3.1 Transformations by dvips

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1 The GFSDIDOT font fmily Alin Aubord Sourire Informtique Genev / Switzerlnd Antonis Tsolomitis Lbortory of Digitl Typogrphy nd Mthemticl Softwre Deprtment of Mthemtics University of the Aegen 9 Jnury 208 Introduction The Didot fmily of the Greek Font Society ws mde vilble for free in utumn 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 (A. Aubord hs done some modifictions to improve the support of the ccented nd other specil chrcters). With the free vilbility of the fonts I hve modified the originl pckge so tht it reflects the chnges occured in the ltest releses by GFS. The pckge supports four encodings: OT, T, TS nd LGR. When some chrcters re missing in the Didot font, they re tken from the font TEX Gyre Pgell nd, for the oblique rule (chrcter 0x20 in OT encoding) used to build the chrcter L with stroke (Ł), from the cmr fmily. All the provided encodings should be firly complete. The greek prt is to be used with the greek option of the Bbel pckge. The Didot fmily in LGR nd OT encoding hs been tested successfuly with Plin TEX too. The fonts re loded with the LATEX commnd (to be given in the premble of LATEX document): \usepckge{gfsdidot}. This commnd lods implicitely the pckge textcomp (Text Compnion) which defines the defult vlues for some chrcters in the NFSS LATEX font selection mechnism. If this behviour is not desired, it is possible to prevent the loding of the textcomp pckge by pssing the option nogfsdidotts when clling the pckge gfsdidot: \usepckge[nogfs DidotTS]{gfsdidot}.

2 The pckge provides new test to check if the pckge textcomp is loded. It cn be used like tht: \ifthenelse{\boolen{gfsdidotts}}% {Wht to do when "textcomp" is loded}% {Wht to do when "textcomp" is not loded} The fonts s relesed contin n itlic version, but its greek prt is just the romn in slnted form. To overcome this problem for the greek prt, we use for itlic nother font by GFS clled Olg. As fr s the ltin prt is 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 choice of pxfonts ws mde on the bsis tht the ltin prt of Didot is bsed on Pltino. Moreover, ll Didot chrcters re scled in the.fd files by fctor of.04 in order to mtch the x-height of pxfonts. 2 Instlltion. Copy the contents of the subdirectories: fm in texmf/fonts/fm/gfs/didot doc in texmf/doc/ltex/gfs/didot enc in texmf/fonts/enc/dvips/gfs/didot mp in texmf/fonts/mp/dvips/gfs/didot tex in texmf/tex/ltex/gfs/didot tfm in texmf/fonts/tfm/gfs/didot type in texmf/fonts/type/gfs/didot vf in texmf/fonts/vf/gfs/didot vpl in texmf/fonts/source/public/gfs/didot/vpl. The vpl files re the sources used to produce vf nd tfm files. In theory, with the utilities tftopl nd vftovp, these files cn be rebuilded. Unfortuntely, the comments re lost nd they re useful with some utilities. 2. In your instlltions updmp.cfg file dd the line: Mp gfsdidot.mp If you use MikTEX (one of the commonly used TEX distribution in Windows) you should dd the following line too: Mp qpl-cs.mp which describe the fonts used for the Czech TeX encoding for font Pgell. This encoding is identicl to the OT encoding for the lower prt (slots under 28) nd it is used to dd the missing chrcter (the dotless j) to the Didot font. 2

3 The file updmp.cfg should not be edited directly under MikTEX but edit it with the commnd initexmf edit-config-file updmp. This commnd cn be typed by opening commnd prompt from the Windows utility provided by MikTEX. 3. Finlly, you hve to refresh the different dtbses by pssing (s the user owning the different TEX trees) the two following commnds: mktexlsr to rebuild the list of ll the files locted in the different texmf tree. updmp-sys to rebuild the mp files for the different bckend drivers (dvips, pdftex etc.). In the subdirectory tools, the Perl script instlldidot.pl utomtizes this process. This script should be clled with two rguments:. the root directory of the distribution of the GFS Didot pckge; 2. the root directory of texmf tree where the GFS Didot pckge should be instlled. You re now redy to use the fonts provided tht you hve reltively modern instlltion tht includes pxfonts. 3 Usge As sid in the introduction the pckge covers both english nd greek (nd most western lnguges if the output encoding T is specified). Greek covers polytonic too through bbel (red the documenttion of the bbel pckge nd its greek option). For exmple, the premble: \documentclss{rticle} \usepckge[english,greek]{bbel} \usepckge[t]{fontenc} \usepckge[iso ]{inputenc} \usepckge{gfsdidot} will be the correct setup for rticles in Greek. For rticles written exclusively in Greek, the line \usepckge[t]{font enc} is not mndtory. This instruction forces the use of the T encoding which is designed to write texts in most of the lnguges of western Europe (including English). However, even if one writes exclusively in English or Greek, the encoding T cn be useful. Without specifying the T encoding, the encoding OT is used. This is the originl encoding of TEX. Unfortuntely this encoding (bsed on 28 chrcters/slots only) is vrible nd it does not mtch the ASCII code for ll chrcters. 3

