.125 138 118]ST 283 | 3434 843 3462 892 DL 284 | [0.125 0 0 0.125 138 118]ST 285 | 3404 796 3434 843 DL 286 | [0.125 0 0 0.125 138 118]ST 287 | 3388 774 3404 796 DL 288 | [0.125 0 0 0.125 138 118]ST 289 | 3372 752 3388 774 DL 290 | [0.125 0 0 0.125 138 118]ST 291 | 3337 709 3372 752 DL 292 | [0.125 0 0 0.125 138 118]ST 293 | 3298 666 3337 709 DL 294 | [0.125 0 0 0.125 138 118]ST 295 | 3249 618 3298 666 DL 296 | [0.125 0 0 0.125 138 118]ST 297 | 3216 589 3249 618 DL 298 | [0.125 0 0 0.125 138 118]ST 299 | 3191 569 3216 589 DL 300 | [0.125 0 0 0.125 138 118]ST 301 | 3172 554 3191 569 DL 302 | [0.125 0 0 0.125 138 118]ST 303 | 3157 543 3172 554 DL 304 | [0.125 0 0 0.125 138 118]ST 305 | 3145 534 3157 543 DL 306 | [0.125 0 0 0.125 138 118]ST 307 | 3136 528 3145 534 DL 308 | [0.125 0 0 0.125 138 118]ST 309 | 3127 522 3136 528 DL 310 | [0.125 0 0 0.125 138 118]ST 311 | 3120 517 3127 522 DL 312 | [0.125 0 0 0.125 138 118]ST 313 | 3114 513 3120 517 DL 314 | [0.125 0 0 0.125 138 118]ST 315 | 3109 510 3114 513 DL 316 | [0.125 0 0 0.125 138 118]ST 317 | 3105 507 3109 510 DL 318 | [0.125 0 0 0.125 138 118]ST 319 | 3101 504 3105 507 DL 320 | [0.125 0 0 0.125 138 118]ST 321 | 3097 502 3101 504 DL 322 | [0.125 0 0 0.125 138 118]ST 323 | 3032 464 3097 502 DL 324 | [0.125 0 0 0.125 138 118]ST 325 | 2968 432 3032 464 DL 326 | [0.125 0 0 0.125 138 118]ST 327 | 2964 430 2968 432 DL 328 | [0.125 0 0 0.125 138 118]ST 329 | 2960 429 2964 430 DL 330 | [0.125 0 0 0.125 138 118]ST 331 | 2956 427 2960 429 DL 332 | [0.125 0 0 0.125 138 118]ST 333 | 2951 424 2956 427 DL 334 | [0.125 0 0 0.125 138 118]ST 335 | 2945 422 2951 424 DL 336 | [0.125 0 0 0.125 138 118]ST 337 | 2938 419 2945 422 DL 338 | [0.125 0 0 0.125 138 118]ST 339 | 2930 416 2938 419 DL 340 | [0.125 0 0 0.125 138 118]ST 341 | 2921 412 2930 416 DL 342 | [0.125 0 0 0.125 138 118]ST 343 | 2909 407 2921 412 DL 344 | [0.125 0 0 0.125 138 118]ST 345 | 2895 402 2909 407 DL 346 | [0.125 0 0 0.125 138 118]ST 347 | 2877 396 2895 402 DL 348 | [0.125 0 0 0.125 138 118]ST 349 | 2853 388 2877 396 DL 350 | [0.125 0 0 0.125 138 118]ST 351 | 2821 378 2853 388 DL 352 | [0.125 0 0 0.125 138 118]ST 353 | 2775 366 2821 378 DL 354 | [0.125 0 0 0.125 138 118]ST 355 | 2703 351 2775 366 DL 356 | [0.125 0 0 0.125 138 118]ST 357 | 2639 343 2703 351 DL 358 | [0.125 0 0 0.125 138 118]ST 359 | 2577 338 2639 343 DL 360 | [0.125 0 0 0.125 138 118]ST 361 | 2515 338 2577 HL 362 | [0.125 0 0 0.125 138 118]ST 363 | 2452 342 2515 338 DL 364 | [0.125 0 0 0.125 138 118]ST 365 | 2419 346 2452 342 DL 366 | [0.125 0 0 0.125 138 118]ST 367 | 2386 351 2419 346 DL 368 | [0.125 0 0 0.125 138 118]ST 369 | 2352 357 2386 351 DL 370 | [0.125 0 0 0.125 138 118]ST 371 | 2318 365 