Antonis Tsolomitis Laboratory of Digital Typography and Mathematical Software Department of Mathematics University of the Aegean

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1 The GFSNEOHELLENIC fot family Atois Tsolomitis Laboratory of Digital Typography ad Mathematical Software Departmet of Mathematics Uiversity of the Aegea 1April Itroductio The NeoHelleic family of the Greek Fot Society was made available forfreedurigthewiterof2006. Thisfotexistedwithacommercial liceseformayyearsbefore. SupportforL A TEXadthebabelpackage was prepared several years ago by the author ad I. Vasilogiorgakis. With the free availability of the fots I have modified the origial package so thatitreflectsthechagesoccuredithelatestreleasesbygfs. The package supports three ecodigs: OT1, T1 ad LGR to the extedthatthefotthemselvescoverthese. OT1adLGRshouldbe fairlycomplete. Thegreekpartistobeusedwiththegreekoptioofthe Babel package. The fots are loaded with usepackage gfseohelleic or usepackage[default] gfseohelleic. The latter oe sets NeoHelleic as the default documet fot. The former defies the eviromet eohelleic ad the commad texteohelleic. For example, while i Greek laguage texteohelleic dokim h produces δοκιμή. x height is adjusted so that it matches with the x height of the cmbright package. This is doe to help with documets requirig mathematics. I this case load cmbright but before gfseohelleic.sty. 1

SupportforL A TEXadthebabelpackage was prepared several years ago by the author ad I. Vasilogiorgakis.

2 2 Istallatio Copy the cotets of the subdirectory afm i texmf/fots/afm/gfs/neohelleic/ Copy the cotets of the subdirectory doc i texmf/doc/latex/gfs/neohelleic/ Copy the cotets of the subdirectory ec i texmf/fots/ec/dvips/gfs/neohelleic/ Copy the cotets of the subdirectory map i texmf/fots/map/dvips/gfs/neohelleic/ Copy the cotets of the subdirectory tex i texmf/tex/latex/gfs/neohelleic/ Copy the cotets of the subdirectory tfm i texmf/fots/tfm/gfs/neohelleic/ Copy the cotets of the subdirectory type1 i texmf/fots/type1/gfs/neohelleic/ Copy the cotets of the subdirectory vf i texmf/fots/vf/gfs/neohelleic/ I your istallatio s updmap.cfg file add the lie Map gfseohelleic.map Refresh your fileame database ad the map file database(for example, ouixsystemsrumktexlsradtherutheupdmap sys(orupdmapo older systems) script as root). Youareowreadytousethefots. 3 Usage As said i the itroductio the package covers both eglish ad greek. Greek covers polytoic too through babel(read the documetatio of the babel package ad its greek optio). For example, the preample documetclass article usepackage[eglish,greek] babel usepackage[iso ] iputec usepackage cmbright usepackage[default] gfseohelleic will be the correct setup for articles i Greek usig NeoHelleic for the mai fot. 3.1 Trasformatios bydvips Other tha the shapes provided by the fots themselves, this package provides a slated small caps shape usig the stadard mechaism provided by dvips. Get slated small caps with scslshape. For example, the code 2

cotets of the subdirectory tfm i texmf/fots/tfm/gfs/neohelleic/ Copy the cotets of the subdirectory type1 i texmf/fots/type1/gfs/neohelleic/ Copy the cotets of the subdirectory vf i

3 textsc small caps textgreek pezokefala ia scslshape textgreek pezokefala ia will give SMALLCAPS The commad textscsl are also provided. 3.2 Alterative characters There are alterative desigs for delta, Epsilo, zeta, Xi ad Omega. Xi has two alteratives. These glyphs are accessed with the followig commads: Default desig Alterative desig Commad δ δ deltaatl Ε Ε Epsiloalt ζ ζ zetaalt Ξ Xioealt Ξ Xitwoalt Ω Ω Omegaalt 3.3 Tabular umbers Tabular umbers(of fixed width) are accessed with the commad tabums. Compare tabums Text fractios Text fractios are composed usig the lower ad upper umerals provided by the fots, ad are accessed with the commad textfrac. For example, textfrac gives-²² ₇. Precomposed fractios are provided too by oehalf, oethird, etc. 3

These glyphs are accessed with the followig commads: Default desig Alterative desig Commad δ δ deltaatl Ε Ε Epsiloalt ζ ζ zetaalt Ξ Xioealt Ξ Xitwoalt Ω Ω Omegaalt 3.

