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−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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−% Tutorial slides on Python.
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−%
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−% Author: Prabhu Ramachandran <prabhu at aero.iitb.ac.in>
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−% Copyright (c) 2005-2009, Prabhu Ramachandran
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−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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−
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−\documentclass[14pt,compress]{beamer}
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−%\documentclass[draft]{beamer}
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−%\documentclass[compress,handout]{beamer}
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−%\usepackage{pgfpages} 
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−%\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm]
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−
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−% Modified from: generic-ornate-15min-45min.de.tex
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−\usepackage[latin1]{inputenc}
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−%\usepackage{times}
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−\usepackage[T1]{fontenc}
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−
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−% Taken from Fernando's slides.
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−\usepackage{ae,aecompl}
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−\usepackage{mathpazo,courier,euler}
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−\usepackage[scaled=.95]{helvet}
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−
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−\usepackage{listings}
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−\lstset{language=Python,
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−
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−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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−% Macros
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−
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−
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−\newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}}  }
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−
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−%%% This is from Fernando's setup.
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−% \usepackage{color}
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−% \definecolor{orange}{cmyk}{0,0.4,0.8,0.2}
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−% % Use and configure listings package for nicely formatted code
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−% \usepackage{listings}
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−% \lstset{
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−%    language=Python,
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−%    basicstyle=\small\ttfamily,
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−%    commentstyle=\ttfamily\color{blue},
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−%    stringstyle=\ttfamily\color{orange},
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−%    showstringspaces=false,
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−%    breaklines=true,
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−%    postbreak = \space\dots
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−% }
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−
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−
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−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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−% Title page
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−\title[]{Numerical Computing with Numpy \& Scipy}
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−
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−\author[FOSSEE Team] {FOSSEE}
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−
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−\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}
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−\date[] {11, October 2009}
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−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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−
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−%\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo}
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−%\logo{\pgfuseimage{iitmlogo}}
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−
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−
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−%% Delete this, if you do not want the table of contents to pop up at
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−%% the beginning of each subsection:
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−\AtBeginSubsection[]
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−{
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−  \begin{frame}<beamer>
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−    \frametitle{Outline}
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−    \tableofcontents[currentsection,currentsubsection]
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−  \end{frame}
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−}
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−
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−\AtBeginSection[]
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−{
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−  \begin{frame}<beamer>
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−    \frametitle{Outline}
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−    \tableofcontents[currentsection,currentsubsection]
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−  \end{frame}
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−}
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−
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−% If you wish to uncover everything in a step-wise fashion, uncomment
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−% the following command: 
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−%\beamerdefaultoverlayspecification{<+->}
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−
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−%\includeonlyframes{current,current1,current2,current3,current4,current5,current6}
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−
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−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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−% DOCUMENT STARTS
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−\begin{document}
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−
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−\begin{frame}
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−  \maketitle
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−\end{frame}
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−
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−\section{Advanced Numpy}
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−\begin{frame}[fragile]
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−  \frametitle{Broadcasting}
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−  \begin{lstlisting}
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−    >>> a = arange(4)
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−    >>> b = arange(5)
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−    >>> a+b #Does this work?
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−    >>> a+3
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−    >>> c = array([3])
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−    >>> a+c #Works!
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−    >>> b+c #But how?
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−    >>> a.shape, b.shape, c.shape
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−  \end{lstlisting}
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−  \begin{itemize}
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−    \item Enter Broadcasting!