2352 357 DL 372 | [0.125 0 0 0.125 138 118]ST 373 | 2283 375 2318 365 DL 374 | [0.125 0 0 0.125 138 118]ST 375 | 2247 386 2283 375 DL 376 | [0.125 0 0 0.125 138 118]ST 377 | 2211 400 2247 386 DL 378 | [0.125 0 0 0.125 138 118]ST 379 | 2175 416 2211 400 DL 380 | [0.125 0 0 0.125 138 118]ST 381 | 2138 434 2175 416 DL 382 | [0.125 0 0 0.125 138 118]ST 383 | 2101 455 2138 434 DL 384 | [0.125 0 0 0.125 138 118]ST 385 | 2063 479 2101 455 DL 386 | [0.125 0 0 0.125 138 118]ST 387 | 2026 506 2063 479 DL 388 | [0.125 0 0 0.125 138 118]ST 389 | 2008 520 2026 506 DL 390 | [0.125 0 0 0.125 138 118]ST 391 | 1990 536 2008 520 DL 392 | [0.125 0 0 0.125 138 118]ST 393 | 1954 570 1990 536 DL 394 | [0.125 0 0 0.125 138 118]ST 395 | 1920 607 1954 570 DL 396 | [0.125 0 0 0.125 138 118]ST 397 | 1903 628 1920 607 DL 398 | [0.125 0 0 0.125 138 118]ST 399 | 1887 649 1903 628 DL 400 | [0.125 0 0 0.125 138 118]ST 401 | 1871 671 1887 649 DL 402 | [0.125 0 0 0.125 138 118]ST 403 | 1856 695 1871 671 DL 404 | [0.125 0 0 0.125 138 118]ST 405 | 1842 719 1856 695 DL 406 | [0.125 0 0 0.125 138 118]ST 407 | 1828 745 1842 719 DL 408 | [0.125 0 0 0.125 138 118]ST 409 | 1816 771 1828 745 DL 410 | [0.125 0 0 0.125 138 118]ST 411 | 1804 799 1816 771 DL 412 | [0.125 0 0 0.125 138 118]ST 413 | 1793 828 1804 799 DL 414 | [0.125 0 0 0.125 138 118]ST 415 | 1783 857 1793 828 DL 416 | [0.125 0 0 0.125 138 118]ST 417 | 1774 888 1783 857 DL 418 | [0.125 0 0 0.125 138 118]ST 419 | 1767 920 1774 888 DL 420 | [0.125 0 0 0.125 138 118]ST 421 | 1761 953 1767 920 DL 422 | [0.125 0 0 0.125 138 118]ST 423 | 1756 986 1761 953 DL 424 | [0.125 0 0 0.125 138 118]ST 425 | 1753 1021 1756 986 DL 426 | [0.125 0 0 0.125 138 118]ST 427 | 1751 1056 1753 1021 DL 428 | [0.125 0 0 0.125 138 118]ST 429 | 1751 1092 1056 VL 430 | [0.125 0 0 0.125 138 118]ST 431 | 1753 1129 1751 1092 DL 432 | [0.125 0 0 0.125 138 118]ST 433 | 1 2 215 d2 0 0 PE 434 | 11150 38483 -6676 -39506 DL 435 | [0.125 0 0 0.125 138 118]ST 436 | 34059 -21235 -30112 26537 DL 437 | [0.125 0 0 0.125 138 118]ST 438 | -39806 4291 40027 -872 DL 439 | [0.125 0 0 0.125 138 118]ST 440 | 17189 -36197 -12768 37982 DL 441 | [0.125 0 0 0.125 138 118]ST 442 | -36227 -16970 35712 18028 DL 443 | [0.125 0 0 0.125 138 118]ST 444 | -26689 -29832 28864 27734 DL 445 | [0.125 0 0 0.125 138 118]ST 446 | 0 0 B 0 0 PE 447 | 1265 3123 64 64 E 448 | [0.125 0 0 0.125 138 118]ST 449 | 5116 3127 64 64 E 450 | [0.125 0 0 0.125 138 118]ST 451 | 2674 1510 64 64 E 452 | [0.125 0 0 0.125 138 118]ST 453 | 2534 1971 64 64 E 454 | [0.125 0 0 0.125 138 118]ST 455 | 1 2 127 d2 0 0 PE 456 | 14496 -37405 -8683 39164 DL 457 | [0.125 0 0 0.125 138 118]ST 458 | 26356 30163 -23056 -32754 DL 459 | [0.125 0 0 0.125 