From: Martin Mares Date: Thu, 8 Oct 2009 13:47:03 +0000 (+0200) Subject: Prednaska o tocich. X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;ds=inline;h=e726bb8d6391faadfaf4366091eb109179a36861;p=ads2.git Prednaska o tocich. --- diff --git a/1-toky/1-toky.tex b/1-toky/1-toky.tex new file mode 100644 index 0000000..fe5b8f9 --- /dev/null +++ b/1-toky/1-toky.tex @@ -0,0 +1,182 @@ +\input lecnotes.tex + +\prednaska{2}{Toky v sítích}{(zapsala Markéta Popelová)} + +\s{Motivaèní úloha:} Rozvod èajovodu do~v¹ech uèeben + +Pøedstavme si, ¾e~by v~budovì fakulty na~Malé Stranì existoval èajovod, který by rozvádìl èaj do~ka¾dé uèebny. Znázornìme si to~na~orientovaném grafu, kde by jeden vrchol pøedstavoval èajovar a~druhý uèebnu, ve~které sedíme. Jak rozvést co nejefektivnìji dostatek èaje do~dané uèebny? + +\figure{toky01.eps}{Èajovod}{2in} + +\s{Definice:} {\I Sí»} je uspoøádaná pìtice $(V,E,z,s,c)$, kde platí: +\itemize\ibull +\:$(V,E)$ je orientovaný graf. +\:$c:E\to{\bb R}_{0}^{+}$ je funkce znázoròující kapacitu hran. +\:$z,s \in V$ jsou dva vrcholy grafu, kterým øíkáme zdroj a~stok (spotøebiè). +\:Graf je symetrický, tedy $\forall u,v \in V: uv \in E \Leftrightarrow vu \in E$ (tuto podmínku si~mù¾eme zvolit bez~újmy na~obecnosti, nebo» v¾dy mohu do~grafu pøidat hranu, která v~nìm je¹tì nebyla, a~dát jí nulovou kapacitu). +\endlist + +\figure{sit.eps}{Pøíklad sítì. Èísla pøedstavují kapacity jednotlivých hran.}{2.5in} + +\s{Definice:} {\I Tok} je funkce $f:E \to {\bb R}_{0}^{+}$ taková, ¾e~platí: +\numlist{\ndotted} +\:Tok po~ka¾dé hranì je omezen její kapacitou: $\forall e \in E : 0\le f(e)\le c(e)$. +\:Kirchhoffùv zákon: $$\forall v \in V \setminus \{z,s\}: \sum_{u: uv \in E}{f(uv)}=\sum_{u: vu \in E}{f(vu)}.$$ Neboli pro~ka¾dý vrchol kromì zdroje a~stoku platí, ¾e~to, co do~nìj pøitéká, je stejnì velké jako to, co z~nìj odtéká. +\endlist + +\s{Poznámka:} Pro~zjednodu¹ení zaveïme speciální znaèení: +\itemize\ibull +\:$f^+(v) = \sum_{u: uv \in E}{f(uv)}$ (to, co do~vrcholu pøitéká) +\:$f^-(v) = \sum_{u: vu \in E}{f(vu)}$ (to, co z~vrcholu odtéká) +\:$f^\Delta(v) = f^+(v) - f^-(v)$ (rozdíl tìchto hodnot) +\endlist +Pak mù¾eme Krichhoffùv zákon zapsat jednodu¹e jako: $$\forall v \in V \setminus \{z,s\}: f^\Delta(v) = 0.$$ + +\s{Poznámka:} Pokud bychom se~chtìli v~definici toku u~bodu 2 vyhnout podmínkám pro~$z$ a~$s$, mù¾eme zdroj a~stok vzájemnì propojit (pak jde o~tzv. cirkulaci). + +\s{Poznámka:} V~angliètinì se~obvykle zdroj znaèí \uv{$s$} a~stok \uv{$t$} (jako source a~target). + +\figure{tok.eps}{Pøíklad toku. Èísla pøedstavují toky po~hranách, v~závorkách jsou kapacity.}{4in} + +\s{Pozorování:} Nìjaký tok v¾dy existuje. V libovolné síti mù¾eme v¾dy zvolit funkci nulovou (po~¾ádné hranì nic nepoteèe). Tato funkce splòuje podmínky toku, a~tedy takovýto nulový tok je zcela korektní. + +\s{Definice:} {\I Velikost toku} $f$ je: $$\vert f\vert:=f^\Delta(s).