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−  \end{itemize}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−  \frametitle{Broadcasting}
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−  \begin{columns}
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−    \column{0.65\textwidth}
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−    \hspace*{-1.5in}
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−    \begin{lstlisting}
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−      >>> a = arange(4)
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−      >>> a+3
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−      array([3, 4, 5, 6])
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−    \end{lstlisting}
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−    \column{0.35\textwidth}
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−    \includegraphics[height=0.7in, interpolate=true]{data/broadcast_scalar}
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−  \end{columns}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−  \frametitle{Broadcasting in 3D}
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−    \begin{lstlisting}
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−      >>> x = ones((3, 5, 1))
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−      >>> y = ones(8)
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−      >>> (x + y).shape
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−      (3, 5, 8)
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−    \end{lstlisting}
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−    \begin{figure}
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−      \begin{center}
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−      \includegraphics[height=1.5in, interpolate=true]{data/array_3x5x8}        
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−      \end{center}
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−    \end{figure}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−  \frametitle{Copies \& Views}
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−  \vspace{-0.1in}
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−  \begin{lstlisting}
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−    >>> a = arange(1,9); a.shape=3,3
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−    >>> b = a
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−    >>> b is a
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−    >>> b[0,0]=0; print a
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−    >>> c = a.view()
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−    >>> c is a
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−    >>> c.base is a
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−    >>> c.flags.owndata
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−    >>> d = a.copy()
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−    >>> d.base is a
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−    >>> d.flags.owndata
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−  \end{lstlisting}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−  \frametitle{Copies \& Views}
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−  \vspace{-0.1in}
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−  \begin{lstlisting}
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−    >>> b = a[0,1:3]
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−    >>> c = a[0::2,0::2]
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−    >>> a.flags.owndata
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−    >>> b.flags.owndata
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−    >>> b.base
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−    >>> c.base is a
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−  \end{lstlisting}
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−  \begin{itemize}
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−  \item Slicing and Striding just reference the same memory
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−  \item They produce views of the data, not copies
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−  \end{itemize}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−  \frametitle{Copies contd \ldots}
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−  \begin{lstlisting}
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−    >>> a = arange(1, 10, 2)
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−    >>> b = a[array([0,2,3])]
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−    >>> b.flags.owndata
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−    >>> abool=a>5
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−    >>> c = a[abool]
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−    >>> c.flags.owndata
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−  \end{lstlisting}
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−  \begin{itemize}
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−  \item Indexing arrays or Boolean arrays produce copies
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−  \end{itemize}
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−\inctime{15}
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−\end{frame}
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−
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−\section{SciPy}
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−\subsection{Introduction}
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−\begin{frame}
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−    {Intro to SciPy}
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−  \begin{itemize}
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−  \item \url{http://www.scipy.org}
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−  \item Open source scientific libraries for Python
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−  \item Based on NumPy
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−    \end{itemize}
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−\end{frame}
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−
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−\begin{frame}
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−  \frametitle{SciPy}
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−  \begin{itemize}
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−  \item Provides:
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−    \begin{itemize}
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−    \item Linear algebra
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−    \item Numerical integration
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−    \item Fourier transforms
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−    \item Signal processing
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−    \item Special functions
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−    \item Statistics
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−    \item Optimization
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−    \item Image processing
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−    \item ODE solvers
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−    \end{itemize}
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−  \item Uses LAPACK, QUADPACK, ODEPACK, FFTPACK etc. from netlib
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−  \end{itemize}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−  \frametitle{SciPy - Functions \& Submodules}
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−  \begin{itemize}
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−    \item All \typ{numpy} functions are in \typ{scipy} namespace
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−    \item Domain specific functions organized into subpackages
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−    \item Subpackages need to be imported separately
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−  \end{itemize}
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−  \begin{lstlisting}
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−    >>> from scipy import linalg
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−  \end{lstlisting}
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−\end{frame}
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−
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−\subsection{Linear Algebra}
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−\begin{frame}[fragile]
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−  \frametitle{Linear Algebra}
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−  \begin{lstlisting}
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−    >>> import scipy as sp
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−    >>> from scipy import linalg
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−    >>> A = sp.array(sp.arange(1,10))
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−    >>> A.shape = 3,3
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−    >>> linalg.inv(A)
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−    >>> linalg.det(A)
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−    >>> linalg.norm(A)
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−    >>> linalg.expm(A) #logm
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−    >>> linalg.sinm(A) #cosm, tanm, ...