138 118]ST 460 | -27466 29256 31777 -24505 DL 461 | [0.125 0 0 0.125 138 118]ST 462 | 0 0 B 0 0 PE 463 | 3512 1084 64 64 E 464 | [0.125 0 0 0.125 138 118]ST 465 | 1 2 B 0 0 PE 466 | -24013 32135 28611 -28120 DL 467 | [0.125 0 0 0.125 138 118]ST 468 | 39997 3191 -40003 3116 DL 469 | [0.125 0 0 0.125 138 118]ST 470 | -33239 -22254 33469 21907 DL 471 | [0.125 0 0 0.125 138 118]ST 472 | 1757 1172 1753 1134 DL 473 | [0.125 0 0 0.125 138 118]ST 474 | 1763 1211 1757 1172 DL 475 | [0.125 0 0 0.125 138 118]ST 476 | 1771 1252 1763 1211 DL 477 | [0.125 0 0 0.125 138 118]ST 478 | 1782 1295 1771 1252 DL 479 | [0.125 0 0 0.125 138 118]ST 480 | 1796 1339 1782 1295 DL 481 | [0.125 0 0 0.125 138 118]ST 482 | 1803 1361 1796 1339 DL 483 | [0.125 0 0 0.125 138 118]ST 484 | 1812 1384 1803 1361 DL 485 | [0.125 0 0 0.125 138 118]ST 486 | 1832 1430 1812 1384 DL 487 | [0.125 0 0 0.125 138 118]ST 488 | 1856 1477 1832 1430 DL 489 | [0.125 0 0 0.125 138 118]ST 490 | 1883 1525 1856 1477 DL 491 | [0.125 0 0 0.125 138 118]ST 492 | 1914 1573 1883 1525 DL 493 | [0.125 0 0 0.125 138 118]ST 494 | 1950 1621 1914 1573 DL 495 | [0.125 0 0 0.125 138 118]ST 496 | 1990 1669 1950 1621 DL 497 | [0.125 0 0 0.125 138 118]ST 498 | 2034 1715 1990 1669 DL 499 | [0.125 0 0 0.125 138 118]ST 500 | 2083 1761 2034 1715 DL 501 | [0.125 0 0 0.125 138 118]ST 502 | 2136 1804 2083 1761 DL 503 | [0.125 0 0 0.125 138 118]ST 504 | 2193 1845 2136 1804 DL 505 | [0.125 0 0 0.125 138 118]ST 506 | 2254 1883 2193 1845 DL 507 | [0.125 0 0 0.125 138 118]ST 508 | 2319 1918 2254 1883 DL 509 | [0.125 0 0 0.125 138 118]ST 510 | 2387 1948 2319 1918 DL 511 | [0.125 0 0 0.125 138 118]ST 512 | 2458 1974 2387 1948 DL 513 | [0.125 0 0 0.125 138 118]ST 514 | 2531 1995 2458 1974 DL 515 | [0.125 0 0 0.125 138 118]ST 516 | 2568 2004 2531 1995 DL 517 | [0.125 0 0 0.125 138 118]ST 518 | 2605 2011 2568 2004 DL 519 | [0.125 0 0 0.125 138 118]ST 520 | 2643 2016 2605 2011 DL 521 | [0.125 0 0 0.125 138 118]ST 522 | 2680 2021 2643 2016 DL 523 | [0.125 0 0 0.125 138 118]ST 524 | 2718 2023 2680 2021 DL 525 | [0.125 0 0 0.125 138 118]ST 526 | 2755 2025 2718 2023 DL 527 | [0.125 0 0 0.125 138 118]ST 528 | 2793 2025 2755 HL 529 | [0.125 0 0 0.125 138 118]ST 530 | 2829 2023 2793 2025 DL 531 | [0.125 0 0 0.125 138 118]ST 532 | 2866 2020 2829 2023 DL 533 | [0.125 0 0 0.125 138 118]ST 534 | 2902 2016 2866 2020 DL 535 | [0.125 0 0 0.125 138 118]ST 536 | 2937 2010 2902 2016 DL 537 | [0.125 0 0 0.125 138 118]ST 538 | 2972 2003 2937 2010 DL 539 | [0.125 0 0 0.125 138 118]ST 540 | 3006 1995 2972 2003 DL 541 | [0.125 0 0 0.125 138 118]ST 542 | 3039 1985 3006 1995 DL 543 | [0.125 0 0 0.125 138 118]ST 544 | 3072 1974 3039 1985 DL 545 | [0.125 0 0 0.125 138 118]ST 546 | 