$$ + +\s{Cíl:} Budeme chtít najít v~zadané síti tok, jeho¾ velikost je maximální. + +\s{Otázka:} Má vùbec smysl mluvit o~maximálním toku? Bude v¾dy existovat? Nevybíráme zde toti¾ z~koneènì mnoha pøípadù a~na~první pohled není jasné, ¾e~supremum mno¾iny v¹ech tokù bude zároveò i~maximum této mno¾iny. + +\s{Odpovìï:} Ano, pro~ka¾dou sí» existuje maximální tok. Toto pomìrnì pøekvapivé tvrzení mù¾eme nahlédnout za~pomoci matematické analýzy. Nástin dùkazu je takový, ¾e~mno¾ina tokù je kompaktní a~velikost toku je spojitá (dokonce lineární) funkce z~mno¾iny tokù do~${\bb R}$. Proto nabývá velikost toku na~mno¾inì v¹ech tokù svého maxima. + +\s{Poznámka:} Pro~na¹e pøípady pøedpokládejme, ¾e~kapacity jsou racionální. Pomìrnì nám to zjednodu¹í práci a~pøíli¹ nám to neublí¾í, nebo» práce s~reálnými èísly je stejnì pro~informatika pomìrnì zapeklitá. + +\s{První øe¹ení:} Hledejme cestu $P$ ze~$z$ do~$s$ takovou, ¾e~$\forall e \in P: f(e) < c(e)$ (po~v¹ech jejích hranách teèe ostøe ménì, ne¾ jim dovolují jejich kapacity). Pak zjevnì mù¾eme tok upravit tak, aby se~jeho velikost zvìt¹ila. Zvolme $$\epsilon := \min_{e \in P} c(e) - f(e).$$ Nový tok $f'$ pak definujme jako $f'(e):=f(e) + \epsilon$. Kapacity nepøekroèíme ($\epsilon$ je nejvìt¹í mo¾ná hodnota, abychom tok zvìt¹ili, ale nepøekroèili kapaicitu ani jedné z~hran cesty $P$) a~Kirchhoffovy zákony zùstanou neporu¹eny, nebo» zdroj a~stok nezahrnují a~ka¾dému jinému vrcholu na~cestì $P$ se~pøítok $f^+(v)$ i~odtok $f^-(v)$ zvìt¹í pøesnì o~$\epsilon$. + +\s{Otázka:} Najdeme takto ov¹em opravdu maximální tok? + +\s{Odpovìï:} Nemusíme. Napø. na~obrázku je vidìt, ¾e~kdy¾ najdeme nejdøíve cestu pøes hranu s~kapacitou 1 (na obrázku tuènì) a~u¾ hodnotu toku na~této hranì nesní¾íme, tak dosáhneme velikost toku nejvý¹e 19. Ale maximální tok této sítì má velikost 20. + +\figure{toky02.eps}{Èísla pøedstavují kapacity jednotlivých hran.}{1.5in} + +Nìkdy je potøeba poslat nìco i~v~protismìu. Definujme si~tedy rezervu hrany. Vyu¾ijeme zde, ¾e~sí» je symetrická. + +\s{Definice:} {\I Rezerva hrany} $uv$ je $r(uv):=c(uv) - f(uv) + f(vu).$ + +Uka¾me si~tedy algoritmus, který rezervu vyu¾ívá a~doka¾me, ¾e~je koneèný a~najde maximální tok ke~ka¾dé racionální síti. + +\s{Algoritmus (Fordùv-Fulkersonùv)} + +\algo +\:$f \leftarrow$ libovolný tok, napø. v¹ude nulový ($\forall e \in E: f(e) \leftarrow 0 $). +\:Dokud $\exists P$ cesta ze $z$ do $s$ taková, ¾e~$\forall e \in P: r(e) > 0$, opakujeme: +\::$\epsilon \leftarrow \min \{r(e) \mid e \in P\}$. +\::Pro v¹echny hrany $uv \in P$: +\:::$\delta \leftarrow min\{f(uv),\epsilon\}$ +\:::$f(vu) \leftarrow f(vu) - \delta$ +\:::$f(uv) \leftarrow f(uv) + \epsilon - \delta$ +\:Prohlásíme $f$ za~maximální tok. +\endalgo + +\s{Problém:} Zastaví se~Fordùv-Fulkersonùv algoritmus? + +\itemize\ibull +\:Pro~celoèíselné kapacity se~v~ka¾dém kroku zvìt¹í velikost toku alespoò o~1. Algoritmus se~tedy zasatví po~nejvíce tolika krocích, jako je nìjaká horní závora pro~velikost maximálního toku -- napø. souèet kapacit v¹ech hran vedoucích do~stoku $$\sum_{u:us \in E}{c(us)}.