4 For instnce the dollr ($) is unvilble in itlic fonts (in the sme slot the english pound is defined). The pipe chrcter ( ) is not correctly printed either. The T encoding is identicl to the ASCII code for ll the printble slots (32 to 27) nd ll the ASCII chrcters re printed s expected. 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: {\itshpe itlics} {\uishpe upright itlics} {\itshpe itlics gin} \textgreek{{\itshpe ἄβγδζξφψῲ} {\uishpe ἄβγδζξφψῲ} {\itshpe ἄβγδζξφψῲ}} will give: \textsc{smll cps \textgreek{πεζοκεφαλαία} } {\scslshpe \textgreek{πεζοκεφαλαία }} itlics upright itlics itlics gin ἄβγδζξφψῲ ἄβγδζξφψῲ ἄβγδζξφψῲ SMALL CAPS The commnds \textui{} nd \textscsl{} re lso provided. 3.2 Tbulr numbers Tbulr numbers (of fixed width) re ccessed with the commnd \tbnums{}. Compre \tbnums{ } Text frctions 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. 4

5 3.4 Additionl chrcters \textbullet \textprgrph \textprgrphlt \creof \numero \estimted \whitebullet \textlozenge \eurocurrency \interrobng \textdgger \textdggerdbl \yencurrency \texteuro 3.5 Alternte chrcters 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 κατάšηλος μετασχηματισμός. 4 Problems The ccents of the cpitl letters should hng in the left mrgin when such letter strts line. TEX nd LATEX do not provide the tools for such feture. However, this seems to be possible with pdftex As this is work in progress, plese be ptient... 5 Smples The next two pges provide smples in english nd greek with mth. This commnd is vilble only if the pckge textcomp is used. Euro is lso vilble in LGR enconding. \textgreek{\euro} gives. 5

6 Adding up these inequlities with respect to i, we get c i d i p + q () since c p i d q i. In the cse p q 2 the bove inequlity is lso clled the Cuchy- Schwrtz inequlity. Notice, lso, tht by formlly defining ( b k q ) / q to be sup b k for q, we give sense to (9) for ll p. A similr inequlity is true for functions insted of sequences with the sums being substituted by integrls. Theorem Let < p < nd let q be such tht / p + / q. Then, for ll functions f, g on n intervl [, b] such tht the integrls b f (t) p dt, b g(t) q dt nd b f (t)g(t) dt exist (s Riemnn integrls), we hve b f (t)g(t) dt ( b ) / p ( b / q f (t) p dt g(t) dt) q. (2) Notice tht if the Riemnn integrl b f (t)g(t) dt lso exists, then from the inequlity b f (t)g(t) dt b f (t)g(t) dt follows tht 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 with endpoints x 0 < x < < x n b. Let x (b )/ n. We hve 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 6

7 Εμβαδόν επιφάνειας από περιστροφή Πρόταση 5. Εστω γ καμπύλη με παραμετρική εξίσωση x g(t), y f (t), t [, b] αν g, f συνεχείς στο [, b] τότε το εμβαδόν από περιστροφή της γ γύρω από τον xx δίνεται B 2π b f (t) g (t) 2 + f (t 2 )dt. Αν η γ δίνεται από την y f (x), x [, b] τότε B 2π b f (t) + f (x) 2 dx Ογκος στερεών από περιστροφή Εστω f : [, b] R συνεχής και R {f, Ox, x, x b} είναι ο όγκος από περιστροφή του γραφήματος της f γύρω από τον Ox μεταξύ των ευθειών x, και x b, τότε V π b f (x)2 dx Αν f, g : [, b] R και 0 g(x) f (x) τότε ο όγκος στερεού που παράγεται από περιστροφή των γραφημάτων των f και g, R {f, g, Ox, x, x b} είναι V π b {f (x)2 g(x) 2 }dx. Αν x g(t), y f (t), t [t, t 2 ] τότε V π t 2 {f (t) 2 g (t)}dt για g(t t ), g(t 2 ) b. 6 Ασκήσεις Ασκηση 6. Να εκφραστεί το παρακάτω όριο ως ολοκλήρωμα Riemnn κατάλληλης συνάρτησης lim n n k e k Υπόδειξη: Πρέπει να σκεφτούμε μια συνάρτηση της οποίας γνωρίζουμε ότι υπάρχει το ολοκλήρωμα. Τότε παίρνουμε μια διαμέριση P n και δείχνουμε π.χ. ότι το U(f, P n ) είναι η ζητούμενη σειρά. Λύση: Πρέπει να σκεφτούμε μια συνάρτηση της οποίας γνωρίζουμε ότι υπάρχει το ολοκλήρωμα. Τότε παίρνουμε μια διαμέριση P n και δείχνουμε π.χ. ότι το U(f, P n ) είναι η ζητούμενη σειρά. Εχουμε ότι n k e k n e + n e n e n n e n + n e 2 n + + n e n n 7