3103 1962 3072 1974 DL 547 | [0.125 0 0 0.125 138 118]ST 548 | 3134 1949 3103 1962 DL 549 | [0.125 0 0 0.125 138 118]ST 550 | 3164 1935 3134 1949 DL 551 | [0.125 0 0 0.125 138 118]ST 552 | 3192 1920 3164 1935 DL 553 | [0.125 0 0 0.125 138 118]ST 554 | 3220 1904 3192 1920 DL 555 | [0.125 0 0 0.125 138 118]ST 556 | 3246 1887 3220 1904 DL 557 | [0.125 0 0 0.125 138 118]ST 558 | 3271 1869 3246 1887 DL 559 | [0.125 0 0 0.125 138 118]ST 560 | 3295 1851 3271 1869 DL 561 | [0.125 0 0 0.125 138 118]ST 562 | 3318 1832 3295 1851 DL 563 | [0.125 0 0 0.125 138 118]ST 564 | 3340 1812 3318 1832 DL 565 | [0.125 0 0 0.125 138 118]ST 566 | 3361 1792 3340 1812 DL 567 | [0.125 0 0 0.125 138 118]ST 568 | 3380 1771 3361 1792 DL 569 | [0.125 0 0 0.125 138 118]ST 570 | 3398 1750 3380 1771 DL 571 | [0.125 0 0 0.125 138 118]ST 572 | 3416 1729 3398 1750 DL 573 | [0.125 0 0 0.125 138 118]ST 574 | 3432 1707 3416 1729 DL 575 | [0.125 0 0 0.125 138 118]ST 576 | 3447 1685 3432 1707 DL 577 | [0.125 0 0 0.125 138 118]ST 578 | 3461 1663 3447 1685 DL 579 | [0.125 0 0 0.125 138 118]ST 580 | 3485 1618 3461 1663 DL 581 | [0.125 0 0 0.125 138 118]ST 582 | 3496 1596 3485 1618 DL 583 | [0.125 0 0 0.125 138 118]ST 584 | 3506 1574 3496 1596 DL 585 | [0.125 0 0 0.125 138 118]ST 586 | 3523 1529 3506 1574 DL 587 | [0.125 0 0 0.125 138 118]ST 588 | 3537 1485 3523 1529 DL 589 | [0.125 0 0 0.125 138 118]ST 590 | 3547 1441 3537 1485 DL 591 | [0.125 0 0 0.125 138 118]ST 592 | 3555 1398 3547 1441 DL 593 | [0.125 0 0 0.125 138 118]ST 594 | 3558 1377 3555 1398 DL 595 | [0.125 0 0 0.125 138 118]ST 596 | 3560 1356 3558 1377 DL 597 | [0.125 0 0 0.125 138 118]ST 598 | 3562 1315 3560 1356 DL 599 | [0.125 0 0 0.125 138 118]ST 600 | 3563 1276 3562 1315 DL 601 | [0.125 0 0 0.125 138 118]ST 602 | 3561 1237 3563 1276 DL 603 | [0.125 0 0 0.125 138 118]ST 604 | 3558 1200 3561 1237 DL 605 | [0.125 0 0 0.125 138 118]ST 606 | 3553 1164 3558 1200 DL 607 | [0.125 0 0 0.125 138 118]ST 608 | 3547 1129 3553 1164 DL 609 | [0.125 0 0 0.125 138 118]ST 610 | 3539 1095 3547 1129 DL 611 | [0.125 0 0 0.125 138 118]ST 612 | 3531 1063 3539 1095 DL 613 | [0.125 0 0 0.125 138 118]ST 614 | 3510 1002 3531 1063 DL 615 | [0.125 0 0 0.125 138 118]ST 616 | 3487 945 3510 1002 DL 617 | [0.125 0 0 0.125 138 118]ST 618 | 3462 892 3487 945 DL 619 | [0.125 0 0 0.125 138 118]ST 620 | 3434 843 3462 892 DL 621 | [0.125 0 0 0.125 138 118]ST 622 | 3404 796 3434 843 DL 623 | [0.125 0 0 0.125 138 118]ST 624 | 3388 774 3404 796 DL 625 | [0.125 0 0 0.125 138 118]ST 626 | 3372 752 3388 774 DL 627 | [0.125 0 0 0.125 138 118]ST 628 | 3337 709 3372 752 DL 629 | [0.125 0 0 0.125 138 118]ST 630 | 3298 666 3337 709 DL 631 | [0.125 0 0 0.125 138 118]ST 632 | 3249 618 3298 666 DL 633 | [0.125 0 0 0.125 138 118]ST 634 | 3216 589 3249 618 DL 635 | [0.125 0 0 0.125 138 118]ST 636 | 3191 569 3216 589 DL 637 | [0.125 0 0 0.125 138 118]ST 638 | 3172 554 3191 569 DL 639 | [0.125 0 0 0.125 138 118]ST 640 | 3157 543 3172 554 DL 641 | [0.125 0 0 0.125 138 118]ST 642 | 3145 534 3157 543 DL 643 | [0.125 0 0 0.125 138 118]ST 644 | 3136 528 3145 534 DL 645 | [0.125 0 0 0.125 138 118]ST 646 | 3127 522 3136 528 DL 647 | [0.125 0 0 0.125 138 118]ST 648 | 3120 517 3127 522 DL 649 | [0.125 0 0 0.125 138 118]ST 650 | 3114 513 3120 517 DL 651 | [0.125 0 0 0.125 138 118]ST 652 | 3109 510 3114 513 DL 653 | [0.125 0 0 0.125 138 118]ST 654 | 3105 507 3109 510 DL 655 | [0.125 0 0 0.125 138 118]ST 656 | 3101 504 3105 507 DL 657 | [0.125 0 0 0.125 138 118]ST 658 | 3097 502 3101 504 DL 659 | [0.125 0 0 0.125 138 118]ST 660 | 3032 464 3097 502 DL 661 | [0.125 0 0 0.125 138 118]ST 662 | 2968 432 3032 464 DL 663 | [0.125 0 0 0.125 138 118]ST 664 | 2964 430 2968 432 DL 665 | [0.125 0 0 0.125 138 118]ST 666 | 2960 429 2964 430 DL 667 | [0.125 0 0 0.125 138 118]ST 668 | 2956 427 2960 429 DL 669 | [0.125 0 0 0.125 138 118]ST 670 | 2951 424 2956 427 DL 671 | [0.125 0 0 0.125 138 118]ST 672 | 2945 422 2951 424 DL 673 | [0.125 0 0 0.125 138 118]ST 674 | 2938 419 2945 422 DL 675 | [0.125 0 0 0.125 138 118]ST 676 | 2930 416 2938 419 DL 677 | [0.125 0 0 0.125 138 118]ST 678 | 2921 412 2930 416 DL 679 | [0.125 0 0 0.125 138 118]ST 680 | 2909 407 2921 412 DL 681 | [0.125 0 0 0.125 138 118]ST 682 | 2895 402 2909 407 DL 683 | [0.125 0 0 0.125 138 118]ST 684 | 2877 396 2895 402 DL 685 | [0.125 0 0 0.125 138 118]ST 686 | 2853 388 2877 396 DL 687 | [0.125 0 0 0.125 138 118]ST 688 | 2821 378 2853 388 DL 689 | [0.125 0 0 0.125 138 118]ST 690 | 2775 366 2821 378 DL 691 | [0.125 0 0 0.125 138 118]ST 692 | 2703 351 2775 366 DL 693 | [0.125 0 0 0.125 138 118]ST 694 | 2639 343 2703 351 DL 695 | [0.125 0 0 0.125 138 118]ST 696 | 2577 338 2639 343 DL 697 | [0.125 0 0 0.125 138 118]ST 698 | 2515 338 2577 HL 699 | [0.125 0 0 0.125 138 118]ST 700 | 2452 342 2515 338 DL 701 | [0.125 0 0 0.125 138 118]ST 702 | 2419 346 2452 342 DL 703 | [0.125 0 0 0.125 138 118]ST 704 | 2386 351 2419 346 DL 705 | [0.125 0 0 0.125 138 118]ST 706 | 2352 357 2386 351 DL 707 | [0.125 0 0 0.125 138 118]ST 708 | 2318 365 2352 357 DL 709 | [0.125 0 0 0.125 138 118]ST 710 | 2283 375 2318 365 DL 711 | [0.125 0 0 0.125 138 118]ST 712 | 2247 386 2283 375 DL 713 | [0.125 0 0 0.125 138 118]ST 714 | 2211 400 2247 386 DL 715 | [0.125 0 0 0.125 138 118]ST 716 | 2175 416 2211 400 DL 717 | [0.125 0 0 0.125 138 118]ST 718 | 2138 434 2175 416 DL 719 | [0.125 0 0 0.125 138 118]ST 720 | 2101 455 2138 434 DL 721 | [0.125 0 0 0.125 138 118]ST 