$$ +\:Pro~racionální kapacity vyu¾ijeme jednoduchý trik -- kapacity vynásobíme spoleèným jmenovatelem a~pøevedeme na~pùvodní pøípad. Uvìdomme si, ¾e~algoritmus nikde kapacity hran nedìlí, tak¾e u¾ zùstanou celoèíselné, a~algoritmus se~tedy zastaví. +\:Na~síti s~iracionálními kapacitami se~algoritmus chová mnohdy divoce, nemusí se~zastavit ale dokonce ani konvergovat ke~správnému výsledku. +\s{K zamy¹lení:} Zkuste vymyslet pøíklad takové sítì. +\endlist + +\s{Otázka:} Vydá algoritmus maximální tok? + +\s{Odpovìï:} Vydá. Abychom si~to dokázali, zaveïme si~øezy a~pou¾ijme je jako certifikát pravdivosti tvrzení, ¾e~algoritmus vydá maximální tok. + +\s{Definice:} {\I Øez} je uspoøádaná dvojice mno¾in vrcholù ($A,B$) taková, ¾e: +\itemize\ibull +\:$A \cap B = \emptyset$. +\:$A \cup B = V$. +\:$z \in A$. +\:$s \notin B$. +\endlist + +\s{Definice:} {\I Hrany øezu} $E(A,B) := E \cap A \times B$. + +\s{Poznámka:} Øezy se~dají definovat více zpùsoby, jedna z~definic je, ¾e~øez je mno¾ina hran grafu takových, ¾e~po~jejich odebrání se~graf rozpadne na~více komponent. Tuto definici splòuje i~ta na¹e, ale ne naopak. + +\s{Definice:} {\I Kapacita øezu} je $$c(A,B) := \sum_{e \in E(A,B)}{c(e)}.$$ + +\s{Definice:} {\I Tok pøes øez} je $$f(A,B) := \sum_{e \in E(A,B)}{f(e)} - \sum_{e \in E(B,A)}{f(e)}.$$ + +\s{Pozorování:} Pro~ka¾dý tok $f$ a~ka¾dý øez $(A,B)$ platí, ¾e~$f(A,B) \leq c(A,B)$. + +\proof +$$f(A,B) = \sum_{e \in (A,B)}{f(e)} - \sum_{e \in E(B,A)}{f(e)} \leq \sum_{e \in E(A,B)}{f(e)} \leq \sum_{e \in E(A,B)}{c(e)} = c(A,B).$$ +\qed + +\s{Lemmátko:} Pro~ka¾dý tok $f$ a~pro~ka¾dý øez $(A,B)$ platí $f(A,B) = \vert f \vert$. + +\proof{Indukcí a~obrázkem} + +Zaèneme s øezem ($V \setminus \{s\}, \{s\}$). Pro~tento øez lemma platí z~definice velikosti toku. + +Dále si~pøedstavme, ¾e~máme ji¾ libovolný øez ($A,B$) a~pøesouváme vrchol $v$ z~$A$ do~$B$. Tedy $A' = A \setminus \{v\}$ a $B' = B \cup \{v\}$. + +Uvìdomme si, ¾e~v¹echny hrany jednoho typu (napø. vedoucí z~$A$ do~$v$) se~chovají stejnì, tak¾e staèí uva¾ovat hrany pouze 4 typù: +\itemize\ibull +\:$\alpha$ -- hrany vedoucí z~$A$ do~$v$ +\:$\beta$ -- hrany vedoucí z~$v$ do~$B$ +\:$\gamma$ -- hrany vedoucí z~$v$ do~$A$ +\:$\delta$ -- hrany vedoucí z~$B$ do~$v$ +\:$\epsilon$ -- hrany øezu stejné pøed pøesunem i~po~pøesunu +\endlist + +\figure{toky03.eps}{Pøesun vrcholu $v$ z~$A$ do~$B$.}{1in} + +Potom pøed pøesunem ($v \in A$) se~$f(A,B)$ skládá z~$\epsilon + \beta - \delta$. Po~pøesunu ($v \in B$) se~$f(A',B')$ skládá z~$\epsilon + \alpha - \gamma$. Rozdíl tìchto hodnot je $\alpha + \delta - \beta - \gamma$. + +Nicménì z Kirchhoffova zákonu o~vrcholu $v$ (co¾ není ani zdroj ani stok) víme, ¾e~$\alpha + \delta - \beta - \gamma = f^\Delta(v) = 0$, nebo» $\alpha + \delta $ je to, co do~$v$ pøitéká a~$\beta + \gamma$ je to, co z~$v$ vytéká. Tedy tok pøes øez pøed pøesunem je stejnì velký jako tok pøes øez po~pøesunu. Pokud tedy lemma platilo pøed pøesunem, musí platit i~po~pøesunu. +\qed + +\s{Dùsledek:} Pro~ka¾dý tok $f$ a~øez ($A,B$) platí, ¾e~$\vert f \vert = f(A,B) \leq c(A,B)$. + +\s{Pozorování:} Pokud najdeme dvojici tok $f$ a~øez $(A,B)$ takovou, ¾e~platí $\vert f \vert = c(A,B)$, tak jsme na¹li maximální tok a~minimální øez. + +\s{Vìta:} Pokud se~Fordùv-Fulkersonùv algoritmus zastaví, tak vydá maximální tok. + +\proof + +Nech» se~Fordùv-Fulkersonùv algoritmus zastaví. Definujme $A = \{v \in V ; \exists $ cesta ze~$z$ do~$v$ jdoucí po~hranách s~$r > 0\}$ a~$B = V \setminus A$. + +Uvìdomme si, ¾e~($A,B$) je øez, nebo» $z \in A$ (ze~$z$ do~$z$ existuje cesta