The GFSDIDOT font fmily Antonis Tsolomitis Lbortory of Digitl Typogrphy nd Mthemticl Softwre Deprtment of Mthemtics University of the Aegen 28 Jnury 2006 Introduction TheDidotfmilyoftheGreekFontSocietywsmdevilbleforfreein utumn 2005. This font existed with commercil license for mny yers before. Support for LTeX nd the bbel pckge ws prepred severl yers go by the uthor nd I. Vsilogiorgkis. With the free vilbility ofthefontsihvemodifiedtheoriginlpckgesothtitreflectsthe chnges occured in the ltest releses by GFS. Thepckgesupportsthreeencodings: OT,TndLGRtotheextendthtthefontthemselvescoverthese. OTndLGRshouldbefirly complete.thegreekprtistobeusedwiththegreekoptionofthebbel pckge. The fonts re loded with \usepckge{gfsdidot}. Thefontssrelesedcontinnitlicversion,butitsgreekprtisjust theromninslntedform. Toovercomethisproblemforthegreekprt, weuseforitlicnotherfontbygfsclledolg.asfrstheltinprtis concerned the itlic chrcters re tken from Didot Itlic(Olg contins no ltin chrcters). The pckge provides lso mtching smll cps shpe for both ltin nd greek including old style numbers. Finlly, the mth symbols re tken from the pxfonts pckge except of course the chrcters tht re lredy provided by Didot nd Olg. The choiceofpxfontswsmdeonthebsisthttheltinprtofdidotis bsed on Pltino. Moreover, ll Didot chrcters re scled in the.fd filesbyfctorof.04inordertomtchthex heightofpxfonts.
2 Instlltion Copy the contents of the subdirectory fm in texmf/fonts/fm/gfs/didot/ Copy the contents of the subdirectory doc in texmf/doc/ltex/gfs/didot/ Copy the contents of the subdirectory enc in texmf/fonts/enc/dvips/gfs/didot/ Copy the contents of the subdirectory mp in texmf/fonts/mp/dvips/gfs/didot/ Copy the contents of the subdirectory tex in texmf/tex/ltex/gfs/didot/ Copy the contents of the subdirectory tfm in texmf/fonts/tfm/gfs/didot/ Copy the contents of the subdirectory type in texmf/fonts/type/gfs/didot/ Copy the contents of the subdirectory vf in texmf/fonts/vf/gfs/didot/ In your instlltions updmp.cfg file dd the line Mp gfsdidot.mp Refresh your filenme dtbse nd the mp file dtbse(for exmple, onunixsystemsrunmktexlsrndthenruntheupdmpscriptsroot). Yourenowredytousethefontsprovidedthtyouhvereltively modern instlltion tht includes pxfonts. 3 Usge As sid in the introduction the pckge covers both english nd greek. Greek covers polytonic too through bbel(red the documenttion of the bbel pckge nd its greek option). For exmple, the premple \documentclss{rticle} \usepckge[english,greek]{bbel} \usepckge[iso-8859-7]{inputenc} \usepckge{gfsdidot} willbethecorrectsetupforrticlesingreek. 3. Trnsformtions by dvips Other thn the shpes provided by the fonts themselves, this pckge provides n upright itlic shpe nd slnted smll cps shpe using the stndrd mechnism provided by dvips. Upright itlics re clled with \uishpe nd slnted smll cps with\scslshpe. For exmple, the code 2