722 | 2063 479 2101 455 DL 723 | [0.125 0 0 0.125 138 118]ST 724 | 2026 506 2063 479 DL 725 | [0.125 0 0 0.125 138 118]ST 726 | 2008 520 2026 506 DL 727 | [0.125 0 0 0.125 138 118]ST 728 | 1990 536 2008 520 DL 729 | [0.125 0 0 0.125 138 118]ST 730 | 1954 570 1990 536 DL 731 | 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-------------------------------------------------------------------------------- 1 | @book {etale, 2 | AUTHOR = {Tamme, G{\"u}nter}, 3 | TITLE = {Introduction to \'etale cohomology}, 4 | SERIES = {Universitext}, 5 | NOTE = {Translated from the German by Manfred Kolster}, 6 | PUBLISHER = {Springer-Verlag}, 7 | ADDRESS = {Berlin}, 8 | YEAR = {1994}, 9 | PAGES = {x+186}, 10 | ISBN = {3-540-57116-7}, 11 | MRCLASS = {14F20}, 12 | MRNUMBER = {1317816 (95k:14033)}, 13 | DOI = {10.1007/978-3-642-78421-7}, 14 | URL = {http://dx.doi.org/10.1007/978-3-642-78421-7}, 15 | } 16 | 17 | @misc{stacks-project, 18 | author = {The {Stacks Project Authors}}, 19 | title = {{\itshape Stacks Project}}, 20 | howpublished = {\url{http://stacks.math.columbia.edu}}, 21 | year = {2013}, 22 | } 23 | 24 | @book {milne, 25 | AUTHOR = {Milne, James S.}, 26 | TITLE = {\'{E}tale cohomology}, 27 | SERIES = {Princeton Mathematical Series}, 28 | VOLUME = {33}, 29 | PUBLISHER = {Princeton University Press}, 30 | ADDRESS = {Princeton, N.J.}, 31 | YEAR = {1980}, 32 | PAGES = {xiii+323}, 33 | ISBN = {0-691-08238-3}, 34 | MRCLASS = {14-02 (14F20 18F99)}, 35 | MRNUMBER = {559531 (81j:14002)}, 36 | MRREVIEWER = {G. Horrocks}, 37 | } 38 | 39 | -------------------------------------------------------------------------------- /sheaf-coh.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/notzeb/all/97969592a92b61f13beb5e90dc945e29e8cd5ebd/sheaf-coh.pdf -------------------------------------------------------------------------------- /sumproduct.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/notzeb/all/97969592a92b61f13beb5e90dc945e29e8cd5ebd/sumproduct.pdf -------------------------------------------------------------------------------- /triangulargrid.eps: -------------------------------------------------------------------------------- 1 | %!PS-Adobe-2.0 EPSF-2.0 2 | %%BoundingBox: 112 108 483 735 3 | %%HiResBoundingBox: 112.500000 108.000000 482.500000 735.000000 4 | %%Creator: Qt 3.3.8b 5 | %%CreationDate: Wed Jun 10 14:54:22 2015 6 | %%DocumentFonts: 7 | 8 | %%EndComments 9 | % EPSF created by ps2eps 1.68 10 | %%BeginProlog 11 | save 12 | countdictstack 13 | mark 14 | newpath 15 | /showpage {} def 16 | /setpagedevice {pop} def 17 | %%EndProlog 18 | %%Page 1 1 19 | %%BeginProlog 20 | % Prolog copyright 1994-2006 Trolltech. 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