délky 0) a~$s \notin B$ (kdyby $s \in B$, tak by musela existovat cesta ze~$z$ do~$s$ s~kladnou rezervou, tudí¾ by ale algoritmus neskonèil, nýbr¾ tuto cestu vzal a~stávající tok vylep¹il). + +Dále víme, ¾e~v¹echny hrany øezu mají nulovou rezervu, neboli $\forall uv \in E(A,B) : r(uv) = 0$ (kdyby mìla hrana $uv$ rezervu nenulovou, tedy kladnou, tak by patøil vrchol $v$ do~$A$). Proto po~v¹ech hranách øezu vedoucích z~$A$ do~$B$ teèe tolik, kolik jsou kapacity tìchto hran a~po~hranách vedoucích z~$B$ do~$A$ neteèe nic, tedy $f(uv) = c(uv)$ a $f(vu) = 0$. Máme tedy øez $(A,B)$ takový, ¾e~$f(A,B) = c(A,B)$. To znamená, ¾e~jsme na¹li maximální tok a~minimální øez. +\qed + +Zformulujme si, co jsme zjistili a~dokázali o~algoritmu pánù Forda a~Fulkersona. + +\s{Vìta:} Pro~sí» s~racionálními kapacitami se~Fordùv-Fulkersonùv algoritmus zastaví a~vydá maximální tok a~minimální øez. + +\s{Vìta:} (Fordova-Fulkersonova) +$$\min_{(A,B)} c(A,B) = \max_f \vert f \vert .$$ +\proof + +Ji¾ víme, ¾e~$\min_{(A,B)} c(A,B) \geq \max_f \vert f \vert$. Staèí tedy dokázat, ¾e~v¾dy existuje tok $f$ a~øez ($A,B$) takové, ¾e~$c(A,B) = \vert f \vert$. Pro~racionální kapacity nám Fordùv-Fulkersonùv algoritmus takový tok (maximální) a~øez (minimální) vydá. Jak je to ale s~reálnými kapacitami? Vyu¾ijeme tvrzení, ¾e~maximální tok existuje v¾dy. Pak mù¾eme spustit Fordùv-Fulkersonùv algoritmus rovnou na~tento maximální tok. Algoritmus se~nutnì ihned zastaví, nebo» neexistuje cesta, která by mìla alespoò jednu hranu s~kladnou rezervu. A~my víme, ¾e~pokud se~algoritmus zastaví, tak vydá minimální øez. Proto i~pro sí» s~reálnými kapacitami platí, ¾e~existuje maximální tok $f$ a~minimální øez $(A,B)$ a $c(A,B) = \vert f \vert$. +\qed + +\s{Vìta:} +Sí» s~celoèíselnými kapacitami má aspoò jeden z~maximálních tokù celoèíselný a~Fordùv-Fulkersonùv algoritmus takový tok najde. + +\proof +Kdy¾ dostane Fordùv-Fulkersonùv algoritmus celoèíselnou sí», tak najde maximální tok a~ten bude zase celoèíselný (algoritmus nikde nedìlí). +\qed + +\s{Aplikace:} Hledání maximálního párování v~bipartitních grafech. + +\s{Definice:} Mno¾ina hran $F \subseteq E$ sevnazývá párování $\equiv \forall e,f \in F : e \cap f = \emptyset$. + +\s{Definice:} Párování je maximální, pokud obsahuje nejvìt¹í mo¾ný poèet hran. + +Mìjme bipartitní graf $G = (V,E)$. Na~nìm hledáme maximální párování. Sestrojme si~sí» takovou, ¾e~vezmeme vrcholy $V$ grafu $G$ a~pøidáme k~nim dva speciální vrcholy $z$ (zdroj) a~$s$ (stok) a~ze~zdroje pøidáme hrany do~v¹ech vrcholù levé partity a~ze~v¹ech vrcholù pravé partity povedeme hrany do~stoku. V¹echny kapacity nastavme na~1. Nyní staèí jen na~tuto sí» spustit Fordùv-Fulkersonùv algoritmus a~a¾~dobìhne, tak prohlásit hrany s~kapacitami 1 za~maximální párování. + +\figure{toky04.eps}{Hledání maximálního párování v~bipartitním grafu.