4 3.5 Additioal characters textbullet textparagraph umero estimated textlozege eurocurrecy textdagger textdaggerdbl yecurrecy Euro is also available i LGR ecodig. textgreek euro gives. 4 Problems Theaccetsofthecapitallettersshouldhagitheleftmargiwhesuch aletterstartsalie. TEXadL A TEXdootprovidethetoolsforsucha feature. However,thisseemstobepossiblewithpdfTEXAsthisisworki progress, please be patiet... 5 Samples Theexttwopagesprovidesamplesieglishadgreekwithmath. 4

4 Problems Theaccetsofthecapitallettersshouldhagitheleftmargiwhesuch aletterstartsalie.

5 Addiguptheseiequalitieswithrespecttoi,weget ci d i 1 p +1 q =1 (1) sice c p i = d q i =1. Ithecasep=q=2theaboveiequalityisalsocalledtheCauchy Schwartz iequality. Notice, also, that by formally defiig ( b k q ) 1/q to be sup b k for q=,wegiveseseto(9)forall1 p. A similar iequality is true for fuctios istead of sequeces with the sums beig substituted by itegrals. TheoremLet1<p< adletqbesuchthat1/p+1/q=1. The, forallfuctiosf,goaiterval[a,b]suchthattheitegrals b a f(t) p dt, b a g(t) q dtad b a f(t)g(t) dtexist(asriemaitegrals),wehave b ( b f(t)g(t) dt a a ) 1/p ( b 1/q. f(t) p dt g(t) dt) q (2) a Notice that if the Riema itegral b a f(t)g(t)dtalsoexists,thefrom theiequality b a f(t)g(t)dt b a f(t)g(t) dtfollowsthat b ( b f(t)g(t) dt a a ) 1/p ( b 1/q. f(t) p dt g(t) dt) q (3) a Proof: Cosider a partitio of the iterval [a, b] i equal subitervals withedpoitsa=x 0 <x 1 < <x =b. Let x=(b a)/. Wehave i=1 f(x i )g(x i ) x = i=1 i=1 f(x i )g(x i ) ( x) 1 p +1 q ( f(x i ) p x) 1/p ( g(x i ) q x) 1/q. (4) 5

A similar iequality is true for fuctios istead of sequeces with the sums beig substituted by itegrals. TheoremLet1<p< adletqbesuchthat1/p+1/q=1.

6 Εμβαδόν επιφάνειας από περιστροφή Πρόταση 5.1 Εστω γ καμπύλη με παραμετρική εξίσωση x = g(t), y=f(t),t [a,b]ανg,f συνεχείςστο [a,b]τότετοεμβαδόναπό περιστροφήτηςγγύρωαπότονxx δίνεται B=2π b a f(t) g (t) 2 +f (t 2 )dt. Ανηγδίνεταιαπότηνy=f(x),x [a,b]τότεb=2π b a f(t) 1+f (x) 2 dx Όγκος στερεών από περιστροφή Εστωf:[a,b] RσυνεχήςκαιR={f,Ox,x=a,x=b}είναιοόγκος απόπεριστροφήτουγραφήματοςτης f γύρωαπότονox μεταξύ τωνευθειώνx=a,καιx=b,τότεv=π b a f(x)2 dx Αν f,g:[a,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=a,x=b}είναι V=π b a {f(x)2 g(x) 2 }dx. Ανx=g(t),y=f(t),t=[t 1,t 2 ]τότεv=π t 2 t 1 {f(t) 2 g (t)}dtγια g(t 1 )=a,g(t 2 )=b. 6 Ασκήσεις Άσκηση 6.1 Να εκφραστεί το παρακάτω όριο ως ολοκλήρωμα Riema κατάλληλης συνάρτησης 1 lim e k k=1 Υπόδειξη: Πρέπει να σκεφτούμε μια συνάρτηση της οποίας γνωρίζουμε ότιυπάρχειτοολοκλήρωμα. ΤότεπαίρνουμεμιαδιαμέρισηP και δείχνουμεπ.χ. ότιτοu(f,p )είναιηζητούμενησειρά. Λύση: Πρέπει να σκεφτούμε μια συνάρτηση της οποίας γνωρίζουμε ότιυπάρχειτοολοκλήρωμα. ΤότεπαίρνουμεμιαδιαμέρισηP και δείχνουμεπ.χ. ότιτοu(f,p )είναιηζητούμενησειρά. Εχουμε ότι 1 k=1 e k = 1 1 e+ e e = 1 e1 + 1 e e 6

Ανηγδίνεταιαπότηνy=f(x),x [a,b]τότεb=2π b a f(t) 1+f (x) 2 dx Όγκος στερεών από περιστροφή Εστωf:[a,b] RσυνεχήςκαιR={f,Ox,x=a,x=b}είναιοόγκος απόπεριστροφήτουγραφήματοςτης f γύρωαπότονox μεταξύ

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Antonis Tsolomitis Laboratory of Digital Typography and Mathematical Software Department of Mathematics University of the Aegean

Antonis Tsolomitis Laboratory of Digital Typography and Mathematical Software Department of Mathematics University of the Aegean The GFSBODONI fot fmily Atois Tsolomitis Lbortory of Digitl Typogrphy d Mthemticl Softwre Deprtmet of Mthemtics Uiversity of the Aege 9 Mrch 2006 Itroductio The Bodoi fmily of the Greek Fot Society ws