%latex2e declaration \documentclass[12pt]{article} \usepackage[cp862]{inputenc} \usepackage[hebrew]{babel} \usepackage{tcourse} \newcommand{\fiber}{\ast} \newcommand{\capacity}{{\sf c}} \newcommand{\period}{{\sf p}} \begin{document} \sethebrew \courseSlogan{ - ''} \exerciseNumberAndDate{1}{18/4/2001} \ques{} $G_1, G_2$ $G_1 \fiber G_2$ $G_1$ -$G_2$.\ \[.S(G_1 \fiber G_2) = S(G_1) \cap S(G_2)\] \ques{} $G=(V,E,L)$ \L{Shannon} - $S=S(G)$ $H$ $(v,v)$, $v \in V$, $G*G$. \begin{enumerate} \item $S$ , $S$ $G$. \item $S$ $H$ . \item $S$ , - $|V|(|V|-1)/2$. \item $S=S(G)$. \end{enumerate} \ques{} $G=(V,E,L)$ $S=S(G)$ - $G$. , , $l$ , \[.\capacity(S^l) = l\cdot \capacity(S)\] {\bf }: ) (: \noindent $\Sigma$ '' $S$ $S_0 = S(G_0)$ '' - $G_0$ $G$ $\capacity(S) = \capacity(S_0)$ ) $S_0$ ?(. \newpage \noindent $\ell_1 < \ell_2 < \cdots < \ell_i < \cdots$ \begin{equation} \lim_{i \rightarrow \infty} \frac{1}{\ell_i} \log_2 |S_0 \cap \Sigma^{\ell_i}| = \capacity(S_0) \end{equation} ) ?(. $m_i = \lceil \ell_i/\ell \rceil$. \begin{eqnarray} \label{eq:sfirst} \capacity(S^\ell) & \ge & \capacity(S_0^\ell) \\ & \ge & \limsup_{i \rightarrow \infty} \frac{1}{m_i} \log_2 |S_0 \cap \Sigma^{m_i \ell}| \\ & \ge & \limsup_{i \rightarrow \infty} \frac{1}{m_i} \log_2 |S_0 \cap \Sigma^{\ell_i}| \\ & \ge & \ell \cdot \lim_{i \rightarrow \infty} \frac{\ell_i}{m_i \ell} \lim_{i \rightarrow \infty} \frac{1}{\ell_i} \log_2 |S_0 \cap \Sigma^{\ell_i}|\\ & = & \ell \cdot \capacity(S_0) \\ \label{eq:slast} % & = & \ell \cdot \capacity(S) \; . & = & \ell \cdot \capacity(S) \; \end{eqnarray} ) ~\L{(\ref{eq:sfirst})--(\ref{eq:slast})}. -$S_0$ -?(. \ques{} $G$ - $\period$. $G^\period$ -$\period$ - . \ques{} $G$ - $\period$ $v$ -$G$. $\period$ )- ( -$G$ $v$. \ques{} $G$ - $\period$, $G_1, G_2, \ldots, G_\period$ - $G^\period$. - $G_i$ : \[.\capacity(S(G_i)) = \period\cdot \capacity(S(G)) \] \goodLuck \end{document}  .