}{2in} + +\bye diff --git a/1-toky/Makefile b/1-toky/Makefile new file mode 100644 index 0000000..e82416b --- /dev/null +++ b/1-toky/Makefile @@ -0,0 +1,3 @@ +P=1-toky + +include ../Makerules diff --git a/1-toky/sit.eps b/1-toky/sit.eps new file mode 100644 index 0000000..f66f5e1 --- /dev/null +++ b/1-toky/sit.eps @@ -0,0 +1,828 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: Ipelib 60027 (Ipe 6.0 preview 27) +%%CreationDate: D:20071018214606 +%%LanguageLevel: 2 +%%BoundingBox: 107 284 460 489 +%%HiResBoundingBox: 107.459 284.641 459.645 488.218 +%%DocumentSuppliedResources: font OXRFMQ+CMR10 +%%+ font GKLBST+CMR12 +%%+ font OKRINM+CMMI12 +%%+ font PVGEOP+CMR17 +%%EndComments +%%BeginProlog +%%BeginResource: procset ipe 6.0 60027 +/ipe 40 dict def ipe begin +/np { newpath } def +/m { moveto } def +/l { lineto } def +/c { curveto } def +/h { closepath } def +/re { 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto + neg 0 rlineto closepath } def +/d { setdash } def +/w { setlinewidth } def +/J { setlinecap } def +/j { setlinejoin } def +/cm { [ 7 1 roll ] concat } def +/q { gsave } def +/Q { grestore } def +/g { setgray } def +/G { setgray } def +/rg { setrgbcolor } def +/RG { setrgbcolor } def +/S { stroke } def +/f* { eofill } def +/f { fill } def +/ipeMakeFont { + exch findfont + dup length dict begin + { 1 index /FID ne { def } { pop pop } ifelse } forall + /Encoding exch def + currentdict + end + definefont pop +} def +/ipeFontSize 0 def +/Tf { dup /ipeFontSize exch store selectfont } def +/Td { translate } def +/BT { gsave } def +/ET { grestore } def +/TJ { 0 0 moveto { dup type /stringtype eq + { show } { ipeFontSize mul -0.001 mul 0 rmoveto } ifelse +} forall } def +end +%%EndResource +%%EndProlog +%%BeginSetup +ipe begin +%%BeginResource: font OXRFMQ+CMR10 +%!PS-AdobeFont-1.1: CMR10 1.00B +%%CreationDate: 1992 Feb 19 19:54:52 +% Copyright (C) 1997 American Mathematical Society. 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/.notdef/one/two/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] +ipeMakeFont +%%BeginResource: font GKLBST+CMR12 +%!PS-AdobeFont-1.1: CMR12 1.0 +%%CreationDate: 1991 Aug 20 16:38:05 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR12) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /GKLBST+CMR12 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 50 /two put +dup 51 /three put +dup 52 /four put +dup 53 /five put +dup 54 /six put +dup 56 /eight put +dup 57 /nine put +readonly def +/FontBBox{-34 -251 988 750}readonly def +currentdict end +currentfile eexec +d9d66f633b846a97b686a97e45a3d0aa052a014267b7904eb3c0d3bd0b83d891 +016ca6ca4b712adeb258faab9a130ee605e61f77fc1b738abc7c51cd46ef8171 +9098d5fee67660e69a7ab91b58f29a4d79e57022f783eb0fbbb6d4f4ec35014f +d2decba99459a4c59df0c6eba150284454e707dc2100c15b76b4c19b84363758 +469a6c558785b226332152109871a9883487dd7710949204ddcf837e6a8708b8 +2bdbf16fbc7512faa308a093fe5cf4e9d2405b169cd5365d6eced5d768d66d6c 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/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] +ipeMakeFont +%%BeginResource: font OKRINM+CMMI12 +%!PS-AdobeFont-1.1: CMMI12 1.100 +%%CreationDate: 1996 Jul 27 08:57:55 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.100) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMMI12) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.04 def +/isFixedPitch false def +end readonly def +/FontName /OKRINM+CMMI12 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 25 /pi put +readonly def +/FontBBox{-30 -250 1026 750}readonly def +currentdict end +currentfile eexec +d9d66f633b846a97b686a97e45a3d0aa0529731c99a784ccbe85b4993b2eebde +3b12d472b7cf54651ef21185116a69ab1096ed4bad2f646635e019b6417cc77b +532f85d811c70d1429a19a5307ef63eb5c5e02c89fc6c20f6d9d89e7d91fe470 +b72befda23f5df76be05af4ce93137a219ed8a04a9d7d6fdf37e6b7fcde0d90b +986423e5960a5d9fbb4c956556e8df90cbfaec476fa36fd9a5c8175c9af513fe +d919c2ddd26bdc0d99398b9f4d03d6a8f05b47af95ef28a9c561dbdc98c47cf5 +5250011d19e9366eb6fd153d3a100caa6212e3d5d93990737f8d326d347b7edc 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR17) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /PVGEOP+CMR17 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 115 /s put +dup 122 /z put +readonly def +/FontBBox{-33 -250 945 749}readonly def +currentdict end +currentfile eexec +d9d66f633b846a97b686a97e45a3d0aa052a014267b7904eb3c0d3bd0b83d891 +016ca6ca4b712adeb258faab9a130ee605e61f77fc1b738abc7c51cd46ef8171 +9098d5fee67660e69a7ab91b58f29a4d79e57022f783eb0fbbb6d4f4ec35014f +d2decba99459a4c59df0c6eba150284454e707dc2100c15b76b4c19b84363758 +469a6c558785b226332152109871a9883487dd7710949204ddcf837e6a8708b8 +2bdbf16fbc7512faa308a093fe5f075ea0a10a15b0ed05d5039da41b32b16e95 +a3ce9725a429b35bad796912fc328e3a28f96fcada20a598e247755e7e7ff801 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--git a/1-toky/tok.eps b/1-toky/tok.eps new file mode 100644 index 0000000..3469634 --- /dev/null +++ b/1-toky/tok.eps @@ -0,0 +1,1143 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: Ipelib 60027 (Ipe 6.0 preview 27) +%%CreationDate: D:20071018214539 +%%LanguageLevel: 2 +%%BoundingBox: 107 274 593 486 +%%HiResBoundingBox: 107.873 274.347 592.993 485.737 +%%DocumentSuppliedResources: font IEYHEQ+CMR10 +%%+ font OLBANC+CMR12 +%%+ font BYMWUK+CMMI12 +%%+ font QRHILV+CMSY10 +%%+ font TLUUTQ+CMR17 +%%EndComments +%%BeginProlog +%%BeginResource: procset ipe 6.0 60027 +/ipe 40 dict def ipe begin +/np { newpath } def +/m { moveto } def +/l { lineto } def +/c { curveto } def +/h { closepath } def +/re { 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto + neg 0 rlineto closepath } def +/d { setdash } def +/w { setlinewidth } def +/J { setlinecap } def +/j { setlinejoin } def +/cm { [ 7 1 roll ] concat } def +/q { gsave } def +/Q { grestore } def +/g { setgray } def +/G { setgray } def +/rg { setrgbcolor } def +/RG { setrgbcolor } def +/S { stroke } def +/f* { eofill } def +/f { fill } def +/ipeMakeFont { + exch findfont + dup length dict begin + { 1 index /FID ne { def } { pop pop } ifelse } forall + /Encoding exch def + currentdict + end + definefont pop +} def +/ipeFontSize 0 def +/Tf { dup /ipeFontSize exch store selectfont } def +/Td { translate } def +/BT { gsave } def +/ET { grestore } def +/TJ { 0 0 moveto { dup type /stringtype eq + { show } { ipeFontSize mul -0.001 mul 0 rmoveto } ifelse +} forall } def +end +%%EndResource +%%EndProlog +%%BeginSetup +ipe begin +%%BeginResource: font IEYHEQ+CMR10 +%!PS-AdobeFont-1.1: CMR10 1.00B +%%CreationDate: 1992 Feb 19 19:54:52 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.00B) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR10) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /IEYHEQ+CMR10 