{\itshpe itlics{\uishpe upright itlics}{\itshpe itlics gin} \textgreek{{\itshpe > bgdzxfy w {\uishpe > bgdzxfy w }{\itshpe > bgdzxfy w }}}} \textsc{smll cps \textgreek{pezokefl i} 023456789}{\scslshpe \textgreek{pezokefl i 023456789}} will give itlics upright itlics itlics gin ἄβγδζξφψῲ ἄβγδζξφψῲ ἄβγδζξφψῲ SMALLCAPS 023456789 023456789 The commnds\textui{} nd\textscsl{} re lso provided. 3.2 Tbulrnumbers Tbulr numbers(of fixed width) re ccessed with the commnd\tbnums{}. Compre 0 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 \tbnums{ 0 2 3 4 5 6 7 8 9 } 0 2 3 4 5 6 7 8 9 3.3 Textfrctions Text frctions re composed using the lower nd upper numerls provided by the fonts, nd re ccessed with the commnd \textfrc{}{}. For exmple,\textfrc{-22}{7}gives-²² ₇. Precomposed frctions re provided too by\onehlf,\onethird, etc. 3.4 Additionlchrcters \textbullet \textprgrph \textprgrphlt \creof \numero \estimted \whitebullet \textlozenge \eurocurrency \interrobng \textdgger \textdggerdbl \yencurrency Euro is lso vilble in LGR enconding.\textgreek{\euro} gives. 3
3.5 Alterntechrcters In the greek encoding the initil thet is chosen utomticlly. Compre: ϑάλασσα but Αθηνά. Other lternte chrcters re not chosen utomticlly. Olg provides double lmbd: š. This cn be ccessed with the commnd\lmbddbl in textmode. For exmple, in LGR encoding \textit{\lmbddbl kt llhlos metsjhmtism;oc} gives αšά κατάλληλος μετασχηματισμός. 4 Problems Theccentsofthecpitllettersshouldhngintheleftmrginwhensuch letterstrtsline. T E XndLAT E Xdonotprovidethetoolsforsuch feture. However,thisseemstobepossiblewithpdfT E XAsthisiswork inprogress,plesebeptient... 5 Smples Thenexttwopgesprovidesmplesinenglishndgreekwithmth. 4
Addinguptheseinequlitieswithrespecttoi,weget c i d i p + () q since c p i d q i. Inthecsep q 2theboveinequlityislsoclledtheCuchy Schwrtz inequlity. Notice,lso,thtbyformllydefining( b k q ) /q tobesup b k forq, wegivesenseto(9)forll p. A similr inequlity is true for functions insted of sequences with the sums being substituted by integrls. TheoremLet<p< ndletqbesuchtht/p+/q. Then,forll functionsf,gonnintervl[,b]suchthttheintegrls b f(t) p dt, b g(t) q dt nd b f(t)g(t) dtexist(sriemnnintegrls),wehve b f(t)g(t) dt ( b ) /p ( b /q f(t) p dt g(t) dt) q. (2) NoticethtiftheRiemnnintegrl b f(t)g(t)dtlsoexists,thenfrom the inequlity b f(t)g(t)dt b f(t)g(t) dtfollowstht b ( b f(t)g(t)dt ) /p ( b /q f(t) p dt g(t) dt) q. (3) Proof: Consider prtition of the intervl[, b] in n equl subintervls withendpoints x 0 <x < <x n b.let x (b )/n.wehve f(x i )g(x i ) x i f(x i )g(x i ) ( x) p + q i ( f(xi ) p x ) /p( g(xi ) q x ) /q. (4) i 5
Εμβαδόν επιφάνειας από περιστροφή Πρόταση5. Εστω γ καμπύλη με παραμετρική εξίσωσηx g(t),y f(t), t [,b] ανg,f συνεχείς στο[,b] τότε το εμβαδόν από περιστροφή της γ γύρω από τονxx δίνεται B 2π b f(t) g (t) 2 +f (t 2 )dt. Αν η γ δίνεται από τηνy f(x),x [,b] τότεb 2π b f(t) +f (x) 2 dx Ογκος στερεών από περιστροφή Εστωf:[,b] RσυνεχήςκαιR {f,ox,x,x b}είναιοόγκοςαπό περιστροφήτουγραφήματοςτηςfγύρωαπότονoxμεταξύτωνευθειών x,καιx b,τότεv π b f(x)2 dx Ανf,g:[,b] Rκαι0 g(x) f(x)τότεοόγκοςστερεούπου παράγεταιαπόπεριστροφήτωνγραφημάτωντωνfκαιg,r {f,g,ox,x,x b}είναι V π b {f(x)2 g(x) 2 }dx. Ανx g(t),y f(t),t [t,t 2 ]τότεv π t 2 {f(t) 2 g (t)}dtγιαg(t t ), g(t 2 ) b. 6 Ασκήσεις Άσκηση 6. Να εκφραστεί το παρακάτω όριο ως ολοκλήρωμα Riemnn κατάλληλης συνάρτησης lim n n k n e k Υπόδειξη: Πρέπει να σκεφτούμε μια συνάρτηση της οποίας γνωρίζουμε ότι υπάρχει το ολοκλήρωμα. Τότε παίρνουμε μια διαμέριση P n και δείχνουμεπ.χ.ότιτοu(f,p n )είναιηζητούμενησειρά. Λύση: Πρέπει να σκεφτούμε μια συνάρτηση της οποίας γνωρίζουμε ότι υπάρχειτοολοκλήρωμα.τότεπαίρνουμεμιαδιαμέρισηp n καιδείχνουμε π.χ.ότιτοu(f,p n )είναιηζητούμενησειρά. Εχουμε ότι n k n e k n n e+ e n n 2 + + n e n n n e n+ n e2 n+ + n en n 6