%%  (c) copyright  2006, 2007
%% Antonis Tsolomitis
%% Department of Mathematics, University of the Aegean
%%
%%  This document can be redistributed and/or modified under the terms
%%  of the LaTeX Project Public License Distributed from CTAN
%%  archives in directory macros/latex/base/lppl.txt; either
%%  version 1 of the License, or any later version.

\documentclass{article}
\usepackage[polutonikogreek,english]{babel}
\usepackage[iso-8859-7]{inputenc}
\usepackage{gfsdidot}

%\renewcommand{\ttdefault}{hlst}

%%%%% Theorems and friends
\newtheorem{theorem}{Θεώρημα}[section]         
\newtheorem{lemma}[theorem]{Λήμμα}
\newtheorem{proposition}[theorem]{Πρόταση}
\newtheorem{corollary}[theorem]{Πόρισμα}
\newtheorem{definition}[theorem]{Ορισμός}
\newtheorem{remark}[theorem]{Παρατήρηση}
\newtheorem{axiom}[theorem]{Αξίωμα}
\newtheorem{exercise}[theorem]{Άσκηση}


%%%%% Environment ``proof''
\newenvironment{proof}[1]{{\textit{Απόδειξη:}}}{\ \hfill$\Box$}
\newenvironment{hint}[1]{{\textit{Υπόδειξη:}}}{\ \hfill$\Box$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%






\title{The \textsc{gfsdidot} font family}
\author{Antonis Tsolomitis\\
Laboratory of Digital Typography\\ and Mathematical Software\\
Department of Mathematics\\
University of the Aegean}
\date {\textsc{28} January \textsc{2006}}


\begin{document}
\maketitle

\section{Introduction}
The Didot family of the Greek Font Society was made available for free
in autumn 2005. This font existed with a commercial license for many
years before. Support for LaTeX and the babel package was prepared
several years ago by the author and I.\ Vasilogiorgakis. With the
free availability of the fonts I have modified the original package
so that it reflects the changes occured in the latest releases by \textsc{gfs}.

The package supports three encodings: OT1, T1 and LGR to the extend
that the font themselves cover these. OT1 and LGR should be
fairly complete. The greek part is to be used with the greek option of
the Babel package.

The fonts are loaded with

\verb|\usepackage{gfsdidot}|.

The fonts as released contain an italic version, but its greek part is
just the roman in slanted form. To overcome this problem for the greek
part, we use for italic another font by \textsc{gfs} called Olga.
As far as the latin part is concerned the italic characters are taken 
from Didot-Italic (Olga contains no latin characters).
The package provides also a matching small caps shape for both latin
and greek including old style numbers.

Finally, the math symbols are taken from the pxfonts package except
of course the characters that are already provided by Didot and Olga.
The choice of pxfonts was made on the basis that the latin part of
Didot is based on Palatino. Moreover, all Didot characters are scaled
in the \verb|.fd| files by a factor of 1.04 in order to match the
x-height of pxfonts.

\section{Installation}

Copy the contents of the subdirectory afm in
texmf/fonts/afm/GFS/Didot/

\medskip

\noindent Copy the contents of the subdirectory doc in
texmf/doc/latex/GFS/Didot/

\medskip

\noindent Copy the contents of the subdirectory enc in
texmf/fonts/enc/dvips/GFS/Didot/

\medskip

\noindent Copy the contents of the subdirectory map in
texmf/fonts/map/dvips/GFS/Didot/

\medskip

\noindent Copy the contents of the subdirectory tex in
texmf/tex/latex/GFS/Didot/

\medskip

\noindent Copy the contents of the subdirectory tfm in
texmf/fonts/tfm/GFS/Didot/

\medskip

\noindent Copy the contents of the subdirectory type1 in
texmf/fonts/type1/GFS/Didot/

\medskip

\noindent Copy the contents of the subdirectory vf in
texmf/fonts/vf/GFS/Didot/

\medskip

\noindent In your installations updmap.cfg file add the line

\medskip

\noindent Map gfsdidot.map

\medskip

Refresh your filename database and the map file database (for example, on Unix systems
run mktexlsr and then run the updmap script as root).

You are now ready to use the fonts provided that you have a relatively
modern installation that includes pxfonts.

\section{Usage}

As said in the introduction the package covers both english and
greek. Greek covers polytonic too through babel (read the
documentation
of the babel package and its greek option). 

For example, the preample

\begin{verbatim}
\documentclass{article}
\usepackage[english,greek]{babel}
\usepackage[iso-8859-7]{inputenc}
\usepackage{gfsdidot}
\end{verbatim}

will be the correct setup for articles in Greek.