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 49 /one put +dup 50 /two put +dup 55 /seven put +dup 57 /nine put +readonly def +/FontBBox{-251 -250 1009 969}readonly def +currentdict end +currentfile eexec +d9d66f633b846a97b686a97e45a3d0aa052a014267b7904eb3c0d3bd0b83d891 +016ca6ca4b712adeb258faab9a130ee605e61f77fc1b738abc7c51cd46ef8171 +9098d5fee67660e69a7ab91b58f29a4d79e57022f783eb0fbbb6d4f4ec35014f +d2decba99459a4c59df0c6eba150284454e707dc2100c15b76b4c19b84363758 +469a6c558785b226332152109871a9883487dd7710949204ddcf837e6a8708b8 +2bdbf16fbc7512faa308a093fe5cf7158f1163bc1f3352e22a1452e73feca8a4 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/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] +ipeMakeFont +%%BeginResource: font OLBANC+CMR12 +%!PS-AdobeFont-1.1: CMR12 1.0 +%%CreationDate: 1991 Aug 20 16:38:05 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR12) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /OLBANC+CMR12 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 40 /parenleft put +dup 41 /parenright put +dup 43 /plus put +dup 49 /one put +dup 50 /two put +dup 51 /three put +dup 52 /four put +dup 53 /five put +dup 54 /six put +dup 56 /eight put +dup 57 /nine put +readonly def +/FontBBox{-34 -251 988 750}readonly def +currentdict end +currentfile eexec +d9d66f633b846a97b686a97e45a3d0aa052a014267b7904eb3c0d3bd0b83d891 +016ca6ca4b712adeb258faab9a130ee605e61f77fc1b738abc7c51cd46ef8171 +9098d5fee67660e69a7ab91b58f29a4d79e57022f783eb0fbbb6d4f4ec35014f +d2decba99459a4c59df0c6eba150284454e707dc2100c15b76b4c19b84363758 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/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /parenleft/parenright/.notdef/plus/.notdef/.notdef/.notdef/.notdef + /.notdef/one/two/three/four/five/six/.notdef + /eight/nine/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] +ipeMakeFont +%%BeginResource: font BYMWUK+CMMI12 +%!PS-AdobeFont-1.1: CMMI12 1.100 +%%CreationDate: 1996 Jul 27 08:57:55 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.100) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMMI12) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.04 def +/isFixedPitch false def +end readonly def +/FontName /BYMWUK+CMMI12 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 25 /pi put +dup 102 /f put +dup 119 /w put +readonly def +/FontBBox{-30 -250 1026 750}readonly def +currentdict end +currentfile eexec +d9d66f633b846a97b686a97e45a3d0aa0529731c99a784ccbe85b4993b2eebde +3b12d472b7cf54651ef21185116a69ab1096ed4bad2f646635e019b6417cc77b +532f85d811c70d1429a19a5307ef63eb5c5e02c89fc6c20f6d9d89e7d91fe470 +b72befda23f5df76be05af4ce93137a219ed8a04a9d7d6fdf37e6b7fcde0d90b +986423e5960a5d9fbb4c956556e8df90cbfaec476fa36fd9a5c8175c9af513fe +d919c2ddd26bdc0d99398b9f4d03d6a8f05b47af95ef28a9c561dbdc98c47cf5 +5250011d19e9366eb6fd153d3a100caa6212e3d5d93990737f8d326d347b7edc 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/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/f/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/w + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] +ipeMakeFont +%%BeginResource: