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-%% (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{txfonts}
-\usepackage[default]{gfsbodoni}
-\usepackage{latexsym,amsfonts}
-%\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{gfsbodoni} font family}
-\author{Antonis Tsolomitis\\
-Laboratory of Digital Typography\\ and Mathematical Software\\
-Department of Mathematics\\
-University of the Aegean}
-\date {\textsc{19} March \textsc{2006}}
-
-
-\begin{document}
-\maketitle
-
-\section{Introduction}
-The Bodoni 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}| or \verb|\usepackage[default]{gfsdidot}|.
-
-
-Tha latter one sets Bodoni as the default document font. The former
-defines the environment \verb|bodoni| and the command
-\verb|\textbodoni|. For example, while in Greek language
-
-\verb|\textbodoni{dokim'h}|
-
-produces
-
-\textgreek{\textbodoni{dokim'h}}.
-
-x-height is adjusted so that it matches with the x-height of the
-\verb|txfonts| package. This is done to help with documents requiring
-mathematics. In this case load \verb|txfonts| \textit{but before} \verb|gfsbodoni.sty|.
-
-\section{Installation}
-
-Copy the contents of the subdirectory afm in
-texmf/fonts/afm/GFS/Bodoni/
-
-\medskip
-
-\noindent Copy the contents of the subdirectory doc in
-texmf/doc/latex/GFS/Bodoni/
-
-\medskip
-
-\noindent Copy the contents of the subdirectory enc in
-texmf/fonts/enc/dvips/GFS/Bodoni/
-
-\medskip
-
-\noindent Copy the contents of the subdirectory map in
-texmf/fonts/map/dvips/GFS/Bodoni/
-
-\medskip
-
-\noindent Copy the contents of the subdirectory tex in
-texmf/tex/latex/GFS/Bodoni/
-
-\medskip
-
-\noindent Copy the contents of the subdirectory tfm in
-texmf/fonts/tfm/GFS/Bodoni/
-
-\medskip
-
-\noindent Copy the contents of the subdirectory type1 in
-texmf/fonts/type1/GFS/Bodoni/
-
-\medskip
-
-\noindent Copy the contents of the subdirectory vf in
-texmf/fonts/vf/GFS/Bodoni/
-
-\medskip
-
-\noindent In your installation's updmap.cfg file add the line
-
-\medskip
-
-\noindent Map gfsbodoni.map
-
-\medskip
-
-Refresh your filename database and the map file database (for example, on Unix systems
-run mktexlsr and then run the updmap-sys (or updmap on older systems) script as root).
-
-You are now ready to use the fonts.
-
-\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[default]{gfsbodoni}
-\end{verbatim}
-
-will be the correct setup for articles in Greek using Bodoni for the
-main font.
-
-\bigskip
-
-\subsection{Transformations by \texttt{dvips}}
-
-Other than the shapes provided by the fonts themselves, this package
-provides a slanted small caps shape
-using the standard mechanism provided by dvips. Get slanted small caps with \verb|\scslshape|.
-For example, the code
-\begin{verbatim}
-\textsc{small caps \textgreek{πεζοκεφαλαία} 0123456789} {\scslshape
- \textgreek{πεζοκεφαλαία 0123456789}}
-\end{verbatim}
-will give
-
-
-\textsc{small caps \textgreek{πεζοκεφαλαία} 0123456789} {\scslshape
- \textgreek{πεζοκεφαλαία 0123456789}}
-
-\noindent The command \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}.
-
-
-
-\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}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% End: