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\title[Exercises]{Exercises}

\date[] {8 November, 2009\\Day 2, Session 5}

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  \titlepage

\begin{frame}{Problem 1.1}

  The aliquot of a number is defined as: the sum of the \emph{proper} divisors of the number. \\For example: 

\center{aliquot(12) = 1 + 2 + 3 + 4 + 6 = 16.}\\

  Write a function that returns the aliquot number of a given number. 

\begin{frame}{Problem 1.2}

  Pair of numbers (a, b) is said to be \alert{amicable} if aliquot number of a is b and aliquot number of b is a.\\

  Example: \texttt{220, 284}\\

  Write a program that prints all four digit amicable pairs.

\inctime{20}

%% \begin{frame}{Problem 2}

%%   Given an empty chessboard and one Bishop placed in any s%quare, say (r, c), generate the list of all squares the Bi%shop could move to.

%% \end{frame}

  \frametitle{Problem Set 2}

  Given a string like, ``1, 3-7, 12, 15, 18-21'', produce the list \\

  \begin{lstlisting}

    [1,3,4,5,6,7,12,15,18,19,20,21]

  \end{lstlisting}

\inctime{10}

\begin{frame} 

  \frametitle{Problem Set 3}

  \begin{description}

    \item[3.1] Count word frequencies in a file.

\end{description}

  \frametitle{Problem set 4}

  Central difference

  \begin{equation*}

  \frac{sin(x+h)-sin(x-h)}{2h}

  \end{equation*}

  In []: x = linspace(0, 2*pi, 100)

  In []: y = sin(x)

  In []: deltax = x[1] - x[0]

  \pause

    \begin{enumerate}

      \item Given this, get the finite difference of sin in the range 0 to 2*pi

    \end{enumerate}

\begin{frame}

  \frametitle{Problem Set 5}

  \begin{itemize}

      \item[5.1] Write a function that plots any regular n-gon given \typ{n}.

      \item[5.2] Consider the logistic map, $f(x) = kx(1-x)$, plot it for

          $k=2.5, 3.5$ and $4$ in the same plot.

\end{itemize}

\begin{frame}[fragile] 

\frametitle{Problem Set 5}

  \begin{columns}

    \column{0.6\textwidth}

    \small{

      \item[3] Consider the iteration $x_{n+1} = f(x_n)$ where $f(x) = kx(1-x)$.  Plot the successive iterates of this process as explained below. 

    \end{itemize}}

    \column{0.35\textwidth}

    \hspace*{-0.5in}

  \includegraphics[height=1.6in, interpolate=true]{data/cobweb}  

\end{columns}

  \frametitle{Problem Set 5.3}

  Plot the cobweb plot as follows:

  \begin{enumerate}

    \item Start at $(x_0, 0)$ ($\implies$ i=0)

    \item Draw a line to $(x_i, f(x_i))$

    \item Set $x_{i+1} = f(x_i)$

    \item Draw a line to $(x_{i+1}, x_{i+1})$

    \item $(i\implies i+1)$ 

    \item Repeat from 2 for as long as you want 

  \end{enumerate}

\inctime{20}