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−  \end{lstlisting}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−  \frametitle{Linear Algebra ...}
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−  \begin{lstlisting}
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−    >>> A = sp.array(sp.arange(1,10))
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−    >>> A.shape = 3,3
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−    >>> linalg.lu(A)
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−    >>> linalg.eig(A)
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−    >>> linalg.eigvals(A)
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−  \end{lstlisting}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−  \frametitle{Solving Linear Equations}
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−  \vspace{-0.2in}
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−  \begin{align*}
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−    3x + 2y - z  & = 1 \\
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−    2x - 2y + 4z  & = -2 \\
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−    -x + \frac{1}{2}y -z & = 0
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−  \end{align*}
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−  To Solve this, 
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−  \begin{lstlisting}
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−    >>> A = sp.array([[3,2,-1],[2,-2,4]
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−                  ,[-1,1/2,-1]])
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−    >>> b = sp.array([1,-2,0])
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−    >>> x = linalg.solve(A,b)
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−    >>> Ax = sp.dot(A,x)
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−    >>> sp.allclose(Ax, b)
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−  \end{lstlisting}
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−\inctime{15}
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−\end{frame}
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−
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−\subsection{Integration}
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−\begin{frame}[fragile]
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−  \frametitle{Integrate}
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−  \begin{itemize}
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−    \item Integrating Functions given function object
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−    \item Integrating Functions given fixed samples
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−    \item Numerical integrators of ODE systems
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−  \end{itemize}
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−  Calculate the area under $(sin(x) + x^2)$ in the range $(0,1)$
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−  \begin{lstlisting}
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−    >>> def f(x):
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−            return sin(x)+x**2
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−    >>> integrate.quad(f, 0, 1)
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−  \end{lstlisting}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−  \frametitle{Integrate \ldots}
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−  Numerically solve ODEs\\
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−  \begin{align*}
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−  \frac{dx}{dt}&=-e^{-t}x^2\\ 
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−           x&=2 \quad at \ t=0
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−  \end{align*}
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−  \begin{lstlisting}
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−>>> def dx_dt(x,t):
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−        return -exp(-t)*x**2
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−>>> t = linspace(0,2,100)
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−>>> x = integrate.odeint(dx_dt, 2, t)
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−>>> plt.plot(x,t)
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−  \end{lstlisting}
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−\inctime{10}
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−\end{frame}
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−
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−\subsection{Interpolation}
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−\begin{frame}[fragile]
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−  \frametitle{Interpolation}
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−  \begin{lstlisting}
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−>>> from scipy import interpolate
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−>>> interpolate.interp1d?
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−>>> x = arange(0,2*pi,pi/4)
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−>>> y = sin(x)
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−>>> fl = interpolate.interp1d(
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−            x,y,kind='linear')
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−>>> fc = interpolate.interp1d(
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−             x,y,kind='cubic')
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−>>> fl(pi/3)
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−>>> fc(pi/3)
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−  \end{lstlisting}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−  \frametitle{Interpolation - Splines}
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−  Plot the Cubic Spline of $sin(x)$
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−  \begin{lstlisting}
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−>>> tck = interpolate.splrep(x,y)
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−>>> xs = arange(0,2*pi,pi/50)
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−>>> ys = interpolate.splev(X,tck,der=0)
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−>>> plt.plot(x,y,'o',x,y,xs,ys)
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−>>> plt.show()
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−  \end{lstlisting}
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−\inctime{10}
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−\end{frame}
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−
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−\subsection{Signal Processing}
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−\begin{frame}[fragile]
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−  \frametitle{Signal \& Image Processing}
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−    \begin{itemize}
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−     \item Convolution
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−     \item Filtering
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−     \item Filter design
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−     \item IIR filter design
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−     \item Linear Systems
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−     \item LTI Representations
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−     \item Window functions
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−    \end{itemize}
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−\end{frame}
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−
+
−\begin{frame}[fragile]
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−  \frametitle{Signal \& Image Processing}
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−  Applying a simple median filter
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−  \begin{lstlisting}
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−>>> from scipy import signal, ndimage
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−>>> from scipy import lena
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−>>> A = lena().astype('float32')
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−>>> B = signal.medfilt2d(A)
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−>>> imshow(B)
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−  \end{lstlisting}
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−  Zooming an array - uses spline interpolation
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−  \begin{lstlisting}
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−>>> b = ndimage.zoom(A,0.5)
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−>>> imshow(b)
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−>>> b = ndimage.zoom(A,2)
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−  \end{lstlisting}
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−    \inctime{5}
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−\end{frame}
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−
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−\begin{frame}[fragile]
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−  \frametitle{Problems}
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−  The Van der Pol oscillator is a type of nonconservative oscillator with nonlinear damping. It evolves in time according to the second order differential equation:
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−  \begin{equation*}
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−  \frac{d^2x}{dt^2}+\mu(x^2-1)\frac{dx}{dt}+x= 0
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−  \end{equation*}
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−  Make a plot of $\frac{dx}{dt}$ vs. $x$.
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−\inctime{30}
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−\end{frame}
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−\begin{frame}{Summary}
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−  \begin{itemize}
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−    \item Advanced NumPy
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−    \item SciPy
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−      \begin{itemize}
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−        \item Linear Algebra
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−        \item Integration
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−        \item Interpolation
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−        \item Signal and Image processing
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−      \end{itemize}
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−  \end{itemize}
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−\end{frame}
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−\end{document}
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−
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−- Numpy arrays (30 mins)
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−    - Matrices
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−    - random number generation.
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−    - Image manipulation: jigsaw puzzle.
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−    - Monte-carlo integration.