font QRHILV+CMSY10 +%!PS-AdobeFont-1.1: CMSY10 1.0 +%%CreationDate: 1991 Aug 15 07:20:57 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMSY10) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.035 def +/isFixedPitch false def +end readonly def +/FontName /QRHILV+CMSY10 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 0 /minus put +dup 112 /radical put +readonly def +/FontBBox{-29 -960 1116 775}readonly def +currentdict end +currentfile eexec +d9d66f633b846a97b686a97e45a3d0aa052f09f9c8ade9d907c058b87e9b6964 +7d53359e51216774a4eaa1e2b58ec3176bd1184a633b951372b4198d4e8c5ef4 +a213acb58aa0a658908035bf2ed8531779838a960dfe2b27ea49c37156989c85 +e21b3abf72e39a89232cd9f4237fc80c9e64e8425aa3bef7ded60b122a52922a +221a37d9a807dd01161779dde7d31ff2b87f97c73d63eecdda4c49501773468a +27d1663e0b62f461f6e40a5d6676d1d12b51e641c1d4e8e2771864fc104f8cbf +5b78ec1d88228725f1c453a678f58a7e1b7bd7ca700717d288eb8da1f57c4f09 +0abf1d42c5ddd0c384c7e22f8f8047be1d4c1cc8e33368fb1ac82b4e96146730 +de3302b2e6b819cb6ae455b1af3187ffe8071aa57ef8a6616b9cb7941d44ec7a +71a7bb3df755178d7d2e4bb69859efa4bbc30bd6bb1531133fd4d9438ff99f09 +4ecc068a324d75b5f696b8688eeb2f17e5ed34ccd6d047a4e3806d000c199d7c +515db70a8d4f6146fe068dc1e5de8bc57036431151ec603c8bcfe359bbd953ad +5f3d9983b036d9202c8fcc4fa88af960e1e49914ec809263862931db14b61eee +6d37a389b488d0b64cfb7da527aaed80494f79a073d895aa287bb47bd5246090 +a76ce91680c1f37e75a8804089ca4d83dbc5044e1eb9714257808bdc7df9d0b5 +fe7274d571b4b44f2e2d98e11b5cf85379f57db353ad8bca58c7047bca299c87 +a55b0066590a3136d60aebcbaeada5b3d618b68c0be7234fe688e597b2706fc4 +2f825d94ca7fea7c6b2c7e3837f5dc05986f4a5c38123e3af71cd2c0332e5a6d +5b89f35f7573055fa88425ac6992079f19b24cc22876fb11fc33bfe08990ebfb +943dcba682e4327772ae01a440c60ba4edcbd5f51e248b34dca93fbd3c560ab6 +fb4110271e9cfc5424d2ee2541071e7946d71d00d7e7fdaffe2f5eff45fd84b0 +c5d3765aac65c022989bcff5621476459905003712cdf63941655009a7b5ed3d +707c641e3e63c7476e88cf3b3247 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +cleartomark +%%EndResource +/F17 /QRHILV+CMSY10 +[ /minus/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /radical/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef + /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ] +ipeMakeFont +%%BeginResource: font TLUUTQ+CMR17 +%!PS-AdobeFont-1.1: CMR17 1.0 +%%CreationDate: 1991 Aug 20 16:38:24 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR17) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /TLUUTQ+CMR17 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 40 /parenleft put +dup 41 /parenright put +dup 43 /plus put +dup 61 /equal put +dup 115 /s put +dup 122 /z put +readonly def +/FontBBox{-33 -250 945 749}readonly def +currentdict end +currentfile eexec +d9d66f633b846a97b686a97e45a3d0aa052a014267b7904eb3c0d3bd0b83d891 +016ca6ca4b712adeb258faab9a130ee605e61f77fc1b738abc7c51cd46ef8171 +9098d5fee67660e69a7ab91b58f29a4d79e57022f783eb0fbbb6d4f4ec35014f +d2decba99459a4c59df0c6eba150284454e707dc2100c15b76b4c19b84363758 +469a6c558785b226332152109871a9883487dd7710949204ddcf837e6a8708b8 +2bdbf16fbc7512faa308a093fe5f075ea0a10a15b0ed05d5039da41b32b16e95 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