\bigskip

\subsection{Transformations by \texttt{dvips}}

Other than the shapes provided by the fonts themselves, this package
provides an upright italic shape and a slanted small caps shape
using the standard mechanism provided by dvips. Upright italics
are called with \verb|\uishape| and slanted small caps with \verb|\scslshape|.
For example, the code
\begin{verbatim}
{\itshape italics {\uishape upright italics} {\itshape italics again}
\textgreek{{\itshape >'αβγδζξφψ`ω| {\uishape >'αβγδζξφψ`ω|} {\itshape >'αβγδζξφψ`ω|}}}}

\textsc{small caps \textgreek{πεζοκεφαλαία} 0123456789} {\scslshape
  \textgreek{πεζοκεφαλαία 0123456789}}
\end{verbatim}
will give 

{\itshape italics {\uishape upright italics} {\itshape italics again}
\textgreek{{\itshape >'αβγδζξφψ`ω| {\uishape >'αβγδζξφψ`ω|} {\itshape
      >'αβγδζξφψ`ω|}}}}

\textsc{small caps \textgreek{πεζοκεφαλαία} 0123456789} {\scslshape
  \textgreek{πεζοκεφαλαία 0123456789}}

\noindent The commands \verb|\textui{}| and \verb|\textscsl{}| are also provided.




\subsection{Tabular numbers}

Tabular numbers (of fixed width) are accessed with the command
\verb|\tabnums{}|. Compare

\begin{tabular}{ll}
\verb+|0|1|2|3|4|5|6|7|8|9|+ & |0|1|2|3|4|5|6|7|8|9|\\
\verb+\tabnums{|0|1|2|3|4|5|6|7|8|9|}+ & \tabnums{|0|1|2|3|4|5|6|7|8|9|}
\end{tabular}


\subsection{Text fractions}

Text fractions are composed using the lower and upper numerals
provided by the fonts, and are
accessed with the command \verb|\textfrac{}{}|.
For example, \verb|\textfrac{-22}{7}| gives \textfrac{-22}{7}.

Precomposed fractions are provided too by \verb|\onehalf|,
\verb|\onethird|, etc.


\subsection{Additional characters}

\begin{center}
\begin{tabular}{|c|c|}\hline
\verb|\textbullet| &\textbullet \\ \hline
\verb|\textparagraph| &\textparagraph \\ \hline
\verb|\textparagraphalt| & \textparagraphalt\\ \hline
\verb|\careof| & \careof\\ \hline
\verb|\numero| & \numero\\ \hline
\verb|\estimated| & \estimated\\ \hline
\verb|\whitebullet| & \whitebullet\\ \hline
\verb|\textlozenge| & \textlozenge\\ \hline
\verb|\eurocurrency| & \eurocurrency\\ \hline
\verb|\interrobang| & \interrobang\\ \hline
\verb|\textdagger| & \textdagger\\ \hline
\verb|\textdaggerdbl| & \textdaggerdbl\\ \hline
\verb|\yencurrency| & \yencurrency\\ \hline
\end{tabular}
\end{center}

Euro is also available in LGR enconding. \verb|\textgreek{\euro}|
gives \textgreek{\euro}. 

\subsection{Alternate characters}

In the greek encoding the initial theta is chosen
automatically. Compare: \textgreek{θάλασσα} but \textgreek{Αθηνά}.
Other alternate characters are not chosen automatically. 

Olga provides a double lambda: \lambdadbl. This can be accessed with the
command \verb|\lambdadbl| in textmode. 

For example, in LGR encoding

\medskip

\verb|\textit{a\lambdadbl'a kat'allhlos metasjhmatism;oc}|

\smallskip

gives 

\smallskip

\textgreek{\textit{a\lambdadbl'a  kat'allhlos metasqhmatism'oc}}.


\section{Problems}


 The
accents of the capital letters should hang in the left margin when such a letter starts a
line.  \TeX\ and \LaTeX\ do not provide the tools for such a
feature. However, this seems to be possible with 
\textlatin{pdf\TeX}
As this is work in progress, please be patient\ldots




\section{Samples}

The next two pages provide samples in english and greek with math.


\newpage

Adding up these inequalities with respect to $i$, we get
\begin{equation} \sum c_i d_i \leq \frac1{p} +\frac1{q} =1\label{10}\end{equation}
since $\sum c_i^p =\sum d_i^q =1$.\hfill$\Box$

In the case $p=q=2$
the above inequality is also called the 
\textit{Cauchy-Schwartz inequality}.

Notice, also, that by formally defining $\left( \sum |b_k|^q\right)^{1/q}$ to be
$\sup |b_k|$ for $q=\infty$, we give sense to (9) for all 
$1\leq p\leq\infty$.


A similar inequality is true for functions instead of sequences with the sums 
being substituted by integrals.

\medskip

\textbf{Theorem} {\itshape Let $1<p<\infty$ and let $q$ be such that $1/p +1/q =1$. Then, 
for all functions $f,g$ on an interval $[a,b]$ 
such that the integrals $\int_a^b |f(t)|^p\,dt$, $\int_a^b |g(t)|^q\,dt$ and
$\int_a^b |f(t)g(t)|\,dt$ exist \textup{(}as Riemann integrals\textup{)},
we have 
\begin{equation}
\int_a^b |f(t)g(t)|\,dt\leq 
\biggl(\int_a^b |f(t)|^p\,dt\biggr)^{1/p}
\biggl(\int_a^b |g(t)|^q\,dt\biggr)^{1/q} .
\end{equation}
}

Notice that if the Riemann integral $\int_a^b f(t)g(t)\,dt$ also exists, then 
from the inequality $\left|\int_a^b f(t)g(t)\,dt\right|\leq 
\int_a^b |f(t)g(t)|\,dt$ follows that
\begin{equation}
\left|\int_a^b f(t)g(t)\,dt\right|\leq 
\biggl(\int_a^b |f(t)|^p\,dt\biggr)^{1/p}
\biggl(\int_a^b |g(t)|^q\,dt\biggr)^{1/q} .
\end{equation}

  

\textit{Proof:} Consider a partition of the interval $[a,b]$ in $n$ equal 
subintervals with endpoints
$a=x_0<x_1<\cdots<x_n=b$. Let $\Delta x=(b-a)/n$.
We have
\begin{eqnarray}
\sum_{i=1}^n |f(x_i)g(x_i)|\Delta x &\leq& 
\sum_{i=1}^n |f(x_i)g(x_i)|(\Delta x)^{\frac1{p}+\frac1{q}}\nonumber\\
&=&\sum_{i=1}^n \left(|f(x_i)|^p \Delta x\right)^{1/p} \left(|g(x_i)|^q 
\Delta x\right)^{1/q}.\label{functionalHolder1}\\ \nonumber
\end{eqnarray}

\newpage\greektext


% $\bullet$ Μήκος τόξου καμπύλης 

% \begin{proposition}\label{chap2:sec1:prop 23}
% Έστω $\gamma$ καμπύλη με παραμετρική εξίσωση $x=g(t)$, $y=f(t)$,
% $t\in [a,\,b]$ αν $g'$, $f'$ συνεχείς στο $[a,\,b]$ τότε η
% $\gamma$ έχει μήκος $S=L(\gamma)=\int_a^b \sqrt{g'(t)^2+f'(t)^2}
% dt$.
% \end{proposition}

\textbullet\ Εμβαδόν επιφάνειας από περιστροφή\\

\begin{proposition}\label{chap2:sec1:prop23-2}
Έστω $\gamma$ καμπύλη με παραμετρική εξίσωση $x=g(t)$, $y=f(t)$,
$t\in [a,\,b]$ αν $g'$, $f'$ συνεχείς στο $[a,\,b]$ τότε το
εμβαδόν από περιστροφή της $\gamma$ γύρω από τον $xx'$ δίνεται \\
$Β=2\pi\int_a^b |f(t)| \sqrt{g'(t)^2+f^{\prime}(t^2)} dt$. \\ Αν η
$\gamma$ δίνεται από την $y=f(x)$, $x\in [a,\,b]$ τότε
$Β=2\pi\int_a^b |f(t)| \sqrt{1+f'(x)^2} dx$
\end{proposition}

\textbullet\ Όγκος στερεών από περιστροφή\\ Έστω $f :
[a,\,b]\rightarrow \mathbb{R}$ συνεχής και $R=\{f, Ox,x=a,x=b\}$
είναι ο όγκος από περιστροφή του γραφήματος της $f$ γύρω από τον
$Ox$ μεταξύ των ευθειών $x=a$, και $x=b$, τότε $V=\pi\int_a^b f
(x)^2 dx$

\textbullet\ Αν $f,g : [a,\,b]\rightarrow \mathbb{R}$ και $0\leq
g(x)\leq f(x)$ τότε ο όγκος στερεού που παράγεται από περιστροφή
των γραφημάτων των $f$ και $g$, $R=\{f,g, Ox,x=a,x=b\}$ είναι \\
$V=\pi\int_a^b\{ f (x)^2-g(x)^2\} dx$.

\textbullet\ Αν $x=g(t)$, $y=f(t)$, $t=[t_1,\,t_2]$ τότε
$V=\pi\int_{t_1}^{t_2}\{ f (t)^2 g'(t)\} dt$ για $g(t_1)=a$,
$g(t_2)=b$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Ασκήσεις}\label{chap2:sec2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercise}\label{chap2:ex1}
Να εκφραστεί το παρακάτω όριο ως ολοκλήρωμα $Riemann$ κατάλ\-ληλης
συνάρτησης\\
$$\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k=1}^{n}\sqrt[n]{e^k} $$
\end{exercise}
%%%%%%%%%
\textit{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.

\bigskip

%%%%%%%%%%%%%%
\textit{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.\\
Έχουμε ότι
\begin{eqnarray}\frac{1}{n}\sum_{k=1}^{n}\sqrt[n]{e^k} =
\frac{1}{n}\sqrt[n]{e}+\frac{1}{n}\sqrt[n]{e^2}+\cdots +
\frac{1}{n}\sqrt[n]{e^n}\nonumber\\
=\frac{1}{n}e^{\frac{1}{n}}+\frac{1}{n}e^{\frac{2}{n}}+\cdots+\frac{1}{n}e^{\frac{n}{n}}\nonumber
\end{eqnarray}




\end{document}

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